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Strategic and transitory models of queueing systems

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Strategic and transitory models of queueing systems

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Content STRATEGIC AND TRANSITORY MODELS OF QUEUEING SYSTEMS by Harsha Honnappa A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (DEPARTMENT OF ELECTRICAL ENGINEERING) December 2014 Copyright 2014 Harsha Honnappa Dedication To My Grandparents ii Acknowledgments To follow knowledge like a sinking star, Beyond the utmost bound of human thought. -Alfred, Lord Tennyson I quoted Tennyson on my statement of purpose when I applied to graduate school to do my Ph.D. While I can’t claim to have gone too far “beyond the utmost bound of human thought”, I did have the opportunity to strike out some new ground in queueing theory. For this, I must thank a number of people and institutions, starting with Rahul Jain my advisor. He encouraged me to work on games in queueing, which in turn led us to study an array of open problems in queueing theory. Working with him has been a great experience, and I’ve learnt a lot, not only how to tackle problems, but also how to be a successful academic. He was also most encouraging in working with other researchers and academics. This led me to contact Amy Ward in the business school about the problems I was working on. Working with Amy has been one of the joys of graduate school, and it has been a privilege to have her as my co-advisor. Besides critiquing my proofs, and forcing me to think through the logic, she has also been very gracious with her time in advising me about being an academic. I look forward to solving the problems that we’ve been discussing together! A big part of these opportunities is the encouraging environment at USC for cross-disciplinary research. I am most grateful to the Electrical Engineering department at USC for encouraging my work by awarding it a best paper (honorable mention) prize and a Ming Hsieh Institute Scholarship. These awards allowed me to showcase my work in other institutions, and obtain valuable iii feedback. During my Ph.D. I also spent more time in the Mathematics department than my own. I must thank every one of my teachers there, for their encouragement and patience with this engineering student, and, in particular, Professor Jianfeng Zhang for many interesting conversations. Amy also introduced me to Peter Glynn at Stanford - my interactions with him over the final year of my Ph.D. has been one of the most eye-opening experiences in my career as an academic. Talking to Peter has given me deeper insight into my own problems, and I’ve also had the wonderful experience of working with him closely on a number of other problems. Despite being extremely busy, he has been generous with his time helping me to navigate the academic job market. I must also thank Vijay Subramanian at the University of Michigan, who has been a sounding board for various research and non-research problems as we navigated the job market at the same time - and hopefully we publish our own paper soon! I must thank Steven Low, Bhaskar Krishnamachari and Michael J. Neely for kindly agreeing to be part of my thesis and qualifying examination committee’s, and for giving me useful feedback on my thesis. Bhaskar was also very generous with his time, and I’ve been the recipient of very good advice from him on being an academic. I also had the great experience of working at the iconic Bell Labs. Getting to spend some time at Bell Labs was, in a sense, a fulfillment of a long-held dream - and I thank Iraj Saniee and Indra Widjaja for a wonderful summer spent there. Thanks are also due to Sasha Stolyar and Marty Reiman, for great conversations that have helped me answer questions that I was unable to myself. I would not have been able to make it though the Ph.D. without the company of many friends - Ananth, Mythili, Nachiketas, Saurov, Arunima, Naumaan in LA, and Karthik, Manish, Akshay for all their help in the Bay Area. I must also thank Pramod, Vinay, Kandarp and Avinash, for friendship that has lasted almost twenty years now, but also for those stirring conversations in Jayanagar 4th Block that helped convince me to get a move on my applications to grad school. But, most of all, I must acknowledge Dileep Kalathil. iv Dileep and I started as Rahul’s first students, and over the course of the last five years, I’ve learnt a lot from him - from tackling mathematical problems to left-wing economics. The mark of a person should surely be their effect on those around them - and Dileep has left an indelible effect on me. Grad school was a great trip with him, and wouldn’t have been half as fun without him. Finally, many thanks to my parents, brother, grandparents, parents-in-law, for their encouragement and support through the years of work. But, above everyone else, is the person who has stood with me through everything - encouraging me to follow my dreams and helping me with the applications to graduate school, moving half-way across the world to the United States to be with me, and listening to too many rants and raves over the past five years - my best friend and partner in life, Ashwini. She has been a constant source of care, encouragement and support. Thanks, Ashwini - and now onto our next adventure! v Contents Dedication ii Acknowledgments iii List of Figures viii Notation x Abstract xi Introduction xii I Transitory Queueing Models 1 1 The Δ (i) /GI/1 Queue 2 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 The Queueing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Fluid Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Diffusion Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.1 Queue Length Process . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Waiting Time and Sample Path Little’s Law . . . . . . . . . . . . . . . . . . 20 1.6 Queue Regimes and States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.6.1 Regimes of ¯ Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.6.2 Sample Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.7 Examples and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.7.1 Uniform Arrival Distribution . . . . . . . . . . . . . . . . . . . . . . . 30 1.7.2 Exponential Arrival Distribution . . . . . . . . . . . . . . . . . . . . 33 1.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 vi 2 Transitory Queueing Theory 45 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1.1 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2 The Transitory Queueing Model . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3 Performance Analysis of Transitory Queueing Systems . . . . . . . . . . . . 50 2.3.1 Large Population Asymptotics of Primitives . . . . . . . . . . . . . . 50 2.3.2 Fluid Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.3.3 Diffusion Approximations . . . . . . . . . . . . . . . . . . . . . . . . 58 2.4 Transitory Traffic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.4.1 The General Δ (i) Model . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.4.2 Conditioned Renewal Model . . . . . . . . . . . . . . . . . . . . . . . 75 2.4.3 Scheduled Arrivals with Epoch Uncertainty . . . . . . . . . . . . . . 81 2.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3 A Generalized Jackson Network of Δ (i) /GI/1 Queues 113 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.1.1 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.2 Transitory Generalized Jackson Network . . . . . . . . . . . . . . . . . . . . 115 3.2.1 Network Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3.2.2 Limits To Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.3 Functional Strong Law of Large Numbers . . . . . . . . . . . . . . . . . . . . 120 3.4 Functional Central Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . 126 II Strategic Users in Transitory Queueing Networks 135 1 The Network Concert Queueing Game 136 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 1.1.1 Fluid Limit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 1.1.2 Game and Equilibrium Concept . . . . . . . . . . . . . . . . . . . . . 138 1.2 The Parallel Network Concert Queueing Game . . . . . . . . . . . . . . . . . 140 1.2.1 Single Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 1.2.2 Multiple Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 1.2.3 Reducing the Price of Anarchy . . . . . . . . . . . . . . . . . . . . . 149 1.3 Tandem Queues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 1.3.1 Single Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 1.3.2 Multiple Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 1.4 Trellis Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 1.5 General Feedforward Networks . . . . . . . . . . . . . . . . . . . . . . . . . . 154 1.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Reference List 183 vii List of Figures 1.1 Anexampleofa Δ (i) /GI/1queuethatwillundergomultiple“regimechanges”. The fluid queue length process is positive on [−T 0 ,τ 1 ) and [τ 2 ,τ 3 ), and 0 on [τ 1 ,τ 2 ) and [τ 3 ,∞). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2 Anexampleofa Δ (i) /GI/1queuethatwillundergomultiple“regimechanges”. The diffusion limit switches between a free Brownian motion (BM), a reflected Brownian motion (RBM), and the zero process. . . . . . . . . . . . . . . . . 19 1.3 An illustration of the various operating regimes of a transitory queueing model. Here, we consider the i.i.d. sampling Δ (i) model. . . . . . . . . . . . 26 1.4 Typical sample paths, mean and variance envelopes of the queue length pro- cess forF uniform over [−20, 40], and exponentially distributed service times with rate μ = 0.03. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.5 Typical sample paths, mean and variance envelopes of the queue length pro- cess for F exponential with parameter λ = 0.1 and exponentially distributed service times with mean rate μ = 0.05. . . . . . . . . . . . . . . . . . . . . . 35 2.1 This x∈C[η,∞) corresponds to the fluid netput process, when F is uniform. 66 2.2 The “population” average arrival distribution function for different values of T. 83 3.1 We study a two node tandem network with uniform arrival time distribution to node 1, and constant service rates μ 1 and μ 2 . . . . . . . . . . . . . . . . . 130 viii 1.1 Example of a multi-stage K×L lattice network, with arbitrary interconnec- tions between adjacent layers. . . . . . . . . . . . . . . . . . . . . . . . . . . 136 1.2 Equilibrium arrival profile of a single population to aK-queue parallel queue- ing network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 1.3 (a) Arrivals from multiple populations into multiple queues. (b) Two parallel queues, and two arriving populations with γ 1 <γ 2 . Population 1 arrives over [−T 0,1 ,T 1 ] at queue 1, and over [−T 0,2 ,T 1 ] at queue 2. However, population 1 need not be served completely until time τ 1 at either queue. Population 2 arrives over [T 1 ,T 2 ] and is completely served at time T 2 . . . . . . . . . . . . 145 1.4 Terminal arrival timeT l as a function ofK l , the number of queues that service population l, plotted for l = 1, 7, 14, and for μτ = 0.1 and μ = 1.0. . . . . . . 148 1.5 An example 2× 2 Lattice network. . . . . . . . . . . . . . . . . . . . . . . . 159 1.6 A Beneš network has 2 n inputs and 2 n outputs. The general network is con- structed from simpler 2× 2 trellis networks, also called a Beneš element. . . 161 1.7 Simple Beneš networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 1.8 Reducing a 2 n × 2 n Beneš network to a 2 n−1 parallel Beneš element network. Step A replaces Layer2 2n− 1 and 2n− 2 by an equivalent parallel element network. Using the induction hypothesis, step B reduces the network further to a two layer network. Finally in step C, the network is reduced to a parallel element network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 ix Notation Unless noted otherwise, all intervals of time are subsets of [−T 0 ,∞), for a given−T 0 ≤ 0. Let D lim :=D lim [−T 0 ,∞)bethespaceoffunctionsx : [−T 0 ,∞)→Rthatareright-continuousat −T 0 , with right and left limits and are either right or left continuous at every pointt>−T 0 . Note that this differs from the usual definition of the spaceD as the space of functions that are right continuous with left limits (cadlág functions), andD⊂D lim . We denote almost sure convergence by a.s. −→ and weak convergence by⇒. The topology of convergence is indi- cated by the tuple (S,m), where S is the metric space of interest and m is the metric that topologizesS. Thus,X n a.s. −→X in (D lim ,U) asn→∞ indicates thatX n ∈D lim converges to X∈D lim uniformly on compact sets (u.o.c.) of [−T 0 ,∞) almost surely. Similarly, X n ⇒X in (D lim ,U) asn→∞ indicates that X n ∈D lim converges weakly to X∈D lim uniformly on compact sets of [−T 0 ,∞). (D lim ,M 1 ) indicates that the topology of convergence is the M 1 topology. We use◦ to denote the composition of functions or processes. The indicator func- tion is denoted by 1 {·} and the positive part operator by (·) + . Finally, all random elements are defined with respect to the canonical space (Ω,F,P), unless noted otherwise. x Abstract Stochastic network theory, and queueing theory in particular, is the bedrock for the anal- ysis and control of resource constrained systems. Such systems are manifest in our world: in healthcare delivery, shared computing, communications and transportation systems, sys- tem operators observe high demand for services necessitating queue management. ’Classi- cal’ queueing theory has largely focused on the analysis of stationary and ergodic models. However, most real world resource allocation systems exhibit time-dependent arrival and service. Further, many systems operate only on a finite time horizon, or system operators are interested in the ’small-time’ or transient behavior of a queueing system. In this dis- sertation, we initiate the development of models of such ’transitory’ queueing systems. Our first contribution is the introduction of several disparate models of multiple server transitory queues. We develop fluid and diffusion approximations, using a mathematical technique called ’Population Acceleration’. Next, we extend this analysis to generalized Jackson net- works. The diffusion approximations are completely unlike the conventional heavy-traffic diffusion approximations. Our second major contribution is the development of game the- oretic models of traffic and routing in generalized Jackson networks. Almost all queueing models assume exogeneous arrivals, routing and service. However, in many situations, like early morning commutes, users are strategic in when they decide to join a service system and which route to take, so that they minimize their sojourn time. We identify the Nash equilibrium traffic and routing profile when users are strategic. xi Introduction This dissertation focuses on the development of a theory of queueing with finite populations of users. There are two parts to the developments: first, we prove fluid and diffusion limit theorems for a class of transitory queueing systems. Second, utilizing these fluid limits, we study a model of strategic users in a network of transitory queues. Part I: Transitory Queueing Models Erlang’s study in 1909 [1], of what came to be known as the M/D/1 queue, initiated the theoretical study of queueing phenomena. He argued that the number of calls arriving at a telephone exchange in a given time interval is Poisson distributed. Subsequent work in queueing theory generalized the Poisson traffic model to renewal, or GI for “general independent”inter-arrivaltime,trafficprocesses. Thisenlargedtheclassofmodelsthatcould be studied while remaining largely tractable analytically as Kingman [2] notes. The early analyses of GI/G/1 queueing models focused on stationary and ergodic queueing systems. Seminal work by Pollaczek, Kendall, Kingman (among others) developed a comprehensive theory of stationary and ergodic queueing systems (see [2] for a chronological perspective of this work.) Ergodic analysis has been hugely important in revealing the “large time” (as t→∞) or steady state behavior of queueing systems. But often, we are interested in transient, or “small time”, analysis of queueing systems. Typically, this is quite messy even for simple xii models such as M/M/1 (see, for example, [3]). One of the more celebrated results in this context is that of Iglehart and Whitt [4] who showed that under the heavy traffic condition, the transient queue length and waiting time are distributionally approximated by reflected Brownian motion processes. This is called the heavy traffic diffusion approximation [5, 6]. In reality, queueing systems often exhibit non-stationarities in their operation, and heavy traffic conditions may not always pertain. For instance, in [7] the authors analyze data at a call center, and show that the traffic is approximately like a non-homogeneous (time-varying) Poisson process [8, 9]. Much of the literature on non-stationary queues has focused on the Markovian case (or M t /M t /· queues). Newell [10, 11, 12, 13] had the earliest, heuristic analysis of such systems. This was expanded upon by [14, 15] via a pointwise stationary approximation (PSA) and formalized in [16, 17, 18] via the uniform acceleration (UA) tech- nique. The key idea behind UA is that by rescaling the arrival and service rates of the system, one obtains a “slowly varying” system whose mixing time is sufficiently small that it reaches stationarity quickly. However, many queueing systems are non-ergodic in nature. This is either because the queue is unstable, or the system is varying rapidly. In such cases, UA/PSA can be a poor approximation. Now, consider queues at post-offices, concert halls, stadiums, retail stores during black friday sales, or scheduled arrivals at a hospital out-patient department. In each instance, there is either a finite number of arriving users, or service is offered for a fixed period of time. These queueing systems are transient, non-stationary and non-ergodic in nature. We call these transitory queueing systems. They are very difficult to analyze via classical queueing theoretic techniques since approximating the system state at any time t by a notional stationary and ergodic process may not be tenable. Moreover, the UA technique is not justified as such models are, in general, non-Markovian, and the arrival and service rates need not vary slowly enough for even local ergodicity to hold. Here, we use an alternative approach that we call the population acceleration (PA) tech- nique. In this technique, the queueing performance metrics are studied by increasing the xiii number of users in a fixed time interval. The PA technique is similar to UA in that the number of users arriving in a small interval goes to infinity but yet distinct, as the time axis is not scaled. We derive functional Strong Law of Large Numbers (fSLLN)/fluid limit and functional Central Limit Theorem (fCLT)/diffusion limit approximations to the per- formance metrics as the population size increases. In Chapter 1, we start with a specific transitory model that we develop in detail called the Δ (i) /GI/1 queue. In the discrete event model, user arrival times are modeled as i.i.d. random variables that are sampled a pri- ori to arrival. Users enter the queue in the order of the sampled arrival times. Note that unlike conventional models in queueing, we are not modeling inter-arrival times, but the actual arrival times of users. Service times are also assumed to be i.i.d. and independent of the arrival times. The direct analysis of the discrete event model is complicated by the fact that the performance metrics are non-Markovian. Instead, here we focus on developing fluid and diffusion approximations to the performance metrics. Recall that the conventional heavy traffic diffusion approximation is a reflected Brownian motion, where the reflection is through the Skorokhod reflection map. In the Δ (i) /GI/1 model the diffusion approximation is a reflected “tied-down” Gaussian process that is a function of a Brownian bridge, and reflected through the directional derivative of the Skorokhod reflection map. This is a fairly unconventional result, and has mostly been observed in the development of diffusion approx- imations for the time-varying Markovian queue [19]. In a time-varying system the queue can switch between various states (overload, underload and critical load), and the diffusion model switches between a free diffusion, a zero process and a reflected diffusion (resp.). This is captured in the directional derivative reflection map. A similar behavior is observed in the Δ (i) /GI/1 queue. As a consequence of the reflection through the directional derivative map, the diffusion approximation also exhibits discontinuities in the limit when the system switches from overload to underload. We also prove what amounts to a sample path Little’s law, giving a linear relationship between the waiting time and queue length processes. xiv The Δ (i) /GI/1 queue is only one example of a whole panoply of finite horizon queueing models. In Chapter 2, we expand the analysis of the Δ (i) /GI/1 queue and consider s- server transitory queues with FIFO service. Our only assumptions on the traffic model now are, Assumption 1: the fluid limit of the arrival process is a cumulative distribution function (i.e., right continuous and nondecreasing with finite total mass), and Assumption 2: the diffusion limit is a tied-down Gaussian process, possibly with continuous sample paths. Under these fairly weak assumptions, we show that the fluid limit is a Skorokhod reflection of the fluid limit of an appropriately defined netput process. It is, in general, time- varying, and switches between ‘overloaded’, ‘underloaded’ and ‘critically loaded’ regimes, as observed for the Δ (i) /GI/1 queue. The diffusion limit of the queue length process is shown to be a combination of a tied-down Gaussian process and an independent Brownian motion process reflected though the directional derivative of the one-dimensional Skorokhod reflection map. We illustrate that this class of queueing models is large, by studying three disparate models of transitory queueing: first, a generalization of the Δ (i) /GI/1 model where the arrival times are only independent. We prove a generalization to the standard Glivenko- Cantelli and Donsker’s theorems in order to develop the fluid and diffusion models. Second, we assume that the arrival times are the event times of a renewal process conditioned on the event of a fixed number of events in a time horizon of interest. Here, the fluid and diffusion approximations emerge as conditioned weak limits. Finally, we also study a model of scheduled arrivals where the actual arrival times are distributed around the scheduled time. In all these cases, we demonstrate that Assumption 1 and Assumption 2 are satisfied. The conventional heavy-traffic analysis of single server queues was extended to open generalized Jackson networks in [20], where it was shown that the diffusion approximation to a K-queue, single class, queueing networks is a multidimensional reflected Brownian motion. The corresponding developments for time-varying systems have not been as direct. The most comprehensive work, utilizing strong approximations in [21], places very strong differentiability assumptions on the arrival and service rate functions at each node. Further, xv the analysis does not extend to non-Markovian networks. Recently, in [22] Mandelbaum and Ramanan prove the existence of a multi-dimensional generalization of the directional derivative reflection map. However, it is also shown that proving weak convergence in general appears to be very difficult. This is due to the fact that, unlike the single-dimensional case, the directional derivative reflected diffusion limit has sample paths that are neither right nor left continuous. We prove weak convergence of the queue length process for networks of Δ (i) /GI/1 queues in Chapter 3, for a simple tandem network with uniform arrival time distributions. This analysis also extends to models with unimodal arrival time distributions. However, it appears almost impossible to easily extend this to more general cases. Diffusion Approximations In Queueing Theory To put the research in this dissertation into perspective, it is useful to take a small tour through the existing results in diffusion approximations in queueing theory. In general, the arrival times of jobs to a queue are not uniformly spaced out and do vary by job. Similarly, the amount of time spent in the server is not the same for each job. Queueing models treat these as random variables, and under various stochastic assumptions on the inter-arrival and service times provide a characterization of the number of jobs waiting (aka, the “queue length”) and how long a job would have to wait for service (aka, the “waiting time” or “workload”). It is natural to think of the inter-arrival and service times as a sequence of tuples:{(u i−1 ,ν i ); i≥ 1}, where u i−1 is the inter-arrival time between the ith and i− 1th job, andν i is the service time of theith job. Thus, one might think of the input to the model as a “marked point process” (see [23]), where the points are the partial sums P i j=1 u j−1 and the marks are the ν i . Inthesimplestqueueingmodels, thearrivalandservicetimetuplesareassumedtoforma stationary sequence. The simplest queueing model here is the so-calledM/M/1 queue, where the inter-arrival and service times are both assumed to be exponential distributed random variables (with different parameters, possibly). As a consequence of these assumptions, the xvi queuelengthprocessisequal(indistribution)toabirth-deathcontinuoustimeMarkovchain. This allows us to characterize the long-term/large time scale (or “stationary”) and short- term/small time scale (or “transient”) behavior of the queue length process. The standard approach to studying the queue length is to obtain analytic solutions to the Kolmogorov Forward Equations, which are relatively easy to obtain and solve for the M/M/1 queue. However, if one moves away from this model the queue length process is no longer Markov. If either the inter-arrival or service times are still assumed to be exponentially distributed (i.e., either an M/G/1 or G/M/1 queue), then there is a sequence of embedded points (the arrival and departure times, respectively) at which the queue length process forms a discrete time Markov chain. This so-called “embedded points” approach can be used to obtain the stationary distribution of the queue length process. However, the transient analysis is still quite difficult and requires an alternative approach. The standard approach for studying the transient performance of queueing models is to approximate the distribution of the queue length process in some well justified manner. The classical result here is the so-called heavy-traffic diffusion approximation by Iglehart and Whitt [4, 24]. This result stems from Kingman’s early result that in a large enough time scale the queue length distribution is well approximated as exponential [25]. In fact, Kingman’s result is specific, showing that the right time-scale is t∝ 1 (1−ρ) 2 , where ρ = λ μ is the load on the system. Now, as ρ→ 1, the stationary distribution is shown to be exponential. In the diffusion approximation, we consider a sequence of queueing models indexed by n≥ 1, and such that the load in the nth system is ρ n . Under the assumption that ρ n ≈ 1− θ √ n , whereθ is given, it is shown that the queue length process process converges in distribution to a reflected Brownian motion (RBM) as n→∞. Note that the stationary distribution of the RBM is well known to be exponential, thus showing that this result is consistent with Kingman’s result. Intuitively, one can think of the O (( √ n(1−ρ n )) −2 ) time scale as being large enough that we accumulate enough statistics for a central limit effect to come into play, which is where the RBM emerges from. Another interpretation of this result is that we are xvii approximating the state of the queue for smallt by what happens in the “long-term”. This is not unreasonable when the inter-arrival and service time tuples form a stationary sequence, or are weakly dependent. In this case, it is possible to argue that the queue length process is in a sense “rapidly mixing”, reaching stationarity in finite time. More generally, most queueing systems do not conform to the stationary assumptions of classical models. For instance, the variance of the queue length process can vary with time and load in the system. A classical example of this behavior is the so-called Joseph effect, where the queue length stays above and below the mean for long stretches of time. This type of behavior can be modeled as either the consequence of long-range dependence (between the inter-arrival and/or service times), or as a result of the arrival and service rates being time dependent (or time inhomogeneous). There is growing body of work that focuses on both of these types of models. The work in this dissertation is strongly related to the second type of models, and in particular, the fluid and diffusion models that we develop are closely related to the models developed in [19, 21] using the uniform acceleration (UA) technique. There the fluid and diffusion models are developed as limits to Poisson arrival and service models, where the time dependent arrival and service rates are scaled by a strictly postive number , as → 0. Intuitively, as → 0, the rate functions are “stretched out” so that they evolve more slowly. At each point in time the state of the queue is approximated by a stationary model, with parameters equal to the underlying model “frozen” at that time instant. The fluid and diffusion approximations emerge from looking at a large enough time scale for law of large numbers and central limit theorem effects to emerge. The population acceleration (PA) approximations developed in this dissertation have similarities to the UA approximations, in that the final approximations have a similar form. Of course, the limits do not emerge from looking at a large time scale (which is held fixed), but as a consequence of scaling up the number of arrivals in the time window of interest. Further, the underlying discrete event models do not model the inter-arrival times, and the service times are scaled in proportion to the total arrival population. The underlying xviii discrete event model displays non-stationarities of the second type above, with the arrival and (potentially) service rate functions being time-dependent. It should be noted that the models developed here are atypical in that we allow arbitrary inter-arrival time dependence. However, the diffusion limits implicitly assume that the inter-arrival times display weak dependence. Part II: Strategic Users in Transitory Queueing Net- works The second part of the thesis is motivated by the following scenario: Users arriving at a concert, a game or at a retail store for Black Friday sales, where arriving before others is preferable, are faced with the dilemma of when to arrive. Should one arrive early before others and wait a while for service to start, or arrive late and wait less, and yet by which time the best seats or deals may already be gone? In such settings, when rational users make strategic decisions of timing, we cannot assume that the arrival process can be modelled by an exogenous renewal process such as a Poisson process. Furthermore, there may be multiple queues (which may start service at different times) and arriving users may have a choice of which queue to join. For instance, users downloading large files from a website often time their downloads to times of day when network congestion is expected to be lower (e.g., late at night.) and moreover, upon arrival, they choose which server to download from. Passengers at airports must not only choose when to arrive, but also which security queue to join upon arrival. Airport security queues also involve more intricate network topologies. Passengers pass through a series of queues (for credentials check, baggage screening and boarding). Thus, users must also choose a route through the network that minimizes their sojourn time. xix We model such scenarios as a single-shot game wherein a finite population of users make simultaneous decisions about time of arrival and routing through a feedforward generalized Jackson queueing network. We call this the network concert queueing game. For a single server queue, a game of strategic arrival timing was introduced in [26, 27]. This was studied in a fluid limit setting that offers significant analytical simplicity and tractability, while still capturing the essential features of the problem. Here, we provide a significant extension to a network of queues. This is much more complicated since when a user arrives influences which route she decides to take (and vice versa). Our equilibrium concept is a generalization of that used in [26, 27]. Queues are assumed heterogeneous with disparate service rates and service start times. We also assume (throughout) that the servers are work conserving with infinite buffers, and that users may queue up before service starts without balking or reneging. Users may belong to different classes, identified by their cost characteristics. Our model of the cost function is a simple linear weighted sum of the sojourn time (or waiting time in the single server case) and the service completion time (sojourn time plus the arrival time), where the weights characterize the user class. The service completion time of a user depends on when she arrived, and maybe considered a proxy for the latter metric. Our first result identifies the equilibrium arrival profile to a network of parallel queues. The fraction of users joining a particular queue is shown to be directly proportional to the fractional service rate on offer at the queue. Further, the price of anarchy is bounded above by 2, implying that the equilibrium is at most 50% worse than an optimal solution. We also extend this to multiple population classes. Next, we show that in the case where the network is fully connected (such as a tandem or trellis network), in equilibrium the network is reduced to an equivalent parallel network of nodes, where the service rates are equal to that of a ’bottleneck’ layer. Finally, in the most general case, the network topology complicates the analysis, and we can only provide an algorithm that provably computes the Nash equilibrium. xx While there has been a lot of work on studying pricing of queueing service (see [28, 29, 30, 31]), games of timing where users choose an arrival time strategically are not so well- understood. The earliest such work is [32] in which a discrete population of users choose the time of their arrival strategically into an ?/M/1 queue, by minimizing the queueing delay. [33] considered a discrete time model where, along the same lines wherein strategic users enter the system to minimize their waiting time, and showed that as the population goes to infinity, there is a unique symmetric Nash equilibrium which yields a discrete time Poisson arrival process. Recently, [34] also considered the same model wherein early bird arrivals were not allowed. Each user wants to minimize their expected waiting time, and thus at equilibrium there is a spike of arrivals at the start time, followed by no arrivals for a finite interval of time. All such work has focused on the single queue case. Problems with similar motivation have been considered in the transportation literature but they have focused on non-queueing theoretic fluid models with delay alone as a cost metric ([35]). In contrast, the framework of [27] considers a more general cost structure that takes into account both the waiting time as well as the service completion time of the user (a proxy for the number of users who arrive before that user) - a significant motivation for users to arrive early in many scenarios. Our contribution is to, for the first time consider the effect of strategic arrivals into general feedforward queueing networks wherein users not only choose their time of arrival but also which route to take within the network. xxi Part I Transitory Queueing Models 1 Chapter 1 The Δ (i) /GI/1 Queue 1.1 Introduction Considern customers who arrive into a single-server queue. Each customer’s time of arrival is modeled as an i.i.d. sample from a distribution F (restrictions on F will be stated later). Let T i be the arrival time of the ith customer, then, customers enter the queue in order of the sampled times: T (1) ≤ T (2) ≤···≤ T (n) . Let Δ (i) := (T (i) −T (i−1) ) then, in Kendall’s notation, this model can be called the Δ (i) /GI/1 queueing model. The analysis of the discrete event model is quite difficult, in general. For instance, when the service process is Poisson, the Kolmogorov forward equations for the joint distribution of the queue length and cumulative arrival processes can be written down, but there is no easy way to obtain general analytical solutions; see Example 1.1. In this chapter, we develop fluid and diffusion approximations to the queue-length process directly as the population size scales to infinity and the service rate is accelerated appropriately (to be defined). We also establish a sample path Little’s Law that links the limit queue-length and virtual waiting time processes under both fluid and diffusion limits. To develop the fluid limits, we use the Glivenko-Cantelli theorem and the functional StrongLawofLargeNumbersforrenewalprocessesalongwiththeSkorokhodreflectionmap- ping theorem. We show that the fluid limit of the queue length process switches between ‘overloaded’, ‘underloaded’, and ‘critically loaded’ regimes as time progresses. The limit- ing diffusion for the queue-length process is derived using a directional derivative reflection mapping lemma. The diffusion process approximation is a reflection of a Brownian bridge 2 process, that arises from the invariance principle related to the Kolmogorov-Smirnov statis- tic, combined with a Brownian motion, that arises from the functional central limit theorem for renewal processes. We also note that our diffusion process convergence results are in Sko- rokhod’s M 1 topology onD lim [0,∞), the space of functions that are right or left continuous at every point, and right continuous at 0. Therestofthischapterisorganizedasfollows. Westartwithabriefreviewoftheexisting literature related to this model. Section 1.2 presents the Δ (i) /GI/1 queueing model and some basic results about fluid and diffusion approximations to arrival and service processes. Section 1.3 develops fluid approximations to the queue length, busy-time and virtual waiting time processes. In Section 1.4, we develop diffusion approximations to these processes. Section 1.5 develops waiting time approximations, as well as a sample path Little’s law. Section 1.6 takes a closer look at the sample paths of the queue length process in various operating regimes. Section 1.7 presents some examples and simulations of queue length process. 1.1.1 Related Literature The form of the diffusion and fluid approximations to the Δ (i) /GI/1 queue parallel that of the well studied M t /M t /1 model in the sense that (1) the fluid limit may switch between overloaded, underloaded, and critically loaded periods, and (2) the diffusion limit arises using a directional derivative for the Skorokhod reflection map. It is thus pertinent to compare those models to the Δ (i) /GI/1 queue. There has been a long-standing interest in non-stationary and time-varying queueing models. One of the first pieces of work is that of Newell [10, 11, 12] who characterized the various operating states of the non-homogeneous Poisson queue as the load factorρ(·) varies with time. The motivation came from transporta- tion networks, and Newell performed a heuristic analysis. Keller [14] provided more formal arguments and showed that the transient distribution at timeτ can be approximated by the stationarymeasureassociatedwithanotionalMarkovchainthathasarrivalandservicerates 3 λ(τ) and μ(τ), respectively. This type of analysis has come to be known as the pointwise stationary approximation (PSA) (see [36] as well). Massey [16, 18] and Massey and Whitt [17] made these arguments more rigorous and showed that Keller’s perturbation approach can be justified as a uniform acceleration (UA) asymptotic expansion of the transient distri- bution. The notion of uniform acceleration comes from the fact that the arrival and service rates are scaled at all time instants by the same parameter , and the expansions arise as → 0. Later, Mandelbaum and Massey [19] developed fSLLN and fCLT results using Strong Approximation techniques for the M t /M t /1 queue, and identified the directional derivative reflection map as the right one to succinctly represent the queue length process diffusion limit in all regimes. This is based on the UA technique which relies on the assumption that the time scales on which the queue can change appreciably is of the order of 1/ for some > 0. This technique has been extensively applied to non-stationary queueing systems with non-homogeneous Poisson input [37]. However, it is not yet clear whether it is also useful for transitory queueing models of the kind we introduce in this dissertation. More closely related is the “Binomial Traffic Model” that Newell [13] introduced. This corresponds to the i.i.d. sampling Δ (i) model. Through heuristic analysis, Newell identifies the limit processes in different regimes. However, these are point-wise and not functional limits, and a weak convergence result is missing. In [38], the authors also identify several scenarios where an i.i.d. sampled Δ (i) traffic model would be meaningful. Some analysis is also presented when the arrival process is approximated to be Markovian. The birth-death transient balance equations are solved numerically, and it is shown that a “deterministic approximation” (i.e., a first order fluid model) is good as the population size increases. We, however, note the the main difficulty in analyzing the Δ (i) /GI/1 model is precisely the lack of Markovian structure. The i.i.d. sampling Δ (i) traffic model was also studied by Louchard [39]. He provided an analysis analogous to Newell [10, 11, 12] and established the diffusion limits at continuity points in certain regimes (overloading and critical loading). However, these are not functional limits. Moreover, the whole difficulty in establishing diffusion limit 4 for the Δ (i) /GI/1 model is precisely the fact there are discontinuities as the limit process switches regimes. 1.2 Preliminaries 1.2.1 The Queueing Model Consider a single server, infinite buffer queue that is non-preemptive, non-idling, and starts empty. Service follows a first-come-first-served (FCFS) schedule. Let n be the customer population size. Customers independently sample an arrival time T i , i = 1,...,n, from a common distribution function F assumed to have support [−T 0 ,T ]⊂R, where T > 0. For simplicity, we assume that F is absolutely continuous with a continuous density function. The customer entry times are the order statistics T (1) ≤ T (2) ≤ ...≤ T (n) of the sampled arrival times. The arrival process is the cumulative number of customers that have arrived by time t: A(t) := n X i=1 1 {T i ≤t} , (1.1) where 1 {·} represents an indicator function. Let{ν i ,i≥ 1} be a sequence of independent and identically distributed (IID) random variables, where ν i represents the service time of the ith customer. Assume that the mean service time Eν i = 1/μ <∞ and the variance of the service times σ 2 := Var(ν i ) <∞, and that the associated CDF G has support [0,∞). Finally, also assume that the sequence is independent of the arrival times T i , i = 1,...,n. Thus, service starts at time t = 0. Let S be the service process, defined as a renewal counting process, so that S(t) :=          0 ∀t∈ [−T 0 , 0), sup{m≥ 1|V (m)≤t}, ∀t≥ 0, (1.2) 5 where V (m) := m X i=1 ν i is the cumulative load from m jobs. Let V (t) := P btc i=1 ν i be the offered load process. The amount of time a customer arriving at time t has to wait for service, is Z(t) :=V (A(t))−B(t)−t1 {t≤0} , (1.3) where B(t) := Z t 0 1 {Q(s)>0} ds ! 1 {t≥0} ,∀t∈ [−T 0 ,∞) (1.4) is the busy time process. Note that this definition of the virtual waiting time varies slightly from the standard definition due to the fact that an arrival at timet< 0 before service starts has to wait an extrat units of time for service to start, which accounts for the−t1 {t≤0} term. LetQ represent the queue length process, including both any customer in service and all waiting customers. This is defined in terms of the arrival and service processes as Q(t) :=A(t)−S(B(t)), ∀t∈ [−T 0 ,∞), (1.5) where B(t) is the busy time process. Finally, the idle time process of the server is I(t) :=t1 {t≥0} −B(t) = Z t 0 1 {Q(s)=0} ds ! 1 {t≥0} ∀t∈ [−T 0 ,∞). (1.6) We note that this model is intractable to exact analysis. As an example, consider the case where the service process is Poisson. Example 1.1 (The Δ i /M/1 Queue) Suppose also that the service times are exponential with rateμ. Then, the joint variable (Q(t),A(t)) is a inhomogeneous continuous time Markov 6 chain. By standard ‘birth-death’ chain arguments, observe that the transient state probability distribution evolves per the following ordinary differential equation: dP (t,k,m) dt =                                                  − (N−m) f(t) 1−F (t) ! P (t,k,m) + (N−m + 1) f(t) 1−F (t) P (t,k− 1,m− 1) if t≤ 0, 0<k≤m≤N − μ + (N−m) f(t) 1−F (t) ! P (t,k,m) +μP (t,k + 1,m) + (N−m + 1) f(t) 1−F (t) P (t,k− 1,m− 1) if t> 0, 0≤k≤m≤N where P (t,k,m) =P(Q(t) =k,A(t) =m) and t∈R. These first-order non-linear ordinary differential equations are not easy to solve analytically. Indeed, this is a problem observed even in the study of the time-varying Markovian queue, see [14, 18] where the authors resort to developing approximations to the solution of similar o.d.e’s. Thus, even the simplest transitory queueing model is not amenable to exact analysis, and we focus on developing large population approximations to the performance metrics. 1.2.2 Basic results We now present known functional strong law of large numbers (FSLLN) and functional central limit theorem (FCLT) or diffusion limits, for the arrival and service processes, as the population size n increases to∞. Let A n :=A be the arrival process associated with the system having population size n. The fluid-scaled arrival process is ¯ A n := A n n . Next, consider an accelerated service process, 7 where the service times (or, equivalently, the service rate) are scaled by the population size n, so that S n (t) :=            0 ∀t∈ [−T 0 , 0), sup ( m≥ 1| P m i=1 ν i n ≤t ) , ∀t≥ 0. The fluid-scaled service process is ¯ S n := S n n . Also, the fluid-scaled offered load process is ¯ V n (t) :=          0 ∀t∈ [−T 0 , 0), P bntc i=1 ν n i , ∀t∈ [0,∞). (1.7) Note that our assumption that ν i ,i≥ 1 is an i.i.d. sequence implies that S n (t) is equivalent to the time-scaled process S(nt) (where n is an arbitrary parameter that increases to infin- ity) used in the conventional heavy-traffic setting. Acceleration, however, provides a nice interpretation to our scaling that we conjecture can potentially be extended to non-i.i.d. settings. Now, the following proposition establishes the fluid limits for these processes. Proposition 1.1 As n→∞, ( ¯ A n (t), ¯ S n (t)1 t≥0 , ¯ V n (t)1 t≥0 ) a.s. −→ (F (t),μt1 {t≥0} , t μ 1 {t≥0} ) in (D 3 lim ,WJ 1 ), (1.8) whereD 3 lim is the three dimensional product space of sample paths. Remarks. 1. The proof of Proposition 1.1 follows easily from standard results and we omit it. The fluid arrival process limit is given by the Glivenko-Cantelli Theorem (see [40]). The fluidlimitsoftheserviceprocessandtheofferedloadprocessfollowfromthefunctionalstrong law of large numbers for renewal processes (see [41]). Joint convergence is a consequence of the independence assumptions between the service times and arrival times. 8 Next, looking at the errors of the fluid-scaled arrival process around the fluid limit, the diffusion-scaled arrival process is ˆ A n (t) := √ n ¯ A n (t)−F (t) ! ∀t∈ [−T 0 ,∞). Similarly, the diffusion-scaled service and offered load processes are ˆ S n (t) := √ n ¯ S n (t)−μt ! , t≥ 0 ˆ V n (t) := √ n ¯ V n (t)− 1 μ t ! , t≥ 0. The following proposition presents the diffusion limits for these processes. Proposition 1.2 As n→∞, ( ˆ A n , ˆ S n , ˆ V n )⇒ W 0 ◦F,σμ 3/2 W◦e,−σμ 1/2 W◦ e μ ! in (D 3 lim ,WJ 1 ), (1.9) where W 0 is the standard Brownian bridge process and W is the standard Brownian motion process, both are mutually independent, e : [0,∞) → [0,∞) is the identity map and◦ represents composition of functions. Remarks. 1. The proof of this proposition follows easily from standard results: The FCLT limit for the diffusion-scaled arrival process, also called the empirical process, is a Brownian bridge by Donsker’s Theorem (see Sections 13 and 16 in [42]). Note that this limit also arises in the study of the invariance principle associated with the Kolmogorov-Smirnov statistic used to compare empirical distributions with candidate ones (see [43] for more detail). The limits for the diffusion-scaled service and offered work processes follow from the FCLT for renewal processes (see Section 16 in [42] and Chapter 5 in [41]). Joint convergence follows from independence. 9 2. Our assumption that the support ofF is compact is largely for technical reasons; viz., the Skorokhodtopologiesrestrictweakconvergencetocompactintervalsofthedomain [−T 0 ,∞). Proving a diffusion approximation that holds for distributions with infinite support would require strong approximation results, and is beyond the scope of the current paper. 1.3 Fluid Approximations Following (1.5) the fluid-scaled queue length process is Q n (t) n = 1 n A n (t)− 1 n S n (B n (t)), (1.10) where B n (t) is the fluid-scaled version of the busy time process (1.4) defined as B n (t) := Z t 0 1 {Q n (s)>0} ds ! 1 {t≥0} . Next, add and subtract the functions F (t), μt1 {t≥0} and μB n (t) to obtain Q n (t) n := A n (t) n −F (t) ! − S n (B n (t) n −μB n (t) ! + F (t)−μt1 {t≥0} ! +μI n (t), whereI n (t) =t1 {t≥0} −B n (t)isthefluid-scaledidletimeprocess. Thus, (1.10)isequivalently Q n (t) := Q n (t) n = ¯ X n (t) +μI n (t), ∀t∈ [−T 0 ,∞), (1.11) where ¯ X n (t) is ¯ X n (t) := A n (t) n −F (t) ! − S n (B n (t)) n −μB n (t) ! + (F (t)−μt1 {t≥0} ). (1.12) In preparation for the main Theorem in this section, recall that the Skorokhod reflec- tion map is a continuous functional (Φ, Ψ) :D lim →D lim ×D lim defined as x7→ Ψ(x) := 10 sup −T 0 ≤s≤t (−x(s)) + , and x7→ Φ(x) := x + Ψ(x), ∀x∈D lim . The continuity of the map with respect to the uniform topology onD lim follows from Theorem 3.1 in [22]. Theorem 1.1 (Fluid Limit) The pair ( ¯ Q n ,μI n ) has a unique representation (Φ( ¯ X n ), Ψ( ¯ X n )) in terms of ¯ X n . Furthermore, as n→∞, ( ¯ Q n ,μI n ) a.s. −→ (Φ( ¯ X), Ψ( ¯ X)) in (D lim ×D lim ,WJ 1 ), where ¯ X(t) = (F (t)−μt1 {t≥0} ). Proof: First note that ¯ Q n (t)≥ 0, ∀t∈ [−T 0 ,∞). It is also true that I n (−T 0 ) = 0 and dI n (t)≥ 0,∀t∈ [−T 0 ,∞). By definition ofI n (t), it follows that R ∞ −T 0 ¯ Q n (t)dI n (t) = 0. Thus, by the Skorokhod reflection mapping theorem (first proved in [44]), the joint process ( ¯ Q n (t), μI n (t)) has a unique reflection mapping representation in terms of ¯ X n (t) as (Φ( ¯ X n ), Ψ( ¯ X n )). Note that by definition of B n (t) ≤ t and from Proposition 1.1, it follows that S n ◦B n n −μB n ! a.s. −→ 0 in (D lim ,J 1 ). Using this and Proposition 1.1 it follows that ¯ X n a.s. −→ ¯ X in (D lim ,J 1 ), where ¯ X := (F (t)− μt1 {t≥0} ). Using the limit derived above and the continuous mapping theorem, it follows that ( ¯ Q n ,μI n ) = (Φ( ¯ X n ), Ψ( ¯ X n )) a.s. −→ (Φ( ¯ X), Ψ( ¯ X)) in (D lim ×D lim ,WJ 1 ). Remarks. 1. ¯ X is the difference between the fluid limits of the arrival and service processes, and is often referred to as the fluid limit of the netput process. 2. Theorem 1.1 shows that the fluid limit of the queue length process is ¯ Q(t) = (F (t)−μt1 {t≥0} ) + sup −T 0 ≤s≤t (−(F (s)−μs1 {s≥0} )) + ,∀t∈ [−T 0 ,∞). 11 ¯ Q(t) t -T0 !1 !2 !3 0 F(t) µt Figure 1.1: An example of a Δ (i) /GI/1 queue that will undergo multiple “regime changes”. The fluid queue length process is positive on [−T 0 ,τ 1 ) and [τ 2 ,τ 3 ), and 0 on [τ 1 ,τ 2 ) and [τ 3 ,∞). ¯ Q can be interpreted as the sum of the fluid netput process and the amount of fluid service capacity lost from the system. As it will be seen below, the time instants where the regulator term sup −T 0 ≤s≤t (−(F (s)−μs1 {s≥0} )) + increases are precisely where the queue idles. 3. Figure 1.1 depicts an example queue length process in the fluid limit, and its dependence on the arrival distribution F and service rate μ. Note that the process switches between beingpositiveandzero, duringthetimetheserveroperates. Wewillinvestigatethisbehavior in detail in Section 1.6. Without formally defining the terms, intuitively it should be clear that on [−T 0 ,τ 1 ) and [τ 2 ,τ 3 ) the queue is ’overloaded’, i.e., the cumulative amount of fluid that has entered exceeds the cumulative amount of fluid served, while on the intervals [τ 0 ,τ 1 ) and [τ 3 ,∞) it is ‘underloaded’. Next, consider the busy time process. It is interesting to observe that B n does not converge to the identity process, in contrast to the conventional heavy-traffic approximation setting. Corollary 1.1 As n→∞, B n a.s. −→ ¯ B in (D lim ,J 1 ) (1.13) where ¯ B(t) := t1 {t≥0} − 1 μ Ψ( ¯ X(t)),∀t∈ [−T 0 ,∞). 12 Proof: By definition, we have B n (t) = t1 {t≥0} −I n (t). This can be rewritten as B n (t) = t1 {t≥0} −I n (t). Using Theorem 1.1, the claim then follows. Note that ¯ B(t) = 0 for all t≤ 0, as Ψ( ¯ X)(t) = 0 on that interval. 1.4 Diffusion Approximations In this section we assume F is absolutely continuous in order to establish the desired limit result. As noted before, this is mainly for simplicity of the analysis. 1.4.1 Queue Length Process Define the diffusion-scaled queue length process as Q n (t) √ n := A n (t) √ n − S n (B n (t)) √ n , ∀t∈ [−T 0 ,∞) (1.14) Introducing the terms √ nμt1 {t≥0} , √ nF (t) and √ nμB n (t) we have Q n (t) √ n = A n (t) √ n − √ nF (t) ! − S n (B n (t)) √ n − √ nμB n (t) ! + √ n(F (t)−μt1 {t≥0} ) + √ nμ(t1 {t≥0} −B n (t)). Recalling the definition of the idle time process Q n / √ n is Q n √ n = ˆ X n + √ n ¯ X + √ nμI n , (1.15) where ˆ X n (t) := A n (t) √ n − √ nF (t) ! − S n (B n (t)) √ n − √ nμB n (t) ! (1.16) = ˆ A n (t)− ˆ S n (B n (t)), ∀t∈ [−T 0 ,∞). 13 Recall from Theorem 1.1 that ¯ X(t) = (F (t)−μt1 t≥0 ) is the fluid netput process. Lemma 1.1 below proves a diffusion approximation to the diffusion-scale refinement ˆ X n (t) as an immediate consequence of Proposition 1.2. Lemma 1.1 As n→∞, ˆ X n ⇒ ˆ X := W 0 ◦F−σμ 3/2 W◦ ¯ B in (D lim ,J 1 ) (1.17) where ¯ B is defined in (1.13), andW 0 andW are independent standard Brownian bridge and standard Brownian motion respectively. Proof: First note that B n (t) ≤ t,∀t ∈ [0,∞), implying that S n ◦ B n ∈ D lim . Using Proposition 1.2, Corollary 1.1 and the random time change theorem (see, for example, Section 17 of [42]), it follows that √ n S n ◦B n n −μB n ! ⇒σμ 3/2 W◦ ¯ B. Now, it follows from Proposition 1.2 that ˆ X n ⇒ ˆ X(t) := W 0 ◦F−σμ 3/2 W◦ ¯ B, thus concluding the proof. Remarks. 1. Note that using a classical time change (see, for example, [45]) it is possible to see that the Brownian bridge is equal in distribution to a time changed Brownian motion, and ˆ X is equal in distribution to a stochastic integral ˆ X(t) d =          R t −T 0 q g 0 (s)d ˜ W s , ∀t∈ [−T 0 ,T ] −σμ 3/2 W ( ¯ B(τ ∗ ∨T )), ∀t>τ ∗ ∨T , (1.18) whereg(t) =F (t)(1−F (t)) +σ 2 μ 3 ¯ B(t), ˜ W is a standard Brownian motion process, τ ∗ := 1 μ and∨ is the max operator. Thus, the process ˆ X can also be interpreted as a time-changed Brownian motion on the interval [−T 0 ,T ], and its sample path is a constant on (T,∞). In the rest of this section, we will use Skorokhod’s almost sure representation theorem [44, 46], and replace the random processes above that converge in distribution by those defined on a common probability space that have the same distribution as the original 14 processes and converge almost surely. The requirements for the almost sure representation are mild; it is sufficient that the underlying topological space is Polish (a separable and complete metric space). We note without proof that the spaceD lim , as defined in this dissertation, is Polish when endowed with the M 1 topology. This conclusion follows from [43]. The authors in [19] also point out that [47] has a more general proof of this fact. We conclude that we can replace the weak convergence in (1.9) by ( ˆ A n , ˆ S n , ˆ V n ) a.s. −→ W 0 ◦F,σμ 3/2 W,−σμ 1/2 W◦ h μ ! in (D lim ,J 1 ), where abusing notation we use the same letters as our original processes. Thus, Lemma 1.1 implies that ˆ X n a.s. −→ ˆ X in (D lim ,J 1 ), as n→∞. TheFCLTtothequeuelengthprocessreliesonthedirectionalderivativeoftheSkorokhod reflection map (Φ, Ψ), defined as sup ∇ ¯ X t (−y)(t) = lim n→∞ Ψ( √ nx +y)(t)− √ nΨ(x)(t), (1.19) pointwise inD lim , where x∈C and y∈C, and∇ x t ={−T 0 ≤ s≤ t|x(s) =−Ψ(x)(t)}, is a correspondence of points upto time t where the fluid netput process achieves an infimum. We can now state and prove our main limit theorem. Let ˜ Y n := √ nμI n − √ nΨ( ¯ X). Theorem 1.2 (Diffusion Limit) The pair ( ˆ Q n , ˜ Y n ) has a unique representation in terms of ˆ X n and √ n ¯ X given by Φ( ˆ X n + √ n ¯ X)− √ n ¯ Q, Ψ( ˆ X n + √ n ¯ X)− √ nΨ( ¯ X) ! , where ¯ Q = ¯ X + Ψ( ¯ X) is the fluid limit of the queue length process. Furthermore, as n→∞ ( ˆ Q n , ˜ Y n )⇒ ( ˆ X + ˜ Y, ˜ Y ) in (D lim ×D lim ,SM 1 ), where ˆ X(t) =W 0 (F (t))−σμ 3/2 W ( ¯ B(t)), and ˜ Y (t) = max s∈∇ ¯ X t (− ˆ X(s))∀t∈ [−T 0 ,∞), and SM 1 is the strong M 1 topology on the product spaceD lim ×D lim . 15 Proof: First, using (1.15), it follows by the Skorokhod reflection mapping theorem that Q n √ n , √ nμI n ! = Φ( ˆ X n + √ n ¯ X), Ψ( ˆ X n + √ n ¯ X) ! . (1.20) Thisimpliesthat ˆ Q n = Q n √ n − √ n ¯ Q = Φ( ˆ X n + √ n ¯ X)− √ n ¯ Q.Usingthefactthat ¯ Q = ¯ X+Ψ( ¯ X) and Φ(x) =x + Ψ(x) for any x∈D lim it follows that ˆ Q n = ˆ X n + √ n ¯ X + Ψ( ˆ X n + √ n ¯ X)− √ n( ¯ X + Ψ( ¯ X)), = ˆ X n + Ψ( ˆ X n + √ n ¯ X)− √ nΨ( ¯ X). (1.21) Next, fromtheexpressionfor √ nμI n in (1.20)itfollowsthat ˜ Y n = Ψ( ˆ X n + √ n ¯ X)− √ nΨ( ¯ X), implying that ˆ Q n = ˆ X n + ˜ Y n . The limit result now follows by use of the following directional derivative reflection mapping lemma which is adapted from Lemma 5.2 in [19], and whose proof can be found in the Appendix. Lemma 1.2 (Directional derivative reflection mapping lemma) Letx andy be real- valued continuous functions on [0,∞), and Ψ(z)(t) = sup 0≤s≤t (−z(s)), for any process z∈ D lim . Let{y n }⊂D lim be a sequence of functions such that y n a.s. → y as n→∞. Then, with respect to Skorokhod’s M 1 topology, ˜ y n := Ψ( √ nx +y n )− √ nΨ(x)−→ ˜ y := sup s∈∇ x t (−y(s)) as n→∞, where∇ x t ={0≤s≤t|x(s) =−Ψ(x)(t)}. Observe that ˜ Y n is exactly in the form of ˜ y n defined in the lemma above. Lemma 1.1 and Lemma 1.2 together imply that ˜ Y n a.s. −→ ˜ Y := max s∈∇ ¯ X · (− ˆ X(s)) in (D lim ,M 1 ). It follows that, ˆ Q n = ˆ X n + ˜ Y n a.s. −→ ˆ X + max s∈∇ ¯ X · (− ˆ X(s)) in (D lim ,M 1 ). It remains to prove that ˆ Q n and ˜ Y n converge jointly in the strongM 1 , orSM 1 , topology. Notice that the joint process can be written as    ˆ Q n ˜ Y n    =    ˆ X n 0    +    Ψ( ˆ X n + √ n ¯ X)− √ nΨ( ¯ X) Ψ( ˆ X n + √ n ¯ X)− √ nΨ( ¯ X)   . 16 The first term on the right hand side converges to ˆ X :=    ˆ X 0    almost surely in (D lim ×D lim ,SM 1 ) by Theorem 12.6.1 of [43], as ˆ X is continuous. The second term converges to ˜ Y :=    ˜ Y ˜ Y    almost surely in (D lim ×D lim ,SM 1 ). Now, by definition ˆ X is a continuous process and does not share any discontinuity points with ˜ Y. Therefore, by Corollary 12.7.1 of [43], the addition operator is continuous, implying that    ˆ Q n ˜ Y n    a.s. →    ˆ X + ˜ Y ˜ Y    in (D lim ×D lim ,SM 1 ). Finally, the weak convergence is a direct implication of the almost sure convergence result, thus concluding the proof. Remarks. 1. Observe that the diffusion limit to the queue length process is a function of a Brownian bridge and a Brownian motion. This is significantly different from the usual limits obtained in a heavy-traffic or large population approximation to a single server queue. For instance, in the G/GI/1 queue, one would expect a reflected Brownian motion in the heavy-traffic setting. In [19] it was shown that the diffusion limit process to the M t /M t /1 queue is a time changed Brownian motion W ( R λ(s)ds + R μ(s)ds), where λ(s) is the time inhomogeneousrateofarrivalprocessandμ(s)isthatoftheserviceprocess, reflectedthrough the directional derivative reflection map used in Lemma 1.2. There are very few examples of heavy-traffic limits involving a diffusion that is a function of a Brownian bridge and a 17 Brownian motion process. However, there have been some results in other queueing models where a Brownian bridge arises in the limit. In [48], for instance, a Brownian bridge limit arises in the study of a many-server queue in the Halfin-Whitt regime. 2. We noted in the remarks after Theorem 1.1 that the fluid limit can change between being positive and zero in the arrival interval for a completely general F. One can then expect the diffusion limit to change as well, and switch between being a ‘free’ diffusion, a reflected diffusion and a zero process. This is indeed the case. Figure 1.2 illustrates this for the example in Figure 1.1. Note that∀t∈ [−T 0 ,τ 1 ) Ψ( ¯ X)(t) =− ¯ X(−T 0 ), implying that the set∇ ¯ X t is a singleton. On the other hand, at τ 1 ∇ ¯ X t ={−T 0 ,τ 1 }. For t∈ (τ 1 ,τ 2 ], Ψ( ¯ X)(t) = 0 = ¯ X(t), implying that∇ ¯ X t = (τ 1 ,t]. On (τ 2 ,τ 3 ), Ψ( ¯ X)(t) = 0, but ¯ X(t)> 0, so that∇ ¯ X t = (τ 1 ,τ 2 ]. Finally, thefluidqueuelengthbecomeszerowhenthefluidserviceprocess exceeds the fluid arrival process in [τ 3 ,∞), implying that Ψ( ¯ X)(t) =−(F (t)−μt) > 0. It can be seen that∇ ¯ X t ={t} in this case. Recall from Corollary 1.1 thatB n converges to a continuous process ¯ B asn→∞. Define the diffusion-scaled busy time process as ˆ B n := √ n( ¯ B−B n ). (1.22) Note that from the definitions of B n (t) and ¯ B(t) it follows that ˆ B n (t) = 0, ∀t < 0. The diffusion limit for this process is given as follows. Corollary 1.2 The diffusion scaled busy time weakly converges to a regulated diffusion pro- cess: ˆ B n ⇒ ˆ B := 1 μ max s∈∇ ¯ X · (− ˆ X(s)), in (D lim ,M 1 ). as n→∞. Proof: Recall that B n (t) = t1 {t≥0} − I n (t). Substituting this and ¯ B from (1.13) in the definition of ˆ B n , and rearranging the expression, we obtain ˆ B n = 1 μ ˜ Y n . A simple application of Theorem 1.2 then provides the necessary conclusion. 18 Figure 1.2: An example of a Δ (i) /GI/1 queue that will undergo multiple “regime changes”. The diffusion limit switches between a free Brownian motion (BM), a reflected Brownian motion (RBM), and the zero process. Observe that B n (t) is approximated in distribution by ˆ B as B n (t) d ≈ ¯ B(t)− 1 √ n ˆ B(t), where Y d ≈ X is defined to be P(Y ≤ x)≈P(X≤ x), and the approximation is rigorously supported by an appropriate weak convergence result. The case of uniform F on [−T 0 ,T ] is instructive and it can be seen that on [−T 0 ,τ) the queue length in the fluid limit is positive. However, as the server starts at time 0, the only interesting sub-interval of [−T 0 ,τ) is [0,τ). Using the appropriate definitions, note that ¯ B(t) = t and ˆ B(t) = 0 for all t ∈ [0,τ), implying that B n (t) = t approximately, though in the non-asymptotic regime B n (t) may be strictly smaller than t. On the other hand, the fluid queue length is zero in (τ,∞) and it follows from definition of Ψ( ¯ X) that 19 ¯ B(t) =t− 1 μ (− ¯ X(t)) = 1 μ F (t) fort∈ (τ,∞). Substituting this expression together with that of ˆ B, and expanding ˆ X, we see that B n (t) d ≈t + 1 μ ( ¯ X(t) + 1 √ n ˆ X(t)) d = 1 μ F (t) + 1 √ n W 0 (F (t))−σμW (F (t)) ! , where the second d = is due to the fact that we used the Brownian motion scaling property. Note that this depends on the arrival distribution F alone. In the fluid limit of the busy time process, we see that ¯ B(t) =F (t)/μ which is the fraction of time from the interval [0,t] that the queue has spent serving. 1.5 Waiting Time and Sample Path Little’s Law Little’s Law is a fundamental tenet of queueing theory, that provides immediate insight into the operation of a queue. While the standard Little’s Law relates averages, in this section we prove a large population asymptotic functional relationship that holds on sample paths of the queue length and workload approximations. One may also view this ’sample path Little’s Law’ as parallel to a snapshot principle in the conventional heavy-traffic setting. First, the accelerated or fluid-scaled virtual waiting time process is Z n (t) = V n n A n (t) n !! − B n (t)−t1 {t≤0} ,∀t∈ [−T 0 ,∞). Proposition 1.3 (Fluid Little’s Law) The fluid-scale workload process is asymptotically related to the queue length fluid limit as n→∞: Z n a.s. −→ ¯ Z := ¯ Q/μ−e in (D lim ,J 1 ), where e :R→ [0,∞) is defined as e(t) :=t1 {t≤0} ∀t∈R . Proof: First note thatZ n (t) can be rewritten asZ n (t) =V n n A n (t) n !! − 1 μ A n (t) n + 1 μ A n (t) n − t1 {t≤0} −B n (t) ! . Proposition 1.1 implies that ¯ V n (t) a.s. −→ t/μ in (D lim ,J 1 ). Now, using the random time change theorem (Theorem 5.3 in [41]) and setting h = A n /n it follows that, as n→∞, V n ◦A n − 1 μ A n n ! a.s. −→ 0 in (D lim ,J 1 ). Using Proposition 1.1 and Corollary 1.1, 20 substituting for ¯ B(t), we have ¯ Z(t) = 1 μ ¯ Q(t)−t1 {t≤0} . Remarks. 1. The term e(t) = t1 {t≤0} accounts for the fact that an arrival at time t < 0 would require−t time units for service to start. Now, consider the diffusion-scale virtual waiting time process given by ˆ Z n (t) = √ n(Z n (t)− ¯ Z(t)) ∀t∈ [−T 0 ,∞). Proposition 1.4 below proves a diffusion approximation to ˆ Z n and relates the sample paths of the limit process to that of ˆ Q. Proposition 1.4 (Diffusion Little’s Law) The diffusion scaled virtual waiting time pro- cess satisfies an FCLT in the limit as n→∞: ˆ Z n ⇒ ˆ Z := 1 μ ˆ Q +σμ 1/2 W◦ ¯ B−σμ 1/2 W◦ F in (D lim ,M 1 ). Proof: Expanding the definition of ˆ Z n (t) and introducing the term 1 μ A n (t) n , we obtain ˆ Z n (t) = √ n V n (A n (t))− 1 μ A n (t) n + 1 μ A n (t) n − F (t) μ + ¯ B(t)−B n (t) ! . Using the Random Time Change Theorem (Section 17 of [42]), Proposition 1.1 and Proposition 1.2 √ n V n ◦A n − 1 μ A n n ! ⇒−σμ 1/2 W◦ F μ in (D lim ,J 1 ). (1.23) Finally, usingthisfact, Proposition1.2andCorollary1.2, itfollowsthat ˆ Z n ⇒ ˆ Z =σμ 1/2 W◦ F μ + 1 μ W 0 ◦F + ˆ B in (D lim ,M 1 ). Note that W and W 0 are independent processes. Adding and subtracting the process σμ 1/2 W◦ ¯ B where W is the Brownian Motion in (1.23), we obtain ˆ Z = 1 μ ˆ Q + σμ 1/2 W◦ ¯ B−σμ 1/2 W◦ F μ ! . Remarks. 1. The limit process in Proposition 1.4 is equal to ˆ Z(t) = 1 μ ˆ Q(t)−σμ 1/2 W ¯ Q(t) μ ! . (1.24) 21 Interestingly, theextradiffusiontermisnon-zeroonlywhenthefluidlimitofthequeuelength process is positive, indicating that it arises from temporal variations in the operating regimes of the queue. To see this, note that the variance of the diffusion term is σ 2 μ E W ( ¯ B(t))− W F (t) μ ! 2 = σ 2 μ ¯ B(t) + F (t) μ − 2 ¯ B(t)∧ F (t) μ ! , where x∧ y := min(x,y). Clearly, the expression on the right-hand side changes depending upon the ratio of the number of users arrived to the number served in the fluid regime at time t. It follows that σ 2 μE W ( ¯ B(t))−W F (t) μ ! 2 =            σ 2 μ F (t) μ − ¯ B(t) ! , F (t) μ ¯ B(t) > 1 σ 2 μ ¯ B(t)− F (t) μ ! , F (t) μ ¯ B(t) ≤ 1. It is easy to see that the first condition above, F (t)/(μ ¯ B(t)) > 1, implies ¯ Q(t)/μ > 0. The second condition, F (t)/(μ ¯ B(t))≤ 1, implies ¯ Q(t) = 0. This in turn, implies (F (t)− μt1 {t≥0} ) + Ψ(F (t)−μt1 {t≥0} ) = 0. Rearranging this expression, it follows that F (t) = μt1 {t≥0} − Ψ(F (t)−μt1 {t≥0} ). Now, using the definition of ¯ B from (1.13) we haveF (t)/(μ ¯ B(t)) = 1. It follows that the diffusion term is equal in distribution to the following (time-changed) Brownian Motion σμ 1/2 W ( ¯ B(t))−W F (t) μ !! d =            σμ 1/2 W F (t) μ − ¯ B(t) ! =σμ 1/2 W ¯ Q(t) μ ! , ¯ Q(t)> 0 σμ 1/2 W ¯ B(t)− F (t) μ ! = 0, ¯ Q(t) = 0. This leads to expression (1.24). 2. We note that ˆ Z can be interpreted as a sample path Little’s Law in the diffusion limit. This result is useful because it provides a sample path relationship between the workload and current queue state. Note that the FCLT of the workload process in a G/GI/1 queue (see Chapter 6 of [41] for details) with arrival rate λ and service rateμ has the form ˜ Z(t) = 1 μ ˆ Q(t) +σμ 1/2 (W ((ρ∧ 1)t)−W (ρt)), where ρ =λ/μ is the traffic intensity function for the 22 G/GI/1 queue, and this is similar to ˆ Z. The extra diffusion term in (1.24) captures the variation of the workload, as the (fluid) queue transitions between various operating states (see Section 1.6 for more details on these states). 3. Another interpretation of the termσμ 1/2 W ( ¯ Q(t)/μ) is that it is in fact the diffusion limit to the service backlog at time t, and the variation in the backlog at each point in time is captured in the term ˆ Q/μ. Suppose that f(t) < μ then the fluid queue length process is zero and the server will idle, and the zero state is recurrent for the queue length process. The workload in the system (for most of the time whenf(t)<μ) should be 0. On the other hand if F (t) = μt, so that the fluid queue length is zero but the server does not idle, it is reasonable to expect that the virtual waiting time is zero for an arrival at time t. However, there is a non-zero probability of the queue being backlogged at time t, and this fact is captured in the term ˆ Q/μ. 1.6 Queue Regimes and States As noted in Section 2.3.3, the diffusion limit for the queue length process is piecewise con- tinuous, with discontinuity points determined by the fluid limit. Indeed, the discontinuity points are precisely where the fluid limit switches between being ‘overloaded’ and either ‘underloaded’ and/or ‘critically-loaded’ regimes. We now provide formal definitions of these notions, in terms of the fluid limit arrival and service processes. We also characterize the sample path of the queue length limit process, and the points at which it has discontinuities. Developments in this section follow the study of the directional derivative limit process in [19]. However, the limit processes and the setting of our model are different, as our limit process is a function of a tied down Gaussian process while in [19] the limit process is a function of a standard Brownian motion. Thus, where necessary, we prove some of the facts about the sample paths. 23 1.6.1 Regimes of ¯ Q It is useful to characterize the state of a queue in terms of a “traffic intensity” measure. For instance, in the case of a G/G/1 queue, the traffic intensity is well defined as the ratio of the arrival rate to the service rate. This definition is inappropriate for the Δ (i) /GI/1 queue, as these systems can be time varying. In [16], a traffic intensity function for the M t /M t /1 queue with arrival rate λ(·) and service rate μ(·) was introduced as the continuous function ρ ∗ (t) := sup 0≤r≤t R t r λ(u)du R t r μ(u)du , t> 0. Note thatρ ∗ follows from the pre-limit model describing the arrival and service processes in the M t /M t /1 queue. For the Δ (i) /GI/1 queue we define the traffic intensity in terms of the fluid limit: ρ(t) :=                    ∞, ∀t∈ [−T 0 , 0] sup 0≤r≤t F (t)−F (r) μ(t−r) , ∀t∈ [0, ˜ T ] 0, ∀t> ˜ T, (1.25) where ˜ T := inf{t> 0|F (t) = 1 and ¯ Q(t) = 0}. Note that we define the traffic intensity to be ∞ in the interval [−T 0 , 0] as there is no service, but there can be fluid arrivals. For example, with F uniform over [−T 0 ,T ], ρ can be shown to be ρ(t) = t∧T t 1 μ(T +T 0 ) ,∀t∈ [0, ˜ T ]. Note that ρ is continuous in time. Now, consider the following obvious definitions of the operating regimes of the fluid Δ (i) /GI/1 queue. Definition 1.1 (Operating regimes.) The Δ (i) /GI/1 queue is (at time t) 1. overloaded if ρ(t)> 1. 24 2. critically loaded if ρ(t) = 1. 3. underloaded if ρ(t)< 1. The operating regimes can also be referenced in terms of the process ¯ Q, which in many instances is more intuitive. The following lemma presents this equivalence. Lemma 1.3 The Δ (i) /GI/1 queue is 1. overloaded at time t if ¯ Q(t)> 0. 2. critically loaded at time t if ¯ Q(t) = 0, ¯ X(t) = Ψ( ¯ X)(t) and there exists an r<t such that Ψ( ¯ X)(t) = Ψ( ¯ X)(s) for all s∈ [r,t]. 3. underloaded at time t if ¯ Q = 0, ¯ X(t) = Ψ( ¯ X)(t) and there exists an r < t such that Ψ( ¯ X)(t)> Ψ( ¯ X)(s) for all s∈ (r,t). The proof of the lemma is in the appendix. Figure 1.3 shows an example of the various operating regimes with the displayed arrival time distribution F and service rate μ> 1/T. Here,BB refers to a Brownian Bridge process andBM refers to a Brownian motion process. Theorem 1.2 proved a diffusion limit to the standardized queue length process, and we have shown that Q n d ≈n ¯ Q + √ n ˆ Q. As noted in the remarks after Theorem 1.2, the queue length process switches between being a ’free’ diffusion BB+BM (when the fluid limit model is overloaded), to a ’reflected’ diffusion R(BB+BM) (when the fluid limit model is critically loaded) and to a ’zero’ process 0 (when the fluid limit model is underloaded). Notice that these regimes correspond to those of a time homogeneous G/G/1 queue. However, since the queue length fluid limit in the Δ (i) /GI/1 queue can also vary with time wealsoidentifythefollowing“finer”operatingstates; thisisanalogoustotheM t /M t /1queue as demonstrated in [19]. In particular, these states are useful in studying the approximation 25 overloaded underloaded critically loaded overloaded underloaded overloaded BB+BM BB+BM R(BB+BM) R(BB+BM) 0 0 Figure 1.3: An illustration of the various operating regimes of a transitory queueing model. Here, we consider the i.i.d. sampling Δ (i) model. to the distribution of queue length process on local time scales. We also note that Louchard [39]identifiedsomeoftheseoperatingregimesinhisanalysis. Thedefinitionsbelowformalize the intuitive presentation in [39]. Definition 1.2 (Operating states.) A transitory queue is at (i) end of overloading at time t if ρ(t) = 1 and there exists an open interval (a,t) or (t,a) such that ρ(r)> 1 for all r in that interval. (ii) onset of critical loading at time t if ρ(t) = 1 and there exists a sequence λ n ↑ t such that ρ(λ n )< 1 for all n. (iii) end of critical loading at timet ifρ(t) = 1, and there exists a sequenceλ n ↑t such that ρ(λ n ) = 1 for all n and a sequence γ n ↓t such that ρ(γ n )< 1 for all n. (iv) middle of critical loading at time t if ρ(t) = 1, and t is in an open interval (a,b), such that sup t∈(a,b) ρ(t)≥ 1 and there exists a sequence λ n ↑t such that ρ(λ n ) = 1 for all n. We illustrate how the limit process can be used to approximate the queue length distri- bution of the exact (pre-limit) model. Our goal is to study this distributional approximation as ¯ Q and ˆ Q vary through the various operating regimes and states as defined above. 26 Theorem 1.3 (Distributional Approximations) The queue length can be approxi- mated in the various operating regimes as follows. (i) Overloaded state. Lett∈ (t ∗ ,τ) be a time instant of overloading in the overloaded interval, where t ∗ := sup∇ ¯ X t and τ := inf{s>t ∗ |ρ(s) = 1}. Then Q n (t) √ n − √ n(F (t)−F (t ∗ )−μ(t−t ∗ ))⇒ ˆ X(t) +X ∗ , as n→∞ whereX ∗ := sup s∈∇ ¯ X t ∗ (− ˆ X(s)). Further, ˜ Z n t := √ n(F (t)−F (t ∗ )−μ(t−t ∗ ))+ ˆ X(t)+X ∗ is the strong solution to the stochastic differential equationd ˜ Z n t = √ n(f(t)−μ)dt+ q g 0 (t)dW t ∀t∈ (t ∗ ,τ) with initial condition ˜ Z t ∗ = ˆ X(t ∗ )−X ∗ , where and g(t) =F (t)(1−F (t)) +σ 2 μ 3 ¯ B(t). (ii) Underloaded state. If t is a point of underloading, i.e. if ρ(t) < 1, then Q n (t) √ n ⇒ 0, as n→∞. (iii) Middle- and End-of-critically-loaded state. An open set of the domain (t ∗ ,τ) is a criti- cally loaded interval, where t ∗ is a point in the onset of critically loaded state and τ a point at the end of critically loaded state, as defined in Definition 1.2. For any t∈ (t ∗ ,τ), let u =t−t ∗ and we have, as n→∞, Q n (t) √ n ⇒ ( ˆ X(t) + sup 0≤s≤u (− ˆ X(s))), where ˆ X(u) d = ˆ X(t)− ˆ X(t ∗ ), and ˆ X(t) d = R t −T 0 q g 0 (s)dW s . (iv) End of overloading state. Let t be a point of end of overloading. Then, for all τ > 0 Q n (t− τ √ n ) √ n ⇒   ˆ X(t) +   sup s∈∇ ¯ X t \{t} (− ˆ X(s))   − (f(t)−μ)τ   + , as n→∞, where f(t) is the density function associated with the fluid limit F. 27 The proof is relegated to the appendix. Remarks: 1. Overloaded regime. (i) Then the approximate distribution is Gaussian with mean F (t)−μt. However, the variance is affected by the fact that the queue may have idled in the past. Recall that the variance is g(t) =F (t)(1−F (t)) +σ 2 μ 3 ¯ B(t), where from Corollary 1.1 ¯ B(t) = 1 {t≥0} − 1 μ Ψ( ¯ X)(t). (ii) We note that this result is analogous to case 5 of Section 4 in [39]. However, in [39], the author notes that no reflection need be applied in an overloaded sub-interval, and proceeds to derive the limit process (in this interval alone) as W 0 ◦F (t)−σμ 3/2 W (t). This is not entirely accurate as the starting state of the process in each new interval of overloading must be factored into the approximation. That is, while∇ ¯ X t is fixed for all t in an overloaded sub-interval, the value sup s∈∇ ¯ X t (− ˆ X(s)) provides the starting state for the diffusion in such an interval. 2. Critically-loaded regime. The queue length process in the critically loaded regime is approximated by a driftless reflected process, with continuous sample paths, with starting state ˆ X(t ∗ ). By the definition of a critically loaded state ρ(t) = 1 at all such points and ∇ ¯ X t “accumulates” the points of critical loading, as t evolves through the critically loaded interval. It follows that the set∇ ¯ X t is the interval (t ∗ ,t]. 3. End of overloading regime. Asnotedinthedefinition, apointtisoneofend-of-overloading if the traffic intensity is 1 at t, and is strictly greater than 1 at all points to the left of it. Here, we are primarily interested in the rate at which the queue empties out asymptotically as overloading ends. Consider a sequence of τ n defined as a sequence of times at which the queue in the nth system first empties out, and define v :=t− τn √ n . Then, from Theorem 1.3 τ n = √ n(t−v)⇒ ˆ X(t) + sup s∈∇ ¯ X t \{t} (− ˆ X(s)) f(t)−μ 28 Thus, it can be seen that the time at which the queue empties out converges to a Gaussian random variable. A similar conclusion was drawn in [39] and in [19] for theM t /M t /1 queue. 1.6.2 Sample Paths We now characterize a typical sample path of the limit process ˆ Q. Proposition 1.5 The process ˆ Q is upper-semicontinuous almost surely. The following proposition summarizes where discontinuities occur in ˆ Q. We note that this is also part of Theorem 3.1 of [19]. Since the proof follows that in [19] we omit it. Proposition 1.6 ˆ Q is discontinuous at time t, with a nonzero probability, if and only if t is the end-point of overloading or critical loading. The set of such points is nowhere dense. Remarks. 1. We note that the queue length limit sample paths for theM t /M t /1 model are alsoupper-semicontinuousasshowninTheorem3.1of[19]. Therethesequenceofconverging processes was shown to be monotone, which easily leads to upper-semicontinuity by Dini’s Theorem. As this monotonicity property does not hold for the corresponding processes in the Δ (i) /GI/1 model, we argue that the sample path is upper-semicontinuous directly from the characterization of the points of continuity and discontinuity in the domain of the sample path. 2. The intuition for the regime switching behavior proved in is easy to see in the case of a uniform arrival distribution with early-bird arrivals, such that the service rate is greater than the value of the density function function. Here, the (fluid) queue is overloaded on the interval [−T 0 ,τ) with the singleton set∇ ¯ X t ={−T 0 }, and underloaded on the interval (τ,∞) with the singleton set∇ ¯ X t ={t}. Atτ itself, there are two points in the set∇ ¯ X t ={−T 0 ,τ}. Thus, there is a discontinuity due to the fact that the set∇ ¯ X t changes from being a singleton on the interval [−T 0 ,τ) to{−T 0 ,τ} at τ. 29 1.7 Examples and Simulations We illustrate the queue length process approximations with uniform and exponential arrival timedistributions. Theformerisinteresting, astheuniformdistributionemergesasthemean field equilibrium arrival profile when arriving users are strategic about when they enter the queue in order to minimize their delay through the queue; see [27, 49]. The exponential distribution case serves to illustrate the fact that many of the conclusions of our theorems can be carried over to infinite support arrival time distributions, though the limit results remain to be fully justified. 1.7.1 Uniform Arrival Distribution The uniform arrival case is particularly simple and illustrates the discontinuities in the limit processes. Recall that∇ ¯ X t is a correspondence that maps each time t to the set of points (upto t) at which the fluid netput process is equal to its infimum at t. Corollary 1.3 Let F be the uniform distribution on [−T 0 ,T ], where−T 0 < 0. Then, ˆ Q(t) =                    W 0 (F (t))−σμ 3 2 W (t), ∀t∈ [−T 0 ,τ) (W 0 (F (τ))−σμ 3 2 W (τ)) + (−(W 0 (F (τ))−σμ 3 2 W (τ))) + , t =τ 0, ∀t∈ (τ,∞), where τ = inf{−T 0 ≤t<∞|F (t) =μt}. 30 Proof: Recall from Theorem 1.2 that ˆ Q = ˆ X +sup s∈∇ ¯ X · (− ˆ X) where ˆ X =W 0 ◦F−σμ 3 2 W◦ ¯ B, and ¯ B is the fluid busy time process. Now, using the definition of∇ ¯ X t , it is easy to deduce that in this case we have ∇ ¯ X t =                    {−T 0 }, ∀t∈ [−T 0 ,τ), {−T 0 ,τ}, t =τ, {t}, ∀ t∈ (τ,∞). Further, Corollary 1.1 yields ¯ B(t) =          t, ∀t∈ [−T 0 ,τ], 0, ∀t∈ (τ,∞). Using these facts the conclusion follows by substitution. The time τ can be interpreted as the first time that the fluid service process catches up with the fluid arrival process. For a uniformF there is at most one such point, but in general there can be many such points. Remarks: 1. A useful way to interpret the discontinuity atτ in Corollary 1.3 is to consider theprocessonthetwosub-intervalsseparatelyandtryto“patch”themtogether. If ˆ Q(τ−) = ˆ X(τ) = ˆ Q(τ) > 0 we should expect a free diffusion path on the interval [−T 0 ,τ], and a reflected process such that the path is 0 on (τ,∞). Furthermore, ˆ Q(τ) becomes the “starting state” for the process on the interval (τ,∞), and the reflection operator is applied an instant after τ. On the other hand, if ˆ Q(τ−) = ˆ X(τ−)≤ 0 we have a free diffusion on [−T 0 ,τ) and the zero process on [τ,∞), i.e., the process drops to zero at τ. Thus, ˆ Q(τ−) provides the starting conditions for the new “regime” of the diffusion, as the process transitions from [−T 0 ,τ) to (τ,∞). 31 2. We note that in [39], a diffusion approximation to the queue length process is derived independently for different operating regimes, and as such does not involve the directional derivative reflection map. These limit results have not been “patched” together to obtain a “process-level” convergence result, which is precisely where the mathematical challenges lie. Note that the nature of the discontinuity at ˆ Q(τ) depends on the the sign of ˆ X(τ). Following [19] it can be shown t is a point of right-discontinuity for a function x∈D lim if x isleft-continuousatt, andx(t−)>x(t+). Ontheotherhand,tisapointofleft-discontinuity if x is right-continuous at t, and x(t+)>x(t−). Corollary 1.4 Let F be the uniform distribution over [−T 0 ,T ], where T 0 > 0, and τ = {−T 0 ≤t<∞|F (t) =μt1 {t≥0} }. Then, for the process ˆ Q in Corollary 1.3, we have (i) [−T 0 ,τ)∪ (τ,∞) are points of continuity. (ii) τ is a point of right-discontinuity, when ˆ X(τ)≥ 0. (iii) τ is point of left-discontinuity, when ˆ X(τ)< 0. The proof is available in the Appendix. Simulations can provide insight into the accuracy of the approximations for various pop- ulation sizes. Consider a uniform arrival distribution over the interval [−20, 40], with service times i.i.d. and exponentially distributed with parameter μ = 0.03. Figures 1.4(a) and 1.4(b) show the sample mean and the sample variance of the (scaled) queue length process forn = 10, 25, 100, 1000 over 10,000 sample runs. Note that asn increases, the sample mean 32 0 0.1 0.2 0.3 0.4 -20 -10 0 10 20 30 40 50 60 E[Q n (t)/n] Time Mean Queue Length Theory n=10 n=25 n=100 n=1000 (a) Sample queue length process mean for n = 10,25,100,1000, averaged over 10000 simulation runs. 0 0.2 0.4 0.6 0.8 1 -20 -10 0 10 20 30 40 50 60 σ 2 n (t) Time Queue Length Variance Theory n=10 n=25 n=100 n=1000 (b) Sample queue length process variance for n = 10,25,100,1000, averaged over 10000 simulation runs. Figure 1.4: Typical sample paths, mean and variance envelopes of the queue length process forF uniform over [−20, 40], and exponentially distributed service times with rateμ = 0.03. approaches the fluid limit, and the sample variance approaches the theoretical variance of the queue length process. For the given F, the latter quantity is σ 2 (t) =                    F (t)(1−F (t)), ∀t∈ [−T 0 , 0] F (t)(1−F (t)) +σ 2 μ 3 t, ∀t∈ (0,τ) 0, ∀t>τ. Observe from Figure 1.4(a) that even for small n, the sample mean is quite close to the fluid limit for t < 0. However, once queueing dynamics come into play, the fluid limit is a good approximation only for n = 100 or larger. A similar effect is manifest for the diffusion limit as well: once service starts, and queueing dynamics come into play, the diffusion limit becomes a reasonably good approximation only for n = 1000 or larger. 1.7.2 Exponential Arrival Distribution AssumeF is an exponential distribution function with parameter λ> 0, so thatF (t) = 1− e −λt and−T 0 = 0. Keep in mind that this is unlike theM/GI/1 queue where the exponential distribution models the inter-arrival times. Recall that the limit results in Theorems 1.1 and 33 1.2 are proved on compact sets of the domain [−T 0 ,∞). Therefore, the limit does not hold simultaneously at all points in the support of F, and proving the FSLLN and FCLT for infinite support distributions is beyond the scope of the current paper. However, observe that the queue length fluid model can be conjectured to be (i) If μ≥λ, then ¯ Q(t) = 0∀t∈ [0,∞). (ii) If μ<λ, then ¯ Q(t) =          (1−e −λt −μt) ∀t∈ [0,τ) 0 ∀t≥τ, where τ := inf{t≥ 0|F (t) = μt} is the last instant the fluid queue length is positive (also known as the makespan). To see this, recall the definition of ¯ Q(t) and notice that if μ≥λ then λe −λt ≤ μ, ∀t > 0. This implies that the queue is underloaded, as defined in Section 1.6.1. On the other hand, if μ < λ, the system shifts from overload to underload, per our definition in Section 1.6.1. It can be shown that τ = 1 λ W − λ μ e − λ μ + 1 μ , where W (·) is the Lambert W-function. To see this, recall that it is the first (strictly positive) solution to e −λt = 1−μt. Substituting in−x =−λt + λ μ , we have xe x =− λ μ e − λ μ . It is well known that this is the defining equation for the Lambert W functionW , implying thatx =W − λ μ e − λ μ . Substituting back for t we obtain the expression for τ. The fluid model allows us to conjecture the corresponding diffusion refinement. Let ˆ Q be the queue length diffusion model. Then, (i) If μ≥λ, then ˆ Q(t) = 0∀t∈ [0,∞). (ii) If μ<λ, then ˆ Q(t) =                    W 0 (F (t))−σμ 3 2 W (t) ∀t∈ [0,τ) (W 0 (F (t))−σμ 3 2 W (t)) + (−W 0 (F (t)) +σμ 3 2 W (t)) + t =τ 0 ∀t∈ (τ,∞). 34 0 0.05 0.1 0.15 0.2 0 10 20 30 40 E[Q n (t)]/n Time Mean Queue Length Theory n=10 n=25 n=100 n=1000 (a) Sample mean queue length for n = 10,25,100,1000, averaged over 30 simulation runs. 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 n 2 (t) Time Queue Length Variance Theory n=10 n=25 n=100 n=1000 (b) Sample variance for n = 10,25,100,1000, averaged over 30 simulation runs Figure 1.5: Typical sample paths, mean and variance envelopes of the queue length process for F exponential with parameter λ = 0.1 and exponentially distributed service times with mean rate μ = 0.05. The “proof” of this is straightforward. Part (i) follows from the fact that the fluid model is underloaded under the same condition. Part (ii) follows from the reasoning in the proof of Corollary 1.3. A little algebra shows that the variance cruve of the diffusion approximation ˆ Q when μ<λ is given by σ 2 (t) =          F (t)(1−F (t)) +σ 2 μ 3 t ∀t∈ [0,τ), 0 ∀t>τ. Let us consider a specific example, where λ = 0.1 and μ = 0.05, in which case it can be verified that τ = 15.9362. Figure 1.5(a) shows that for even low values of n, the fluid limit is a very good approximation to the observed mean queue length. Similarly, Figure 1.5(b) shows that the variance of the diffusion limit is a reasonable approximation to the variance of the queue length in the (accelerated) discrete event system. 35 We also note a very interesting connection between random graph theory and the Δ (i) /GI/1 queue, brought to our notice by J.S.H. van Leeuwaarden in a personal com- munication. Specifically, he has shown that the excursions of the queue length process in the discrete event system, observed at the departure times of jobs, also measures the size of the connected components of a random graph with n vertices. [50] shows that in the “large graph” limit (i.e., asn→∞), the connected components in a (nearly) critical Erdös- Rényi random graph (see [51] for details on these terms) can be related to the excursions of a Brownian motion on a parabola by a weak convergence limit result linking the two. This type of result is also intimately connected with the question of the final size of an epidemic in a critical random graph; see [52, 53] where the distribution of the final size in a critical Susceptible-Infected-Recovered (SIR) epidemic model is studied. Using a Taylor series expansion on the fluid limit of the queue length, it can be shown that for small t and ignoring terms of order 3 and higher, the diffusion approximation is a Brownian excursion on a parabola. This connection with the Δ (i) /GI/1 queue might provide a new framework to study the final size distribution of other epidemic models in the critical regime. 1.8 Appendix Proof of Lemma 1.2. Rewrite ˜ y n as ˜ y n = (Ψ( √ nx +y n )− Ψ( √ nx +y))− (Ψ( √ nx +y)− √ nΨ(x)). Now, using the fact that the Skorokhod reflection map is Lipschitz continuous under the uniform metric (see Lemma 13.4.1 and Theorem 13.4.1 of [43]) we have (Ψ( √ nx+y n )−Ψ( √ nx+y))≤ky n −yk, wherek·k is the uniform metric. It follows that ˜ y n ≤ky n −yk + (Ψ( √ nx +y)− √ nΨ(x)), Now, by Theorem 9.5.1 of [46] we know that as n→∞ (Ψ( √ nx +y)− √ nΨ(x)) a.s. → ˜ y, in (D lim ,M 1 ). 36 Using this result, and the fact that by hypothesis y n converges to y in (D lim ,J 1 ) we have ˜ y n a.s. → ˜ y, in (D lim ,M 1 ). Proof of Lemma 1.3. First, suppose ¯ Q(t) > 0. It follows that ¯ F (t)−μt > inf −T 0 ≤s≤t ( ¯ F (s)−μs) = w where the latter equality follows because the queue starts empty at time 0, and the fluid netput is positive before time 0 (Note that we ignore the positive part operator in the definition of Ψ, as the systems starts empty at time−T 0 ). Now, let t ∗ = sup{0≤ s≤ t|( ¯ F (s)−μs) = inf 0≤s≤t ( ¯ F (s)−μs)} be the point at which the infimum is achieved, on the right hand side. It follows that ¯ F (t)−μt> ¯ F (t ∗ )−μt ∗ , in turn yielding ρ(t) = sup 0≤s≤t ¯ F (t)− ¯ F (s) μ(t−s) > 1. Next, suppose ¯ Q(t) = 0, ¯ X(t) = Ψ( ¯ X)(t) and there exists an r<t such that Ψ( ¯ X)(t) = Ψ( ¯ X)(s) for all s∈ [r,t]. It follows that ¯ F (t)−μt =− sup −T 0 ≤s≤t (−( ¯ F (s)−μs)), implying there exists a point r ∗ ∈ [0,t] such that ¯ F (t)−μt = ¯ F (r ∗ )−μr ∗ . This, in turn, implies that sup 0≤s≤t ¯ F (t)− ¯ F (s) μ(t−r) ≥ ¯ F (t)− ¯ F (r ∗ ) μ(t−r ∗ ) = 1. However a simple contradiction argument shows that sup 0≤s≤t ¯ F (t)− ¯ F (s) μ(t−r) > 1 is impossible, implying that sup 0≤s≤t ¯ F (t)− ¯ F (s) μ(t−r) = 1. 37 Finally, consider case (iii). We have,∀r<t, −( ¯ F (t)−μt) = sup −T 0 ≤s≤t (−( ¯ F (s)−μs))> sup −T 0 ≤s≤r (−( ¯ F (s)−μs)). It follows that−( ¯ F (t)−μt)>−( ¯ F (r)−μr), implying 1> ¯ F (t)− ¯ F (r) μ(t−r) ∀r∈ [0,t). Proof of Theorem 1.3. (i) Overloaded regime. Proof: First, note that τ is the first instant of an end of overloading phase, and the current overloaded phase ends at τ. In the overloaded state ¯ Q(t) > 0, implying that Ψ( ¯ X)(t) is a constant. Using the definition of∇ ¯ X t it follows that Ψ( ¯ X)(t) =− ¯ X(t ∗ ), and ¯ Q(t) = ¯ X(t)− ¯ X(t ∗ ) = ( ¯ F (t)− ¯ F (t ∗ )−μ(t−t ∗ )). Next, from Theorem 1.2, it is obvious that Q n (t) √ n d ≈ ˜ Z n t . Next, from Remark 1 after Lemma 1.1, ˆ X(t)− ˆ X(t ∗ ) = R t t ∗ q g 0 (s)dW s , which can be seen to be a diffusion process that starts from 0 at t ∗ . Noting that∇ ¯ X t does not change on the interval (t ∗ ,τ), it follows that X ∗ = sup s∈∇ ¯ X t ∗ {− ˆ X(s)) is a fixed random variable, and ˜ Z n t has an initial condition ˜ Z n t ∗ = ˆ X(t ∗ )−X ∗ . It is straightforward to see that ˜ Z · n is the strong solution to the mentioned SDE, since it is adapted to the filtration generated by W. (ii) Underloaded regime. This result is immediate from the definition of the limit processes. 38 (iii) Middle- and End-of critically-loaded state. Proof: For any t ∈ (t ∗ ,τ) we have ¯ Q(t) = 0. From the weak convergence result in Theorem 1.2 we have Q n (t) d ≈ n ¯ Q(t) + √ n ˆ Q(t), and expanding the definition of ˆ Q it follows that Q n (t) d ≈ √ n( ˆ X(s) + sup s∈∇ ¯ X t (− ˆ X(s))). Using the fact that Ψ( ¯ X)(t) =w =− ¯ X(t)∀ t∈ (t ∗ ,τ) in a critically loaded regime, it follows that∇ ¯ X t = (t ∗ ,t] for t∈ (t ∗ ,τ). Thus, we have Q n (t) d ≈ √ n( ˆ X(s) + sup t ∗ <s≤t (− ˆ X(s))). Let u =t−t ∗ . Then, after a change of variables we obtain Q n (u +t ∗ ) d ≈ √ n( ˆ X(u +t ∗ ) + sup 0≤s<u (− ˆ X(s))). SinceW 0 is a Brownian Bridge process, the strong Markov property of Brownian motion ([45]) implies that ˆ X(u +t ∗ )− ˆ X(t ∗ ) = ˆ X(u). Substituting this into the expression above we see that we have, Q n (u +t ∗ ) = Q n (u) + ˆ X(t ∗ ), where ˆ X(t ∗ ) is the starting state of the process in the middle-of-critically loaded state. A simple change of variables will provide the desired result. A similar argument will hold for the end-of-critical loading state as well. (iv) End of Overloading state. Proof: By definition for any τ > 0, t− τ √ n is a point of overloading. Therefore Q n (t− τ √ n ) √ n = ˆ X n (t− τ √ n )+ √ n(F (t− τ √ n )−μ(t− τ √ n ))+Ψ( ˆ X n + √ n ¯ X)(t− τ √ n )− √ nΨ( ¯ X)(t− τ √ n ). Without loss of generality, we assume that service started when the queue was in the overloaded state, so that Ψ( ¯ X)(t− τ √ n ) = 0. Now, using the fact the derivative f exists, the mean value theorem implies the existence of a point ˜ t∈ [t− τ √ n ,t] such that F (t− τ √ n ) = F (t)−f( ˜ t) τ √ n . Adding and subtracting the term f(t)τ/ √ n to the expression above we have F (t− τ √ n ) =F (t)−f(t) τ √ n +f(t) τ √ n −f( ˜ t) τ √ n . 39 Substituting this into the expression for Q n above, and introducing the term ˆ X n (t), we obtain Q n (t− τ √ n ) √ n = ˆ X n (t− τ √ n )− ˆ X n (t) + ˆ X n (t) + √ n(F (t)−μt)− (f(t)−μ)τ +Ψ( ˆ X n + √ n ¯ X)(t− τ √ n ) + (f(t)−f( ˜ t)) τ √ n . Now, using Lemma 1.1 and the continuity of the limit process we see that ˆ X n (t− τ √ n )− ˆ X n (t)⇒ 0. Further, sincef is bounded by virtue of being defined on a finite interval we have τ(f(t)−f( ˜ t))/ √ n→∞ asn→∞. Next, consider the term ˆ Z(t) := ˆ X n (t)+ √ n(F (t)−μt)+ Ψ( ˆ X n + √ n ¯ X)(t− t √ n ). Let δ> 0 be sufficiently small, so that the following decomposition of the expression above holds, ˆ Z n (t) = sup −T 0 ≤s<t−δ ( ˆ X n (t) + √ n(F (t)−μt)− ˆ X n (s)− √ n ¯ X(s)) ∨ sup t−δ≤s≤t− τ √ n ( ˆ X n (t) + √ n(F (t)−μt)− ˆ X n (s)− √ n ¯ X(s)). Lett ∗ = sup{∇ ¯ X t \{t}}. ConsiderthefirsttermontheRHSabove,andcallit ˆ Z n 1 (t). Sincethe queue is overloaded beforet no points are “added” to the correspondence∇ ¯ X t , it follows from the definition of an end of overloading point that (F (t)−μt) =−Ψ( ¯ X)(t)≡−Ψ( ¯ X)(t ∗ +δ). This, in turn, provides ˆ Z n 1 (t) = ˆ X n (t) + Ψ( ˆ X + √ n ¯ X)(t ∗ +δ)− √ nΨ( ¯ X)(t ∗ +δ). Using Lemma 1.2, it follows that ˆ Z n 1 (t)⇒ ˆ X(t) + sup s∈∇ ¯ X t \{t} (− ˆ X(s)) as n→∞, followed by letting δ→ 0. Next, consider the second term ˆ Z n 2 (t) = sup t−δ≤s≤t− τ √ n ( ˆ X n (t) + √ n(F (t)−μt)− ˆ X n (s)− √ n ¯ X(s)) ≤ sup t−δ≤s≤t− τ √ n ( ˆ X n (t)− ˆ X n (s)) + sup t−δ≤s≤t− τ √ n √ n( ¯ X(t)− ¯ X(s)) ≤ sup t−δ≤s≤t ( ˆ X n (t)− ˆ X n (s)) + sup t−δ≤s≤t− τ √ n √ n( ¯ X(t)− ¯ X(s)). 40 For large n, as the queue is overloaded at t− τ √ n it follows that ˆ Z n 2 (t)≤ sup t−δ≤s≤t ( ˆ X(t)− ˆ X(s)) + √ n( ¯ X(t)− ¯ X(t− τ √ n )). Again, by the mean value theorem √ n( ¯ X(t)− ¯ X(t− τ √ n )) = √ n(F (t)−F (t− τ √ n )−μ τ √ n ) = √ n(f(t)−μ) τ √ n + (f(t)−f( ˜ t))τ, where ˜ t∈ [t− τ √ n ,t]. Since, ˜ t→ t as n→∞, by the (right) continuity of f it follows that f(t)−f( ˜ t)→ 0 as n→∞. Then it follows by an application of Lemma 1.1 (and using the Skorokhod’s almost sure representation) that lim n→∞ ˆ Z n 2 (t)≤ ˆ X(t) + sup t−δ≤s≤t (− ˆ X(s)) + (f(t)−μ)τ. On the other hand, for a lower bound, using the mean value theorem again, we have ˆ Z n 2 (t)≥ ˆ X n (t)− ˆ X n (t− τ √ n ) + (f(t)−μ)τ + (f(t)−f( ˜ t))τ. Once again, using the continuity of f, the almost sure representation theorem and Lemma 1.1, and noting the continuity of the limit process ˆ X, we have lim n→∞ ˆ Z n 2 (t)≥ (f(t)−μ)τ a.s. Now, using the limits derived for ˆ Z n 1 and ˆ Z n 2 it follows that Q n (t− τ √ n ) √ n =⇒ −(f(t)−μ)τ + sup s∈∇ ¯ X t \{t} ( ˆ X(t)− ˆ X(s))∨ (f(t)−μ)τ =   ˆ X(t) + sup s∈∇ ¯ X t \{t} (− ˆ X(s))− (f(t)−μ)τ   + . 41 Proof of Proposition 1.5. The proof is a consequence of the following lemma, which consolidates Lemmas 6.5, 6.6 and 6.7 in [19]. The lemma characterizes the points of discontinuity (and continuity) of the process ˜ Y (t) = sup s∈∇ ¯ X t (− ˆ X(s)) in relation to the correspondence∇ ¯ X t . We do not prove these conditions, but direct the interested reader to [19]. Lemma 1.4 A point t∈ [−T 0 ,∞) is characterized as follows. (i) Continuity Conditions. The following are equivalent: 1. t is a continuity point. 2. t ∈ ∇ ¯ X t = {t}, or t 6∈ ∇ ¯ X t , or t ∈ ∇ ¯ X t 6= {t} and t is not isolated in∇ ¯ X t and ∇ ¯ X t ⊆∇ ¯ X u for some u>t. (ii) Right-discontinuity Conditions. The following are equivalent: 1. t is a point of right-discontinuity. 2. t∈∇ ¯ X t 6={t} and∇ ¯ X u ⊆ (t,u]∀ u>r. 3. ˜ Y (t) = ˜ Y (t−)> ˜ Y (t+) =− ˆ X(t). (iii) Left-discontinuity Conditions. The following are equivalent: 1. t is a point of left-discontinuity. 2. t∈∇ ¯ X t 6={t} and t is isolated in∇ ¯ X t . 3. ˜ Y (t) = ˜ Y (t+) =− ˆ X(t)> ˜ Y (t−). 42 A point of right-discontinuity can be seen to be left-continuous, coupled with an ordering on the right and left limits, such that ˜ Y (t−) > ˜ Y (t+). Similarly, a point of left-discontinuity is right-continuous, and the limits are ordered such that ˜ Y (t+) > ˜ Y (t−). Using these definitions, we proceed to prove the upper-semicontinuity of the limit process. Proof: [Proposition 1.5] By definition, ˆ X is continuous, and it suffices to check that a sam- ple path of the component ˜ Y (t) = sup s∈∇ ¯ X t (− ˆ X(s)) is upper-semicontinuous. To see this, consider the pullback of the level set ˜ Y −1 [a,∞) ={t∈ [−T 0 ,∞)| ˜ Y (t)≥ a}. It suffices to check that this is a closed set; see [54]. Let{τ n }⊆{t∈ [−T 0 ,∞)| ˜ Y (t)≥a} be a sequence of points such thatτ n →τ asn→∞, whereτ∈ [−T 0 ,∞) is an arbitrary point in the domain of ˜ Y. Thus, if > 0, then there exists an n 0 ∈N such that∀ n≥n 0 , ≥τ−τ n ≥−. If τ is a continuity point, then the conclusion is obvious. On the other hand, suppose that τ is a left-discontinuity point. By part (iii) of Lemma 1.4 it follows that ˜ Y (τ−)< ˜ Y (τ+) = ˜ Y (τ). By the definition of a left-discontinuity there exits an interval [t ∗ ,τ), wheret ∗ = sup∇ ¯ X τ \{τ}, on which ˜ Y is (locally) continuous. Fix δ > 0, then there exists an η > 0 such that if τ− −t≥−η, then δ≥ ˜ Y (τ−)− ˜ Y (t)≥−δ. If is small enough, then there exists n 0 such that∀ n≥ n 0 , τ−−τ n >−η. It follows that δ≥ ˜ Y (τ n )− ˜ Y (τ−)≥ a− ˜ Y (τ−), implying that ˜ Y (τ−)≥ a−δ. Since δ is arbitrary, it follows that ˜ Y (τ−)≥ a, in turn implying that ˜ Y (τ)≥ 0. Thus, τ∈ ˜ Y −1 [a,∞). Next, suppose that τ is a right-discontinuity point. Then, from part (ii) of Lemma 1.4 we have ˜ Y (τ) = ˜ Y (τ−) < ˜ Y (τ+). Furthermore, for any u > τ, we have∇ ¯ X u ⊆ (τ,u] implying that these are continuity points (by part (i) of Lemma 1.4). Using an argument similar to that for a left-discontinuity, on points to the right of τ, it follows that ˜ Y (τ)≥a. This implies that the pullback set ˜ Y −1 [a,∞) is closed. As{τ n } is an arbitrary sequence in ˜ Y −1 [a,∞) it is necessarily true that ˜ Y is upper-semicontinuous. 43 Proof of Corollary 1.4. The proof of the corollary depends on Lemma 1.4 above. Proof: [Corollary 1.4] Recall that ˆ Q = ˆ X + ˜ Y, where ˜ Y (t) = sup s∈∇ ¯ X t (− ˆ X(s)). The proof of (i) follows directly from part (i) of Lemma 1.4. Next, recall from the proof of Corollary 1.3 that∇ ¯ X τ ={−T 0 ,τ}. Thus, τ is isolated in the set and it follows that part (iii) of Lemma 1.4 is satisfied. On the other hand, recall that∇ ¯ X t ={t}⊂ (τ,t], ∀t > τ, and τ can also be a point of right-discontinuity, by part (ii) of Lemma 1.4. Thus, τ is one or the other depending on the path of ˆ X. If ˆ X(τ)< 0 then ˜ Y (τ+) = ˜ Y (τ)> ˜ Y (τ−) and τ is a point of left-discontinuity. Otherwise, if ˆ X(τ)≥ 0, then s ˜ Y (τ) = ˜ Y (τ−) = 0 > ˜ Y (τ+) and τ is a point of right-discontinuity. 44 Chapter 2 Transitory Queueing Theory 2.1 Introduction In this chapter we show that the Δ (i) /GI/1 model introduced in Chapter 1 is only one specific example of a fairly broad class of ’transitory queueing models’. As noted in the introduction, we define a queueing model as ’transitory’ if its arrival process satisfies the following assumptions: Assumption 1: the fluid limit of the arrival process is a cumulative distribution function (i.e., right continuous and nondecreasing with finite total mass), and Assumption2: thediffusionlimitisatied-downGaussianprocess,possiblywithcontinuous sample paths. Here, we study three disparate traffic models that satisfy these assumptions. (i) The Δ (i) Traffic model: In the basic Δ (i) model, we assume that the arrival times of users are sampled independently from an identical distribution. Here, we consider the generalized Δ (i) traffic model wherein the arrival time of each user is independently sampled from non- identical distributions. Thus, the unordered arrival times now form a triangular array. To studythismodel, weprovidegeneralizationsoftheGlivenko-CantelliandDonsker’sTheorem for triangular arrays. We do so via a generalization of Hahn’s Central Limit Theorem [55] to non-identically distributed processes. Using the notion of a random distribution function introduced by Dubins and Freedman [56] we identify the fluid and diffusion limits for the generalized theorems, thus proving that the generalized Δ (i) traffic model satisfies Assumptions 1 and 2. 45 (ii) The Conditioned Renewal Process model: The Δ (i) traffic model seems very natural, particularly to those uninitiated in queueing theory. And yet, queueing theory has mostly focused on the renewal process traffic model. A key question is how are the two related, if at all? Thus, we next introduce the conditioned renewal arrival process model. Herein, the arrivals happen according to a renewal process but we condition on there being n arrivals by some time T. When the renewal process is Poisson, it is well known that the joint distribution of the arrival times is Uniform when conditioned (which is the Δ (i) arrival model with the Uniform distribution). From this, it is easy to conclude that the arrival processes are also equal in distribution. It is well known that a conditioned renewal process is not distributionally equivalent to a model with i.i.d. sampling from some distribution F. Herein, we show that, in fact, the conditioned renewal arrival process is asymptotically distributionally equivalent to a Δ (i) arrival model with some distribution F as n→∞, in the sense that both processes converge to the same weak limit, namely a Brownian bridge process. (iii) Scheduled Arrivals with Epoch Uncertainty model: Many queueing scenarios involve scheduled arrivals at appointment times, e.g., arrivals at a doctor’s office. And yet, there is randomness in the actual arrival time, around the scheduled time. The earliest reference to such a model of traffic is in [57], where it is referred to as “a regular arrival process with unpunctuality.” There is an increasing interest in such queueing models which are not amenable to analysis via known queueing theoretic methods. We model randomness in arrival times around the scheduled times as being uniformly distributed over a small interval. We show that even though the sampling model is different from the Δ (i) model, this is also a transitory queueing model that satisfies Assumptions 1 and 2, and has the same weak limit as the Δ (i) /GI/1 queueing model. This confluence of asymptotes for some disparate but natural models for ‘transitory’ queueing phenomena is an interesting coincidence. In fact, one may potentially construct other models as well that satisfy Assumptions 1 and 2. In many of these, the fluid and 46 diffusion limits will be the same as for the Δ (i) /GI/s queueing model. It is worth mentioning here that the Δ (i) /GI/1 model arose as an equilibrium model in [49, 27] where a finite population of users were considered to be strategically picking their arrival times. Thus, in some sense the Δ (i) /GI/1 queueing model can be considered canonical to the study of transitory queues just as the M/GI/1 and G/G/1 queueing models are to the study of stationary queueing systems. In this chapter we establish fluid and diffusion limits for the queue length process for the whole class of such models, under much more liberal assumptions on the arrival time distributions and the queue length limit process than Chapter 1. We do this by using the population acceleration technique. These results also have interesting implications for mean field models in games. Thepaperisorganizedasfollows. Westartwithabriefreviewofsomerelevantliterature. Section 2.2 introduces and defines a general transitory queueing model. Section 2.3 develops fluid and diffusion limits for performance metrics of transitory queueing models that satisfy Assumptions 1 and 2. Section 2.4 introduces three transitory queueing models that satisfy Assumptions 1 and 2, and show that the fluid and diffusion limits for all three coincide. 2.1.1 Related Literature The existing literature on transitory and non-stationary models was covered in Section 1.1.1. The results in this section are of a more fundamental nature, however. The generalized Δ (i) traffic model is closely related to generalizations of the Glivenko-Cantelli and Donsker’s Theorems. The main results here are [58] and [59]. However, these results only prove existence of ’generalized’ limits, while we can construct the limit processes under relatively mild restrictions. The conditioned Poisson process has been studied in some detail, with a lot of attention being paid to the order statistics property; see [60] and [61]. To the best of our knowledge, weak limits for conditioned renewal processes have not been considered before. However, these theorems are related to results on weakly dependent sequences [62, 63]. The 47 results on scheduled arrivals with epoch uncertainty have not been observed before in the literature. However, there has been long running interest in studying such models starting with [57], and as recent as [64]. However, none of these models consider a finite horizon/finite population setting. 2.2 The Transitory Queueing Model There is a finite population of N customers that arrive to an infinite buffer for service. The service opens at time 0; however, some customers arrive beforehand. The earliest possible arrival time is−T 0 ≤ 0. The random vector T := (T 1 ,T 2 ,··· ,T N )∈ [−T 0 ,∞) N represents the arrival times of theN customers. We assume all elements of T are finite with probability 1. The cumulative number of arrivals up to time t is A(t) = n X i=1 1 {T i ≤t} . (2.1) We call the A(0) customers that arrive before the service opens early birds. The s servers process the arrivals in a first-come-first-served manner. The servers are non-idling and service is non-preemptive. The ith customer to receive service from server j∈{1,...,s} has processing time ν j,i , which has CDF G and support [0,∞). The s i.i.d. infinite sequences of processing times{ν j,i ,i≥ 1} are mutually independent of each other and of the arrival epochs T. The service time mean is 1/μ :=Eν j,i <∞, and the variance is σ 2 := Var(ν j,i )<∞. The number of potential service completions if server j was busy in all of [0,t] is given by the renewal counting process S j , defined as S j (t) := sup{m≥ 1|V j (m)≤t},∀t∈ [0,∞), (2.2) where V j (m) := P m i=1 ν j,i . 48 Now, letQ represent the queue length process, including any customers in service and all buffered customers. The sample paths of Q are defined in terms of those of the arrival and service processes as Q(t) :=A(t)− s X j=1 S j (B j (t))≥ 0, ∀t∈ [−T 0 ,∞), (2.3) whereB j (t) is the busy time of serverj, defined as the amount of time serverj spent serving jobs in the interval [0,t]. Let B(t) := P s j=1 B j (t) be the total busy time of the queue. When s = 1, it follows that B 1 (t) := R t 0 1 {Q(s)>0} ds, for all t> 0; however, in general the characterization of each B j is complex, and depends on how arriving customers that find more than one server idle are routed. We do not provide an explicit representation for B j . Instead we provide conditions that must be satisfied by restricting when servers can be idle. Our analysis applies to any routing policy that satisfies those conditions. For a concrete example, the reader may assume that when an arriving customer finds more than one server idle, that arrival is equally likely to be served by any of the idle servers. The cumulative idle time of server j∈{1, 2,...,s} in [0,t] is I j (t) :=t1 {t≥0} −B j (t) ∀t∈ [−T 0 ,∞). Note that it is natural to track only how much idle time each server has had since the service opened at time 0, despite the fact that customers may have been waiting before time 0, when the servers were “off-duty”. The total cumulative idle time of all the servers I(t) := s X j=1 I j (t) must satisfy I(0) = 0, I is non-decreasing, and I(t) increases only if Q(t)≤s. Our objective is to characterize the time-dependent queue-length distribution in our tran- sitory queueing model However, the queue-length process is non-Markovian in general, which 49 makes the analysis very difficult. Our approach is to develop asymptotic approximations for the queue-length process as the population size N becomes large. 2.3 Performance Analysis of Transitory Queueing Sys- tems In this section we present an analysis of the queue length performance metric of the queueing model presented in Section 2.2, using the population acceleration technique. In Section 2.3.1 we first establish the large population asymptotics for the arrival and service processes for a generic transitory queueing model. Next, we establish fluid approximations for the queue length by proving a fSLLN Theorem in Section 2.3.2. Finally, we establish a fCLT for the queue length process and discuss its implications by considering a special case in Section 2.3.3 . 2.3.1 Large Population Asymptotics of Primitives We consider a sequence of systems indexed by n∈N. The customer population size in the nth system is N n , and we assume N n →∞ as n→∞. Our convention is to superscript all processes and quantities associated with the nth system by n. The arrival times in the nth system are T n := (T n 1 ,T n 2 ,··· ,T n Nn ), and the cumulative arrival processA n is as defined in (2.1) withT n i replacingT i . Our analysis requires that the empirical arrival distribution A n /N n is well-behaved as n becomes large. Assumption 1 There exists a probability distribution functionF that has compact support such that the following holds. (a) The arrival process satisfies a functional Strong Law of Large Numbers: A n := A n N n a.s. → ¯ F in (D lim ,J 1 ), as n→∞. 50 (b) The arrival process satisfies a functional Central Limit Theorem: ˆ A n := q N n A n N n − ¯ F ⇒ ˜ W in (D lim ,J 1 ) as n→∞, where ˜ W is a zero mean Gaussian process with known covariance function, that is tied-down to 0 at−T 0 and T 1 := inf{t : ¯ F (t) = 1}. Forexample, whenN n =nand (T n 1 ,...,T n n )arei.i.d. samplesfromauniformdistributionon [0, 1], the Glivenko-Cantelli theorem guarantees that Assumption 1(a) holds with F (t) = t for t ∈ [0, 1], and, from Donsker, Assumption 1(b) holds with ˜ W a standard Brownian Bridge (see [65, 45] for a formal definition of a Brownian Bridge process and Theorem 13.1 in [42] for a statement of Donsker’s result). However, Assumption 1 is also satisfied in much greater generality, and we explore this in Section 2.4. In particular, Assumption 1 holds when arrival times are sampled from different distributions, for a conditioned renewal arrival model, and for a scheduled arrival model. In all the models we have investigated, the limit ˜ W is concentrated onD⊂D lim . With a small loss of generality, we assume thatP( ˜ W∈D) = 1 for the remainder of this paper. The service times in the nth system are small; specifically, the service times of the ith arrival to server j in the nth system is ν n j,i := ν j,i N n , i = 1, 2,..., for each j∈{1,...,s}, so that V n j (m) = P m i=1 ν j,i /N n , and (1.2) defines S n j with V n j replacing V j . Furthermore, the fluid-scaled service process is S n (t) := 1 N n s X j=1 S n j (t), t≥ 0, 51 and the diffusion-scaled service process is ˆ S n (t) := q N n S n (t)−sμt , t≥ 0. Our analysis requires that the arrival and service processes, when appropriately scaled, jointlysatisfyafunctionalstronglawoflargenumbersandafunctionalcentrallimittheorem. The multi-dimensional result is shown to hold in the weak J 1 topology WJ 1 onD lim ×D lim ; see Chapter 11 of [43] for details. Proposition 2.1 As n→∞, the fluid-scaled arrival and service processes jointly satisfy ( ¯ A n (t), ¯ S n (t)1 t≥0 ) a.s. −→ ( ¯ F (t),sμt1 {t≥0} ) in (D lim ×D lim ,WJ 1 ), (2.4) and the diffusion-scaled arrival and service processes jointly satisfy ( ˆ A n , ˆ S n )⇒ ˜ W,W ! in (D lim ×D lim ,WJ 1 ), (2.5) where W (t) =          σμ 3/2 P s j=1 W j (t) t≥ 0 0 t< 0 is the sum of independent standard Brownian motion processes W j , jointly independent of ˜ W and e : [0,∞)→ [0,∞) is the identity map. The proof of Proposition 2.1 (see below) follows from Assumption 1 and standard results on renewal processes, except for the subtlety that those results are usually proved inD instead ofD lim . In particular, the following technical Lemma, used repeatedly throughout this paper, is useful to show (2.5). Its proof can be found in the appendix. Lemma 2.1 (Technical Lemma) LetD lim andD represent the Borel σ-algebra generated by the J 1 topology onD lim andD (resp.) (i) Let x be a random element taking values in 52 the spaceD, whereD⊂D lim . Then, the measure induced by x on (D,D) can be extended to (D lim ,D lim ). (ii) Let{x n }, n≥ 1 be a collection of random elements inD, such that x n ⇒x in (D,J 1 ) as n→∞. Then, x n ⇒x in (D lim ,J 1 ) as n→∞. Proof: [Proposition 2.1] First note that by Assumption 1 and the assumed independence of the arrival and service processes, it is enough to show the convergence of S n (t)1 t≥0 and ˆ S n . These convergences hold in (D,J 1 ) by the functional strong law and functional central limit theorems for renewal processes (see, for example, Chapter 5 of [41] and Theorem 7.3.2 and Corollary 7.3.1 in [43]), and the continuity of the addition operator with respect to the J 1 topology when the processes are continuous. Finally, the convergence in (D lim ,J 1 ) in (2.4) is immediate since the measure induced by the limits concentrates degenerately on the fixed sample paths of the limit process inD⊂D lim , and in (2.5) is immediate from Lemma 2.1. The transitory queueing system model having customer population size N n (in the nth system) is defined as in Section 2, except thatA n replaces A in (1) andS n replaces S in (2). Then, the queue-length processQ n evolves as in (3) withA n andS n replacing A and S, and the busy time processB n replacing B. The busy time processB n is defined through the idle time process I n accordingly. 2.3.2 Fluid Approximations We first derive the fluid limit for the queue-length process, and then the limit to the busy time process. Recall the definition of the queue length process in (2.3). The fluid-scaled queue length process is ¯ Q n (t) := Q n (t) N n = 1 N n A n (t)− 1 N n s X j=1 S n j (B n j (t)), (2.6) 53 where B n j (t) is server j’s fluid-scaled busy time process. Centering the right hand side of (2.6) by adding and subtracting the corresponding fluid-scaled processes, and introducing the process (sμt∀t≥ 0) we obtain ¯ Q n (t) = A n (t) N n − ¯ F (t) ! − s X j=1 S n j (B n j (t)) N n −μB n j (t) ! + ¯ F (t)−sμt1 {t≥0} ! + s X j=1 μI n j (t), whereI n j (t) =t1 {t≥0} −B n j (t) is the fluid-scaled idle time process. ¯ Q n is equivalently written as ¯ Q n (t) = ¯ X n (t) + ¯ Y n (t), ∀t∈ [−T 0 ,∞), where ¯ X n (t) := A n (t) N n − ¯ F (t) ! − s X j=1 S n j (B n j (t)) N n −μB n j (t) ! + ¯ F (t)−sμt1 {t≥0} ! and ¯ Y n (t) := s X j=1 μI n j (t) =μI n (t). In preparation for the main theorem in this Section, recall that the one-dimensional Skorokhod reflection map is a (Lipschitz) continuous functional under the uniform metric, (Φ, Ψ) :D lim →D lim ×D lim defined as x7→ Ψ(x)(t) := sup −T 0 ≤s≤t (−x(s)) + ∀t∈R and x7→ Φ(x)(t) :=x(t) + Ψ(x)(t), ∀x∈D lim and ∀t∈R The Skorokhod reflection map satisfies the following properties. The proof of claim (i) is part of Theorem 3.1 of [22], while (ii) is a standard property of the Skorokhod reflection map and a proof can be found in [41, 43]. Proposition 2.2 (i) Ψ(·) is continuous with respect to the uniform topology onD lim . (ii) Ψ(x)(t) is non-decreasing in t. Further, for any other pair of processes (z,y) ∈ (D lim ,D lim ) such that z = x +y≥ 0, y is non-decreasing, y(0) = 0 and y increases only if 54 z(t)≤k, for k > 0, the following relations hold: Φ(x−k)(t)≥z(t)−k≥ Φ(x)(t)−k and Ψ(x−k)(t)≥y(t)≥ Ψ(x)(t). Clearly, ify is such that it does not increase when z > 0, then the inequalities in (ii) match each other. This is called the dynamic complementarity property of the one-dimensional Skorokhod reflection map. Therefore, (ii) defines an approximate dynamic complementarity property. Theorem 2.1 (Fluid Limit) The pair ( ¯ Q n , ¯ Y n ) jointly converges as n→∞, ( ¯ Q n , ¯ Y n ) a.s. −→ (Φ( ¯ X), Ψ( ¯ X)) in (D lim ×D lim ,WJ 1 ), where ¯ X(t) = ( ¯ F (t)−sμt1 {t≥0} ). Proof: First note that ¯ Q n (t)≥ 0, ∀t∈ [−T 0 ,∞). It is also true that I n (−T 0 ) = 0 and dI n (t)≥ 0, ∀t∈ [−T 0 ,∞). From (ii) in Proposition 2.2 it follows that ¯ Q n (t)≥ Φ( ¯ X n )(t) and ¯ Y n (t)≥ Ψ( ¯ X n )(t). By definition, B n j (t) ≤ t for all j = 1,...,N and from (2.4) in Proposition 2.1, it follows that P s j=1 S n j ◦B n j n −μB n j a.s. −→ 0 in (D lim ,J 1 ). Therefore, by applying (2.4) in Proposition 2.1 to the arrival process it follows that ¯ X n a.s. −→ ¯ X in (D lim ,J 1 ). As a consequence of the limit derived above and the continuity of the reflection map from (i) of Proposition 2.2 we have lim inf n→∞ ( ¯ Q n , ¯ Y n ) ≥ lim n→∞ (Φ( ¯ X n ), Ψ( ¯ X n )) = (Φ( ¯ X), Ψ( ¯ X)) in (D lim ×D lim ,WJ 1 ) a.s. Next, using the upper bound in (ii) of Proposition 2.2 we have the relation ( ¯ Q n , ¯ Y n ) ≤ (Φ( ¯ X n − s Nn ) + s Nn , Ψ( ¯ X n − s Nn )). As s is fixed, it is obvious from the continuity of the reflection map that lim sup n→∞ ( ¯ Q n , ¯ Y n ) ≤ (Φ( ¯ X), Ψ( ¯ X)) a.s. in (D lim ×D lim ,WJ 1 ) This concludes the proof. Remarks 1. Theorem 2.1 shows that the fluid limit of the queue length process is ¯ Q(t) = ( ¯ F (t)−sμt1 {t≥0} )+sup −T 0 ≤p≤t (−( ¯ F (p)−sμp1 {p≥0} )) + ,∀t∈ [−T 0 ,∞). ¯ Q can be interpreted 55 as the sum of the fluid netput process and the potential amount of fluid lost from the system. Suppose that service started with some workload in the system at time 0 and that ( ¯ F (t)−sμt1 {t≥0} )< 0fort> 0,sothatthefluidserviceprocesshas“caughtup”andexceeded the cumulative amount of fluid arrived in the system by timet (for simplicity assumet> 0). Let ¯ f represent the density function associated with the distribution function ¯ F (if ¯ F has a discontinuity at some point t, then f(t) := f(t−)+f(t+) 2 ). Suppose ¯ f(t)−sμ< 0, implying that the netput process is decreasing att. In this case, sup −T 0 ≤p≤t (−( ¯ F (p)−sμp1 {p≥0} )) + = −( ¯ F (t)−sμt). This is the amount of extra fluid that could have been served, but is now lost. Figure 1.1 depicts an example queue length process in the fluid limit for the special case in Section 2.3.3, and its dependence on F and μ. In particular, notice that the process switches between being positive and zero, during the time the queue operates. In particular, observe that these ‘regimes’ correspond to when the queue is overloaded, underloaded and critically loaded. It is important to note that in any transitory queueing model, the (fluid) limit system can experience these changes unlike a G/G/1 queue. Formally, these regimes can be codified by defining a ‘load factor’ ρ in terms of the fluid limit system as follows: ρ(t) :=                    ∞, ∀t∈ [−T 0 , 0] sup 0≤r≤t ¯ F (t)− ¯ F (r) μ(t−r) , ∀t∈ [0, ˜ T ] 0, ∀t> ˜ T, (2.7) where ˜ T := inf{t > 0| ¯ F (t) = 1 and ¯ Q(t) = 0}. Note that we define the traffic intensity to be∞ in the interval [−T 0 , 0] as there is no service, but there can be fluid arrivals. Based on this, we can now define the regimes of the transitory queueing model. Definition 2.1 (Operating regimes.) The transitory queue is (i) overloaded if ρ(t)> 1. 56 (ii) critically loaded if ρ(t) = 1. (iii) underloaded if ρ(t)< 1. Thus, in Figure 1.1 the queue is overloaded between [−T 0 ,τ 1 ] and (τ 2 ,τ 3 ] and critically loaded between (τ 1 ,τ 2 ]. We refer the reader to Section 1.6 for further details. It is interesting to observe that the busy time of the queue, B n , does not converge to the identity process in contrast to the limit for the GI/GI/1 queue in the heavy-traffic approximation setting. The following corollary characterizes the busy time fluid limit. Corollary 2.1 The fluid scaled busy time process B n := P s j=1 B n j satisfies a fSLLN as n→ ∞: B n a.s. −→ ¯ B in (D lim ,J 1 ) (2.8) where ¯ B(t) := st1 {t≥0} − 1 sμ Ψ( ¯ X(t)),∀t∈ [−T 0 ,∞). Proof: By definition, we have B n (t) =st1 {t≥0} −I n (t) =st1 {t≥0} − ¯ Y n (t) μ . Theorem 2.1 now implies the limit. Note that ¯ B(t) = 0 for allt≤ 0, as Ψ( ¯ X)(t) = 0 on that interval. It is important to keep in mind that ¯ B is the total busy time of the entire queueing system. For each server, on the other hand, we can prove the following existence result. Corollary 2.2 For every j = 1,...,s, there exists a function ¯ B j ∈C such that B n j a.s. → ¯ B j in (D,J 1 ). Proof: Without loss of generality, lett, s ∈ [0, 1]. By definition,B n j is a uniformly bounded sequence of functions (on the given compacta) and |B n j (t)− B n j (s)| ≤ |t− s|, for any such pair t, s. Thus, B n j is uniformly Lipschitz, implying equicontinuity. Then, by the Arzela-AscoliTheoremthesequence{B n j }issequentiallycompact, sothatalimitexists. 57 An exact characterization of ¯ B j depends on the routing policy. However, this is not required for the rest of our analysis. 2.3.3 Diffusion Approximations Next, we derive the diffusion limit for the queue-length process in Section 2.3.3. In Section 2.3.3, we specialize this result to a specific instance of a transitory queueing system. Next, we show by a counterexample by convergence in the J 1 topology is not possible in general, in Section 2.3.3. Finally, we leverage our main result to develop the diffusion limit for the busy time process. Queue Length Process Define the diffusion-scaled queue length process as Q n (t) √ N n := A n (t) √ N n − s X j=1 S n j (B n j (t)) √ N n , ∀t∈ [−T 0 ,∞) (2.9) Rewriting this expression by introducing the term √ N n sμt1 {t≥0} and centering the terms on the right hand side Q n (t) √ N n = A n (t) √ N n − q N n ¯ F (t) ! − s X j=1 S n j (B n j (t)) √ N n − q N n μB n j (t) ! + q N n ( ¯ F (t)−sμt1 {t≥0} ) + q N n s X j=1 μ(t1 {t≥0} −B n j (t)). Using the definition of the idle time process √ N n I n j (t) = √ N n (t1 {t≥0} −B n j (t)), we can express Q n / √ N n as Q n √ N n = ˆ X n + q N n ¯ X + ˆ Y n (2.10) 58 where ˆ X n (t) := A n (t) √ N n − q N n ¯ F (t) ! − s X j=1 S n j (B n j (t)) √ N n − q N n μB n j (t) ! (2.11) = ˆ A n (t)− s X j=1 ˆ S n j (B n j (t)), ∀t∈ [−T 0 ,∞), and ˆ Y n := q N n s X j=1 μI n j . (2.12) Recall from Theorem 2.1 that ¯ X(t) = ( ¯ F (t)−μt1 t≥0 ) is the fluid netput process. We can think of ˆ X n as a diffusion refinement of the netput process. Lemma 2.5 in the Appendix proves that ˆ X n converges weakly to a Gaussian process ˆ X as a direct consequence of (2.5) in Proposition 2.1. In the rest of this section, we will use Skorokhod’s almost sure representation theorem [44,46]andreplacetherandomprocessesabovethatconvergeindistributionbythosedefined on a common probability space that have the same distribution as the original processes and converge almost surely. The requirements for the almost sure representation are mild; it is sufficient that the underlying topological space is Polish (a separable and complete metric space). We note without proof that the spaceD lim , as defined in this paper, is Polish when endowed with the J 1 topology. This conclusion follows from Chapter 12.8 of [43] and the fact that the proof there extends easily toD lim . The authors in [19] also point out that [47] has a more general proof of this fact. We conclude that we can replace the weak convergence in (2.5) by ( ˆ A n , ˆ S n ) a.s. −→ ˜ W,W ! in (D lim ×D lim ,WJ 1 ), where abusing notation we denote the new limit random processes by the same letters as the old ones. This implies that in Lemma 2.5 ˆ X n a.s. −→ ˆ X in (D lim ,J 1 ), as n→∞. 59 Our goal is to establish diffusion limits for the centered queue length process ˆ Q n (t) := q N n Q n (t) N n − ¯ Q(t) ! , (2.13) and the process ˜ Y n (t) := ˆ Y n (t)− q N n Ψ( ¯ X)(t). We achieve this by using part (ii) of Proposition 2.2 to bound (Q n (t)/ √ N n , ˆ Y n ) in terms of ˆ X n and ¯ X, and then establish the limit as n→∞. The limit for each is proved in the weaker M 1 topology, as opposed to the more common U or J 1 topologies as convergence to the directional derivative reflection map (Lemma 2.6 presented in the Appendix; see also [66] where a version of this theorem is proved in a special case) in general holds in (D lim ,M 1 ). In fact, in Proposition 2.4 below, a counterexample is provided that shows that the limit result is not achievable in the stronger J 1 topology, in general. Recall that (Φ, Ψ) is the Skorokhod reflection map. The directional derivative of the Skorokhod reflection map is defined below. Definition 2.2 (Directional Derivative Reflection Map) Let x∈D and y∈D. For fixed t∈ [0,∞) sup s∈∇ x,L t (−y(s−))∨ sup s∈∇ x,R t (−y(s)) := lim a→∞ Ψ(ax +y)(t)−aΨ(x)(t), (2.14) is the directional derivative of Ψ and∇ x,L t :={s≤t|x(s−) =−Ψ(x)(t)}, is a correspondence of points up to time t where the left limits of x achieve an infinimum and∇ x,R t :={s≤ t|x(s+) =−Ψ(x)(t)}, is a correspondence of points up to time t where the right limits of x achieve an infinimum. Theorem 9.3.1 of [46] proves the (pointwise) existence of the limit. In establishing our main result, we use Theorem 9.5.1 of [46] and the Lipschitz continuity of the reflection map (in 60 one-dimension) to prove the queue length diffusion limit in the M 1 topology. Of course, M 1 convergence is stronger than pointwise, in general. Theorem 2.2 (Diffusion Limit) (i) The diffusion scaled process ˜ Y n converges to a direc- tional derivative of the Skorokhod reflection regulator map: ˜ Y n ⇒ ˜ Y in (D lim ,M 1 ) (2.15) as n→∞, where ˜ Y (t) = sup s∈∇ ¯ X,L t (− ˆ X(s−))∨ sup s∈∇ ¯ X,R t (− ˆ X(s))∀t∈ [−T 0 ,∞) with∇ ¯ X,· t as in Definition 2.2. (ii) The diffusion scaled queue length process ˆ Q n converges to a reflected process, asn→∞: ˆ Q n ⇒ ˆ X + ˜ Y in (D lim ,M 1 ), (2.16) where ˆ X(t) = ˜ W (t)− P s j=1 σ j μ 3/2 j W j ( ¯ B j (t)). Proof: First, using (2.10) and the lower bound in (ii) of Proposition 2.2 we have Q n √ N n , ˆ Y n ! ≥ Φ( ˆ X n + q N n ¯ X), Ψ( ˆ X n + q N n ¯ X) ! . (2.17) This implies that ˆ Q n = Q n √ N n − q N n ¯ Q≥ Φ( ˆ X n + q N n ¯ X)− q N n ¯ Q. (2.18) Recall from Theorem 2.1 that ¯ Q = ¯ X + Ψ( ¯ X). Substituting this expression into (2.18), and using the fact that Φ(x) =x + Ψ(x) for x∈D lim , we have ˆ Q n ≥ ˆ X n + q N n ¯ X + Ψ( ˆ X n + q N n ¯ X)− q N n ( ¯ X + Ψ( ¯ X)), = ˆ X n + Ψ( ˆ X n + q N n ¯ X)− q N n Ψ( ¯ X). (2.19) 61 Next, utilizing the expression for ˆ Y n in (2.17), and letting ˜ Y n := ˆ Y n − √ N n Ψ( ¯ X), we have ˜ Y n ≥ Ψ( ˆ X n + q N n ¯ X)− q N n Ψ( ¯ X). (2.20) Therefore, ˆ Q n , ˜ Y n ≥ ˆ X n + ˜ Y n , Ψ( ˆ X n + q N n ¯ X)− q N n Ψ( ¯ X) . (2.21) Next, using the upper bound in (ii) of Proposition 2.2 we have Q n √ N n , ˆ Y n ! ≤ Φ ˆ X n + q N n ¯ X− s √ N n ! + s √ N n , Ψ ˆ X n + q N n ¯ X− s √ N n !! . Now, using the centering arguments used in the lower bound we have ˆ Q n , ˜ Y n ≤ ˆ X n + Ψ ˆ X n + q N n ¯ X− s √ N n ! − q N n Ψ( ¯ X), Ψ ˆ X n + q N n ¯ X− s √ N n ! − q N n Ψ( ¯ X) ! . (2.22) The limit process follows by use of the directional derivative reflection mapping lemma, Lemma 2.6 in the Appendix. Using the fact that ˆ X n a.s. → ˆ X in (D lim ,J 1 ), together with the lemma, it follows that lim inf n→∞ ˜ Y n ≥ ˜ Y := sup s∈∇ ¯ X,L · (− ˆ X(s))∨ sup s∈∇ ¯ X,R · (− ˆ X(s)). Consequently, from (2.21) we have lim inf n→∞ ˆ Q n ≥ ˆ X + ˜ Y in (D lim ,M 1 ) a.s. as n→∞. Similarly, from (2.22), and using Lemma 2.5 and Lemma 2.6 again, we have lim sup n→∞ ˜ Y n ≤ ˜ Y in (D lim ,M 1 ) a.s. as n → ∞. Then, using the upper bound on ˆ Q n in (2.22), we have lim sup n→∞ ˆ Q n ≤ ˆ X + ˜ Y in (D lim ,M 1 ) a.s. as n→∞. Combined with the lower bound above, this proves convergence of the sample paths almost surely. Finally, the weak convergence in the statement of the theorem follows by the fact that the pre-limit processes are equal in distribution to our original processes. Remarks. 1. Observe that the diffusion limit to the queue length process is a function of a Gaussian bridge process and a Brownian motion process. This is significantly different from 62 the usual limits obtained in a heavy-traffic or large population approximation to a single server queue. For instance, in the GI/G/1 queue, one would expect a reflected Brownian motion in the heavy-traffic setting. In [19] it was shown that the diffusion limit process to the queue length process of aM t /M t /1 queue is a time changed Brownian motionW ( R λ(s)ds + R μ(s)ds), where λ(s) and μ(s) (resp.) are the time inhomogeneous mean arrival rate and mean service rate (resp.), reflected through the directional derivative reflection map used in Lemma 2.6. There are very few examples of heavy-traffic limits involving a diffusion that is a function of a Gaussian bridge and a Brownian motion process; see example 3 of [4]. A Special Case To illustrate the difference between the population acceleration diffusion limit with the RBM observedfortheGI/G/1queue, wepresentacorollarytoTheorem2.2when ˜ W isaBrownian Bridge process. A Brownian Bridge limit arises, for instance, when the arrival times T n,i are sampled in an i.i.d. manner from some distribution function F. We assume that F has compact support [−T 0 ,T ], where T 0 > 0 to allow for early bird arrivals, and that the queue has a single server. This queue was comprehensively studied in [66], where we call this model a Δ (i) /GI/1 queue. Notice that in this case, ¯ F =F. Proposition 2.3 If T n,i are i.i.d. samples from distribution function F, then ¯ A n a.s. → F in (D lim ,J 1 ) and ˆ A n ⇒W 0 ◦F in (D lim ,J 1 ) as n→∞. Proof: We first fix F to the uniform distribution function on [0, 1]. The fluid limit follows by the standard Glivenko-Cantelli; see Theorem 2.4.7 of [40]. Theorem 16.4 of [42] proves 63 that ˆ A n ⇒ W 0 in (D,J 1 ). To prove that convergence holds in (D lim ,J 1 ), we must first show that the Brownian Bridge is well defined on the larger space. But, this is a direct consequence of part (i) of Lemma 2.1. Next, by part (ii) of Lemma 2.1, it follows that ˆ A n ⇒ W 0 in (D lim ,J 1 ). Finally, let F be any arbitrary cumulative distribution function. SinceW 0 concentrates onC⊂D, Corollary 1 to Theorem 5.1 of [42] implies the final result. This proposition allows us to state a diffusion limit for the queue length process of the Δ (i) /GI/1 queue, as a corollary of Theorem 2.2. Corollary 2.3 Let ˜ W =W 0 ◦F be a time changed Brownian Bridge process. Then, ˆ X(t) d = Z t −T 0 q g 0 (s)d ˇ W s , ∀t∈ [−T 0 ,∞) (2.23) where g(t) =F (t)(1−F (t)) +σ 2 μ 3 ¯ B(t) and ˇ W is a Brownian motion process on the same underlying sample space (Ω,F,P). Further, the queue length diffusion limit process is ˆ Q(t) = ˆ X(t) + sup s∈∇ ¯ X t (− ˆ X(s)) ∀t∈ [−T 0 ,T ], (2.24) where∇ ¯ X t :={0≤s≤t| ¯ X(s) =−Ψ( ¯ X)(t)} and ¯ X :=F (t)−μt is absolutely continuous. Proof: By Lemma 2.5 it follows that ˆ X =W 0 ◦F−W◦ ¯ B. By a classical time change (see, for example, [45]) W 0 ◦F is equal in distribution to a time changed Brownian motion, and ˆ X is equal in distribution to the stochastic integral (2.23). The diffusion function g(t) can be easily verified. The expression for ˆ Q now follows by substitution in (2.16). Note that therightandleftcorrespondences∇ ¯ X,R t ,∇ ¯ XL t coincidesince ¯ X isabsolutelycontinuous. Remarks. 1. In Section 1.4.1, in fact, we prove joint convergence of ( ˆ Q n , ˜ Y n ), in the strong M 1 topology, for the Δ (i) /GI/1 model. 64 Why M 1 , and not J 1 ? We now discuss why we establish the diffusion limit in the space (D lim ,M 1 ), and why it can’t hold in the space (D lim ,J 1 ) in general. This section can be skipped on a first reading without any loss of continuity, though we encourage the reader to read it for a better understanding of Theorem 2.2. There are several equivalent definitions of convergence in the M 1 topology (the inter- ested reader is directed to [44, 46, 43] for an in-depth study.) A simple characterization of convergence in M 1 for processes with range in R is the following involving the num- ber of visits to a strip [α,β]⊂ R in an interval [t 1 ,t 2 ]⊂ [η,∞). Let y ∈D (orD lim ) and suppose there are N + 1 points t 1 ≤ t (0) < t (1) < ... < t (N) ≤ t 2 such that either y(t (0) )≤ α,y(t (1) )≥ β,y(t (2) )≤ α,···, or y(t (0) )≥ β,y(t (1) )≤ α,y(t (2) )≥ β,... Then, there are N visits to the strip in [t 1 ,t 2 ]. Let ν [α,β] [t 1 ,t 2 ] (y)7−→N be the number of visits to the strip [α,β] in [t 1 ,t 2 ] by the function y. Definition 2.3 summarizes this characterization [46]. Definition 2.3 (Convergence in M 1 ) Let y,y n be elements ofD and d M 1 (·,·) the M 1 metric. Then, d M 1 (y n ,y)−→ 0 as n→∞ if and only if ν [α,β] [t 1 ,t 2 ] (y n )−→ν [α,β] [t 1 ,t 2 ] (y) as n→∞. Convergence in the J 1 topology, on the other hand, can be seen as a “relaxation” of the definition of convergence in the uniform metric topology. Specifically, letz n ,z be elements of the spaceD lim [η,∞). FixT∈ [η,∞) that is a continuity point ofz, and letk·k be the local uniform metric on the interval [η,T ]. Define Λ to be the set of all non-decreasing continuous homeomorphisms from [η,T ] to itself. Then, convergence in J 1 can be defined as follows. Definition 2.4 (Convergence in J 1 ) There exists a sequence{λ n }⊆ Λ such thatkλ n − ek−→ 0 as n→∞, where e is the identity map, d J 1 (z n ,z)−→ 0 as n→∞ if and only if kz n ◦λ n −z◦ek +kλ n −ek−→ 0 as n→∞. 65 It is well known that the M 1 topology is weaker than the U (uniform) or J 1 topologies, and processes converging in M 1 need not converge in U or J 1 . As already stated, the diffusion limit for the queue length process is obtained in the spaceD lim when endowed with the M 1 topology because the directional derivative reflec- tion mapping lemma (Lemma 2.6) that we use yields convergence in the M 1 topology alone. Intuitively, the reason the convergence result holds only inM 1 is that asymptoticallyy n con- verges to a continuous process, and it is well known that continuous processes can converge to discontinuous limits only in the M 1 topology. To make this intuition concrete, we give a counterexample that shows that convergence in J 1 is not possible in this case. It will suffice to show that for some > 0 at least one of the terms in the expression d J 1 (z n ,z) =kz n ◦λ n −z◦ek +kλ n −ek exceeds . Define the process ˜ y n , ˜ y n = Ψ( q N n x +y)− q N n Ψ(x), where x is the function in Figure 2.1, and y is a Brownian motion. We show that there is a non-empty set of points in the vicinity of τ where the normed distance d J 1 (˜ y n , ˜ y)>, for any > 0. Recall that ˜ y = sup s∈∇ x · (−y(s)). The next proposition formalizes this argument. x τ t η 0 Figure 2.1: This x∈C[η,∞) corresponds to the fluid netput process, when F is uniform. Proposition 2.4 (Non-convergence in J 1 ) Let x be the function in Figure 2.1,{y n }⊂ D lim [η,∞) and y∈C[η,∞) is a Brownian motion, such that y n a.s. → y in (D lim [η,∞),U). Then, the process ˜ y n = Ψ( √ N n x +y n )− √ N n Ψ(x) does not converge to ˜ y in the J 1 topology as n→∞. 66 The proof is available in the Appendix. Thus, we see that the process ˜ y n does not converge to the directional derivative of the reflection map in the J 1 topology (and hence even the uniform topology), necessitating the use of the M 1 topology. This result clearly implies that ˜ Y n does not necessarily converge to ˜ Y in theJ 1 topology. Thus, we have a situation where the limit process is discontinuous and the limit result must be proved in the M 1 topology in full generality. Busy Time Process Recall from Section 2.3.2 that the fluid-scaled busy time process B n (t) converges to a con- tinuous process ¯ B(t) asn→∞, in Corollary 2.1. Now, define the diffusion-scaled busy time process as ˆ B n := q N n ( ¯ B−B n ). (2.25) Note that from the definitions ofB n (t) and ¯ B(t) it follows that ˆ B n (t) = 0,∀t< 0. As might be expected, the diffusion refinement displays the same non-stationarity observed above. Corollary 2.4 The diffusion scaled busy time process converges to a (directional derivate) reflected Gaussian process as n→∞, ˆ B n ⇒ ˆ B := 1 μ max s∈∇ ¯ X · (− ˆ X(s)), in (D lim ,M 1 ). Proof: Recall that B n (t) = st1 {t≥0} −I n (t). Substituting this and ¯ B from (2.8) in the definition of ˆ B n , and rearranging the expression, we obtain ˆ B n = 1 μ ˜ Y. A simple application of Theorem 2.2 then provides the necessary conclusion. Observe that B n (t) is approximated in distribution by ˆ B as B n (t) d ≈ ¯ B(t)− 1 √ Nn ˆ B(t), where Z n d ≈Z is defined to beP(Z n ≤x)≈P(Z≤x), and the approximation is rigorously 67 supported by an appropriate weak convergence result. Expressing the result in this manner exposes the fact that the probability that the queue idled at any time up to t> 0 is P(I n (t)> 0)≈P( ˆ B(t)> √ N n ¯ I(t)). Note that the approximation is most accurate when the system is such that it started with some workload at time t = 0. 2.4 Transitory Traffic Models In this section, we study three different traffic models for transitory queueing systems, all of which satisfy Assumption 1. We note that these models are very different in nature and there could be many more that satisfy the assumptions. Recall that the collection of arrival epochs T n := (T n,1 ,T n,2 ,··· ,T n,Nn ) is an instance of a finite point process. We assume, without loss of generality, that the support of the arrival epochs is [0, 1] in this section. We give a brief description of the models before continuing to a detailed description of each. First, we model the arrival epochs as independent random variables. Customers enter the queue in order of the sampled arrival times, where each arrival time is sampled from a customer dependent distribution. As noted before in Proposition 2.3 we studied a special case of this model in [66], where the arrival epochs were assumed to be i.i.d. We call this the general Δ (i) traffic model, for reasons elaborated on below. See Section 2.4.1. Classical queueing theory has focused extensively on modeling traffic by renewal pro- cesses. In our second model, we assume that the joint distribution of the arrival epochs is determined by conditioning the arrival epochs of a renewal point process on an appropriate set of interest. We prove fluid and diffusion limits for this model, and establish a close connection with the asymptotics of the Δ (i) model. See Section 2.4.2. 68 Finally, we consider a model of scheduled traffic with uncertainty, where the realized arrival epoch is different from the scheduled epoch, due to the fact that users sometimes arrive before or after the scheduled time. We model this variation between the realized and scheduled arrival times by a uniformly distributed random variable with zero mean. We present fluid and diffusion limits to this model as the population size tends to infinity. In particular, it is most interesting that this model can be reduced to the general Δ (i) model. See Section 2.4.3. 2.4.1 The General Δ (i) Model Independent arrival epochs are a natural assumption to impose, especially while modeling a large number of customers who independently take decisions on when to arrive at a queue. In general, the marginals need not be identically distributed, as individuals may have differing assessments on when to arrive at a queue. We call this the general Δ (i) traffic model. Cus- tomers enter the queue in order of the sampled arrival times, so that the inter-arrival times are the difference of order statistics, hence the term Δ (i) . In this section, we comprehensively study the population acceleration (PA) fluid and diffusion limits for this model. Assume that N n is a deterministic, non-decreasing sequence of natural numbers repre- senting the population size in the nth system. Recall that T n is an instance of the finite point process, determining the arrival epochs. Without loss of generality of the domain, let F n,i : [0, 1]→ [0, 1], for each i∈{1,...,N n }, represent the arrival time distribution of customeri in thenth system. Notice that{F n,i , i∈{1,...,N n }}∀n∈N forms a triangular array of distribution functions. As noted above the joint distribution of T n is of product form, or formallyP(T n ∈ Π Nn i=1 [0,t i ]) = Π Nn i=1 F n,i (t i ), for any Borel set Π Nn i=1 [0,t i ]⊂ [0, 1] Nn . Intuitively, one might expect that any fluid limit to the arrival process (2.1) would be an average over the individual sampling distributions. We first formalize this notion, by placing the following restriction on theF n,i . LetK := [0, 1] represent the index set of customers, and (K,B(K),m) represent the sample space of the indices, whereB(K) is the Borel σ-algebra 69 onK and m is the Lebesgue measure. LetL [0,1] be the space of all distribution functions with support [0, 1]. Following [56], we define a random distribution function as a mapping Υ :K→L [0,1] . Thus, (F s (t) := Υ(s)(t), t∈ [0, 1]), is the distribution function of customers. Clearly, Υ induces a sample space (L [0,1] ,B(L [0,1] ),P) whereB(L [0,1] ) is the Borel σ-algebra containing the weak- ∗ topology onL [0,1] andP = m◦ Υ −1 is the measure induced on the spaceL [0,1] . The average distribution function ¯ F is now well defined in relation toP as ¯ F (t) := R F∈L [0,1] F (t)dP(F ) = R [0,1] Υ(s)(t)m(ds) = R 1 0 F s ds. Notice that Υ is a measure-valued stochastic process with domain [0, 1] and rangeL [0,1] . It is useful to view Υ in the fol- lowing sense: it represents a summary of the beliefs of all the possible customers (in the universe of these models) who may choose to arrive per the distribution they choose from L [0,1] . While the total order property ofK plays no role in our description of the population of customers, it is not unusual to expect that customers “close” to each other, in the sense of the Euclidean norm onK, should have similar beliefs. Thus, we impose the condition that Υ satisfieskΥ(ω 1 )− Υ(ω 2 )k≤ K|ω 1 −ω 2 |, for any ω 1 ,ω 2 ∈K and K <∞ is some given constant. In particular, in the simplest case where Υ(ω) = δ F (ω) for all ω∈K and some F∈L [0,1] (i.e., in an i.i.d. sampling model), this condition is satisfied automatically. Note that this not the usual sample path continuity of a stochastic process, but is instead a constraint on the variation of the sample path. Recall that (2.1) implies that the cumulative arrival process in the nth system is A n (t) := P Nn i=1 1 {T n,i ≤t} ∀t∈ [0, 1]. The fluid-scaled arrival process is simply ¯ A n := A n Nn , and the diffusion-scaled arrival process is ˆ A n := √ N n ¯ A n − ¯ F n . Customers enter the queue in the order of sampled times. Thus, the inter-arrival times in the Δ (i) arrival model are the differences of the ordered arrival times, so that τ (n,i) =T (n,i) −T (n,i−1) , where T (n,0) = 0. Without loss of generality, assume customer i corresponds to the point i/N n ∈ (0, 1]. In order to establish large population fSLLN and fCLT’s, we need some “control” on the average distribution function ofrowninthearray, definedas ¯ F n (t) := 1 Nn P Nn i=1 F n,i (t),∀t∈ [0, 1].We 70 start by proving some useful properties of this average distribution function. The following lemma shows that ¯ F n converges to ¯ F as n→∞. Lemma 2.2 There exists a distribution function ¯ F such that ¯ F n (t)→ ¯ F (t) := Z K F p (t)m(dp), (2.26) uniformly on [0, 1] as n→∞. A straightforward calculation shows that the covariance function of the arrival process in the nth row of the array is K n (s,t) :=E[ ˆ A n (s) ˆ A n (t)] = 1 Nn P Nn i=1 F n,i (s∧t)−F n,i (s)F n,i (t). The following lemma shows that K n has a well defined limit as n→∞. Lemma 2.3 There exists a function K(s,t) such that, K n (s,t) = 1 N n Nn X i=1 F n,i (s∧t)−F n,i (s)F n,i (t)→K(s,t) := Z K (F p (s∧t)−F p (s)F p (t))m(dp), (2.27) as n→∞, uniformly for all s,t∈ [0, 1]. The proofs of Lemma 2.2 and 2.3 are in the appendix. The fSLLN theorem is a generalization of the Glivenko-Cantelli Theorem [40] to triangu- lar arrays of non-identically distributed random variables. In Theorem 2.3 we show that the normalized arrival process ¯ A n converges to ¯ F inD lim , uniformly on compact sets of [0,∞). We prove this result by demonstrating the uniform convergence of the sample paths of the empirical distribution. We make the reasonable assumption that none of the distribution functions ¯ F n share discontinuity points in the support. This implies that the limit ¯ F is (almost surely) continuous, allowing us to prove convergence in the uniform metric. 71 Theorem 2.3 (Glivenko-Cantelli Theorem for Triangular Arrays) The fluid scaled arrival process ¯ A n = A n Nn satisfies a functional strong law of large numbers, ¯ A n a.s. → ¯ F in (D lim ,U), as n→∞. The proof is presented in the appendix. Remarks. Versionsofthistheoremhavebeenprovedintheliteratureandwedrawattention, in particular, to Theorem 1 of [58]. There it was shown that ¯ A n and ¯ F n converge to the same limit point in the Prokhorov metric ([42, 43]) onL [0,1] . However, this result does not explicitly identify the fluid limit process. Our construction of the empirical distribution via random distribution functions allows us to do this. A requirement in the proof of Theorem 2.3 is that the sequence{N n ,n≥ 1} must satisfy P ∞ n=1 1 N 2 n <∞, implying that N n =O(n 1+δ ) for δ≥ 0, and this will play a role in the proof of the fCLT. As a consequence of our construction of the empirical distribution function space and Lemma 2.2, we can explicitly identify the Gaussian limit process in our setting. We first present the fCLT, and then elaborate on the diffusion limit process. Theorem 2.4 (Empirical Process Limit for Triangular Arrays) The centered arrival process ˆ A n satisfies a functional central limit theorem, ˆ A n ⇒ ˜ W in (D lim ,U), as n → ∞, where ˜ W is a mean zero Gaussian process with covariance function K(s,t) defined in (2.27) and continuous sample paths. The proof can be found in the appendix. 72 Remarks. Our proof is a generalization of Hahn’s Central Limit Theorem (Theorem 2 in [55]) to nonidentically distributed random elements ofD lim . We also draw attention to Theorem 1.1 of [59] that proves the existence of an empirical process limit for triangular arrays, under the sufficient condition that an appropriate covariance function exists. How- ever, it does not specifically identify the Gaussian process limit, something that is crucial for this paper. Once again, our identification of the empirical distribution function space with random distribution functions enables this identification. The covariance structure of the process ˜ W is interesting in itself, and we make the follow- ing observations. First, notice that the covariance function is an average of the covariance functions of the Brownian Bridge processes W 0 ◦F p (with p∈K) where W 0 is a standard Brownian Bridge. In a sense, these are Brownian Bridge processes associated with empirical processes of random samples from the function F p . Second, differentiating the expression for K(t,t) with respect to t we have dK(t,t) dt = R K (f p (t)− 2f p (t)F p (t))m(dp), where f p is the density (or at least the right-derivative) of the distribution function F p . This is the average of the infinitesimal variance of the Brownian Bridges W 0 ◦F p . Recall that the infinitesimal mean and variance of a diffusion process are defined as E[X(t+h)−X(t)|X(t)=x] h → μ(t,x) and E[|X(t+h)−X(t)| 2 |X(t)=x] h →σ 2 (t,x) ash→ 0 (resp.). For the Brownian Bridge processW 0 ◦F p , it is well known that the infinitesimal mean and variance are (for a fixed p∈K) μ p (t,y) = −yf p (t) 1−F p (t) σ 2 p (t,y) = f p (t). 73 Further, it can be shown that the mean and variance of the Brownian Bridge satisfies the following o.d.e’s: d dt E h W 0 ◦F p (t) i = E h μ p (t,W 0 ◦F p (t)) i = −f p (t) 1−F p (t) E[W 0 ◦F p (t)] = 0 d dt Var W 0 ◦F p (t) = E h σ 2 (t,W o ◦F p (t)) i + 2E h W 0 ◦F p (t)×μ p (t,W 0 ◦F p (t)) i = f p (t)− 2f p (t)F p (t). Comparing the variance derivative above with dK(t,t) dt , we conjecture that the ˜ W is a Gaus- sian diffusion process with infinitesimal generator equal to the average of the infinitesimal generators of the Brownian Bridges W 0 ◦F p . However, we have not been able to verify that the process is Markov with respect to its natural filtration to make a definitive conclusion. Aparticularcaseofinterestiswhenthe{T n,i }arei.i.d. drawnfromacommoncontinuous distributionF∈L [0,1] . This result, of course, is the standard fCLT for the empirical process (see [42, 67, 43] for a deeper exposition). Corollary 2.5 For each n≥ 1, let{T n,i , i = 1,...,N n } be a triangular array of i.i.d. random samples drawn from a distribution F. Then, as n→∞ ˆ A n ⇒W 0 ◦F in(D,U). Here W 0 is the standard Brownian Bridge process defined on the common sample space. A formal proof of this result is standard and omitted (see Chapter 13 of [42]). However, it is also straightforward to see this from Theorem 2.4 by setting F p =F for allp∈ [0, 1]. It can be readily verified that the Gaussian process ˜ W is equal in distribution to a Brownian Bridge process. 74 2.4.2 Conditioned Renewal Model The most common traffic model assumed in the queueing theory literature is the renewal model. In this section, we consider a model of traffic for transitory queueing systems that is related to renewal traffic models. Specifically, we allow the arrival process to be a renewal process conditioned on the even that N n arrivals occur in some finite time horizon. First, we recall that there is a strong connection between the i.i.d. Δ (i) arrival process and the conditioned Poisson renewal process. Next, we show that even though this property is not satisfiedbyrenewalprocesses ingeneral, asymptoticallyweobtainthe samefSLLNandfCLT limit processes as the population size scales to infinity. This appears to be a new result and should be of wider interest. Without loss of generality we fix N n =n in this section. Relation Between Conditioned Poisson and Δ (i) Models Consider a renewal point process (M(t),t ≥ 0) defined with respect to (Ω,F,P). Let (λ(t),t≥ 0) be an integrable, non-negative function defined to be the arrival rate of M. Therefore, Γ(t) := R t 0 λ(s)ds is the mean cumulative arrival process. We first note the fol- lowing. Lemma 2.4 Let Γ : [0,∞)→ [0,∞) be the mean cumulative arrival process of a Poisson process. Then, for a fixed T > 0, F (t) := Γ(t) Γ(T ) ∀t∈ [0,T ], (2.28) is a continuous probability distribution function. It is straightforward to verify this result and we omit a proof. The ordered statistics (OS) property of point processes provides the connection between the i.i.d. Δ (i) and Poisson processes. 75 Definition 2.5 (Property OS) Conditioned on {M(T ) = n}, the event epochs (T 1 ,...,T n ), are distributed as the ordered statistics of n independent and identically dis- tributed random variables with distribution F (t), for t∈ [0,T ]. By Theorem 1 of [60], M possesses theOS property if and only if it is a Poisson process (see [61] as well). Notice that this distributional relationship is true for every n≥ 1. By Kolmogorov’s Extension Theorem, there exists a stochastic process ˆ M n (t) such that for any partition 0<t 1 <···<t d <T and (x 1 ,··· ,x d )∈R d ,P( ˆ M n (t 1 )≤x 1 ,..., ˆ M n (t d )≤x d ) =P 1 √ n (M(t 1 )−nF (t 1 ))≤x 1 ,..., 1 √ n (M(t d )−nF (t d ))≤x d |M(T ) =n ! . The OS property implies that we can easily obtain an fCLT for the conditioned Poisson process. Theorem 2.5 The sequence of processes{ ˆ M n }, n≥ 1, satisfies a functional central limit theorem, ˆ M n ⇒W 0 ◦F in (D,U), as n→∞, where W 0 is a standard Brownian Bridge process defined on the same sample space as M The proof is simple and in the appendix. The implication of this theorem is that the conditioned Poisson and Δ (i) traffic models are equivalent in distribution. In fact, verifying that an observed traffic sequence satisfies theOS property is sufficient to conclude that the arrival process is Poisson. A thorough study of statistical tests for this purpose is presented in [9]. Remarks 1. It is important to note that this limit result fundamentally differs from the standard diffusion limit for non-homogeneous Poisson processes, which we review here. Let N(·) be a unit rate Poisson process. Then, M(t) d = N( R t 0 λ(s)ds) is a non-homogeneous 76 Poisson process. The diffusion approximation to this process is developed by scaling the compensated (Martingale) process ˆ N(t) := M(t)− R t 0 λ(s)ds in an appropriate manner. The commonly accepted approach is the uniform acceleration method developed in [19], where the rate function λ(s) is scaled by a constant > 0, so that we obtain the scaled process ˆ N (t) :=N 1 R t 0 λ(s)ds − 1 R t 0 λ(s)ds. In [19], the Strong Approximation Theorem (see [68]) is used to prove that the sample paths of ˆ N converges to those of a standard Brownian motion as → 0. Recall that the strong approximation theorem implies that ˆ N (t) = 1 √ W (t) +o 1 √ a.s. as → 0. When λ(s) = λ for all s≥ 0, and = 1/n for n∈N, it is straightforward to see that the standard Poisson process diffusion approximation is a special case of this approach (see also [69] for an overview of strong approximation methods applied to queueing theory). Now, contrast this limit with Theorem 2.5. Note that, by definition, ˆ M n is not equivalent to the compensated Martingale process ˆ N (when = 1/n), as it is defined with respect to the conditioned measure on the set{M(T ) =n}, and not the full measureP. Furthermore, the limit can only hold in the weak sense, as the strong approximation only applies to processes with independent increments. This is not a condition satisfied by ˆ M n . It appears that obtaining a strong approximation (or rate of convergence) result for the conditioned process is an open problem, and of independent interest. Functional Limit Theorems of Conditioned Renewal Processes Theorem 1 of [60] clearly shows that a non-Poisson renewal process does not satisfy the OS property. However, in this subsection we prove that the conditioned renewal process in fact converges to a Brownian Bridge process when scaled appropriately. For simplicity, we assume that the renewal process is time-homogeneous, but the results extend easily to the general case. Without loss of generality we also assume thatλ(t) = 1 for allt≥ 0. First, we recall the definition of a finitely exchangeable sequence. 77 Definition 2.6 Let{X 1 ,...,X n } be a collection of random variables defined with respect to the sample space (Ω,F,P). Then, this collection is said to be finitely exchangeable if {X 1 ,··· ,X n } D ={X π(1) ,··· ,X π(n) }, where π :{1,...,n}→{1,...,n} is a permutation function on the index of the collection. Renewalprocessessatisfytheexchangable (orE)property, assummarizedinthefollowing proposition. Proposition 2.5 (Property E) Let ξ i : Ω→ R + i∈ N be a sequence of i.i.d. positive random variables defined with respect to (Ω,F,P), such thatM(t) := sup{k> 0| P k l=1 ξ l ≤t}, for all t > 0 is the associated renewal counting process. Then, the finite collection Ξ n := (ξ 1 ,...,ξ n ) is finitely exchangeable under the measure conditioned on the event{M(T ) =n}, for T <∞ fixed. A proof of this fact is in the appendix. Notice that finitely exchangeable random vari- ables are not infinitely exchangeable and important results such as de Finetti’s Theorem are unavailable. To prove the functional limit theorems for the counting processes, we will first prove that certain scaled partial sums of the exchangeable inter-arrival times (conditioned) have Gaussian process weak limits. However, in contrast to classical functional central limit theorem results, the conditioning increases the complexity of the problem significantly, since for each n the random variables exist on different (but related) probability sample spaces. The result for the counting process will follow by a random time-change argument. Functional Strong Law of Large Numbers Consider a triangular array of random vari- ables Ξ n := (ξ n,i , i = 1,...,n) and n≥ 1, defined as the inter-arrival times of the renewal events associated with a sequence of independent and indistinguishable renewal processes, {M n , n≥ 1}. By Proposition 2.5, we know that Ξ n is an exchangeable array of random variables conditioned on the event{M n (T ) =n}. The limit results are proved with respect to a conditional measure ¯ P that we construct in Section 2.5 in the Appendix. This can 78 be skipped on a first reading by accepting the premise that such a measure exists. In the ensuing, any reference to Ξ n is to be interpreted with respect to the conditional measure ¯ P. Let μ n := E¯ P [ξ n,i ] = E[ξ n,i |M n (T ) = n] be the conditioned mean of the inter-arrival periods; the exchangeable property implies that these random variables are identically dis- tributed. Ourfirstresultisafunctionalstronglawforpartialsumsoftheserandomvariables. Theorem 2.6 Let ¯ S n (t) := P bntc l=1 ξ n,l ∈D lim ,∀t∈ [0, 1]. Then, ¯ S n ¯ P−a.s. → e in (D lim ,U) as n→∞, where e : [0, 1]→ [0, 1] is the identity function. This is an intuitively satisfying result, that provides strong evidence that an fCLT along the lines of Theorem 2.5 is satisfied by a conditioned renewal process. Functional Central Limit Theorem Consider the standardized random variables, {φ n,l , l = 1,...,n} defined with respect to Ξ n : φ n,l := ξ n,l −μ n √ n . The next theorem characterizes the sequence φ n,l and shows that the partial sums of these random variables satisfy a functional central limit theorem. Theorem 2.7 Let{φ n,l , l = 1,...,n}, n≥ 1, be the triangular array of random variables defined above and ˆ M n (t) := P dnte i=1 φ n,i ∈D lim and ∀t∈ [0, 1]. Then, the random variables (φ n,1 ,...,φ n,n ) are exchangeable and satisfy: (i) P n l=1 φ n,l ¯ P → 0, (ii) max 1≤l≤n |φ n,l | ¯ P → 0, (iii) P n l=1 φ 2 n,l ¯ P → 1, and (iv) ˆ M n ⇒W 0 in (D lim ,U), as n→∞, where W 0 is a standard Brownian Bridge process. 79 Proof: The exchangeability of φ n,i follows directly by the fact that ξ n,i is exchangeable. (i), (ii) and (iii) are proved in Proposition 2.7 in the appendix. Then, by Theorem 24.2 of [42] ˆ M n ⇒W 0 in (D,U). The extension to (D lim ,U) follows from Lemma 2.1. The conditions in Theorem 2.7 are natural in the context of the conditioned limit result we seek. Note that the conditioned limit result is akin to proving a diffusion limit for a tied- down random walk (see [70, 71]). The first condition here enforces a type of “asymptotic tied down” property. The second condition is a necessary and sufficient condition for the limit process to be infinitely divisible (see [72] for more on this). The third condition is necessary to ensure that the Gaussian limit, when t = 1, has variance 1. Similar conditions have been observed to be sufficient to prove central limit theorems for dependent random variables (see, in particular, [62, 63]). The final step in describing the limit behavior of the conditioned renewal process is to obtain a result that parallels Theorem 2.5, and show that the “counting” counterpart of the partial sum process also converges to a Brownian Bridge. To that end, we start with a definition. Definition 2.7 (Counting Process) L n (t) := sup{0≤m≤n| ¯ S n ( m n ) := P m l=1 ξ n,l ≤t} is the standard counting counterpart to the partial sum process. Now, the main theorem of this section proves that the counting process satisfies Assump- tion 1. Theorem 2.8 (i) ¯ L n := Ln n a.s. → e in (D lim ,U) as n→∞, where e : [0,∞)→ [0,∞) is the identity map. (ii) √ n ¯ L n −e ⇒−W 0 in (D lim ,U) as n→∞, where W 0 is the Brownian Bridge limit process observed in Theorem 2.7. Part (i) shows that the conditioned renewal traffic model converges to the uniform dis- tribution function on [0, 1], in a large population limit. Part (ii), in turn, proves that the 80 diffusion scaled conditioned renewal traffic model satisfies Assumption (b) in Assumption 1. The proof is a consequence of the random time change theorem (see chapter 17 of [42]), and relegated to the appendix. This is an intuitively satisfying result as we know that the Poisson renewal model certainly satisfies the same limit. Further, as a result of Theorem 2.8, it is obvious that the large population approximations to the performance metrics of a “conditioned renewal” transitory queueing model is asymptotically equal in distribution to an equivalent Δ (i) transitory queueing model. 2.4.3 Scheduled Arrivals with Epoch Uncertainty Traffic scheduled to arrive at regular intervals is a common occurrence in many service systems. Often, schedules are made for a finite period of time, and the traffic pattern is transitory in nature. For example, hospital outpatient units schedule patients at particular times during the day (typically 8AM to 8PM). Another classic example of scheduled traffic is air traffic arrivals. However, while the arrivals may be scheduled, it is often the case that there is some randomness in the realized arrival time: users can arrive a little before or after the scheduled arrival time. The earliest description of a model of such arrival behavior appears in [57] where it was introduced as “a regular arrival process with unpunctuality”. Recent work in [64] studied this model with heavy tailed uncertainty and demonstrated convergence to a fractional Brownian motion with Hurst index < 1/2. In this section, we present a novel and intuitive model of scheduled arrivals with uncertainty on a finite interval, and demonstrate its connection with the Δ (i) traffic model. For simplicity, let the population size be N n = n∈ N, and without loss of generality we assume that arrivals take place over the interval [0, 1] at equal intervals. For simplicity we assume the first arrival is scheduled at time 0, and the last one at time 1. The jth user arrives at time τ n,j := j/n; for simplicity, assume 0 = τ n,1 ≤ τ n,2 ···≤ τ n,n = 1. Let ξ n,i be a random variable uniformly distributed on the interval [−T,T ], where T is a constant to be defined. Then, the realized arrival time of user j is modeled as T n,j :=τ n,j +ξ n,j . Users 81 can potentially enter the service system in the intervalT := [−T, 1 +T ], and the cumulative number of arrivals by time t∈T is A n (t) := P n i=1 1 {T n,i ≤t} . We now argue that this arrival process satisfies Assumption 1. Analogous to Section 2.4.1, consider the average distribution function ¯ F n (t) := 1 n E[A n (t)] = 1 n n X i=1 F (t−τ n,i ). Note that the summands are not the same distribution function, as the mean of T n,i is τ n,i . This is analogous to the definition of the average distribution function in Lemma 2.2. Our first result argues that there exists a functional limit to ¯ F n as n→∞. Proposition 2.6 Let ¯ F n be the average distribution function, for a given n≥ 1. Then, for a fixed T∈ [0, 0.5] and t∈T ¯ F n (t) a.s. → ¯ F (t) :=                            (t+T ) 2 4T , −T≤t≤T, t, T <t≤ 1−T, t +T 2T − t 2 −T 2 2T + (t−T ) 2 4T − 1 4T + (t−T ), 1−T <t≤ 1 +T, uniformly on [0, 1] as n→∞. The proof is available in the appendix. Figure 2.2 depicts ¯ F for different values of T. Notice that the support of the population mean distribution function depends on the value of T, and the larger the value of T, the earlier and later arrivals can occur to the system (obviously). Interestinglyenough, onecanalsoshowthatthelimitpopulation(or mean field) distribution is an average over the individual distribution functions for each user. Following 82 0 0.25 0.5 0.75 1 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 F - (t) Time T=0.1 T=0.2 T=0.4 T=0.5 Figure 2.2: The “population” average arrival distribution function for different values of T. Section 2.4.1, letK := [0, 1] represent the universe of all possible users to the queueing system. Then, for p∈K, we know that F p (t) := t−p +T 2T for p−T≤t≤p +T, (2.29) is the arrival distribution function of customerp. That is, customerp arrives at time p, with a uniform uncertainty distribution centered at p. The following corollary shows that this population average coincides with the distribution ¯ F in Proposition 2.6, when T∈ [0, 0.5]. The proof is a simple integration argument, and is relegated to the appendix. Corollary 2.6 Let F p defined in (2.29) be the arrival distribution associated with user p∈K. Then, Z 1 0 F p (t)m(dp) =                            (t+T ) 2 4T , −T≤t≤T, t, T <t≤ 1−T, t +T 2T − t 2 −T 2 2T + (t−T ) 2 4T − 1 4T + (t−T ), 1−T <t≤ 1 +T, 83 where m(·) is the Lebesgue measure on the setK. This is precisely the condition that needs to be satisfied for the generalized Glivenko- Cantelli result to be true in Theorem 2.3. LetF n,i (t) :=F (t−τ n,i ) and extend the support of F n,i (t) to the interval [−T, 1+T ] such thatF n,i is zero outside the interval [−T + i Nn ,T + i Nn ]. The scheduled arrival model is, therefore, a special case of the general Δ (i) traffic model. We claim the following theorem as an immediate consequence of Theorem 2.3. Theorem 2.9 The fluid-scaled arrival process satisfies a functional strong law of large num- bers: ¯ A n a.s. → ¯ F in (D lim ,U), as n→∞. Similar arguments as above we can also prove that the sample covariance function also converges to the average covariance, as obtained in Lemma 2.3. Since the arguments are similar, we skip the proof and present the final result. The diffusion-scaled arrival process is denoted ˆ A n (t) := √ n A n (t) n − 1 n P n i=1 F(t−τ n,i ) . Theorem 2.10 The centered arrival process ˆ A n satisfies a functional central limit theorem, ˆ A n ⇒ ˜ W in (D lim ,U), as n→∞, where ˜ W∈C is a zero mean Gaussian process with covariance function K(s,t), as obtained in Lemma 2.3. As a final note, observe that there is an important distinction between the scheduled arrival and the Δ (i) pre-limit traffic models. In the latter, the realized arrival times are the ordered statistics of the sampled arrival times, while in the former this is not the case. However, the natural (partial) ordering of the real numbers is all that is required to establish 84 the functional limits, and in the limit as n→∞ any difference between these models is “washed out”. The results in this and the previous section strongly indicate that, in some sense, the Δ (i) traffic model is canonical to the study of transitory queueing systems. In the next section we focus on a deeper study of sample path approximations suggested for such models. 2.5 Appendix Proof of Lemma 2.1 (i). D is a subspace under the relative topology τ r :={A∩D|A∈ τ lim }, where τ lim is the topology induced by the J 1 metric onD lim . Then, it follows thatD =D r :={D∩A|A∈ D lim }. To see this, note thatD r contains all possible open sets ofD (since these are elements of D lim ) and D r is a σ-algebra. This implies that D ⊆ D r since the Borel σ-algebra is the smallest to contain all open subsets ofD. In the opposite direction, by sinceD is a subspace, the injection map ι :D→D lim is a homeomorphism, implying that for any Borel set A∈D lim , ι −1 (A) is Borel inD. But, the inclusion map is clearly ι −1 (A) = A∩D by definition. ThereforeD r ⊆D. This implies that for any D∈D lim , P(x∈D) =P(x∈D∩D) is well defined since x is defined with respect to (D,D). This extends the definition of the measure induced by x to D lim . (ii) Now, let G⊂D lim be any closed subset. Then, we have P(x n ∈G) =P(x n ∈G∩D) +P(x n ∈G∩D c ) =P(x n ∈G∩D), whereD c is the complement set. Let E := G∩D, and note thatD\E = D∩E c = D∩G c . This implies thatD\E is open in the relative topology onD since G c is open in 85 τ lim . Now, using the fact that x n ⇒ x in (D,J 1 ) and part (iii) of Theorem 2.1 of [42] we have lim sup n→∞ P(x n ∈G∩D)≤P(x∈G∩D). This implies that lim sup n→∞ P(x n ∈G)≤P(x∈G∩D) =P(x∈G), where the last equality follows by the fact thatx concentrates onD. By part (i) of Theorem 2.1 of [42], it follows that x n ⇒x in (D lim ,J 1 ) as n→∞. Statement and Proof of Lemma 2.5 Lemma 2.5 As n→∞, ˆ X n ⇒ ˆ X := ˜ W−W◦ ¯ B in (D lim ,J 1 ) (2.30) where ¯ B is defined in (1.13), and ˜ W and W are mutually independent Gaussian bridge and Brownian motion processes respectively, as defined in Proposition 2.1 Proof: Fix j ∈ {1,...,N}. Recall that B n j (t) ≤ t,∀t ∈ [0,∞), implying that S n j ◦ B n j ∈ D lim . Using (2.5) in Proposition 2.1, Corollary 1.1 and the ran- dom time change theorem (see, for example, Section 17 of [42]), it follows that √ N n S n j ◦B n j Nn − μB n j ! ⇒ σμ 3/2 W j ◦ ¯ B j . Now, it follows from Proposition 2.1 and the weak limit just proved that ˆ X n ⇒ ˆ X(t) := ˜ W− P s j=1 σμ 3/2 W j ◦ ¯ B j d = ˜ W−W◦ ¯ B, where the final equality follows from the fact that the sum of independent Brownian motions is equal in distribution to a Brownian motion. 86 Statement and Proof of Lemma 2.6. This result is a consequence of Theorem 9.5.1 of [46]. A version of this result is also proved in [66]. Lemma 2.6 (Directional derivative reflection mapping lemma) Let x ∈ D and y ∈ C be real-valued functions on [0,∞), and Ψ(z)(t) = sup 0≤s≤t (−z(s)), for any pro- cess z∈D lim . Let{y n }⊂D lim be a sequence of functions such that y n a.s. → y as n→∞. Then, with respect to Skorokhod’s M 1 topology, ˜ y n := Ψ( √ N n x +y n )− √ N n Ψ(x)−→ ˜ y := sup s∈∇ x,L t (−y(s))∨ sup s∈∇ x,R t (−y(s)) as n→∞, where∇ x,· t are defined in Definition 2.2. Rewrite ˜ y n as ˜ y n = (Ψ( √ nx +y n )− Ψ( √ nx +y))− (Ψ( √ nx +y)− √ nΨ(x)). Now, using the fact that the Skorokhod reflection map is Lipschitz continuous under the uniform metric (see Lemma 13.4.1 and Theorem 13.4.1 of [43]) we have (Ψ( √ nx +y n )− Ψ( √ nx +y))≤ky n −yk, wherek·k is the uniform metric. It follows that ˜ y n ≤ky n −yk + (Ψ( √ nx +y)− √ nΨ(x)), Now, by Theorem 9.5.1 of [46] we know that as n→∞ (Ψ( √ nx +y)− √ nΨ(x)) a.s. → ˜ y, in (D lim ,M 1 ). Using this result, and the fact that by hypothesis y n converges to y in (D lim ,U) we have ˜ y n a.s. → ˜ y, in (D lim ,M 1 ). 87 Proof of Proposition 2.4. Recall that ˜ y n converges to ˜ y in the M 1 topology, from Lemma 2.6. Now, consider a path of y that is non positive at τ. Thus, the limit process ˜ y has a discontinuity at τ such that ˜ y(τ) > ˜ y(τ+). Note that the process path is left continuous at τ. Assume that ˜ y(τ)− ˜ y(τ+) > δ > 0, and it follows that ˜ y(τ) =−y(τ) > δ (since ˜ y(τ+) = 0). Fix an > 0 such that δ > . Now, by the continuity of y, there exists η > 0 such that sup t∈[τ−η,τ+η] |y(t)− y(τ)| ≤ 4 . Then, there also exists a n 0 such that for all n > n 0 , 0≥− √ nx(t)>− e 4 for t∈ [τ−η,τ]. Then, for any t∈ [τ−η] it follows that − √ nx(t)−y(t) +y(τ)>− 2 . This implies that− √ nx(t)−y(t)>δ− 2 > e 2 , since δ>. It follows that Ψ( √ nx +y)> 2 for all time points t∈ [τ−η,τ]. Thus, it cannot be the case that uniform convergence is possible on any compact set of [−T 0 ,∞). Furthermore, consider any sequence{λ n }⊆ Λ. Then, for large n, by assumption, λ n is uniformly close to the identity map. Thus, any distortion introduced by the homeomorphism will be minimal, and the same argument will show that it cannot be the case that, for any fixed > 0,k˜ y n ◦λ n − ˜ y◦ek≤ 2 for large n, and there is a set of points determined by η (due to the continuity of y) where it is the case that|(˜ y n ◦λ n )(t)− (˜ y◦e)|> 2 . 88 Proof of Lemma 2.2 For each n ∈ N we have F n,i = Υ i Nn , i = 1,...,N n . There- fore, (2.26) can be rewritten as ¯ F n (t) = 1 Nn P Nn i=1 Υ i Nn (t), and we prove that 1 Nn P Nn i=1 Υ i Nn (t)− R [0,1] Υ(s)(t)m(ds) [0,1] → 0asn→∞. Noticethat ¯ F (t)isaRiemann- Stieltjes integral with respect to the Lebesgue measure. Therefore, it is natural to view ¯ F n as a Riemann-Stieltjes (pre-limit) sum. For a fixed t∈ [0, 1], therefore, we show that the Riemann sums converge to the Riemann-Stieltjes integral. Let M i (t) := sup Υ(x)(t) and m i (t) := inf Υ(x)(t) for all x ∈ h i−1 Nn , i Nn i and i = 1,...,N n . We define the “upper” and “lower” Riemann sums as (respectively) U n (t) := P Nn i=1 M i i Nn − i−1 Nn andN n (t) := P Nn i=1 m i i Nn − i−1 Nn . Clearly, for every> 0 and large enough n, U n (t)−N n (t)< due to the Lipschitz continuity property assumed for Υ. This is tantamount to showing that the bound holds for at least one possible partition of [0, 1]. Then, by Theorem 6.6 of [73], it follows that the limit exists and is equal to ¯ F (t). The Lipschitz continuity property implies that the limit clearly holds for all t∈S, implying uniform convergence. Proof of Lemma 2.3 The first summation in the definition ofK n (s,t) = 1 Nn P Nn i=1 F n,i (s∧t)− 1 Nn P Nn i=1 F n,i (s)F n,i (t) converges to R K F p (s∧t)m(dp) as n→∞ by Lemma 2.2. The second summation converges as well by using the same Riemann-Stieltjes summation argument used in the proof of Lemma 2.2, and the limit is R K F p (t)F p (s)m(dp). 89 Proof of Theorem 2.3 First, fix t∈ [0, 1] and > 0. Consider P ¯ A n (t)− ¯ F n (t) > = P 1 N n Nn X i=1 1 {T n,i ≤t} − Υ(i/N n )(t) > ! ≤ 1 4 N 4 n E Nn X i=1 1 {T n,i ≤t} − Υ(i/N n )(t) 4 = 1 4 N 4 n Nn X i=1 E 1 {T n,i ≤t} − Υ(i/N n )(t) 4 + 12 4 N 4 n Nn−1 X i=1 Nn X j=i+1 E 1 {T n,i ≤t} − Υ(i/N n )(t) 2 E 1 {T n,j ≤t} − Υ(j/N n )(t) 2 ≤ 1 4 N 4 n Nn X i=1 Υ(i/N n )(t)(1− Υ(i/N n )(t)) + 12 4 N 4 n Nn X i=1 Υ(i/N n )(t)(1− Υ(i/N n )(t)) ! 2 , (2.31) where the last inequality follows due to the fact that the terms that remain in the expan- sion ofE P Nn i=1 1 {T n,i ≤t} − Υ(i/N n )(t) 4 are P Nn i=1 E|1 {T n,i ≤t} − Υ(i/N n )(t)| 2 and cross prod- ucts P Nn−1 i=1 P Nn j=i+1 E|1 {T n,i ≤t} − Υ(i/N n )(t)| 2 E|1 {T n,j ≤t} − Υ(j/N n )(t)| 2 . From Lemma 2.3 it follows that (2.31) is bounded above by C 4 N 2 n . Therefore, by the Borel-Cantelli Lemma, P ∞ n=1 P ¯ A n (t)− ¯ F n (t) > <∞, implying that P ¯ A n (t)− ¯ F n (t) > i.o. = 0. Combin- ing this result with Lemma 2.2 proves that ¯ A n converges to ¯ F almost surely pointwise. Next, consider a uniform partition of the support [0, 1], and suppose j−1 M ≤t≤ j M , where j = 1,...,M and M is the size of the partition. Then, for fixed n, ¯ A n j−1 M ≤ ¯ A n (t)≤ ¯ A n j M , implying that 1 Nn P Nn i=1 1 {T n,i ≤j−1/M} −F n,i (j− 1/M) ≤ 1 N n Nn X i=1 1 {T n,i ≤t} −F n,i (t) + 1 N n Nn X i=1 (F n,i (j/M)−F n,i (j− 1/M)) ≤ 1 N n Nn X i=1 1 {T n,i ≤j/M} −F n,i (j/M) . 90 For each M, there exists n M such that for all n≥ n M |F n,i (j/M)−F n,i (j− 1/M)|≤ 1 M . Further, for > 0, there exists n 0 M such that for all n≥ max(n M ,n 0 M ), 1 N n Nn X i=1 1 {T n,i ≤k} −F n,i (k) <, where k =j− 1/M or j/M. It follows that sup t∈[0,1] 1 N n Nn X i=1 1 {T n,i ≤t} −F n,i (t) < 2( + 1 M ). Since is arbitrary, letting M→∞ the desired result follows. Proof of Theorem 2.4 We first prove pointwise convergence by verifying the sufficiency of the Lyapunov Central Limit Theorem (Theorem 7.3 [42]). Fix t∈ [0, 1] and let δ> 0, and consider P Nn i=1 E|F n,i (t)− 1 {T n,i ≤t} | 2+δ P Nn i=1 F n,i (t)(1−F n,i (t)) 2+δ . Dividing the numerator and denominator by 1/N 2+δ n , note that the denominator converges to ( ¯ F (t)) 2+δ as a consequence of Lemma 2.2. Consider the numerator alone, 1 N 2+δ n Nn X i=1 E|F n,i (t)− 1 {T n,i ≤t} | 2+δ ≤ 2 δ N 2+δ n Nn X i=1 F n,i (t)(1−F n,i (t)), which tends to 0 as n→∞. The Lyapunov CLT implies that P Nn i=1 (F n,i (t)− 1 {T n,i ≤t} ) q P Nn i=1 F n,i (t)(1−F n,i (t)) = P Nn i=1 (F n,i (t)− 1 {T n,i ≤t} √ N n × √ N n q P Nn i=1 F n,i (t)(1−F n,i (t)) ⇒N (0, 1). 91 By Lemma 2.3 it follows that P Nn i=1 (F n,i (t)−1 {T n,i ≤t} √ N n ⇒ ˜ W (t) := q K(t,t)N (0, 1). Next, using the Cramer-Wold device it is straightforward to argue that ( ˆ A n (t 1 ),..., ˆ A n (t k ))⇒ ( ˜ W (t 1 ),..., ˜ W (t k )) where (t 1 ,...,t k )∈ [0, 1] k for all k∈N. Finally, we verify the sufficiency of Theorem 15.6 of [42] to show that ˆ A n ⇒ ˜ W in (D,J 1 ). To ease the notation, let X n,i (t) := (1 {T n,i ≤t} −F n,i (t)). By Chebyshev’s inequality, for any λ> 0 and t 1 ≤t≤t 2 ∈ [0, 1], λ 4 P(| ˆ A n (t)− ˆ A n (t 1 )|≥λ,| ˆ A n (t)− ˆ A n (t 2 )|≥λ) ≤ E h ( ˆ A n (t 1 )− ˆ A n (t)) 2 ( ˆ A n (t 1 )− ˆ A n (t)) 2 i . = 1 N 2 n E   Nn X i=1 (X n,i (t)−X n,i (t 1 )) 2 Nn X i=1 (X n,i (t 2 )−X n,i (t)) 2   = 1 N 2 n E                     Nn X i=1 |X n,i (t)−X n,i (t 1 )| 2 Nn X i=1 |X n,i (t 2 )−X n,i (t)| 2 + 2 X i<j (X n,i (t 2 )−X n,i (t))(X n,j (t 2 )−X n,j (t)) Nn X l=1 |X n,l (t)−X n,l (t 1 )| 2 + 2 X i<j (X n,i (t)−X n,i (t 1 ))(X n,j (t)−X n,j (t 1 )) Nn X l=1 |X n,l (t 2 )−X n,l (t)| 2 + 4 X i<j (X n,i (t)−X n,i (t 1 ))(X n,j (t)−X n,j (t 1 )) X i<j (X n,i (t 2 )−X n,i (t))(X n,j (t 2 )−X n,j (t))                     ≤ 1 N n Nn X i=1 [(F n,i (t)−F n,i (t 1 ))(F n,i (t 2 )−F n,i (t))(1−F n,i (t) +F n,i (t 1 ))(1−F n,i (t 2 ) +F n,i (t))] 1/2 + 2 1 N 2 n X i<j [(F n,i (t)−F n,i (t 1 ))(F n,j (t 2 )−F n,j (t))(1−F n,i (t) +F n,i (t 1 ))(1−F n,j (t 2 ) +F n,j (t))] + 4 X i<j (F n,i (t)−F n,i (t 1 ))(F n,i (t 2 )−F n,i (t))(F n,j (t)−F n,j (t 1 ))(F n,j (t 2 )−F n,j (t)) ≤ C where C ≥ 8 and the bound is true for all t 2 > t 1 . Theorem 15.6 of [42] shows that if P(| ˆ A n (t)− ˆ A n (t 1 )| ≥ λ,| ˆ A n (t)− ˆ A n (t 2 )| ≥ λ) ≤ (G(t 2 )− G(t 1 )) 2α , where G is a non-decreasing function on [0, 1] and α > 1/2, then ˆ A n converges weakly to a limit in (D,J 1 ). Therefore, ˆ A n ⇒ ˜ W as n → ∞. The convergence in (D lim ,J 1 ) follows by an 92 application of part (ii) of Lemma 2.1. Finally, by part (ii) of Theorem 1.1 of [59] we know that ˜ W has continuous sample paths, implying that A n ⇒ ˜ W in (D lim ,U), thus completing the proof. Proof of Theorem 2.5 Let T :={T i , i = 1,...,n} be a collection of i.i.d. random variables, with distribution functionF (defined in (2.28)). Let A n (t) := P n i=1 1 {T i ≤t} and ˆ A n (t) := √ n A n (t) n −F (t) be the empirical process associated with T. For a fixed n≥ 1 and x∈R we have P( ˆ M n (t)≤x|M(T ) =n) =P(M(t)≤x √ n +nF (t)|M(T ) =n). TheOS property implies that M(t)| {M(T )=n} d =A n (t). Proposition 2.3 implies that P(M(t)≤x √ n +nF (t)|M(T ) =n) = P(A n (t)≤x √ n +nF (t)) = P( ˆ A n (t)≤x) ⇒ (W 0 ◦F )(t) proving the pointwise convergence of the process ˆ M n . It is also well known that for any 0 < t 1 < ··· < t d < T, P(M(t 1 ) = n 1 ,...,M(t d ) = n d |M(T ) = n) = P(A n (t 1 ) = n 1 ,...,A n (t d ) = n d ), where n 1 +···n d = n, so the fact that ( ˆ A n (t 1 ),..., ˆ A n (t d ))⇒ (W 0 ◦F )(t 1 ),··· , (W 0 ◦F )(t d )) implies that the finite dimensional distrbutions of ˆ M n converge to the same limit. The tightness of ˆ M n is implied directly by that of ˆ A n , so by Theorem 8.1 of [42] the theorem is proved. 93 Proof of Proposition 2.5 Let{x 1 ,...,x n }⊂ [0,T ] be such that 0≤ x 1 < x 2 <··· < x n ≤ T. Consider the measure of the event{ξ 1 ∈dx 1 ,··· ,ξ n ∈dx n }∈F, P (ξ 1 ∈dx 1 ,··· ,ξ n ∈dx n |M(T ) =n) = P ((ξ 1 ∈dx 1 ,··· ,ξ n ∈dx n ),M(T ) =n) P (M(T ) =n) . Recall that{M(T ) =n} ={ P n l=1 ξ l ≤T < P n+1 l=1 ξ l }, implying that we have: P ((ξ 1 ∈dx 1 ,··· ,ξ n ∈dx n ),M(T ) =n) P (M(T ) =n) = P (ξ 1 ∈dx 1 ,...ξ n ∈dx n , P n l=1 ξ l ≤T, P n l=1 ξ l +ξ n+1 >T ) P (M(T ) =n) . Using the fact that under the measure P, ξ i are i.i.d. random variables, it follows that the measure of the joint event is invariant under any permutation of the firstn random variables. That is, if π(·) is a permutation of{1,...,n}, then we have P ((ξ 1 ∈dx 1 ,··· ,ξ n ∈dx n ),M(T ) =n) P (M(T ) =n) = P (ξ π(1) ∈dx 1 ,...ξ π(n) ∈dx n , P n l=1 ξ π(l) ≤T, P n l=1 ξ π(l) +ξ n+1 >T ) P (M(T ) =n) , which is equal to P n (ξ π(1) ∈ dx 1 ,...ξ π(n) ∈ dx n ). Next, suppose that π(·) is a per- mutation of {1,...,n + 1}. Then, it is possible that the event P n i=1 ξ π(l) > T, since ξ n+1 > T− P n l=1 ξ l > 0, conditionally on{M(T ) = n}. Thus, Ξ n cannot be extended to a larger collection of exchangeable random variables, implying that it is finitely exchangeable. 94 Conditioned Renewal Process: Lemmata Sample Space Construction We assume that the underlying sample space Ω,F,P is rich enough to support a sequence of (jointly) independent stochastic processes{M n }, n≥ 1, such that they are each indistin- guishable from M. That is, for any n≥ 1,P(M n (t)6=M(t),∀t≥ 0) = 0. For a fixed n≥ 1 andT > 0, wedefinetherestrictedsamplespace, (Ω n ,F n ,P n ), where Ω n = Ω∩{M n (T ) =n}, F n := σ{A∩{M n (T ) = n} : A∈F} and P n (B) := P (B) P (Mn(T )=n) for any B∈F n . Clearly {{M(T ) =n} n≥ 1} forms a partition of Ω. The following claim shows that this property (P−a.s.) extends to the collection{Ω n }. Lemma 2.7 {Ω n } n≥ 1 forms a P−a.s. partition of Ω. Proof: For a fixed n≥ 1 and m∈N,{M n (T ) =m}, forms a partition of Ω, as do{M(T ) = m}. ByassumptionM n andM areindistinguishablefromeachother. Itisstraightforwardto deduce thatP({M n (T ) =m}Δ{M(T ) =m}) = 0, implying that{M n (T ) =m} ={M(T ) = m}P−a.s. for every m∈N. Now, consider the collection of sets{{M n (T ) = n}} n≥ 1. For brevity, let A n := {M n (T ) =n} and B n :={M(T ) =n}. We have (∪ n≥1 {M n (T ) =n}) Δ (∪ n≥1 {M(T ) =n}) = (∪ n≥1 A n ) Δ (∪ n≥1 B n ) = ∪ l≥1 (∩ n≥1 (B l ∩A c n )). By the assumption of indistinguishability,P(B l ∩A c l ) = 0, implying thatP(∩ n≥1 (B l ∩A c n )) = 0, for every l≥ 1. Therefore, P (∪ n≥1 {M n (T ) =n}) Δ (∪ n≥1 {M(T ) =n}) = 0. 95 By virtue of the fact that∪ n≥1 {M(T ) = n} = Ω it follows that{{M n (T ) = n}}, n≥ 1 forms a partition of ΩP−a.s. Next, we construct a new product space from the restricted sample spaces (Ω n ,F n ,P n ) as follows. Let, ¯ Ω := Ω 1 × Ω 2 ×···, so that A⊂ ¯ Ω =A 1 ×A 2 ×··· for sets A n ⊂ Ω n . The product σ−algebra, ¯ F :=F 1 ⊗F 2 ⊗··· is the σ−algebra generated from cylinder sets of the type R ={(ω 1 ,ω 2 ,··· )∈ ¯ Ω|ω i 1 ∈ A i 1 ,··· ,ω i k ∈ A i k }, where (i 1 ,...,i l ) is an arbitrary subset of N of size k≥ 1 and A in ∈F n . The existence of such a product σ−algebra is well-justified by Proposition 1.3 in [74]. Finally, we define ¯ P(R) = Π k i=1 P i l (A i l ), for the cylinder sets. This extends to ¯ P =P 1 ×P 2 ×···, which is the natural product measure on the measure space ( ¯ Ω, ¯ F), by standard arguments showing that the measure is countably additiveon ¯ F. ThedefinitionoftheLebesgueintegralonthespace ( ¯ Ω, ¯ F, ¯ P)nowfollowsfrom standard definitions of integration on product spaces. However, we introduce some notation to ease our burden. In particular, consider a function defined in the following manner: ¯ X :=X× Π l6=n I {Ω l } , whereX is measurable and integrable with respect to (Ω n ,F n ,P n ), and I {·} is the indicator function. Then E¯ P [ ¯ X] = Z Ωn XdP n Z Π l6=n I {Ω l } dP l is well-defined, and we write this asE¯ P [X], where it is to be understood that the integration is actually of ¯ X. 96 Preliminary Lemmata Armed with the new product space, we can now proceed to the proof of the diffusion limit. We define the collection of random variables, Ξ n := (ξ n,1 ,··· ,ξ n,n ) to be “conditioned” inter-arrival times of the renewal process M n provided that M n (t) = sup{k≥ 0| k X i=1 ξ n,i ≤t}, fort≤T and P n i=1 ξ n,i ≤T. From Proposition 2.5 it follows that Ξ n forms an exchangeable array, under the measure P n . Furthermore, under the measure ¯ P, Ξ n and Ξ m (m6= n) are independent of each other (by the definition of the product measure). The following lemma characterizes the mean and variance of the inter-arrival times, for a fixed n. Intuitively, one should expect the mean and variance to decrease to 0 asn→∞, as there are a larger number of variables being packed into a fixed interval (T is fixed, of course). This characterization is important in proving the diffusion limit. We start with a simple lemma. Lemma 2.8 Let{f n } be a sequence of non-negative, measurable, functions. Let P be a given measure. Assume: a) R f n dP→ 0 as n→∞, b)|f n |≤C <∞. Then, lim n→∞ f n = 0 as n→∞. Proof: lim Z f n dP DCT = Z limf n dP 0 = Z limf n dP, from whence the conclusion follows easily. The proof of the asmptotic negligibility of the mean and variance follow as consequences of Lemma 2.8. 97 Lemma 2.9 For ξ n,j ∈ Ξ n , (i) μ n :=E[ξ n,j |M n (T ) =n]→ 0 as n→∞. (ii) E[|ξ n,j −μ n | 2 |M n (T ) =n]→ 0 as n→∞. Proof: (i) Conditioned on{M n (T ) =n},{ξ n,j } j≤n are exchangeable, implying that they have the same distribution. Thus, E[ξ n,1 |M n (T ) =n] = 1 n E[S n |M n (T ) =n], where S n = P n i=1 ξ n,i . By the definition of conditioned expectation: Z 1 {Mn(T )=n} S n dP = Z 1 {Sn≤T≤S n+1 } S n dP ≤ TP(S n ≤T <S n+1 )≤TP(S n ≤T ). Note: we interpret S n+1 as the sum of n + 1 inter-event times in the nth unconditioned system. By assumption, E P [ξ n,j ] = μ > 0, implying (by SLLN or Second Borel Cantelli Lemma) thatS n →∞a.s. asn→∞. This implies that lim n→∞ P(S n ≤T ) = 0. Now, since S n > 0, it follows from Lemma 2.8 that lim n→∞ E[S n |M n (T ) =n] = 0. Finally, 1 n E[S n |M n (T ) =n]≤E[S n |M n (T ) =n]→ 0. 98 (ii) follows by a similar argument. From part (i) we have convergence in measure, using Markov’s inequality: for any > 0, P n (ξ n >) =P(ξ n >|M n (T ) =n) ≤ E[ξ n |M n (T ) =n] → 0, as n→∞. Next, to see that the variance converges to 0 as well, consider the following: recall thatP n is the conditional probability measure (this is regular, in fact). Then, we have, E[(ξ n,j −μ n ) 2 |M n (T ) =n] = Z (ξ n,j −μ n ) 2 dP n = Z {ξ n,j >} (ξ n,j −μ n ) 2 dP n + Z {ξ n,j ≤} (ξ n,j −μ n ) 2 dP n ≤ 2T 2 P n (ξ nj >) + 2 P n (ξ nj ≤) +μ 2 n < (2T 2 + 2) for largen. Note that here, we used the fact that under the measureP n the random variables are bounded byT, as well as the convergence in measure noted above. The same argument, in fact, extends to any r-th mean. It is straightforward to see that Lemma 2.9 is true under measure ¯ P. The next theorem proves a rate of convergence for the mean inter-arrival times in the conditioned systems. Lemma 2.10 Let M be a renewal process with renewal time distribution F. Let μ n be the mean inter-arrival time, when the process is conditioned to have n events by time T. Then, (i) μ n → 0 and (ii) √ n|1−nμ n |→ 0 as n→∞. Proof: Let F :R + → [0, 1] be the inter-arrival time distribution, defining a renewal process M(t). Assume that f(t) := dF (t) dt is well defined. Then, the conditional intensity function (CIF) of M(t) is λ ∗ (t) :=E[N(dt)|H t ] = f(t)dt 1−F (t) ≥ 0, whereH t is the filtration generated by M. The integrated CIF, Λ ∗ (t) = R t 0 λ ∗ (s)ds is the martingale compensator so that M(t)− 99 Λ ∗ (t) is a Martingale process. Theorem 7.4.I of [23] shows that M(t) = N(Λ ∗ −1 (t)) is a unit rate Poisson Process. That is, if{T i } is a realization of the event times of process M, then{ ˜ T i = Λ ∗ (T i )} is a realization of those of a unit rate Poisson process. As Λ ∗ is non- decreasing, it follows that{ ˜ T n+1 > Λ ∗ (T )≥ ˜ T n } if and only if{T n+1 > T≥ T n }, implying that{M(T ) =n} ={N(Λ ∗ (T )) =n}. Let ξ i =T i −T i−1 be the inter-arrival time random variable. Then, P(ξ 1 >u|M(T ) =n) =P (φ 1 ≥ Λ ∗ (u)|N(Λ ∗ (T )) =n), where φ 1 = Λ ∗ (ξ 1 ). Consider the latter conditioned probability, and recall that a Poisson process satisfies the Ordered Statistics property. It follows that P (φ 1 ≥ Λ ∗ (u)|N(Λ ∗ (T )) =n) = 1− Λ ∗ (u) Λ ∗ (T ) ! n . As the inter-arrival times are exchangeable (when conditioned by{N(Λ ∗ (T )) =n}) they are also identically distributed, so the ensuing arguments hold true for any inter-arrival time φ i i = 1,...,n. The conditional distribution implies that ˜ μ n := Z Λ ∗ (T ) 0 P (φ 1 ≥ Λ ∗ (u)|N(Λ ∗ (T )) =n)dΛ ∗ (u) = 1 n + 1 . Equivalently, after a time change, we have Z T 0 P(ξ 1 >u|M(T ) =n)λ ∗ (u)du = 1 n + 1 . 100 However, we are interested in the asymptotics of the closely related integral μ n := R T 0 P(ξ 1 >u|M(T ) =n)du. Therefore, consider (n + 1) μ n − 1 (n + 1) = (n + 1) Z T 0 P(ξ 1 >u|M(T ) =n)(1−λ ∗ (u))du (2.32) ≤ K(n + 1) Z T 0 P(ξ 1 >u|M(T ) =n)du (2.33) = K Z T 0 (n + 1) 1− Λ ∗ (u) Λ ∗ (T ) ! n du, (2.34) where the inequality follows as the CIF is bounded on compact intervals. Since 0≤ Λ ∗ (u) Λ ∗ (T ) ≤ 1 for every 0 < u≤ T, it follows that (n + 1) 1− Λ ∗ (u) Λ ∗ (T ) n → 0 as n→∞. Then, using (the reverse) Fatou’s Lemma we have lim sup n→∞ (n + 1) μ n − 1 (n + 1) ≤ Z T 0 lim sup n→∞ (n + 1) 1− Λ ∗ (u) Λ ∗ (T ) ! n du = 0. Thus, Lebesgue almost everywhere on [0,T ], we have μ n ∼ 1 n+1 , so that μ n → 0 as n→∞. This immediately implies that √ n|1−nμ n |→ 0. Proof of Theorem 2.6 Consider the interval [0, 1], and consider | bntc X l=1 ξ n,l −t|≤| bntc X l=1 (ξ n,l −μ n )| +|bntcμ n −t|. ThesecondtermontheRHStendsto 0,asaconsequenceofLemma2.10(itisstraightforward to see that{ξ n,l } n l=1 satisfies the conditions of the theorem under the space ( ¯ Ω, ¯ F, ¯ P)). For the first term, consider the martingale sequence z n,l := (ξ n,l −μ n )−E[(ξ n,l −μ n )|F n,l−1 ], 101 whereF n,l :=σ{(ξ n,1 −μ n ),..., (ξ n,l−1 −μ n ), P n i=l (ξ n,i −μ n )}. Using the fact that{ξ n,l } are exchangeable it is easy to deduce that n X i=j (ξ n,i −μ n ) = n X i=j E[(ξ n,i −μ n )|F n,j−1 ] = (n−j + 1)E[ξ n,j −μ n |F n,j−1 ]. This implies that z n,l = (ξ n,l −μ n ) + 1 n−j+1 P n i=l (ξ n,i −μ n ) =ξ n,l − 1 n−l+1 P n i=l ξ n,l . Using the fact that ξ n,l ∈ [0, 1], under the measure ¯ P, it follows that n X l=1 1 n−l + 1 n X i=l ξ n,l ≤ n−1 X l=0 1 n−l n X j=l+1 ξ n,j ≤ 2 n− n−1 X l=0 1 n−l ! ≈ 2 (n− logn +o(n)). By definition we have, for any > 0, {ω∈ ¯ Ω :| n X l=1 ξ n,l −μ n |}⊆{ω∈ ¯ Ω :| n X l=1 z n,l |>−nμ n − 2(n + logn +o(n))}. Thus, it suffices to bound the latter event, which we do using the Azuma-Hoeffding inequal- ity for Martingale differences. First, let us recall the statement of the Azuma-Hoeffding inequality. Let Z l , 1≤l≤n, be a martingale difference sequence defined with respect to a sample space (S,S,P), such that|Z l |≤c l , for some set of constants{c 1 ,...,c n }. Then, for all > 0, P max 1≤k≤n k X l=1 Z l > ! ≤ exp − 2 2 P n l=1 c 2 l ! . Returning to the problem at hand, we have, for any > 0 ¯ P(| n X l=1 Z n,l |>−nμ n − 2(n + logn +o(n)))≤ 2 exp(− (−nμ n − 2(n + logn) 2 4n ), 102 where|z n,l |≤ 2. Now, all that is required is a lower bound on the exponentiated expression. Let f(,n) := 1 4n (−nμ n − 2(n + logn) 2 . Multiplying and dividing by n 2 on the RHS, we obtain, f(,n) = n 4 2 + (μ n + 2)(1− 4/n) + 8( logn n ) 2 + 4 logn n − 4 logn n (1−/n) ! ≥ n 4 2 + (1− 4/n)(1/n + 2 + 4 logn n ) + 8( logn n ) 2 ! , where in the last step we make use of the fact that 1 n > μ n . For large enough n such that n> 4 we have f(,n)≥ n 2 4 + 2κ n , where κ := log 4. It follows that exp(−f(,n))≤ exp(− n 2 4 ) exp(− 2κ n )≤ exp(− n 2 4 ). This implies that for any > 0 ∞ X n=1 ¯ P(| n X l=1 (ξ n,l −μ n )|>)≤ ∞ X n=1 ¯ P(| n X l=1 Z n,l |>−nμ n − 2(n + logn +o(n)))<∞. Thus, by the First Borel-Cantelli Lemma, ¯ P(| P n l=1 (ξ n,l −μ n )| > )i.o.) = 0. Therefore, P n l=1 (ξ n,l −μ n )→ 0 a.s. as n→∞. Clearly, this holds true for any t∈ [0, 1], so that P bntc l=1 (ξ n,l −μ n )→ 0 a.s. as n→∞. By standard arguments it follows that the limit holds uniformly on [0, 1] concluding the theorem. 103 Statement and Proof of Proposition 2.7 Proposition 2.7 The triangular array{φ n,l , l = 1,...,n} n≥ 1 satisfies the following properties: (i) P n l=1 φ n,l P → 0 as n→∞. (ii) max 1≤l≤n |φ n,l |→ 0 as n→∞. (iii) P n l=1 φ 2 n,l P → 1 as n→∞. Proof: (i) The proof follows by using the definition of ¯ P. Fix > 0, and consider the following: ¯ P(| n X l=1 φ n,l |>) = ¯ P(| n X l=1 φ n,l |>,M n (T ) =n) = ¯ P(| n X l=1 ξ n,l −nμ n |> √ n,M n (T ) =n) = ¯ P( n X l=1 ξ n,l > √ n +nμ n ,M n (T ) =n) + ¯ P( n X l=1 ξ n,l <− √ n +nμ n ,M n (T ) =n). The first equality follows by the fact that under ¯ P{M n (t) = n}, for every n≥ 1, are full measure sets. Recalling that{M n (T ) =n} ={ P n l=1 ξ n,l ≤T < P n l=1 ξ n,l +ξ n,n+1 }, it follows that for any ω∈A n :={ P n l=1 ξ n,l > √ n +nμ n ,M n (T ) =n)} we have T≥ n X l=1 ξ n,l > √ n +nμ n . Now, using the fact that ξ n,l are exchangeable (for a fixed n), it follows directly that nμ n = E¯ P [ P n l=1 ξ n,l ]≤ T, under the measure ¯ P. Therefore, nμ n is uniformly bounded (for every n ≥ 1). It follows that for a given T, there exists a n T such that for every n > n T , 104 √ n +nμ≥ T. As > 0 is arbitrary, asymptotically, A n is an impossible event. Next, consider B n :={ P n l=1 ξ n,l <− √ n +nμ n ,M n (T ) =n}. Similar arguments show that − √ n +nμ n > n X l=1 ξ n,l ≥ 0. Clearly as n → ∞, − √ n + nμ n → −∞, as nμ n is uniformly bounded. Since > 0 is arbitrary, B n too is (asymptotically) an impossible event. It follows that φ n,l P → 0 as n→∞. (ii) The proof is elementary. First, using the union bound we have, for a fixed > 0, ¯ P(max 1≤l≤n |φ n,l |>)≤ n X l=1 ¯ P(|φ n,l |>). Using the fact that the random variables φ n,l , l ≤ n are exchangeable, they are also (marginally) identically distributed. Thus, ¯ P(max 1≤l≤n |φ n,l |>) ≤ n ¯ P(|φ n,1 |>) ≤ n E¯ P |ξ n,l −μ n | 2 n 2 = σ 2 n 2 , where the latter expression follows by an application of Chebyshev’s inequality under the ¯ P measure. As noted before, σ 2 n → 0 as n→∞. As > 0 is arbitrary, (ii) is proved. (iii) The proof is more involved, requiring the construction of a martingale difference sequence, and then an appeal to the Azuma-Hoeffding inequality for a tight bound on the martingale difference sequence. Note that φ 2 n,l , 1≤l≤n, is an exchangeable sequence. Let Z n,l :=φ 2 n,l −E¯ P [φ 2 n,l |F n,l−1 ], where{F n,l } is a filtration defined with respect to φ 2 n,l as F n,l =σ(φ 2 n,1 ,...,φ 2 n,l−1 , n X i=l φ 2 n,i ). 105 Now, consider the conditional expectation in the definition of Z n,l . Notice that we have, n X i=j φ 2 n,i = E¯ P [ n X i=j φ 2 n,i |F n,j−1 ] = E¯ P [ n X i=j φ 2 n,j |F n,j−1 ] = (n−j + 1)E[φ 2 n,j |F n,j−1 ]. The penultimate equation follows from the fact that φ 2 n,l are exchangeable, from which the last equality follows by the fact that they are also identically distributed. This implies that E¯ P [φ 2 n,j |F n,j−1 ] = 1 n−j + 1 n X i=j φ 2 n,i Thus, the martingale difference sequence Z n,l has the compact representation Z n,l =φ 2 n,l − 1 n−l + 1 n X i=l φ 2 n,i . In order to obtain a bound on P n l=1 φ 2 n,l − 1, we first bound P n l=1 1 n−l+1 P n i=l φ 2 n,i from above. Recall that φ n,l ≤ 2T/ √ n, so that n X l=1 1 n−l + 1 n X i=l φ 2 n,i = n−1 X l=0 1 n−l n X j=l+1 φ 2 n,j ≤ 4T 2 n n−1 X l=0 1− 1 n−l ≈ 4T 2 − 4T 2 logn n + o(n) n . Therefore, it follows that n X l=1 Z n,l ≥ n X l=1 φ 2 n,l − 4T 2 + 4T 2 logn n − o(n) n . 106 Now, consider the event{ω∈ ¯ Ω| P n l=1 φ 2 n,l − 1≥}, where> 0. From the bound, it follows that n X l=1 Z n,l ≥ n X l=1 φ 2 n,l − 4T 2 + 4T 2 logn n ≥ + 1− 4T 2 + 4T 2 logn n . Using the Azuma-Hoeffding inequality as described above, it follows that ¯ P( n X l=1 Z n,l ≥ + 1− 4T 2 + 4T 2 logn n )≤ exp − ( + 1− 4T 2 + 4T 2 logn n ) 2 n× 64T 4 n 2 ! . (ignoring the o(n) term). The bound in the numerator on the RHS follows by the fact that |Z n,l |≤|φ 2 n,l | + 1 n−l + 1 n X j=l |φ 2 n,j |≤ 2|φ 2 n,l |, and noting that φ 2 n l ≤ 2T 2 /n. Considering the expression being exponentiated on the RHS, it is a straightforward exercise to see that as n→∞, the expression tends to∞, in turn implying that ¯ P( n X l=1 φ 2 n,l − 1≥)≤ ¯ P( n X l=1 Z n,l ≥ + 1− 4T 2 + 4T 2 logn n )→ 0, as n→∞. Next, note that since φ 2 n,l ≥ 0 for all 1≤ l≤ n, it follows that P n l=1 Z n,l ≤ P l n=1 φ 2 n,l . Clearly, ¯ P( n X l=1 φ 2 n,l < 1−)≤ ¯ P( n X l=1 Z n,l < 1−). Using the Azuma-Hoeffding inequality again, we have ¯ P( n X l=1 Z n,l < 1−)≤ exp − (1−) 2 n× 64T 4 n 2 ! . 107 Asn→∞, the LHS tends to 0, exponentially fast. Thus, it follows that| P n l=1 φ 2 n,l − 1| P → 0 as n→∞, completing the proof. Proof of Theorem 2.8 We first state a couple of useful lemmata. We will find it useful to work with a relaxed form of ¯ N n : Definition 2.8 (Relaxed Counting Process) ˜ N n (t) := sup{p∈ [0, 1]| ¯ S n (p)≤t}. This process has the useful description as the fraction of arrivals by time t. Clearly both ¯ N n , ˜ N n ∈D lim [0, 1], and are asymptotically close as the following proposition shows. k·k refers to the sup norm over [0, 1]. In the remainder of the section we work with the process ˜ N n . Lemma 2.11 k ¯ N n − ˜ N n k→ 0 as n→∞. Proof: Fix p∈ [0, 1). Clearly, m n ≤p< m + 1 n , for some m = 0, 1,...,n− 1. Suppose that t∈ [0, 1], | ¯ N n (t)− ˜ N n (t)| = ˜ N n (t)− m n , for somem that ist dependent. Using the upper bound we have| ¯ N n (t)− ˜ N n (t)|< 1 n , which is independent of t. The conclusion follows. To complete the analysis we also require a definition of the inverse function of ¯ S n . 108 Definition 2.9 (Partial Sum Inverse) ¯ S −1 n (t) := inf{p∈ [0, 1]| ˜ N n (p)>t}. The partial sum inverse and the relaxed counting process are related by the expression: ¯ S −1 n (t) = ˜ N n ( ˜ N −1 n ( ˜ N n (t))), where ˜ N −1 n (t) := inf{p∈ [0, 1]| ˜ N n (p) > t}. Clearly, the partial sum inverse and the counting process must be close, asymptotically (as ¯ S n converges to a continuous process in the limit). The following lemma shows that this is indeed the case. Lemma 2.12 (i)k ¯ N n − ¯ S −1 n k→ 0 as n→∞. (ii) √ n( ¯ N n − ¯ S −1 n )→ 0 as n→∞. Proof: (i) Fix t∈ [0, 1]. By definition it follows that ¯ S n ( ˜ N n (t))≤t and ¯ S n ( ¯ S −1 n (t))>t (and ¯ S n ( ¯ S −1 n (t)−)≤t). Thus, for any> 0, ¯ S n ( ˜ N n (t) +)>t. In particular, ˜ N n (t) + 1 n ≥ ¯ S −1 n (t). Since ¯ S n is non-decreasing (since the increments ξ n,l ≥ 0), it follows that 1 n ≥ ¯ S −1 n (t)− ˜ N n (t)≥ 0, where the last inequality follows by definition. Combining this expression with Lemma 2.11, the conclusion follows. (ii) The result is an obvious corollary of the argument for part (i). We now present the proof of Theorem 2.8. By an application of Theorem 7.8.1 of [46], the fSLLN in Theorem 2.6 implies the con- vergence of the corresponding inverse function, ¯ S −1 n toe, and by part (i) of Lemma 2.12 the convergence of the counting process ¯ N n . This proves part (i) of Theorem 2.8. Next, note that 1 √ n bntc X l=1 ξ n,l − √ nt = 1 √ n bntc X l=1 (ξ n,l −μ n ) + √ n (bntcμ n −t)⇒W 0 (t), 109 u.o.c. of [0,∞) a.s. as n→∞, where the latter term of the second expression converges to 0: √ n (bntcμ n −t)→ 0 as n→∞. Theorem 7.8.2 of [46] implies that the fCLT above implies the convergence of the scaled and centered inverse process, and hence the counting process by part (ii) of Lemma 2.12 Proof of Proposition 2.6 Fix n≥ 1 and t∈ [0, 1]. Define j ∗ n := inf{j = 0,...,n :t∈ (j/n−T,j/n +T ]} as the first arrival index such thatt is in the support ofF n,j ∗ n , andj ∗∗ n := sup{j = 0,...,n :t≥j/n−T} as the largest index such that t is greater than the lower bound of the support of F n,j ∗∗ n . By the definition of the infimum and supremum, for any > 0, j ∗ n n − <t−T≤ j ∗ n n and (2.35) j ∗∗ n n <t +T≤ j ∗∗ n n +, (2.36) so that as n→∞, j ∗ n n →t−T and j ∗∗ n n →t +T. Now, let t∈ [−T,T ], then using the definition of j ∗ n and j ∗∗ n , 1 n n+1 X i=1 F t− i− 1 n = 1 n j ∗∗ n X i=1 F t− i− 1 n = 1 n j ∗∗ n X i=1 t− i−1 n +T 2T = 1 2T ( (t +T ) j ∗ n − 1 n − 1 n 2 j ∗∗ n (j ∗∗ n − 1) 2 ) → (t +T ) 2 4T , 110 where the second equality follows by the fact that j ∗ n = 0 since t∈ [−T,T ), and j ∗∗ n < n follows from the fact that T ∈ [0, 0.5]. The limit follows by the limit argument presented above for j ∗∗ n . Next, fix t∈ (1−T, 1 +T ]. In this case, j ∗∗ n =n and we have 1 n n+1 X i=1 F t− i− 1 n = 1 n n+1 X i=j ∗ n t +T− i−1 n 2T + j ∗ n n = 1 2T ( (t +T ) n + 1−j ∗ n n − 1 n 2 " (n + 1)(n + 2) 2 − j ∗ n (j ∗ n + 1) 2 #) + j ∗ n n → t +T 2T − t 2 −T 2 2T + (t−T ) 2 4T − 1 4T + (t−T ), where the first equality follows by the fact that for all indices j < j ∗ n the value of the distribution function at this t is 1 (in effect, this t is outside the support of these arrival distributions). The limit, of course, is straightforward from those of j ∗ n . Finally, for t∈ [T, 1−T ] we have 1 n n+1 X i=1 F t− i− 1 n = j ∗ n n + 1 n j ∗∗ n X i=j ∗ n t +T− i−1 n 2T = j ∗ n n + t +T 2T j ∗∗ n −j ∗ n n − 1 2Tn 2 j ∗∗ n (j ∗ n ∗ +1) 2 − j ∗ n (j ∗ n − 1) 2 ! → t, where first equality follows from the fact that, for large enough n, j ∗ n > 1 and j ∗ n ∗<n. The rest of the argument is a consequence of the limits for j ∗ n and j ∗∗ n . This completely characterizes the limit for a fixed t. Uniform convergence can be shown by a similar argument as in the proof of Lemma 2.2. 111 Proof of Corollary 2.6 Fix t∈ [−T,T ). Notice that only users indexed by p∈ [0,t +T ] can possibly arrive in this interval. Therefore, for such a t, we have Z 1 0 F p (t)m(dp) = Z t+T 0 t−p +T 2T dp = (t +T ) 2 4T . This is matches the expression found in Proposition 2.6. Next, let t∈ [T, 1−T ]. In this instance, users indexed by p∈ [0,t−T ) and p∈ (t +T, 1] cannot arrive in this interval. However, those indexed in the former interval will have arrived by time t. This implies that Z 1 0 F p (t)m(dp) = Z t−T 0 m(dp) + Z t+T t−T t−p +T 2T dp = t, agreeing with Proposition 2.6. Finally, for t ∈ (1− T, 1 + T ], only arrivals with index p∈ [t−T, 1] can arrive in this interval and, furthermore, all other arrivals will have happened by time t. This implies that, Z 1 0 F p (t)m(dp) = Z t−T 0 dp + Z 1 t−T t−p +T 2T dp = t +T 2T − t 2 −T 2 2T + (t−T ) 2 4T − 1 4T + (t−T ). This completes the proof. 112 Chapter 3 A Generalized Jackson Network of Δ (i) /GI/1 Queues 3.1 Introduction Single class open queueing networks, in our definition, consist of a number of infinite buffer, FIFO, single server queues (a.k.a. ’nodes’) interconnected by customer routing. On com- pletion of service at a particular node, a customer is routed to another node or exits the network altogether. These queueing networks are called as ‘Generalized Jackson Networks’. Here, we are interested in a variation of such networks, called ’transitory’ generalized Jackson networks. These are models of service and communication networks that operate over a finite time horizon, or the system operator is interested in short-time behavior of such networks. Further, traffic and service in these queueing networks can also exhibit time inhomogeneties. For example, in datacenters and hospitals demand exhibits very obvious time-of-day and day-of-week effects. In these circumstances, standard analyses of the per- formance of generalized Jackson networks will not suffice, and one must resort to a purely transient, or ‘transitory’, analysis of the network. The transient analysis of the performance of generalized Jackson networks is non-trivial. The usual approach is to develop heavy-traffic diffusion approximations to the performance metrics. However, in time inhomogeneous and finite horizon networks this approach will not work, since conventional heavy-traffic approximations cannot be obtained. Instead, we develop a ’population acceleration’ approximation. Thus, in effect, we are considering a sequence of queueing networks with ’n’ jobs arriving at each node, in the nth network. To 113 the best of our knowledge, this is the first time transitory networks have been studied in depth. The queue length fluid limit is shown to be equal to the oblique reflection of the difference of the arrival and service processes (or the ’netput’ process). The diffusion limit turns out to be complicated, and is a reflection of a multidimensional diffusion bridge process - however, the reflection is through a directional derivative of the oblique reflection of the fluid netput in the direction of the diffusion limit of the netput process. This is a highly non-standard result. Indeed, it is only in the recent past that Mandelbaum and Ramanan [22] have investigated the existence of a directional derivative to the oblique reflection map. Leveraging the results of [22], we can only establish that the diffusion scaled sample paths converge in a pointwise manner for an arbitrary transitory generalized Jackson network. This is due to the fact that the directional derivative limit can have sample paths with discontinuitiesthatareneitherrightnorleftcontinuous. Thus,establishingweakconvergence in the M 1 topology, for instance, is not possible in general. Instead, we focus on the case of a tandem network, with uniform and unimodal arrival time distribution functions. In this case, we show that the discontinuities in the limit are either right or left continuous, and hence establish M 1 convergence. The rest of the chapter is organized as follows. We start with a description of the tran- sitory generalized Jackson network model in Section 3.2, and we develop fluid and diffusion approximations to the network primitives. In Section 3.3, we develop functional strong law of large numbers approximations to the queueing equations, and identify the fluid model corresponding to the transitory network. We identify the diffusion network model in Section 3.4, and establish a weak convergence result for a tandem network with unimodal arrival time distribution. 114 3.1.1 Related Literature Reiman first established the heavy-traffic diffusion approximation to open generalized Jack- son networks in [20]. In particular, the diffusion approximation is shown to be a multi- dimensional reflected Brownian motion in the non-negative orthant, reflected through the oblique reflection map. Such reflection maps have come to be called as Harrison-Reiman maps following the work in [75]. Chen and Mandelbaum [76, 77] characterize homogeneous fluid networks, as well as establishing fluid and diffusion approximations to these networks. The analysis of non-stationary and time inhomogeneous queueing systems is non-trivial in general. For single server queues, see [14, 18, 19] among others. In [66, 78] we develop fluid and diffusion models of transitory single server queues. For networks of queues, [21] develop strong approximations to queueing networks with nonhomogeneous Poisson arrival and ser- vice processes. In [79], the authors study the offered load process in a bandwidth sharing network, with nonstationary traffic and general bandwidth requirements. More recently, Liu and Whitt [80] study a network of non-Markovian fluid queues with time-varying traffic and customer abandonments. To be precise, they consider a (G t /M t /s t +GI t ) m /M t network with m nodes, time-varying arrivals, staffing and abandonments, and inhomogeneous Poisson ser- vice and routing, and characterize the performance of the network as a direct extension of the single-server queue case. 3.2 Transitory Generalized Jackson Network Consider a single class queueing network withK single server FIFO nodes. Each node starts service at some fixed time (which could be different from the other nodes) and offers service with a random, but bounded, service time with finite second moment. Assume every job is served independently of the others and that the servers are non-preemptive and non-idling. Jobs transit to another node or exits the network entirely on completion of service at a 115 particular node. We assume that the routing matrix satisfies a so-called H-R condition, so it has a spectral radius of less than one. (Ω,F,P)isanappropriateprobabilityspaceonwhichonecandefinetherequisiterandom variables. LetK :={1,...,K} be the set of nodes in the the network, andE⊂K the set of nodes where exogeneous traffic enters the network. Each node inE receives n jobs in a finite time horizon [0,T ]. We assume a very general model of the traffic: for each node in l∈K\E, set to T l,m =∞ for each m≤ n. Let T m := (T 1,m ,...,T K,m ) ,m≤ n, represent the tuple of arrival times of the mth job to each of the nodes. Assume that{T m ;m = 1,...,n} forms a sequence of independent random vectors, and that each component of the vector is independent. Let a m (t) := (1 {T 1,m ≤t} ,..., 1 {T K,m ≤t} )∈D K [0,∞), then A n (t) := P n m=1 a m (t) = (A n 1 (t),...,A n K (t))∈D K [0,∞) is the vector of cumulative arrival processes to the nodes inK. Clearly, A n l (t) = 0 for every n≥ 1 and l∈K\E. The service time of each arrival is assumed to be independent of the others. Let ν k i : Ω→ (0,∞) represent the service time of theith arrival to thekth node in the network, and let m k = 1/μ k be the mean service time. Suppose that service starts at time T s,k ≥ 0 in the kth node. Then, the number of service completions by timet at nodek isS k (t) :=sup{m≤ n| P m i=1 ν k i ≤ t−T s,k , and t ≥ T s,k }. The tuple S(t) := (S 1 (t),...,S K (t)) ∈ D K [0,∞) represents the service processes in the network. By assumption, the components of this vector are independent. On completion of service at node i, the job can join node j with probability p i,j ≥ 0 , i,j∈{1,...,K}, or exit the network with probability 1− P j p i,j . Thus, the routing matrix P := [p i,j ] is substochastic. Note that, we also allow feedback of jobs to the same node; i.e., p i,i ≥ 0. Let φ k i : Ω→{1,...,K}∀k∈{1,...,K} and∀i∈N such that φ k i = j implies that theith job at nodek will be routed to nodej. For brevity, let Γ k i =e φ k i , wheree i istheithunitvector. TherandomvectorR l (n) := P n i=1 Γ l i representsthecumulativenumber of jobs that have been routed out from node i. The kth component of R l (n) represents the number of departures from node l to node k, out of the n arrivals to that node. R(n) := 116 (R 1 (n),...,R K (n)) is aK×K matrix whose columns are the routing vectors from the nodes in the network. 3.2.1 Network Parameters Let Q k (t) = E k (t)−D k (t) be the queue length sample path at node k, where E k (t) := A k (t) + P K l=1 R k l (S l (B l (t))) is the total number of jobs arriving at nodek in the interval [0,t] and D k = S k (B k (t)) is the cumulative departure process. Here, B k (t) := R t 0 1 {Q k (s)>0} ds is the total busy time of the server, which follows consequent to the non-idling assumption. Therefore, the queue length process is Q k (t) :=A k (t) + K X l=1 R k l (S l (B l (t)))−S k (B k (t)); (3.1) LetV k (m) be the cumulative service time requirement of m arrivals to the kth node. As defined earlier, let ν k i be the workload presented by the ith arrival to the kth node in the network. Then, by definition, V k (m) := m X i=1 ν k i . By assumption the nodes in the network are non-idling. Thus,V k (0) = 0. Now, the instanta- neous workload measured in units of time, at nodek at timet is a function of the cumulative service time requirement of all arrivals at the nodek, including both arrivals from the exter- nal stream and from internal routing, and the amount of time the server has been busy upto the instant of interest. Thus, we have, Z k (t) :=V k (A k (t) + K X l=1 R l,(k) (S l (B l (t))))−B k (t). 117 3.2.2 Limits To Primitives The network primitives are the arrival process A(t), the service process S(t) and the routing matrix R(n). We follow the widely used ’continuous mapping approach’ to establish fluid and diffusion approximations for the network performance metrics as a function of those for the primitives. Our first result develops limits for the arrival process. Recall that{T m } is an i.i.d. sequenceofrandomvectors, andthatthecomponentsoftherandomvectorsareindependent. Let the marginal distribution of T i,m be F i (t) :=P(T i,m ≤t), and F(t) := (F 1 (t),...,F K (t)) the vector of these marginals (not the joint distribution). Recall that a standard Brownian Bridge process, W 0 , is equal in distribution to W (t)−tW (1) for all t∈ [0, 1], where W is a standard Brownian motion. Consider an extension of this definition to multiple-dimensions. Definition 3.1 Let W be aK-dimensional correlated Brownian motion process with covari- ance matrix ρ. Then W 0 (t) := W(t)−tW(1) for all t∈ [0, 1] is a K-dimensional Gaussian Bridge process, with EW 0 (t) = 0 and E[W 0 (t)W 0 (s)] =t(1−s)ρ, when s<t. For notational simplicity, we abuse the usual notation of composition of functions so that W 0 ◦ F = (W 0 1 ◦F 1 ,...,W 0 K ◦F K ). Proposition 3.1 Let{T m },m≤ n and n≥ 1, form a triangular array of i.i.d. random vectors with independent components, and for t∈ [0,T ] a m (t) := (1 {T 1,m ≤t} ,..., 1 {T K,m ≤t} ). Then, (i) 1 n P n m=1 a m (t)− F(t) a.s. → 0 in (C K ,U) as n→∞. (ii) ˆ A n := √ n 1 n P n m=1 a m − F ⇒ W 0 ◦ F in (C K ,U) as n→∞, where W 0 ∈C K [0,∞) is aK-dimensional Gaussian Bridge process andk·k = sup 1≤k≤K |· |. 118 Proof: The first part of the theorem is straightforward by using the union bound: for a fixed t, P( sup 1≤k≤K | 1 n n X m=1 1 {T k,m ≤t} −F k (t)|>)≤ K X k=1 P(| 1 n n X m=1 1 {T k,m ≤t} −F k (t)|>) and then use the First Borel-Cantelli Lemma to establish the pointwise convergence. Con- vergence in (C,U) now follows by standard arguments. Now, let ˆ A n = ( ˆ a 1 n ,..., ˆ a K n ). By the independence assumption, it is straightforward to conclude that for any linear combination of the components, P K k=1 t k ˆ a k ⇒ P K k=1 t k W 0 k ◦F k in (C,U) as n→∞, owing to the fact that each individual component converges and the continuity of the additon operator in (C,U). The joint convergence now follows by the Cramer-Wold Theorem. Next, consider the service process. For each n≥ 1 and k ∈K, consider a function μ n k : [0,∞)→ [0,∞), such that μ n k ∈ L 1 and M n k (t) := R t 0 μ n k (s)ds has a well defined limit M k (t),andconvergenceisuniformoncompactsets(u.o.c.) of [0,∞). LetS n k : Ω×[0,∞)→N represent the service process, such that ES n k (t) = M n k (t) for any t≥ 0. Thus, μ n k (t) is the ’service rate process’, associated with an (accelerated) system with n arriving jobs. Let S n := (S n 1 ,...,S n K ) be the tuple of service processes in the network, and as noted before we assume that the processes are independent of each other. This type of a counting process can be constructed, for instance, by population accelerating the service times (ν n i,1 ,...,ν n i,K ) := (ν i,1 /n,...,ν i,K /n), and then observing S n (t) as defined in Section 3.2. The functional limits to the service process are presented below. Proposition 3.2 Let S n form a sequence of stochastic processes such that E[S n (t)] = M n (t) := (M n 1 (t),...,M n K (t)) := ( R t 0 μ n 1 (s)ds,..., R t 0 μ n K (s)ds) for all t ≥ 0 and Cov(S n (t), S n (s)) is well defined. Furthermore,∃M∈C[0,∞) such that M n → M u.o.c. as n→∞. Then, (i) S n n − M n a.s. → 0 in (C K ,U) as n→∞. 119 (ii) ˆ S n (t) := √ n S n n − M ⇒ W◦ M in (C K ,U) as n→∞, where W := (W 1 ,...,W K ) is a K−dimensional Brownian motion process with covariance matrix ρ. Such limits can be observed in the case of non-homogeneous Poisson processes, for instance. Finally, recall the routing matrix R(n) as defined in Section 3.2. We assume that the entries in R(n) satisfy ER j i (n) =p j,i for all n≥ 1, and that P = [p i,j ]. Proposition 3.3 Let R(n) form a stationary sequence of random matrices such that 1 n R j i (n)→p j,i a.s. as n→∞. Then, (i) 1 n R(nt)− P 0 e a.s. → 0 in (C K×K ,U)as n→∞, where e : [0,∞)→ [0,∞) is the identity function. (ii) ˆ R n (t) := √ n 1 n R(nt)− P 0 e ⇒ ˆ R, in (C K×K ,U) as n → ∞. Here, ˆ R = (W 1 ,..., W K ), where W i is a K-dimensional Brownian motion process. The proof of this claim is in Chapter 7 of [41]. As a final note, we claim the following joint convergence result. Proposition 3.4 Assume that for eachn≥ 1, A n , S n and R(n) are mututally independent. Then, (i) 1 n A n , 1 n S n (t), 1 n R(nt) a.s. → (F, M, P 0 e) in (C K ×C K ×C K×K ,U) as n→∞. (ii) ˆ A n , ˆ S n , ˆ R n ⇒ W 0 ◦ F, W◦ M, ˆ R in (C K ×C K ×C K×K ,U) as n→∞. Here, the joint convergence is a direct consequence of the independence of the arrival, service and routing processes. 3.3 Functional Strong Law of Large Numbers Next, we develop the fluid approximation to the system parameters leveraging the Contin- uous Mapping Theorem. We start with the queue length process and show that an appro- priately centered and scaled version of the process satisfies the Oblique Reflection Mapping 120 Theorem (see Theorem 7.2 in [41]). Following (3.1), the fluid-scaled queue length process at nodek isQ n k (t) = A n k (t) + P K l=1 R k l (nS n l (B n l (t)))−S n k (B n k (t)). After centering the right hand side, Q n k (t) n = 1 n A n k (t)−F k (t) + 1 n K X l=1 " R k l (nS n l (B n l (t)))−p l,k S n l (B n l (t)) # + K X l=1 p l,k " S n l (B n l (t)) n − Z B n l (t) 0 μ n l (s)ds # +F k (t)− Z t 0 μ n k (s)ds 1 {t≥T s,k } + K X l=1 p l,k Z t 0 μ n l (s)ds 1 {t≥T s,l } + (1−p k,k ) Z t B n k (t) μ n k (s)ds + X l6=k p l,k Z t B n l (t) μ n l (s)ds − S n k (B n k (t)) n + Z B n k (t) 0 μ k (s)ds. (3.2) Note that I n k (t) := t1 {t≥T s,k } −B n k (t) = ( R t T s,k 1 {Q n k (s)=0} ds)1 {t≥T s,k } is the idle time process, whichmeasurestheamountoftimein [T s,k ,t]thatthenodeisnotservingjobs(i.e., thequeue is empty). While this expression appears formidable, the analysis is simplified significantly by the fact that the queue length process Q n satisfies a reflection mapping theorem. We first recall the definition of the definition of the Oblique Reflection Problem, and the map that solves it. Note that an M-matrix is a square matrix with spectral radius less than one. Theorem 3.1 [Oblique Reflection Problem] Let R be a K×K M-matrix. Then, for every x∈D K 0 ={x∈D K :x(0)≥ 0}, there exists a unique pair (y,z) inD 2K satisfying z = x + Ry≥ 0, dy ≥ 0 and y(0) = 0, z j dy j = 0, j = 1,...,J. The process (z,y) := (Φ(x), Ψ(x)) is the Oblique Reflection Map, where Φ(x) =x + RΨ(x) and R is called a reflection matrix. 121 Note that, in general, if G is a nonnegative M-matrix then so is R≡ I− G (Lemma 7.1 of [41]). Notice that Q n k can be decomposed as the sum of ¯ X n k and ¯ Y n k . ¯ X n k (t) = 1 n A n k (t)−F k (t) + 1 n K X l=1 " R k l (nS n l (B n l (t)))−p l,k S n l (B n l (t)) # + K X l=1 p l,k " S n l (B n l (t)) n − Z B n l (t) 0 μ n l (s)ds # +F k (t)− Z t 0 μ n k (s)ds 1 {t≥T s,k } + K X l=1 p l,k Z t 0 μ n l (s)ds 1 {t≥T s,l } − S n k (B n k (t)) n + Z B n k (t) 0 μ n k (s)ds and (3.3) ¯ Y n k (t) = (1−p k,k ) Z t B n k (t) μ n k (s)ds− X l6=k p l,k Z t B n l (t) μ n l (s)ds. (3.4) The following lemma shows that the queue length satisfies the Oblique Reflection Map- ping. Lemma 3.1 Let ¯ X n (t) = ( ¯ X n 1 (t),..., ¯ X n K (t)) 0 ∈D K 0 , where ¯ X n k (t) k∈{1,...,K} is defined in (3.3), ¯ Q n (t) = 1 n (Q n 1 (t),...,Q n K (t)) 0 ∈D K 0 and ¯ Y n (t) = (Y n 1 (t),...,Y n K (t)) 0 ∈D K 0 . Then, ( ¯ Q n (t), ¯ Y n (t)) = (Φ( ¯ X n (t)), Ψ( ¯ X n (t))). Proof: First, note that ¯ Q n (t) = ¯ X n (t) + (I− P 0 ) ¯ Y n (t). Further, P is a non-negative (substochastic) matrix with spectral radius less than unity and, therefore, an M-matrix. It follows that I− P 0 is also an M-matrix. By definition, we have Q n k (t)≥ 0∀k∈{1,...,K}, and Y n k (t) is non-decreasing and its slope is strictly positive when Q n k (t) is zero. Thus, the conditions of Theorem 3.1 are satisfied and the conclusion follows. Next, we establish a functional strong law of large numbers result for (3.3). Lemma 3.2 Let ¯ X n (t) = ( ¯ X n 1 (t),..., ¯ X n K (t))∀t∈ [0,∞). Then, ¯ X n (t) a.s. → ¯ X(t) := ( ¯ X 1 (t),..., ¯ X K (t)) in (C K ,U) as n→∞, 122 where, ¯ X k (t) =F k (t)− Z t 0 μ k (s)ds 1 {t≥T s,k } + K X l=1 p l,k Z t 0 μ l (s)ds 1 {t≥T s,l } . (3.5) Proof: The result follows by an application of part (i) of Proposition 3.4 to (3.3). Not- ing that B n k (t) ≤ t, the Random Time Change Theorem (Theorem 5.5, [41]) implies that, 1 n S n k (B n k (t))− R B n k (t) 0 μ n k (s)ds → 0 u.o.c. a.s. as n → ∞∀t ∈ [0,∞). Similarly, 1 n R k l (nS n k (B n k (t)))−p l,k S n k (B n k (t)) → 0 u.o.c. a.s. as n→∞∀t∈ [0,∞). Applying these results to (3.3) it follows that ¯ X n k (t)→ ¯ X k (t) u.o.c. a.s. as n→∞. We can now establish the functional strong law of large numbers limit for the queue length process. Theorem 3.2 Let ¯ X n (t) and ¯ X(t) be as defined in (3.3) and (3.5) respectively. Then, ( ¯ Q n (t), ¯ Y n (t)) satisfy Theorem 3.1 and, as n→∞, ( ¯ Q n (t), ¯ Y n (t)) a.s. → (Φ( ¯ X(t)), Ψ( ¯ X(t))) in (C K ×C K ,U). Proof: It follows by Lemma 3.1 that ( ¯ Q n (t), ¯ Y n (t)) satisfy the Oblique Reflection Mapping Theorem. Therefore, ( ¯ Q n (t), ¯ Y n (t)) ≡ (Φ( ¯ X n (t)), Ψ( ¯ X n (t))). Now, the reflec- tion regulator map, Ψ(·), is Lipschitz continuous under the uniform metric (Theorem 7.2, [41]). By the Continuous Mapping Theorem and Lemma 3.2 it follows that, (Φ( ¯ X n (t)), Ψ( ¯ X n (t)))→ (Φ( ¯ X(t)), Ψ( ¯ X(t))) u.o.c. a.s. as n→∞,∀t∈ [0,∞). Note that neither Theorem 3.1 nor Theorem 3.2 provide an explicit functional form for the reflection regulator Ψ(·). It can be shown that the regulator map is the unique fixed point,y ∗ ∈D K , of the mapπ(x,y)(t) := sup 0≤s≤t [−x(s)+Gy(s)] + ∀t∈ [0,∞), whereG is an M-matrix. Note that the sup in the definition of the regulator is applied to every dimension of ¯ X(t) simultaneously. The following corollary shows that the reflection map and fluid limit 123 of the queue length process for a parallel node queueing network is particularly simple and an obvious generalization of that of a single node system. Corollary 3.1 Consider a K-node parallel queueing network. The fluid limit to the queue length process is ( ¯ Q n (t), ¯ Y n (t)) a.s. → (Φ( ¯ X(t)), Ψ( ¯ X(t))) in (C K ×C K ,U) as n→∞, with Ψ( ¯ X(t)) = sup 0≤s≤t [− ¯ X(s)] + . Proof: Note that for a parallel queueing network P = 0. Therefore, the ’fixed point’ of the map π(·,·) is simply sup 0,≤s≤t [−x(s)] + . It follows that the regulator map of the fluid scaled queue length process is Ψ( ¯ X n (t)) = sup 0≤s≤t [− ¯ X n (s)] + . It follows by Theorem 3.2 that Ψ( ¯ X n (t))→ sup 0≤s≤t [− ¯ X(s)] + and Φ( ¯ X n (t))→ ¯ X(t) + Ψ( ¯ X(t)) u.o.c. a.s. as n→∞. Now, consider a tandem network with uniform arrival distribution with compact support. Corollary 3.2 Consider a tandem queueing network with P =    0 0 1 0   , and R = I− P T =    1 0 −1 1   . Let F = F 1 be uniform over the interval [−T 0 ,T ] where T 0 ,T > 0, and assume that μ 1 and μ 2 are the fixed service rates. Then, the fluid limit to the queue length process is ( ¯ Q n , ¯ Y n ) a.s. → (Φ( ¯ X), Ψ( ¯ X)) in (C K ×C K ,U) as n→∞, where ¯ X := (X 1 ,X 2 ) T = ((F 1 − μ 1 e), (μ 1 −μ 2 )e) T , Φ( ¯ X)(t) =    X 1 +Y 1 X 2 +Y 2 −Y 1   , Y 1 (t) = sup 0≤s≤t (−X 1 (s)) + , Y 2 (t) = sup 0≤s≤t (−X 2 (s) +Y 1 (s)) + = sup 0≤s≤t [−X 2 (s) + sup 0≤r≤s (−X 1 (r)) + ] + − sup 0≤s≤t (−X 1 (s)) + and e :R→R is the identity map. The proof is straightforward by substitution and we omit it. Next consider the fluid limit process for the busy time process, when the service process is stationary so that μ k (t) =μ k for all t≥ 0 and all k∈K. 124 Theorem 3.3 Let ¯ B n (t) = (B n 1 (t),...,B n K (t)). Then, as n→∞, ¯ B n (t) a.s. → t− MΨ( ¯ X(t)) in (C K ,U). (3.6) Here, t = (t1 {t≥T s,1 } ,...,t1 {t≥T s,K } ) and M =diag(1/μ 1 ,..., 1/μ K ). Proof: By definition ¯ B n (t) = t1− ¯ I n (t), where ¯ I n (t) = (I n 1 (t),...,I n K (t)) 0 . Recalling the definition of the process ¯ Y n (t) it is straightforward to see that ¯ I n (t) = (I− P 0 ) −1 ¯ Y n (t) for all t≥ 0. Therefore, ¯ B n (t) = t− (I− P 0 ) −1 ¯ Y n (t) Theorem 3.2 implies that, as n→∞, ¯ B n (t)→ t− Ψ( ¯ X(t)) u.o.c. a.s., ∀t∈ [0,∞). The final system parameter of interest is the virtual waiting time process. Here, we assume that the service rates are constant at every node in the network, allowing us to establish a snapshot principle relating the queue length and waiting time sample paths. First, the fluid scaled workload process in the kth node is given by, W n k (t) :=V n k A n k (t) + K X l=1 1 n R n l,(k) (S n l (B n l (t))) ! −B n k (t)−t1 {t≤T s,k } . (3.7) Let M = diag 1 μ 1 ,..., 1 μ K represent the diagonal matrix of expected service times. The functional strong law of large numbers process for the virtual waiting time is summarized in the following theorem. Theorem 3.4 Let W n := (W n 1 ,...,W n K ) be the waiting time process in the network. Then, W n a.s. → M ¯ Q in (C K ,U) as n→∞. 125 Proof: Adding and substracting appropriate centering terms, it is straightforward to see that W n k (t) = V n k A n k (t) + K X l=1 1 n R n l,(k) (S n l (B n l (t))) ! − 1 μ k " A n k (t) n + K X l=1 1 n R n l,(k) (S n l (B n l (t))) # + 1 μ k " A n k (t) n −F k (t) # + 1 μ k " K X l=1 1 n R n l,(k) (S n l (B n l (t)))−p l,k (S n l (B n l (t))) # + " F k (t) + K X l=1 p l,k (S n l (B n l (t))) # −B n k (t)−t1 {t≤T s,k } . Now, leveraging Theorem 3.3 and Proposition 3.4, the conclusion follows. 3.4 Functional Central Limit Theorems Next, consider the second order refinement to the fluid limit. Unlike the heavy traffic limits for generalized Jackson networks (see [41, 20]), the FCLT in the transitory case isn’t a reflected Brownian motion (RBM) process (reflected through the Oblique Reflection Map). Instead, the reflection map is the directional derivative of the Oblique Reflection of ¯ X (from Lemma 3.2) in the direction of a (non-RBM) diffusion process ˆ X, to be defined below. A similar result was observed in the case of single server queues in Chapters 1 and 2. In that case, the directional derivative reflection map was explicitly characterized. However, in general this is not possible in the network case. As before, let R be aK×K M-matrix, P T := I−R, andx∈C K 0 . Under the hypothesis of Theorem 3.1, there exist (z,y) := (Φ(x), Ψ(x))∈C K ×C K such that z = x + Ry, y is non-decreasing (in all components) and y j grows only when z j is zero (for all j = 1,...,K). Recall the definition of the directional derivative of the oblique reflection map adapted from [22]. 126 Definition 3.2 Given (x,χ)∈C K 0 ×C K , the directional derivative of the oblique reflection map Φ(x) =x +RΨ(x) in the direction of χ is the pointwise limit of Δ n χ (x) := √ n Φ χ √ n +x ! − Φ(x) ! ∈C K n≥ 1 as n→∞. Theorem 1.1 (ii) of [22] identifies the limit process. We state this as a lemma for com- pleteness. Lemma 3.3 (Mandelbaum and Ramanan) If (x,χ) ∈ C K 0 ×C K then the directional derivative limit Δ χ (x) exists and convergence in Definition 3.2 is uniform on compact subsets of continuity points of the limit Δ χ (x). Further, if (z,y) solve the oblique reflection problem for x then Δ χ (x) =χ + Rγ(x,χ), where γ :=γ(x,χ) lies inD K usc and is the unique solution to the system of equations γ i (t) =                    0 if t∈ (0,t i l ), sup s∈∇ i t [−χ i (s) + [P T γ] i (s)] + if t∈ [0,t i u ], sup s∈∇ i t [−χ i (s) + [P T γ] i (s)] t>t i u , (3.8) for i = 1,...,K, where∇ i t :={s∈ [0,t]|z i (s) = 0 and y i (s) = y i (t)}, t i l := inf{t≥ 0 : z i (t) = 0} and t i u := inf{t≥ 0 :y i (t)> 0}. Returning to the queue length process, the diffusion scale process is ˆ Q n := √ n ¯ Q n − ¯ Q ∈ D K . Similarly, the second order refinement to the ’netput’ process is ˆ X n := √ n ¯ X n n − ¯ X ∈D K . Using Proposition 3.4, and the fact that the limit processes have sample paths inC K , the following Lemma is straightforward to establish. 127 Lemma 3.4 Let ¯ X n := ( ¯ X n 1 ,..., ¯ X n K ) ∈ D K 0 , where ¯ X n k is defined in (3.3), and ¯ X := ( ¯ X 1 ,..., ¯ X K )∈C K with ¯ X k defined in (3.5). Then, ˆ X n ⇒ ˆ X := ( ˆ X 1 ,..., ˆ X K )∈ (C K ,U) as n→∞, where ˆ X k := W 0 k ◦F k −W k ◦ R t 0 μ k (s)ds +hW k ◦M, 1i,h·,·i is the inner product operator and 1 is the one-dimensional vectors of ones and W k is the kth row of the product PM. Recall from Lemma 3.1 that ¯ Q n = ¯ X n + RΨ( ¯ X n ), and from Theorem 3.2 that ¯ Q = ¯ X + RΨ( ¯ X). It follows that ˆ Q = √ n ¯ X n + RΨ( ¯ X n )− ¯ X− RΨ( ¯ X) = Δ n ˆ X n ( ¯ X). The diffusion limit is a consequence of Lemma 3.4 and the limit toγ( ¯ X, ˆ X n ). Notice that ˆ X∈C K , in which case Lemma 3.3 characterizes Δ ˆ X ( ¯ X), the directional derivative limit. Our first lemma relates Δ ˆ X ( ¯ X) with Δ n ˆ X n ( ¯ X). Lemma 3.5 Let Δ n ˆ X ( ¯ X) and Δ n ˆ X n ( ¯ X) be defined as in Definition 3.2. Then, Δ n ˆ X n ( ¯ X)− Δ n ˆ X ( ¯ X) a.s. → 0 as n→∞, wherek·k is the supremum norm. Proof: First, simplifying the expression on the left hand side, it is straightforward to see that Δ n ˆ X n ( ¯ X) = ˆ X n + R √ n Ψ ˆ X n √ n + ¯ X − Ψ( ¯ X) . By Lemma 3.4 and the Skorokhod representation theorem, it follows thatk ˆ X n − ˆ Xk a.s. → 0 as n→∞. The lemma follows once it is shown thatk √ n Ψ ˆ X n √ n + ¯ X − Ψ ˆ X √ n − ¯ X k a.s. → 0 as n→∞. 128 Chen and Whitt [81] show that the oblique reflection map and the reflection regulator are Lipschitz continuous with respect to the uniform metric topology. Therefore, √ n Ψ ˆ X n √ n + ¯ X ! − Ψ ˆ X √ n − ¯ X !! ≤ K √ n ˆ X n √ n + ¯ X− ˆ X √ n − ¯ X , = Kk ˆ X n − ˆ Xk. The lemma now follows as a consequence of Lemma 3.4 and the continuity of the addition operator under the uniform metric. Lemma 3.5 frees us to concentrate on the process Δ n ˆ X ( ¯ X). Lemma 2 in [66] (an extension of Theorem 3.2 in [19]) proves the existence of the limit in the single dimensional case, where P = 0, and also extends to convergence in M 1 . In the multi-dimensional case, however, this is not straightforward, as shown by [22]. We direct the reader to Theorem 1.2 of [22] which encapsulates the various necessary conditions for discontinuities in the sample paths of the directional derivative limit Δ ˆ X ( ¯ X). First, given (z,y) as the solution to the ORP forx∈C K define, for each t∈ [0,∞), O(t) := {i∈{1,...,K} :z i (t)> 0}, U(t) := {i∈{1,...,K} :z i (t) = 0, Δy i (t+)6= 0, Δy i (t−)6= 0}, C K (t) := {1,...,K}\[O(t)∪U(t)], EO(t) := {i∈C K (t) :∃δ> 0 such that z i (s)> 0∀s∈ (t−δ,t)}, SU(t) := {i∈C K (t) : Δz i (t−) = 0, Δz i (t+)6= 0.} When x = ¯ X,O(t) is the set of nodes in the network that are overloaded at time t,U(t) is the set of underloaded nodes,C K (t) the set of critically loaded nodes,EO(t) is the set of critically loaded queues that are at the end of overloading andSU(t) is the set of critically loaded nodes that are at the start of underloading. Note that the definitions of overloading, 129 t F 1 μ 1 μ 2 Figure 3.1: We study a two node tandem network with uniform arrival time distribution to node 1, and constant service rates μ 1 and μ 2 . underloading and critical loading conform to the standard notions for G/G/1 queues, as noted in [66]. Theorem 1.2 of [22] gives necessary conditions so that, in general, the sample paths of the directional derivative can have both a right and left discontinuity att∈ [0,∞). This is a consequence of the H-R matrix R. As a result, the sample paths of Δ ˆ X ( ¯ X) lie in ¯ D K lim and establishing M 1 convergence in this space is non-trivial. Recall that the standard description ofM 1 convergence is through the graphs of the functions - which can be described via linear interpolations inD K andD K l,r . However, in ¯ D K lim no such simple description exists (see Chapter 12 of [43] and Chapter 6, 8 of [46] for further details on these issues). Given the inherent difficulty in establishing a general result, we focus on the simple tandem network, where the arrival time distribution is uniform on the interval [−T 0 ,T ] and T 0 ,T > 0; see Figure 3.4. For simplicity, we will also assume that all queues start empty, which implies that t 1 l =−T 0 and t i l = 0 for i∈{1,...,L} in the definition of (3.8). Theorem 3.5 Consider a tandem queueing network with P =    0 1 0 0   , and R = I− P 0 =    1 0 −1 1   . Assume that F = F 1 is uniform over [−T 0 ,T ], and service rate at node 1 is μ 1 and at node 2 μ 2 . Then, ˆ Q n ⇒ ˆ Q := Δ ˆ X ( ¯ X) in (D K lim ,SM 1 ) as n→∞, where ˆ X := (W 0 1 ◦F 1 −σ 1 μ 3/2 1 W 1 ), (σ 1 μ 3/2 1 W 1 −σ 2 μ 3/2 2 W 2 ) , ¯ X = ((F 1 −μ 1 e), (μ 1 −μ 2 )e) T ande :R→ R is the identity map. Proof: 130 Recall that F (t) = t+T 0 T +T 0 for all t∈ [−T 0 ,T ]. We consider three subcases and establish the weak convergence result for each of them separately. (i) Let μ 1 <μ 2 . Then, ¯ Q 1 (t) =          (F (t)−μ 1 t1 {t≥0} ) ∀t∈ [−T 0 ,τ 1 ), 0 ∀t∈ [τ 1 ,∞), (3.9) and ¯ Q 2 (t) = 0∀t≥ 0, where τ 1 := inf{t> 0|F (t) =μ 1 t}. These follow as a consequence of Corollary 3.2, and noting that ¯ X = (F (t)−μ 1 e, (μ 1 −μ 2 )e). Thus, we have ∇ 1 t :=                    {−T 0 } ∀t∈ [0,τ 1 ), {−T 0 ,τ 1 } t =τ 1 , {t} ∀t>τ 1 , and (3.10) ∇ 2 t := {t}∀t∈ [0,∞). (3.11) Thus, node 1 is inO(t) for allt∈ [−T 0 ,τ 1 ),C K (t) fort =τ 1 and inU(t) fort>τ 1 , and node 2 is inU(t) for all t. The limit process ˆ Q has a discontinuity only in the first component at ˆ Q 1 (τ 1 ) = ˆ X 1 (τ 1 )+ max{0,− ˆ X 1 (τ 1 )}. Note that ˆ Q 1 (τ 1 −) = ˆ X 1 (τ 1 ) and ˆ Q 1 (τ 1 +) = 0, implying that ˆ Q 1 has either a right or left discontinuity at τ 1 . If ˆ X 1 (τ 1 )≥ 0 then ˆ Q 1 (τ 1 ) = ˆ X 1 (τ 1 ) = ˆ Q 1 (τ 1 −) > ˆ Q 1 (τ 1 +) = 0 and has a right discontinuity. Else, if ˆ X 1 (τ 1 )< 0 then ˆ Q 1 (τ 1 ) = 0 = ˆ Q 1 (τ 1 +)> ˆ Q 1 (τ 1 −) and has a left discontinuity. Thus, the limit process ˆ Q has sample paths inD K lim . The proof of convergence for ˆ Q n = ( ˆ Q n,1 , ˆ Q n,2 ) in this case is simple. First, Theorem 2 of [66] shows that ˆ Q n,1 ⇒ ˆ Q 1 := ˆ X 1 + sup 1 s∈∇· (− ˆ X(s)) in (D K lim ,M 1 ) as n→∞, and ˆ Q n,2 ⇒ 0 in (D K lim ,M 1 ). Recall that Disc( ˆ Q 1 ) and Disc( ˆ Q 2 ) are the (respective) sets of discontinuity point, and it is obvious thatDisc( ˆ Q 1 )∩Disc( ˆ Q 2 ) =φ. Therefore, by Corollary 6.7.1 of [46], 131 ˆ Q n,1 + ˆ Q n,2 ⇒ ˆ Q 1 in (D K lim (R),M 1 ) asn→∞. Consequent to Theorem 6.7.2, it follows that ˆ Q n ⇒ ˆ Q := ( ˆ Q 1 , 0) T in (D K lim ,SM 1 ) as n→∞. (ii) Let μ 1 >μ 2 . Then, ¯ Q 1 and∇ 1 t follow (3.9) and (3.10) (resp.). ¯ Q 2 on the other hand, is more complex now: ¯ Q 2 (t) =                    (μ 1 −m 2 )t ∀t∈ [0,τ 1 ], (F 1 (t)−μ 2 t) ∀t∈ [τ 1 ,τ 2 ], 0 ∀t>τ 2 , where τ 2 := inf{t>τ 1 :F 1 (t) =μ 2 t} (note that τ 2 >τ 2 since μ 1 >μ 2 ). It follows that ∇ 2 t =                    {0} ∀t∈ [0,τ 2 ), {0,τ 2 } t =τ 2 , {t} ∀t>τ 2 . It follows that node 2 is inO(t) for all t∈ [0,τ 2 ),C K (t) at t =τ 2 andU(t) for t>τ 2 . The diffusion limit ˆ Q := ( ˆ Q 1 , ˆ Q 2 ) has discontinuities in both components. For node 1, if ˆ X 1 (τ 1 )≥ 0 then ˆ Q 1 (τ 1 ) has a right discontinuity, while ˆ X 1 (τ 1 ) < 0 then ˆ Q 1 (τ 1 ) has a left discontinuity. Similary, if ˆ X 2 (τ 2 )≥ 0 then ˆ Q 2 (τ 2 ) has a right discontinuity, and if ˆ X 2 (τ 2 )< 0 it has a left discontinuity. It follows that ˆ Q has sample paths inD K lim . Furthermore, it is clear that Disc( ˆ Q 1 )∩Disc( ˆ Q 2 ) =φ. Therefore, the weak convergence result follows by the same reasoning as in part (i). (iii) Assume μ 1 =μ 2 . Once again, ˆ Q 1 and∇ 1 t follow (3.9) and (3.10) (resp.). On the other hand, for node 2 ˆ Q 2 = 0, but unlike case (i), the queue is empty but the server operates at full capacity till τ 1 , and then enters underload. Thus, ∇ 2 t =          [0,t] ∀t∈ [0,τ 1 ], {t} ∀t>τ 1 . 132 It is clear that node 2 switches fromC K (t) in [0,τ 1 ] toU(t) for t > τ 1 . Furthermore, at τ 1 itself, the node is inSU(t) (the regulator is flat to the left ofτ 1 and increasing to the right). The diffusion limit, once again, has discontinuities in both components. However, it is clear that Disc( ˆ Q 1 ) =Disc( ˆ Q 2 ) ={τ 1 }. For anyT >−T 0 , it is straightforward to see that ( ˆ Q 1 (t)− ˆ Q 1 (t−))( ˆ Q 2 (t)− ˆ Q 2 (t−))≥ 0 for all−T 0 ≤t≤T : clearly, for anyt<τ 1 , ˆ Q i ,i = 1, 2 are both continuous. On the other hand, at τ 1 , ˆ Q 1 (τ 1 )≥ ˆ Q 1 (τ 1 −) and ˆ Q 2 (τ 1 ) = ˆ Q 2 (τ 1 −). Finally, for any t>τ 1 , ˆ Q 1 (τ 1 ) = ˆ Q 1 (τ 1 −) and ˆ Q 2 (τ 1 ) = ˆ Q 2 (τ 1 −). Now, by Theorem 6.7.3 of [46], it follows that ˆ Q n,1 + ˆ Q n,2 ⇒ ˆ Q 1 + ˆ Q 2 in (D K lim (R),M 1 ) as n→∞. Then, by Theorem 6.7.2 of [46], ˆ Q n ⇒ ˆ Q in (D l,r ,SM 1 ) as n→∞. This concludes the proof. Theorem 3.5 shows that in the case of a tandem network, with uniform arrival time distribution, the weak convergence result can be established in the spaceD K lim and in the SM 1 topology. In fact this result is true, if F 1 is unimodal such that node 1 is overloaded in the initial phase (i.e., in the interval [−T 0 ,τ 1 ), with T 0 ≥ 0 now). We capture this fact in the following lemma. Without loss of generality, we will assume that T 0 = 0. Lemma 3.6 Let F 1 be a unimodal distribution function with finite support [0,T ], and con- sider a tandem queue as defined in Theorem 3.5. Then, ˆ Q n ⇒ ˆ Q := Δ ˆ X ( ¯ X) in (D K lim ,SM 1 ) as n→∞, where ˆ X := W 0 1 ◦F 1 −σ 1 μ 3/2 1 W 1 , (σ 1 μ 3/2 1 W 1 −σ 2 μ 3/2 2 W 2 ) T , ¯ X = (F 1 −μ 1 e, (μ 1 −μ 2 )e) T and e :R→R is the identity map. The proof follows that of Theorem 3.5 and is omitted. Note that the compact support assumptionisrequired,duetothefactthatweproveweakconvergenceovercompactintervals of time (see Section 7.2 of [66] for a discussion on this point). 133 Turning to a multi-stage tandem queue, the proof for Theorem 3.5 is quite difficult to adapt,asthenumberofcasesballoonswiththenumberofstages. UsingLemma3.3,however, we can state the pointwise diffusion limit. We conjecture that this limit will be the diffusion limit to the queue length process of a multi-stage tandem network. Theorem 3.6 Consider a L node tandem queueing network with P =            0 1 0 ··· 0 0 0 1 ··· 0 . . . ··· . . . 0 0 0 ··· 0            , with R = I− P 0 . Assume that F = F 1 is uniform over [−T 0 ,T ], and service rates are μ i for i∈{1,...,L}. Then, ˆ Q n ⇒ ˆ Q := Δ ˆ X ( ¯ X) in (D lim ,SM 1 ) as n→∞, where ˆ X = ( ˆ X 1 ,..., ˆ X L ) with ˆ X 1 = W 0 1 ◦F 1 −σ 1 μ 3/2 1 W 1 and ˆ X i = σ i−1 μ 3/2 i−1 W i−1 −σ 2 μ 3/2 i W i for i∈ {2,...,L}, ¯ X = ( ¯ X 1 ,..., ¯ X L ) with ¯ X 1 =F 1 −μ 1 e and ¯ X i = (μ i−1 −μ i )e for i∈{2,...,L} and e :R→R is the identity map. It’s useful to unpack the reflection term in ˆ Q. Recall that Δ ˆ X ( ¯ X) = ˆ X + Rγ( ¯ X, ˆ X), where the reflection term γ := (γ 1 ,...,γ L ) is defined in (3.8). Using P from the Theorem, it’s straightforward to see that γ 1 (t) = sup s∈∇ 1 t [− ˆ X 1 (s)] ∀t ∈ [−T 0 ,∞) and γ i (t) = sup s∈∇ 1 t [− ˆ X i (s) +γ i−1 (s)]∀t∈ [−T 0 ,∞). Finally, using R = I− P 0 , it follows that ˆ Q 1 (t) = ˆ X 1 (t) +γ 1 (t) and Q i (t) = ˆ X i (t) +γ i (t)−γ i−1 (t) for i∈{2,...,L} 134 Part II Strategic Users in Transitory Queueing Networks 135 Chapter 1 The Network Concert Queueing Game 1.1 Introduction 1.1.1 Fluid Limit Model Consider a transitory queueing network with K single server FIFO queues. In this chapter we consider the fluid model presented in Section 3.3 of Part I. . . . . . . . . . L1 L2 LL } Single Stage Figure 1.1: Example of a multi-stageK×L lattice network, with arbitrary interconnections between adjacent layers. This paper is focused on feedforward generalized Jackson queueing networks, which sim- plifies the description to a large extent. In particular, the most general network topology we assume has a “lattice” network structure, as depicted in Figure 1.1. We define a Lattice Network as follows. Definition 1.1 A K×L lattice network is one that can be arranged in L layers of K node parallel networks, with arbitrary interconnections between adjacent layers. A K× 2 lattice sub-network will be called a stage. 136 Notice that any general feedforward network can be arranged as aK×L lattice network for some K and L by introducing “dummy nodes”, i.e., queues with infinitely fast servers so that they do not introduce any delay. LetL l represent the set of nodes in layer l where l = 1,...,L. Nodes inL l are represented by the tuple (l,i), i = 1,...,K. Interconnections between the layers are described by stage-wise binary incidence matrices G l = [e (l,i),(l+1,j) ], wheree (l,i),(l+1,j) = 1 if there is a connection between the nodes (l,i) and (l + 1,j), and zero otherwise. Let G l (l,i) :={(l + 1,j)∈L l+1 |e (l,i),(l+1,j) = 1} be the set of nodes in layer l + 1 that node (l,i) connects to. Similarly, H l (l,i) :={(l + 1,j)∈L l−1 |e (l+1,j),(l,i) = 1} is the set of nodes in layer l− 1 that connect to node (l,i). Let F := F 1 = F (1,i) ; (1,i)∈L 1 ∈C(−∞,∞) be the exogeneous arrival process at layer 1, where C(−∞,∞) is the space of absolutely continuous functions on (−∞,∞); the support of F is defined as supp(F) = ∪ (1,i)∈L 1 supp(F (1,i) ). By definition, it follows that R supp(F) P i∈L 1 F (1,i) (t)dt = 1. Let p (l,i),(l+1,j) (t) denote the routing probability for a non-atomic user to go from node (l,i) to node (l + 1,j) at time t. In the fluid regime, this is just the fraction of fluid leaving (l,i) ∈L l to (l + 1,j) ∈L l+1 at time t and so P (l+1,j)∈G l (l,i) p (l,i),(l+1,j) (t) = 1. For brevity, denote p (l,i) = (p (l,i),(l+1,j) ; (l + 1,j)∈ G l (l,i) ), p l = p (l,i) ; (l,i)∈L l and p = (p 1 ,..., p L ). Let F (l,i) denote the cumulative amount of fluid that has entered node (l,i) in layer l. Denote F l = F (l,i) ; (l,i)∈L l ∈C[0,∞) K , which we term fluid arrival functions; letsupp(F l ) =∪ K i=1 supp(F (l,i) ) be the support of F l . The (fluid) departure rate from (l,i)∈L l at timet is dB (l,i) (t) dt = dF (l,i) (t) dt ∧μ (l,i) , where the derivativeisthefluidarrivalrateandμ (l,i) istheservicerate. Thedepartureratefollowsfrom the cumulative departure functionB (l,i) (t) =F (l,i) (t)−Q (l,i) (t) =F (l,i) (t)−(F (l,i) (t)−μ (l,i) t)− sup 0≤s≤t (−F (l,i) (s)+μ (l,i) s) + , whereQ (l,i) (t) = (F (l,i) (t)−μ (l,i) t)+sup 0≤s≤t (−F (l,i) (s)+μ (l,i) s) + is the queue length at time t. Let Y (l,i) (t; (p l−1 )) := P (l−1,j)∈H l−1 (l,i) R t 0 p (l−1,j),(l,i) (s)dB (l−1,j) (s) denote the fluid limit of the cumulative arrival process to node (l,i) from nodesL l−1 , by time t≥ 0. 137 In Chapter 3 Part I, we prove a snapshot principle that relates the sample paths of the queue length process with those of the workload process W(t) = (W (l,i) (t;F (l,i) ); 1≤ l≤ L, 1≤ i≤ K), where we have explicitly stated the dependence on the cumulative arrival functionF (l,i) . Recallthattheworkloadprocesstimeispreciselytheamountoftimeittakesa user arriving at time t to exit the node. The snapshot principle states that W(t) = MQ(t), where M := diag(μ −1 1 ,...,μ −1 K ) is a diagonal matrix of inverse service rates, and Q(t) is the vector of queue lengths at all nodes in the network. Making use of the expression for Q (l,i) , the delay at node (l,i) is W (l,i) (t;F (l,i) ) := sup 0≤s≤t F (l,i)(t) −F (l,i)(s) μ (l,i) − (t−s) , and the departure time of a non-atomic user arriving at time t is τ (l,i) (t) :=t +W (l,i) (t;F (l,i) ). Of particular interest to us is the sojourn time process; i.e., the amount of time it takes a user arriving at time t to exit the network. This value depends upon the route taken through the network, and depends on the workload at each node along the route. The sojourn time for any path σ containing the nodes ((l,i l );l = 1,...,L) is recursively defined byD (l,i l ) (t;F (l,i l ) ) :=W (l,i l ) (t;F (l,i l ) )+D (l+1,i l+1 ) (τ (l,i l ) ;Y (l+1,j) (τ (l,i) (t); p l )),forl = 1,...,L−1. For simplicity, let the sojourn time of taking route σ starting from node (1,i 1 )∈L 1 be D(t,σ) :=D (1,i) (t;F (1,i) (t)). 1.1.2 Game and Equilibrium Concept We assume a continuum of users arriving at the network, each indexed by a number s in [0, 1] (i.e., the total volume of users is one unit). We consider a one-shot game, where users arriving at the network choose a time to arrive and the route to take through the network a priori so that they minimize a cost function that trades-off the sojourn time and the exit time. LetF s = (F s , p s ) denote the decisions by user s, where F s = (F s 1 ,...,F s K ) is a joint randomization over the time of arrival to each node, and p s is the vector of routing probabilities in aK×L lattice network. Let ¯ F(t) := R 1 0 F s (t)ds and ¯ p(t) := R 1 0 p s (t)ds denote the aggregate arrival and routing profile given individual profiles. Furthermore, denote ¯ F = ( ¯ F, ¯ p)andassumeitisgiven. Then, foragivenrouteσ containingnodes ((l,i l );l = 1,...,L), 138 the cost faced by a non-atomic user arriving at queue (1,i) with i 1 =i at time t is given by C ¯ F (t,σ) = (α +β)D ¯ F (t,σ) +βt, where D ¯ F (t,σ) is the sojourn time of the user along route σ, when the strategy profile is ¯ F. Let a users pick an arrival time according to distribution G s and a routing probability p s . Then, the expected cost faced by it can easily be computed from the above, and will be denoted by C ¯ F (G s ; p s ). Definition 1.2 A multi-strategy{F s = (F s , p s ) : s∈ [0, 1]} is a Nash equilibrium, if (i) ¯ F(t) = R 1 0 F s (t)ds and ¯ p(t) = R 1 0 p s (t)ds are well-defined for each t, and (ii) for any user s, C ¯ F (F s ; p s )≤C ¯ F (G s ; q s ), ∀(G s , q s ). We will specifically be interested in symmetric mixed strategies, i.e., equilibria wherein all players use the same mixed strategies. In that case, equilibrium strategies can be given a specific meaning, and have the following further characterization just as in [27] for the single player-single queue concert arrival game. Definition 1.3 A symmetric strategy profileF ∗ = (F ∗ , p ∗ ) is an equilibrium (strategy) pro- file if (F ∗ , p ∗ ) is a minimizer of the expected cost C F ∗(t; p) at every time τ in the support of F ∗ (denotedT ∗ ), i.e., C F ∗(τ; p ∗ )≤C F (t; p), ∀F = (F, p), ∀τ∈T ∗ , −∞<t<∞. This implies that at a symmetric equilibrium, the cost of arriving at a timet is minimized by arrival strategy F ∗ and following routing policy p ∗ . Note that it implies that at equilibrium an arriving user will see a constant cost in the support of F ∗ . If this were not the case, there would be a time τ 0 which will be a minima, and each user would then have an incentive to arrive at that time, and hence it can’t be an equilibrium. The above definition is for a single class of users arriving into a general feedforward net- work. This definition will be specialized for the parallel queueing network we will consider 139 first. We will also study the multiple population case in Section 1.2.2, each being char- acterized by constants (α i ,β i ). Rather than present the equilibrium notion with multiple populations for a general feedforward network, we will later provide a specialized definition for the parallel network case we study. For simplicity, we express the cost of choosing route σ at time t with symmetric strategies as C σ (t) = C F (t,σ), suppressing the strategy profile in the notation. 1.2 The Parallel Network Concert Queueing Game 1.2.1 Single Population Inthissection, weconsideraparallelnetworkofK queues. Weconsiderasinglepopulationof users with cost characteristics (α,β) choosing mixed strategies, i.e., probability distributions over arrival times, and look for a mixed-strategy Nash equilibrium profile of the non-atomic game, in the fluid regime. Specializing the cost function from Section 1.1.2, the cost of entering queue k is C k (t) = α ¯ W k (t) +β ¯ t c k , where ¯ t c,k = t + ¯ W k (t) and ¯ W k (t) is the waiting time in queuek. From Section 1.1.1,W k (t) = 1 μ k ( ¯ X k (t)+Ψ( ¯ X k (t))−(t−T s,k )1 {t≤T s,k } ), with ¯ X k (t) =F k (t)−μ k (t−T s,k )1 {t≥T s,k } , where, as before,F k (t) is the aggregate arrival profile to queuek, andT s,k ≥ 0 is the time at which queuek starts service. Without loss of generality, we assume that server 1 starts service atT s,1 = 0, and that the queues 1,··· ,K start in that order. The game is particularly simple in a parallel queueing network, as there is no routing decision to be made other than which queue to join - which is captured in the equilibrium (arrival) profile F ∗ := (F ∗ 1 ,...,F ∗ K ). such that C k (t) :=C F ∗(t,k) =c, ∀k = 1,...,K, where c is a constant, in the support of F ∗ . Note that this is a specialization of definition 1.3 to the parallel network case with slightly modified notation (which is easier to work with here). We note that in recent literature, these have been called mean field equilibria [82]. 140 We denote the time of first arrival into queue l by−T 0,l , the time of last arrival into any queue by T l , and the time the last user served departs from queue l by T fl . Their values will be determined from equilibrium analysis. The next two Lemmas help in finding the equilibrium arrival profile. Lemma 1.1 At equilibrium, all queues finish serving users at the same time instant. We define a sub-interval in the arrival profile support to be an idle arrival interval if the density function associated with the arrival distribution function is zero. Lemma 1.2 The equilibrium arrival distribution has no idle arrival intervals in its support. Lemma 1.2 implies that W k (t) = F k (t) μ k −t +T s,k , for all k = 1,...,K. It follows from Lemmas 1.1 and 1.2 that at equilibrium, with a homogeneous population, the last arrivals into any queue should all happen at the same instant, and this time coincides with the instant at which the service process catches up with the backlog; that is,T fl =T l = T. Let C(t) := (C 1 (t),...,C K (t)), then, using the lemmata above, it follows that the cost function for a (non-atomic) user arriving at time t is C(t) =        (α +β)( F 1 (t) μ 1 +T s,1 )−αt . . . (α +β)( F K (t) μ K +T s,K )−αt        . (1.1) Arrival Distribution Let the arrival profile at queue k at equilibrium be F ∗ k (t) with support on [−T 0,k ,T ], and define p 0,k := F ∗ k (T ) and γ = α/(α +β). Due to space constraints, we note without proof that any equilibrium arrival profile F is absolutely continuous (see [27] for the argument for a single server). We now derive the equilibrium arrival profile illustrated in Figure 1.2. 141 Theorem 1.1 Assume that T s,K < 1+ P K k=1 μ k T s,k P K k=1 μ k . Then, the unique equilibrium arrival profile is F ∗ = (F ∗ 1 ,··· ,F ∗ K ) with F ∗ k (t) = p 0,k (t+T 0,k ) T +T 0,k with support [−T 0,k ,T ], where T = 1+ P K k=1 μ k T s,k P K k=1 μ k and−T 0,k = (1− 1 γ )T + T s,k γ , and p 0,k = μ k P K l=1 μ l (1− P l6=k μ k (T s,k −T s,l )). F K −T 0,1 0 −T 0,2 t F 1 p 0,1 p 0,2 p 0,K F(t) T −T 0,K . . . F 2 Figure 1.2: Equilibrium arrival profile of a single population to a K-queue parallel queueing network. Remark. 1. We assume T s,K < T (given in Theorem 1.1) for convenience. If T s,l > 1+ P l−1 k=1 μ k T s,k P l−1 k=1 μ k for some queue l ∈{2,··· ,K}, then at equilibrium no users would arrive at queues{l,...,K}. 2. Notice that p 0,k can be interpreted as the external routing probability of a user joining queue k on arrival. Price of Anarchy Define the social cost of arrival profileF as Γ(F ) = P K k=1 R C F k (t)dF k (t). Let Γ opt denote the optimal social cost over all arrival profiles, and Γ eq (F ∗ ) the social cost at equilibrium F ∗ . It is to be expected that Γ eq (F ∗ ) will be greater than Γ opt . The inefficiency of the equilibrium arrival profile can be characterized by the price of anarchy (PoA), η = sup F ∗ Γeq (F ∗ ) Γopt , where the supremum is over all equilibria. Note that the equilibrium arrival profile is unique. 142 Theorem 1.2 The price of anarchy of the network concert queueing game is given by η = 2(1 + P K k=1 μ k T s,k ) 1 + P K k=1 P K l=1 μ k μ l T s,l (T s,k −T s,l ) + 2 P K k=1 μ k T s,k ! . Corollary 1.1 The price of anarchy η is upper bounded by 2. Remarks. 3. It is easy to see that the upper bound is achieved if T s,k = 0. This is not surprising, as a set of parallel queues that start service at the same instant operate like a single server queue with effective service capacity P K k=1 μ k . This implies that strategic arrival behavior appears to produce a “resource pooling effect”, which is typically not observed in parallel queueing networks. 4. Surprisingly, Corollary 1.1 implies that staggering the start times of the queues can reduce the PoA (even though it may increase the social cost), and induce arrival behavior closer to the optimum. 5. It is possible to sharpen the bound. Consider the special case where all queues have the same service rate μ K and start at times spaced τ apart. Then, the PoA expression reduces to η = 2 +μτ(K− 1) 1 +μτ(K− 1)− μ 2 τ 2 12 (K 2 − 1) . (1.2) An easy lower bound on this expression follows from the fact that 1 +μτ(K− 1) > 1 + μτ(K− 1)− μ 2 τ 2 12 (K 2 − 1), which after simple algebra (and using some elementary facts) yields 2>η> 1 + 1 1+μτ(K−1) > 4/3. 1.2.2 Multiple Populations In the parallel network case, we can also consider N populations of users with population j users having cost characteristics (α j ,β j ), and identify the unique Nash equilibrium arrival profile. For simplicity, we will assume that there are an equal number of users of each population type (and in the fluid limit, has volume 1). LetF jk (t) denote the arrival strategy 143 of population j at queue k, and F = (F 1 ,··· ,F K ) denotes the (aggregate) arrival profile. Denote F j = (F jl ,l = 1,··· ,K) and F = (F j ,j = 1,··· ,N) as the strategy profile. The service completion time for a population j user arriving at time t at queue k is given by t c j,k =t +W j,k (t), where W j,k (t) is the virtual waiting time. Thus, the cost for a population j user arriving at time t at queue k under arrival distribution F is C j,k (t) := C j,F (t,k) = α j W j,k (t) +β j (t +W j,k (t)). We now define (symmetric) mixed strategy Nash equilibrium profiles for the non-atomic/population game. Definition 1.4 A strategy profileF is an equilibrium strategy profile if for each population j, F j is a minimizer of the corresponding cost functions C jk at each queue k at every time τ in the support of F jk (denotedT jk ), i.e.,∀j,∀k, C j,F (τ,k)≤C j,F (t,k) ∀τ∈T jk , −∞<t<∞. NotethatthisisanextensionofDefinition1.3tothemultiplepopulationcase, butspecialized to the parallel queueing network case. The equilibrium condition captures the fact that for each population, the equilibrium profile must minimize its cost into any queue at any time. Furthermore, it is also equivalent to our earlier condition that all queues with a positive flow of population j must have equal cost, i.e., ∀l,k, C j,F (t,l)1{F jl (∞) > 0} = C j,F (t 0 ,k)1{F jk (∞) > 0}, for all t and t 0 in the support of (the Lebesgue measure corresponding to) F jl and F jk , respectively. Our first result proves an interesting self-organization property at equilibrium because it manifests a “population-level” outcome as a result of individual (“user-level”) strategizing. Lemma 1.3 Suppose that γ 1 <γ 2 ···<γ N . Then, at equilibrium population i users arrive before population j users for i < j. Furthermore, the arrivals are over disjoint intervals, without any gaps. A very rough intuition for why a population with smallerγ prefers to arrive before one with larger γ is that those with smaller γ “don’t like” waiting but “don’t mind” coming later. 144 α1, β1 αN, βN μ1 μK f1,2(t) 0 −T0,1 T1 τ1 −T0,2 T1 τ1 T2 T2 Ts,2 t t F1(t) p2,1 p1,1 F2(t) p2,2 p1,2 Q1(t) Q2(t) f2,1(t) f2,2(t) f1,1(t) Figure 1.3: (a) Arrivals from multiple populations into multiple queues. (b) Two parallel queues, and two arriving populations with γ 1 < γ 2 . Population 1 arrives over [−T 0,1 ,T 1 ] at queue 1, and over [−T 0,2 ,T 1 ] at queue 2. However, population 1 need not be served completely until timeτ 1 at either queue. Population 2 arrives over [T 1 ,T 2 ] and is completely served at time T 2 . Arrival Distribution Denote by−T 0,k , the time of the first arrival into queue k, and T 0 :=−T 0,1 , the time of the very first arrival to the network. Then, from Lemma 1.3, users from populationi arrive in the interval [T i−1 ,T i ]. Let τ i denote the time the last user of population i is served, and τ 0 = 0. Obviously, T i ≤ τ i , with equality only if i = N since at equilibrium the last (non-atomic) user to arrive into the network has no incentive to arrive before his service time. Figure 1.3 (b) provides a simple illustration of this for two populations arriving at two parallel nodes. Define J i ={1≤ k≤ K : T s,k ∈ [τ i−1 ,τ i ]}. Then, population i users are served by the queuesJ i = S i j=1 J j , where queues J i first serve population i before any other. Consider l∈ J i with aggregate arrival profile F l (t) = P N n=i F n,l (t), where by Lemma 1.3, F n,l has support [T n−1 ,T n ]. Notethat P l∈J i p i,l = P i j=1 P k∈J j p i,k = 1. Wenowderivetheequilibrium arrival profile for each population. Theorem 1.3 Suppose γ i < γ i+1 , ∀i and T s,k < T s,k+1 , ∀k. (i) Then, the unique equilibrium arrival profile for population i at queue k ∈ J j , j < i, is dF ∗ i,k (t) = γ i μ k , ∀t ∈ [T i−1 ,T i ], and at queue l ∈ J i is dF ∗ i,l (t) = γ i μ l ∀t ∈ [−T 0,l ,T i ] where 145 T N = 1 P N i=1 P k∈J i μ k N + P N i=1 P k∈J i μ k T s,k , and T i−1 = T i − p i,k γ i μ k for i = 1,··· ,N− 1, k∈J j and j <i. For l∈J i , the arrival interval is [−T 0,l ,T i ], where−T 0,l =T i − p i,l γ i μ l . (ii) Furthermore, equilibrium routing probability for l∈J i ,i≥ 1 is p i,l = μ l P i j=1 P k∈J j μ k i− i X j=1 X k∈J j μ k (T s,l −T s,k ) ! , (1.3) and for k∈J j , j <i, and i≥ 2 is p i,k = μ k P i j=1 P k∈J j μ k 1− X l∈J i μ l μ k ( i−1 X q=j p q,k )− X l∈J i μ l (T s,k −T s,l ) ! . (1.4) Theorem 1.3 shows that at equilibrium the arriving populations self-organize in ascending order of γ i and there are no gaps in the arrival profile. Furthermore, the queues operate at full capacity till all arriving users have been served. Price of Anarchy We now compute the price of anarchy for the multiple populations case. Define the social cost at equilibrium with arrival profile F, as Γ eq (F ) = P N i=1 Γ eq,i (F ), where Γ eq,i (F ) = P i−1 j=1 P k∈J j R T i T i−1 C i,k (t)dF i,k (t) + P l∈J i R T i −T 0,l C i,l (t)dF i,l (t). At equilibrium, the cost of arrival for population i is uniform over the support of its arrival profile, and moreover is the same at all queues that the population chooses to arrive at. Thus, C i,l (t) = c i , some constant. Then, if F is an equilibrium arrival pro- file Γ eq,i (F ) = c i P i−1 j=1 P k∈J j R T i T i−1 dF i,k (t) + P l∈J i R T i −T 0,l dF i,l (t) = c i P l j=1 P k∈J j p 0,k = c i , since P i j=1 P k∈J j p 0,k = 1. The aggregate equilibrium social cost is given by Γ eq (F ) = P N i=1 JΓ eq,i (F ) = P N i=1 c i . Lete 1 (i) denote the “first” queue inJ i , i.e., the queue with the earliest service start time T s,l , l∈ J i . At equilibrium we have C i,e 1 (i) (T i ) = (α i +β i )( p i,e 1 (i) μ e 1 (i) +T s,e 1 (i) )−α i T i ≡ c i , where, p i,e 1 (i) is the fraction of population i users routed to queue e 1 (i). Now, let e 1 (1) 146 be the very first queue to start service (and serve population 1 first). Without loss of generality, let T s,e 1 (1) = 0. For population i the cost of arrival at any queue inJ i is the same over the arrival interval, and it follows that C i,e 1 (i) (T i ) = C i,e 1 (1) (T i ), which implies that p i,e 1 (i) μ e 1 (i) + T s,e 1 (i) = P i j=1 p j,e 1 (1) μ e 1 (1) . Further, using the recursive definition of T i , T i = T N − P N j=i+1 p j,e 1 (1) γ j μ e 1 (1) . Substituting for p i,e 1 (i) μ e 1 (i) +T s,e 1 (i) and T i in C i (T i ), we obtain Γ eq = N X j=1 α j 1 γ i i X j=1 μ e 1 (i) μ e 1 (1) p j,e 1 (1) ! −μ e 1 (i) T s,e 1 (i) − T N − N X j=i+1 p j,e 1 (1) γ j μ e 1 (1) !! . (1.5) Now, from Theorem 1.3 we have an expression forT N in terms of the exogeneous parameters of the network. Substituting that into (1.5) we obtain an expression for Γ eq that is, unfor- tunately, quite messy for the general case. Below, we illustrate this expression for a much simpler special case. Next, we note that the optimal arrival profile would be for each non-atomic user to arrive right at the instant of service. In this case, there is no waiting time and the cost of arrival at time t is β i t, for a user of population i. Let π :N→N be a permutation on the set of populations such that β π(1) >···>β π(N) . In the optimal arrival profile, populations should arrive in the order π(1),π(2),··· ,π(N). A key observation is that since the size of each population is the same, the set of queues J i that serve population i at equilibrium, will serve population π(i). LetJ ∗ i be the set of queues that serve populationπ(i) in the optimal arrival profile. This is because there are no gaps in the optimal and equilibrium arrival profiles. Thus, for given queue start times, T s,k , if population i in the equilibrium arrival profile is replaced by population π(i), the set of queues that served populationi will serve populationπ(i), whose users arrive over [T i−1 ,T i ], which is the equilibrium arrival interval for population i. Thus, the optimal social cost is Γ opt = N X i=1 Γ opt,π(i) = 1 2 N X i=1 β π(i) i−1 X j=1 X k∈J j μ k ((T ∗ i ) 2 − (T ∗ i−1 ) 2 ) + X l∈J i μ l ((T ∗ i ) 2 −T 2 s,l ) ! , (1.6) 147 where T i = i+ P i j=1 P k∈J j μ k T s,k P i j=1 P k∈J j μ k , and this is precisely the time at which population i finishes service at equilibrium. We can substitute for T i in (1.6), which yields a fairly complicated expression for Γ opt and the price of anarchy, η = Γ eq /Γ opt , in terms of the exogeneous parameters. To get some insight into the price of anarchy, η, we illustrate it for a special case where the service rate offered by every queue is the sameμ> 0 and the start time of thekth queue is τ(k− 1), for some τ > 0. Let the number of queues that serve the first l populations to arrive beK l . Then, the instant at which populationπ(l) (or populationl, at equilibrium) is served out is given by T l = l+ P K l k=1 μτ(k−1) P l k=1 μ = l+ μτ 2 K l (K l −1) μK l . Note that K l is unknown a priori, but can be easily calculated. Suppose μτ < 1, and relax K l ∈ [1,K] to take real values. Then, the following Lemma shows that T l is a convex function of K l . The optimal value of K l then is the nearest integer to the optimal real value computed. Let [x] denote the nearest integer to the real number x. Figure 1.4: Terminal arrival time T l as a function of K l , the number of queues that service population l, plotted for l = 1, 7, 14, and for μτ = 0.1 and μ = 1.0. 148 Figure 4 plotsT l as a function ofK l , the number of queues that serve populationl. Note that T l is a convex function of K l and has a minima. We establish this fact formally in Lemma 1.4. Lemma 1.4 Suppose μτ < 1. Then, T l is a convex function of K l ∈ (1,K]. Further, it achieves a minimum at K ∗ l = [ q 2l μτ ]. Lemma 1.4 shows that the number of queues that will serve population l, K l , is propor- tional to √ l, when μτ < 1. 8. If (l− 1)≤ μτ < l, then, there will be more than one queue that serves population l, and at most one that serves all populations with index less than l. In this case as well, the number of queues that serve population l can be found by solving a convex optimization problem. Now, we characterize the price of anarchy in the special case when every queue in the network serves at the same rate, and service start times are equi-spaced. Theorem 1.4 Suppose that each queue offers the same service rate μ > 0, and queue k starts service at time τ(k− 1) with τ > 0. Then, the price of anarchy η is η = τ 2 {(1− 1 γ N ) P N l=1 α l + P N l=1 β l } μ 2 P N l=1 β l τ 2 12 q 2 μτ ( √ l− √ l− 1) + 2τ 2 3 ( 2 μτ ) 3/2 (l √ l− (l− 1) √ l− 1)− τ μ ! ≤ 2. 1.2.3 Reducing the Price of Anarchy Corollary 1.1 showed that the PoA is bounded above by 2 for a single population arriving at a parallel queueing network. And ifτ ∗ is the value ofτ that minimizesη in (1.2), thenτ ∗ is the optimal staggering for start times. In fact, the worst staggering for start times from a PoA perspective is for all queues to start at the same. Further, Remark 4 indicates that staggering the start times can reduce the PoA but can’t get rid of the inefficiency completely. 149 Some other mechanisms to reduce PoA were introduced in [27] for the single-server set- ting. Similar mechanisms can also be introduced in a multi-server setting. In fact, the basic ideas are the same: In the single population setting, if one segments the population into L parts, then the PoA will reduce to 1 + 1/L. This can be done by one of three ways: (i) temporal segmentation: serve only segmentl of the population in some interval [τ l−1 ,τ l ] , (ii) priority assignement: give populationl priority in that same interval, or (iii) time-dependent tarrifs: charge different price in different intervals. The segmentation intervals and prices can easily be determined as in the single server case. For multiple populations too, we can segment each population into L segments. Now, the segmentation intervals must be set such that in the interval [τ 0,i−1 ,τ 0,i ] (with τ 0,0 = 0), populationj≤i users are served, or have priority over populationj >i users. This interval is further segmented into L parts (and each population is also segmented into L segments) such that in the firstl segments, only populationj <i users and populationi users belonging to the first l segments are served, or have priority. This reduces the PoA to 1 + 1/L. A pricing scheme that achieves the same outcome can also be devised similarly. In each case, as L goes to infinity, PoA reduces to one. 1.3 Tandem Queues 1.3.1 Single Population Consider a K node tandem queueing network where users first arrive into queue 1, queue k feeds queue k + 1 and the users exit the network from queue K. Let S ={1,...,K} be the set of nodes in the network. Queue k offers service rate μ k . For simplicity, assume that all servers start service at time T s,k = 0. The cost function is C(t) = (α +β)D(t) +βt, where D is the fluid limit of the sojourn time through the network and t is the time of arrival. Simplifying the expression for the sojourn time from Section 1.1.1, ifW k is the delay at node 150 k thenanarrivalattexitsattimeτ k =W k (t)+t. ItfollowsthatD(t) =W 1 (t)+ P K k=2 W k (τ k ). The main result of this section is that the equilibrium in a tandem network is determined by a so-called “bottleneck” node. Definition 1.5 A node i> 1 will be called a bottleneck if μ i <μ i−1 . Node 1 will always be denoted as a bottleneck. The procedure to determine the set of bottleneck nodes is simple: start backwards from node K, and check the bottleneck condition. Let B = {b 1 ,...,b L } ⊆ S be the set of bottlenecknodesidentified. IteratingoverB andcheckingthebottleneckconditionswillyield yet another subset of bottlenecks (to the bottlenecks). This procedure can be continued till the subset identified does not change from one iteration to the next. This converged subset of service rates is the “bottleneck subnetwork”, and the service rates increase from higher indexed nodes to lower indexed nodes in the identified subset i.e., μ b i >μ b i+1 . By an abuse of notation, let B represent the final bottleneck subnetwork, andB ={μ b 1 ,...,μ b L } the service rates of these nodes, where L =|B|. We present our main result next. Theorem 1.5 (i) The unique equilibrium arrival profile to a K node tandem queueing net- work is given by F ∗ (t) = γ¯ μ b (t +T 0 ), ∀t∈ [−T 0 ,T ], where γ = α/(α +β), ¯ μ b = minB, T = 1 ¯ μ b and T 0 = β α 1 ¯ μ b . (ii) The price of anarchy (PoA) of the tandem network is 2. 1.3.2 Multiple Populations Now, suppose that there areN populations of users, with (α j ,β j ) the cost characteristics of population j users. The Nash equilibrium concept in Definition 1.4 continues to hold true. It is also easy to see that the self-organization property observed in Lemma 1.3 is true in the tandem network as well (the network topology plays no role in the determination of this property). Thus, the multiple population equilibrium profile is precisely equivalent to that of the single server case observed in [27], but it is determined by the bottleneck node following 151 Theorem 1.5. As a consequence, we skip the details of this analysis and move on to study more complicated network topologies. 1.4 Trellis Network The network topologies considered thus far have been particularly simple, with the “network effects” being minimal. Next, we consider a fully connected network topology, that we call a Trellis Network, where it is possible to characterize the equilibrium arrival profile completely. Trellis Networks can be seen as models of fully connected telecommunications grids, such as a forwarding subnet in a computer network, or a plain-old-telephone network. We also note, upfront, that we only consider a single population of strategic users as our focus is on the derivation of the equilibrium in a network with route choice, and considerations of multiple populations only complicate the expressions. Many of the insights obtained earlier in Section 1.2.2 carry over automatically. Definition 1.6 A K×L feedforward trellis network is one that can be arranged in L layers of K node parallel networks, with every node in a layer connected to every node in the immediate successive layer. Figure 1.5(a) depicts an example trellis network. Let μ k,l be the service rate of node k in layer l. First consider the 2× 2 trellis network depicted in Figure 1.5(b). The analysis of the arrival and routing profiles in this relatively simple network topology is illustrative of the results in the more general setting. Proposition 1.1 Consider a 2× 2 trellis network, with service rates (μ 1,1 ,μ 1,2 ) in Layer 1 and (μ 2,1 ,μ 2,2 ) in Layer 2. (i) The equilibrium routing profile between the layers is given by Case A: if μ 1,1 +μ 1,2 >μ 2,1 +μ 2,2 , then p 2,1 =p 1,1 = μ 2,1 μ 2,1 +μ 2,2 , and is unique. Case B: if max(μ 2,1 ,μ 2,2 ) < μ 1,1 +μ 1,2 ≤ μ 2,1 +μ 2,2 , then p 2,1 = p 1,1 = μ 2,1 μ 2,1 +μ 2,2 , and is 152 . . . . . . . . . (a) Afeedforwardtrellisnetwork. Nodesineach layer are connected to every node in only the immediate downstream layer of nodes. p1,1 p2,2 p2,1 p1,2 1,1 1,2 2,2 2,1 (b) A fully connected 2× 2 element trellis net- work. Thenodesinlayeriaredenoted (i,1)and (i,2), where i = 1,2. p j,k is the routing profile from node j of layer 1 to node k of layer 2. unique. Case C: if min(μ 2,1 ,μ 2,2 ) < μ 1,1 + μ 1,2 < max(μ 2,1 ,μ 2,2 ), then (p 1,1 ,p 2,1 ) ∈ {p ∈ [0, 1] 2 |μ 1,1 p 1,1 +μ 1,2 p 2,1 ≤ min(μ 2,1 ,μ 2,2 )}. (ii) The unique equilibrium arrival profile is F ∗ (t) = γμ b (t +T 0 ), ∀t∈ [−T 0 ,T ], where μ b = min(μ 1,1 +μ 1,2 ,μ 2,1 +μ 2,2 ), T 0 = β α T and T = 1 μ b . The external routing probabilities are given by p 1,j = μ 1,k μ 1,1 +μ 1,2 , j = 1, 2. (iii) There exists a two node parallel queueing network that has the same equilibrium arrival profile, and has service rates ˜ μ 1 =μ 1,1 min(1, μ 2,1 +μ 2,2 μ 1,1 +μ 1,2 ) and ˜ μ 2 =μ 1,2 min(1, μ 2,1 +μ 2,2 μ 1,1 +μ 1,2 ). (iv) The price of anarchy is 2. Observe that the routing profile is, in general, not unique even in this simple setting. However, the bottleneck formulation we observed in the tandem network does still play a role in determining the equilibrium arrival profile, however, now we must consider a bottleneck layer, as opposed to a single node. Theorem 1.6 below derives the equilibrium arrival profile to a general K×L trellis network. For the purposes of this theorem, let p (l,m),(l+1,n) be the internal routing probability between node (l,m) and (l + 1,n) in the trellis. Theorem 1.6 Consider aK×L trellis network with service ratesS l = (μ k,l ; k = 1,··· ,K), at layer l for l = 1,··· ,L. (i) The equilibrium internal routing profile between the layers is 153 given by Case A’: if P K k=1 μ l,k > P K k=1 μ l+1,k , then p (l,m),(l+1,n) = μ l+1,n P K k=1 μ l+1,k m,n = 1,...,K, and is unique. Case B’: if max(μ l+1,k ,...,μ l+1,K )< P K k=1 μ l,k < P K k=1 μ l+1,k , then p (l,m),(l+1,n) = μ l+1,n P K k=1 μ l+1,k for all m,n = 1,...,K, and is unique. Case C’: if μ l+1,π(k) < P K k=1 μ l,k < μ l+1,π(k+1) , where π : {1,...,K} → {1,...,K} is a permutation of service rates in increasing order, then the K× (K− 1)-dimensional vec- tor of routing profiles (p (l,k),(l+1,1) ,...,p (l,k),(l+1,K−1) ) K k=1 ∈ {p ∈ [0, 1] K×K−1 |hp,μ π(j) i ≤ μ l+1,π(j) , ∀π(j)≤ π(k)}, a convex polytope, where μ π(j) = [0, 0,...,μ 1 ,μ 2 ,...,μ K , 0,..., 0] with non-zero entries in components (π(j) +i + (K− 1)π(j− 1)) K i=1 . (ii) The unique equilibrium arrival profile is F ∗ (t) = γ¯ μ b (t +T 0 ), ∀t∈ [−T 0 ,T ], where γ = α/(α +β), ¯ μ b = min{ P K k=1 μ 1,k ,..., P K k=1 μ L,k }, T = 1/¯ μ b and T 0 = β α T. The external routing probabilities are given by p 1,j = μ 1,j P K k=1 μ 1,k , j = 1,...,K. (iii) The price of anarchy is 2. 1.5 General Feedforward Networks Finally, we arrive at the problem of determining the equilibrium profile in a general feed- forward network. As it should be clear by now, the increasing complexity of the network topology makes the determination of a closed-form expression for the equilibrium progres- sively more difficult. Indeed, in the most general setting, we can only provide an algorithm for computing the equilibrium. The argument is based on the fact, as noted in Section 1.1.1, that a general feedforward network can be modeled as a Lattice Network. We consider an arrival and routing game wherein a player’s (mixed) strategy is to pick a probability distribution F (according to which it’s time of arrival is picked) and routing probabilities p. As noted before, we look for a Nash Equilibrium Profile (NEP) (F ∗ (t), p ∗ (t)) of the arrival game, which is also a Nash equilibrium in symmetric strategies. Of course, 154 F ∗ and p ∗ are tightly coupled, making the equilbrium analysis of the lattice network much more complicated since the adjacent layers are not fully connected. This lack of symmetry makes it difficult to obtain a closed form expression for the equilibrium. Notice that in the case of a Trellis Network, all routes in a network stage are equivalent at equilibrium, which is not the case in the Lattice Network. Thus, we give an algorithm that for a fixed topology can provably compute the NEP. We view the arrival and routing game as an L-stage extensive form game of imperfect and complete information. In the first stage game, (non-atomic) users arriving at nodesL 1 choose an arrival strategy profile F ∗ . In the subsequent L− 1 stages, the users pick routing probabilities in such a way that a Wardrop equilibrium p ∗ results. Thus, the equilibrium that we derive is a sequential (subgame perfect) Nash equilibrium. Of course, it is a Nash equilibrium of the one-shot game as well. Routing Game. We work in a “recursive” manner, and first fix an arrival process F, and consider the routing game. This is a network flow game with cost functions equal to the total sojourn time on paths that are used. By Corollary 18.10 of [83], it is known that if the cost functions are convex and differentiable (conditions satisfied by the sojourn time functions), then there is an equivalence between equilibria and optimal “flows” (i.e., flows that minimize the total social cost) in population routing games. Using Proposition 18.11 of [83], this implies that the routing game can be viewed as a potential game. We first consider a single-stage lattice network (consisting of only two lay- ers). A potential function for the corresponding game is given by Φ(t, p 1 ; W 2 ) := P (2,j)∈L 2 R Y (2,j) (t;p 1 ) 0 W (2,j) (t;x)dx, t ≥ 0, where, as before, Y (2,j) (t; p 1 ) := P (1,i)∈H 1 (2,j) R t 0 p (1,i),(2,j) (s)dB (1,i) (s), andB (1,i) is the departure process from (1,i)∈L 1 . Since this is a potential game, the Nash (or Wardrop) equilibrium is given by the optimizer of the 155 following program; here, F 1 is the fluid arrival process at nodesL 1 and W 2 is the vector of delays at nodesL 2 . Denote for each t∈supp(F 1 ), Single(F 1 , W 2 ) : min p 1 Φ(t, p 1 ; W 2 ) s.t. B (1,i) (t) = X (2,j)∈G 2 (1,i) Z t 0 p (1,i),(2,j) (s)dB (1,i) (s) ∀(1,i)∈L 1 . Note that the solution to the above depends on t. We solve it for each t to get p ∗ 1 . The constraint in the program is the usual flow conservation rule. The following proposition is standard and we do not prove it (see [83] for a proof easily adapted to Single). Proposition 1.2 (i) There exists a solution p ∗ 1 to the problem Single. (ii) If p ∗ (1,i),(2,j) (t) > 0, then W (2,j) (t;Y (2,j) (t; p ∗ 1 ))≤ W (2,k) (t;Y (2,j) (t; p ∗ 1 )), for any(2,k)∈ L 2 , k6=j implying that any optimal solution to the Problem Single is a Wardrop Equilib- rium. Part (i) is a consequence of the compactness of the feasible region, and part (ii) is implied by the first order optimality conditions of the problem. The implication of part (ii) is that if a path from (l,i) is being used, it must have minimal cost compared to any other path at time t. The computation of the Wardrop equilibrium in anL-layer lattice network is much more complicated since we allow arbitrary inter-layer connections. However, usingSingle we can compute a Nash equilibrium for any L-layer Lattice network in a recursive manner. To fix the idea, Single is solved iteratively at each layer of the network (working back from layer L− 1) with the total downstream sojourn time as the new cost function. The total downstream sojourn time on a path containing ((l,i), (l + 1,j)) is D ∗ (l,i) (t;F (l,i) (t)) = W (l,i) (t;F (l,i) (t)) + D ∗ (l+1,j) (τ (l,i) (t);F (l,i) (t)), where D ∗ (l+1,j) (τ (l,i) (t);F (l,i) (t)) := D (l+1),j (τ (l,i) (t);Y (l+1,j) (t, p ∗ l )) for any (l + 1,j) ∈ H l (l,i) is the 156 Wardrop equilibrium delay at node (l + 1,j), and W (l,i) is the waiting time at node (l,i), that depends on the arrival process F l to nodesL l . Note that D (L,j) (t;Y (L,j) (t, p ∗ l )) = W (L,j) (t;Y (L,j) (t, p ∗ l )). Recall that, W L (t; Y L (t)) := (W (L,j) (t;Y (L,j) (t, p ∗ l ); (L,j) ∈ L L ). Algorithm 1 formally describes the steps involved in computing the Wardrop equilibrium in an L-stage Lattice network. Algorithm 1 WardropEquilibrium(F 1 (t),t∈ [0,∞)) 1: procedureBackward Iteration: 2: for all l =L− 1 to 1 do a) If l = L − 1, D ∗ L−1 (t; F L−1 ) = W L (t; Y L ); else, set D ∗ l (t; F l ) := (D ∗ (l,i) (t,F (l,i) (t)); (l,i) ∈ L l ), where D ∗ (l,i) (t;F (l,i) (t)) = W (l,i) (t;F (l,i) (t)) + D ∗ l+1 (τ (l,i) (t);F (l,i) (t)). 3: π ∗ l (F l (t)) = arg min p Single(F l (t), D ∗ l (t; F l )) ∀ F l (t)∈C[0,∞) K 4: D ∗ l (t, ;F (l,i) (t)) =D (l+1,j) (t;Y (l+1,j) (t; p ∗ l )) for any (l + 1,j)∈L l+1 . 5: end for 6: end procedure 7: procedureForward Iteration 8: for all l = 1 to L− 1 do 9: If l > 1, set F l (t) = (F (l,i) (t); (l,i) ∈ L l ), F (l,i) (t) := Y (l,i) (t; (p ∗ l−1 )) = P (l−1,j)∈H l−1 (l,i) R t 0 p ∗ (l−1,j),(l,i) (s)dB (l−1,j) (s) 10: p ∗ l (t) =π ∗ l (F l (t)) 11: end for 12: end procedure In the Backward Iteration procedure, we start from the penultimate network layer L− 1 and compute the equilibrium routing probability between layers L− 1 and L, for an arbitrary arrival process F L−1 at layer L− 1 and downstream sojourn time being the delay at nodesL L . Note that we solved this problem in Single. At the next iteration, consider arrival process F L−2 at layer L− 2. Then, the corre- sponding total downstream delay is the sum of the delay in the nodesL L−1 and the Wardrop equilibrium delay in the last layer nodesL L . The optimal delay in the last stage is computed with respect to the arrival process Y L−1 at layer L− 1. The optimization problem now has the same structure as Single for a 2-layer network. Solving it yields the optimal routing 157 between layersL−2 andL−1. We can iterate backwards in this manner to find the optimal routing probabilities for the entire network, for arbitrary arrival processes at layer 1. In Forward Iteration, we collect the equilibrium routing profiles corresponding to F 1 . At layer 1, the arrival process is just F 1 , and p ∗ 1 is the equilibrium routing profile. In the next iteration, first the arrival process to layer 2 is calculated using p ∗ 1 and the departure process B 1 from nodesL 1 (second part of step 9). The routing profile now is p ∗ 2 . Iterating forward up to layer L− 1, we obtain the Wardrop equilibrium routing profiles for the entire network corresponding to F 1 . Proposition 1.3 argues that p ∗ := (p ∗ 1 ,..., p ∗ L−1 ) is a Wardrop equilibrium corresponding to the arrival profile F 1 . Proposition 1.3 Consider aK×L lattice network with arrival profile F 1 at layer 1. Then, the routing probabilities p ∗ computed by Algorithm 1 is a Wardrop equilibrium. The proof can be found in the Appendix. Arrival Timing Game. Now, we consider the time of arrival as a decision that a non-atomic user makes along with the routing decisions. On arrival, routing decisions are made such that they result in a Wardrop equilibrium, as computed above. Let F ∗ denote the symmetric arrival strategy that all non-atomic users pick. Recall from Section 1.5 that the total cost of arriving at time t, and traversing pathσ := ((l,i l );l = 1,...,L), isC F,σ (t) = (α+β)D (1,i 1 ) (t)+βt, where F is an arbitrary arrival profile andD (1,i 1 ) is the total sojourn time for non-atomic users arriving at time t. At equilibrium, the downstream sojourn time must be the same regardless of the path chosen through the network. Hence, let D ∗ (t) represent the Wardrop equilibrium sojourn time and C ∗ (t) the corresponding arrival cost. We now argue the existence and essential uniqueness of the Nash equilibria (i.e., the equilibrium cost is unique) in Theorem 1.7, and also provide some insight into the nature of the equilibrium. 158 1,1 1,2 2,1 2,2 Figure 1.5: An example 2× 2 Lattice network. Theorem 1.7 (i) An equilibrium arrival profile F ∗ := (F ∗ (1,i) ; (1,i) ∈ L 1 ) exists and is essentially unique. (ii) supp(F ∗ (1,i) ) are equal for all (1,i) ∈ L 1 . (iii) There exists an equivalent K-node parallel queueing network at equilibrium, with service rate 1 ˜ μ (1,i) (t) = 1 μ (1,i) + 1 F ∗ (1,i) (t)μ (1,i) sup 0≤s≤t (−F ∗ (1,i) (s) +μ (1,i) s) + 1 F ∗ (1,i) (t) D ∗ (τ (1,i) (t)) at node (1,i)∈L l , where t∈supp(F ∗ ). The proof can be found in the Appendix. Example We now demonstrate the application of Algorithm 1 to compute a Nash equilibrium (F ∗ , p ∗ ) in the simple 2× 2 network in Figure 1.5. Algorithm 1 has only one iteration, and p(·) := p (1,1)(2,1) (·) is the only optimization parameter. Denote the arrival process by F 1 = (F (1,1) ,F (1,2) ) (not necessarily an equilibrium). Then, the first order optimality conditions of Single imply that every Wardrop equilibrium p(·) satisfies μ (2,1) +μ (2,2) μ (2,1) μ (2,2) R t 0 p(s)dB (1,1) (s) = 1 μ 2,2 (B (1,1) (t) +B 1,2 (t), where B (1,i) (t) = R t 0 dF (1,i) (s) ds ∧μ (1,i) ds is the cumulative departure process from (1,i)∈L 1 . Next, by part (i) of Theorem 1.7 we know that an equilibrium arrival profile F ∗ , exists. Using the Wardrop equilibrium condition that the equilibrium cost of all the paths are equal, it is straightforward to show that, corresponding to F ∗ , p = min μ 1,1 +μ 1,2 μ 2,1 +μ 2,2 μ 2,1 μ 1,1 , 1 . Further, using part (iii) of Theorem 1.7, the equivalent parallel queueing network allows us to use the proof technique of Theorem 1.1 to conclude that this arrival profile is uniform. 159 Recall that we derived the equilibrium arrival and routing profiles for the Trellis network by exploiting the symmetry of the network topology, in Section 1.4. Applying Algorithm 1 to derive the equilibrium in this case is more involved, but one can show with some effort that the same equilibrium follows. We leave it to the interested reader to verify this. Beneš Network As a final (complicated) example, consider strategic arrivals to a Beneš network. The Beneš networkarchitecturehasbeenusedextensivelytomodeltraditionalcircuit-switchednetworks such as telephone networks. Recently, it has also been studied as a potential architecture for large data centers where servers and routers may need to be interconnected. Thus, we now study equilibrium behavior in a Beneš network. A Beneš network has 2 n inputs and 2 n outputs, where n = 1, 2,···. Every input node is connected to every output node through 2 n−1 distinct paths through the network. Figure 1.6(a) depicts the basic 2×2 “Beneš network element” (henceforth just called ‘the element’). A 4× 4 Beneš network is constructed using 6 of these elements as shown in Figure 1.7(b). A 2 n × 2 n network is constructed using 2 n−1 Beneš elements and 2 n−1 × 2 n−1 Beneš networks as depicted in Figure 1.6(b). We emphasize that a Beneš network is not a fully connected feedforward network, nor is it a trellis network. However, it has a self-smilar structure that can be exploited to recursively find the equilibrium arrival profile and routing probabilities through the network. We introduce some notation: A 2 n × 2 n Beneš network has 2n− 1 layers and 2 n−1 rows of 2× 2 Beneš elements. We identify a particular element by the tuple (k,i), where 1≤ k≤ 2n− 1 and 1≤ i≤ 2 n−1 . The service rates of the queues in element (k,i) are μ k,i j , j = 1, 2, 3, 4, and the service rate ratio is R k,i := μ k,i 1 +μ k,i 2 μ k,i 3 +μ k,i 4 . Also denote ˇ R k,i := μ k,i 1 +μ k,i 2 ˇ μ k,i 3 +ˇ μ k,i 4 , where ˇ μ k,i j , j = 3, 4 is defined below. To simplify the computations (and without loss of any generality), we will assume that μ k,i 1 +μ k,i 2 >μ k,i 3 +μ k,i 4 for all elements (k,i). 160 1 2 3 4 (a) A 2×2 Beneš element. 2 n-1 2 n-1 . . . . . . (b) A feedforward 2 n ×2 n Beneš network. Figure 1.6: A Beneš network has 2 n inputs and 2 n outputs. The general network is con- structed from simpler 2× 2 trellis networks, also called a Beneš element. Proposition 1.4 Consider a 2 n × 2 n Beneš packet network. There exists an equivalent parallel element network, with 2 n−1 elements, that offers the same equilibrium sojourn time as the Beneš network, with service rates expressed recursively as follows: If 1≤k≤n, ˇ μ k,i 3 = μ k+1, i+2 n−k −1 2 n−k 2−mod(i,2) ˇ R k+1, i+2 n−k −1 2 n−k μ k+1, i+2 n−k −1 2 n−k 2−mod(i,2) ˇ R k+1, i+2 n−k −1 2 n−k + μ k+1, i+2 n−k −1 2 n−k +2 n−k−1 2−mod(i,2) ˇ R k+1, i+2 n−k −1 2 n−k +2 n−k−1 , (1.7) ˇ μ k,i 4 = μ k+1, i+2 n−k −1 2 n−k +2 n−k−1 2−mod(i,2) ˇ R k+1, i+2 n−k −1 2 n−k +2 n−k−1 μ k+1, i+2 n−k −1 2 n−k 2−mod(i,2) ˇ R k+1, i+2 n−k −1 2 n−k + μ k+1, i+2 n−k −1 2 n−k +2 n−k−1 2−mod(i,2) ˇ R k+1, i+2 n−k −1 2 n−k +2 n−k−1 . (1.8) On the other hand, if n<k≤ 2n− 1 and i + 2 n−k ≤ 2 n−k+1 , ˇ μ k,i 3 = μ k+1,(i−1)2 2n−1−k +1 1 ˇ R k+1,(i−1)2 2n−1−k +1 μ k+1,(i−1)2 2n−1−k +1 1 ˇ R k+1,(i−1)2 2n−1−k +1 + μ k+1,(i−1)2 2n−1−k +2 1 ˇ R k+1,(i−1)2 2n−1−k +2 , (1.9) ˇ μ k,i 4 = μ k+1,(i−1)2 2n−1−k +2 1 ˇ R k+1,(i−1)2 2n−1−k +2 μ k+1,(i−1)2 2n−1−k +1 1 ˇ R k+1,(i−1)2 2n−1−k +1 + μ k+1,(i−1)2 2n−1−k +2 1 ˇ R k+1,(i−1)2 2n−1−k +2 , (1.10) or, if n<k≤ 2n− 1 and i + 2 n−k > 2 n−k+1 , ˇ μ k,i 3 = μ k+1,(i−1)2 2n−1−k +1 2 ˇ R k+1,(i−1)2 2n−1−k +1 μ k+1,(i−1)2 2n−1−k +1 2 ˇ R k+1,(i−1)2 2n−1−k +1 + μ k+1,(i−1)2 2n−1−k +2 2 ˇ R k+1,(i−1)2 2n−1−k +2 . (1.11) ˇ μ k,i 4 = μ k+1,(i−1)2 2n−1−k +2 2 ˇ R k+1,(i−1)2 2n−1−k +2 μ k+1,(i−1)2 2n−1−k +1 2 ˇ R k+1,(i−1)2 2n−1−k +1 + μ k+1,(i−1)2 2n−1−k +2 2 ˇ R k+1,(i−1)2 2n−1−k +2 . (1.12) Finally, the service rate ratio for nodes in Layer 2n− 1 is ˇ R 2n−1,i =R 2n−1,i . 161 Remarks. 1. We provide an expression that is recursive in definition. Thus, to find the precise expression, one would work backwards from Layer 2n− 1 to Layer 1. 2. Notice that the network structure is reflected/mirrored in two different ways. The inter- connections between the layers are mirrored around Layer n. Also, the interconnections between the first 2 n−2 rows are reflected in the bottom 2 n−2 rows. The somewhat convoluted expressions for the routing and service rates is due to this “reflected” structure. The main result of this section is that there exists an equivalent parallel node queueing networktoa 2 n ×2 n Benešnetworkthathasthesamedelay/sojourntimethroughthenetwork atequilibrium. Thissignificantlyreducesthecomplexityofcomputingtheequilibriumarrival profile. Further, the recursive procedure to determine the equivalent parallel node network automatically finds the equilibrium routing profiles in every Beneš element of the network. Theorem 1.8 (i) There exists an equivalent parallel queue network to a 2 n × 2 n Beneš network at equilibrium, with service rates ˜ μ 1,i j =μ 1,i j ˇ μ 1,i 3 + ˇ μ 1,i 4 μ 1,i 1 +μ 1,i 2 , j = 1, 2 and i = 1,..., 2 n , (1.13) where ˇ μ 1,i k k = 3, 4, is defined in (1.7) and (1.8) of Proposition 1.4. (ii) The equilibrium arrival profile is given by, F ∗ (t) =γ( 2 n−1 X i=1 ˜ μ 1,i 1 + ˜ μ 1,i 2 )(t +T 0 ) = t +T 0 T +T 0 , ∀t∈ [−T 0 ,T ], (1.14) where, γ =α/(α +β), 1/T = ( P 2 n−1 i=1 ˜ μ 1,i 1 + ˜ μ 1,i 2 ) and T 0 = β α T. (iii) The price of anarchy is 2. 162 1.6 Appendix Proof: (Lemma 1.1) For simplicity, consider only two queues. To see that T f1 =T f2 , note that at equilibrium, the costs at each of the queues must be equal at all times. Assume that T f1 < T f2 . Then a user arriving into queue 2 at any time t∈ (T f1 ,T f2 ] must experience a higher cost than if she had simply joined queue 1 (which is now idle). It follows that this arrival profile cannot be an equilibrium. Similarly, T f2 6< T f1 . Thus, we must have T f1 =T f2 . Proof: (Lemma 1.2) We prove this by contradiction. In light of the fact that the cost of arriving at any of the queues is the same, it suffices to prove the assertion in the case of a single queue alone. Let the service start time, T s = 0, and suppose the equilibrium arrival profile F is such that there is a countable set of disjoint intervals{T i } in its support such that the server idles in these intervals, for a given service rate μ. LetT idle =∪ n i=1 T i be the idle times of the server. Consider a point t ∗ ∈T idle and t6∈T idle > 0. Recall that ¯ X(s) = F (s)−μs for all s > 0. By definition of the idle times, we have ¯ Q(t ∗ ) = 0, implying ¯ X(t ∗ ) =−Ψ( ¯ X(t)) and C(t ∗ ) =βt ∗ . Consider C(t)−C(t ∗ ) = (α +β)W (t) +βt−βt ∗ = α +β μ ¯ X(t) + Ψ( ¯ X(t))− ¯ X(t ∗ )− Ψ( ¯ X(t ∗ )) +β(t−t ∗ ) = α +β μ F (t)−F (t ∗ )−μ(t−t ∗ ) + Ψ( ¯ X(t))− Ψ( ¯ X(t ∗ )) +β(t−t ∗ ). Now, since ¯ Q(t)> 0, it follows that Ψ( ¯ X(t))>− ¯ X(t). Thus, we have C(t)−C(t ∗ ) > α +β μ (F (t)−F (t ∗ ))− α +β μ (F (t)−F (t ∗ )) +β(t−t ∗ ) = β(t−t ∗ ). 163 If t>t ∗ , clearly C(t)>C(t ∗ ), implying that F cannot be an equilibrium arrival profile and we have shown a contradiction. This implies that no equilibrium arrival profile can have idle intervals in its support. Proof: (Theorem 1.1) Note that the cost function is unbounded as t goes to±∞. Thus, at equilibrium the arrival profile must have bounded support. Let the support of the arrival profile to queue l be [−T 0,l ,T ]. Now, at equilibrium, the cost of arriving at queue l is the same at any time in this arrival interval. Thus, C l (T ) =C l (−T 0,l ), from which we get p 0,l =γμ l (T +T 0,l ). (1.15) Next, the equilibrium expected cost of arrival is the same at any queue and at any time in their respective arrival intervals. From Lemma 1.1, we know that the time of last arrival at any queue is the same for all queues. Thus, C l (T ) = C k (T ), for any l,k, and using P l6=k p 0,l = 1−p 0,k , we get P l6=k μ l ( p 0,k μ k +T s,k −T s,l ) = 1−p 0,k , rearranging which we get that the equilibrium probability of routing to queue k upon arrival is p 0,k = μ k P K l=1 μ l (1− P l6=k μ k (T s,k −T s,l )). Now, from Lemma 1.2, we have that μ l (T−T s,k ) = p 0,k since the population size has been normalized to 1. Substituting for p 0,k , we get T = 1+ P K k=1 μ k T s,k P K k=1 μ k . Now, it follows from equation (1.15) that−T 0,l = T− p 0,l γμ l . Substituting for T and p 0,l we get−T 0,l = 1 P K k=1 μ k (1− 1 γ )(1 + P K k=1 μ k T s,k ) + T s,l γ P K k=1 μ k which simplifies to−T 0,l = (1− 1 γ )T + T s,l γ . Finally, equating the cost of arrival at queue l at any time t with that at T 0,l gives (α +β) F ∗ l (t) μ l −αt = αT 0,l , which yields F ∗ l (t) = μ l γ(t +T 0,l ) = p 0,l (t+T 0,l ) T +T 0,l , an equilibrium arrival profile at queue l. We now argue uniqueness. First, note that for a given T s,l , the terminal service time T is unique. Let F be another equilibrium profile with supportT i for F i , where we can take T i = (−∞,T ]. Now, the cost of arriving at T is C F i (T ) = βT = C F i (t) for each t∈T i . 164 This is the same as under the profile F ∗ . Thus, F i (t) = F ∗ i (t) onT i ∩ [−T 0,i ,T ]. Since F ∗ i (−T 0,i ) = 0 and F ∗ i (T ) = p 0,i , F has total measure 1 at T and is absolutely continuous, it follows that F i (t) =F ∗ i (t) on [−T 0,i ,T ]. Proof: (Theorem 1.2) Let the equilibrium cost at queue l be C l (t) = c ≡ αT 0,1 for all t ∈ [−T 0,l ,T ]. The equilibrium social cost under profile F ∗ is given by J eq = P K k=1 R C F k (t)dF k (t) = c P K k=1 R dF k (t) = c(p 0,1 +··· +p 0,K ) = c. Substituting for c, we have J eq =αT 0,1 =β 1+ P K k=1 μ k T s,k P K k=1 μ k . Now, the socially optimal outcome would be for each non-atomic user to arrive just at the instant of service, with zero waiting. In this case, the instantaneous cost would be C opt (t) =βt. Thus, the optimal arrival profile is given by dF opt (t) = l X k=1 μ k dt, for T s,l <t≤T s,l+1 . It is straightforward to see that the time of last arrival (and service) is T opt = T = 1+ P K k=1 μ k T s,k P K k=1 μ k . From this, the optimal social cost can be computed as J opt = Z T s,2 0 βμ 1 tdt + Z T s,3 T s,2 β(μ 1 +μ 2 )tdt +··· + Z T T s,K β K X k=1 μ k ! tdt, = β 2 T 2 K X k=1 μ k − K X k=1 μ k T 2 s,k ! = β 2 P K k=1 μ k 1 + K X k=1 K X l=1 μ k μ l T s,l (T s,k −T s,l ) + 2 K X k=1 μ k T s,k ! . Using this along with the expression forJ eq derived above, we get the expression forη. Proof: (Corollary 1.1) We will show that J eq ≤ 2J opt . Consider the dif- ference J eq − 2J opt and simplify the expression to obtain J eq − 2J opt = β P K k=1 μ k P K k=1 μ k P K k=1 μ k T 2 s,k − P K k=1 μ k T s,k P K k=1 μ k T s,k + 1 . Recalling that T = 1+ P K k=1 μ k T s,k P K k=1 μ k , we can replace the last term on the R.H.S. above and simplify to get 165 J eq − 2J opt = β P K k=1 μ k P K k=1 μ k P K k=1 μ k T s,k (T s,k −T ) . Now, we know from the statement of Theorem 1.1 that T >T s,k ∀k. Therefore, it follows that J eq ≤ 2J opt . Proof: (Lemma 1.3) Let T s,k = 0, for all queues k. The general case will follow easily from the ensuing argument. First note that there can be no gaps in any equilibrium arrival profile, F k = P N i=1 F i,k . If there were, then, any arriving non-atomic user right after the gap can arbitrarily improve its cost by arriving just before such a gap, implying this arrival profile is not an equilibrium. The cost of arriving at queue k is constant, for a given population i over the arrival interval. Differentiating C i,k (t), we have 0 = (α i +β i )( 1 μ k ∂F i,k (t) ∂t )−α i , from which we have that the arrival density, f i,k (t) =μ k γ i . Now, let (t 1 ,t 3 ) be an arbitrary interval, and suppose that t 1 <t 2 <t 3 , such that some populationj users arrive in (t 1 ,t 2 ] and only populationi users arrive in (t 2 ,t 3 ). Consider the cost of arrival for population j, C j,k (t) = (α j +β j ) F k (t) μ k −α j t, for t∈ (t 1 ,t 3 ). As (t 1 ,t 2 ] is in the support ofF j,k , it follows that the cost of arrival is constant over this interval, and it can be evaluated at t 2 . Now, let > 0 be small enough so that t,t +∈ (t 2 ,t 3 ). Evaluating the cost of arrival at these points and taking the difference of the resulting expressions we obtain C j,k (t+)−C j,k (t) = (α j +β j ) P N l=1 F l,k (t+)− P N l=1 F l,k (t) μ k −α j . By assumption, there are only arrivals from population i in the sub-interval (t 2 ,t 3 ), and it follows that F l,k (t +) =F l,k (t) for all l6=i. This implies that C j,k (t +)−C j,k (t) = (α j +β j ) (F i,k (t+)−F i,k (t)) μ k −α j . Divide through by and let→ 0 to obtainC 0 j,k (t) = (α j +β j ) f i,k (t) μ k −α j , wheref i,k (t) is the density of the arrival profileF i,k (t), which was shown to bef i,k (t) =μ k γ i , fort∈ (t 2 ,t 3 )⊂ SuppF i,k . Substituting for f i,k it follows that C 0 j,k (t) = (α j +β j )(γ i −γ j ). By assumption, γ i <γ j , so that C 0 j,k (t)< 0. This implies that the cost of arrival for population j is strictly decreasing over the interval (t 2 ,t 3 ), and it must be less than C j,k (t 2 ). Clearly, (t 1 ,t 2 ] cannot be in the support of F j,k in equilibrium. This implies that population j cannot arrive before population i, if i<j. 166 Proof: (Theorem 1.3) Note that population i is served by queues J l ,l≤ i. The expected cost for a populationi user to arrive at queuek∈J i isC i,k (t) = (α i +β i )W k (t) +β i t fort∈ [T i−1 ,T i ] fork∈J l ,l<i , and fort∈ [−T 0,k ,T i ] fork∈J i . Recall that,F k (t) = P N j=i F j,k (t), where F i,k (t) has support [−T 0,k ,T i ] and F jk (t) has support [T j−1 ,T j ] for each j > i. The virtual waiting time process at queue k is given by W k (t) = 1 μ k F k (t)− (t−T s,k ). Now, note that at equilibrium, the expected cost for a population i user to arrive at any queues k,l∈J i has to be the same. Thus, from C i,k (T i ) =C i,l (T i ), we get that p i,k μ k +T s,k = p i,l μ l +T s,l . (1.16) Let p≤ q < i be two populations that arrive prior to population i. Then, for any k∈ J p and m∈ J q , k < m and at equilibrium the expected cost of arriving into queues k and m for a population i user has to be the same. Thus, from C i,k (T i ) =C i,m (T i ), we get 1 μ k (p p,k +p p+1,k +··· +p i,k ) +T s,k = 1 μ m (p q,m +p q+1,m +··· +p i,m ) +T s,m . (1.17) First, considerp<q. Then, for a populationq user,C q,k (T q ) =C q,m (T q ) for queuesk,m as above, which yields 1 μ k (p p,k +p p+1,k +··· +p i,k ) +T s,k = 1 μm (p q,m ) +T s,m . Substituting this into (1.17) we have, by induction that p i,m μ m = p i,k μ k , ∀k∈J p , m∈J q , p<q<i. (1.18) Now consider p =q. Then, for any k,l∈J p we have C i,k (T i ) =C i,l (T i ), implying 1 μ k (p p,k + ··· +p i,k ) +T s,k = 1 μ l (p p,l +··· +p i,l ) +T s,l . It follows from (1.16) that p p,k μ k +T s,k = p p,l μ l +T s,l . Thus, by induction, we have p i,l μ l = p i,k μ k , ∀k,l∈J p , p<i. (1.19) 167 Next, let population j < i, and k∈ J j and l∈ J i are two queues that serve population i. Once again by the equilibrium conditions we have C i,k (T i ) = C i,l (T i ). Simplifying the expression we obtain p i,k = μ k μ l p i,l +μ k (T s,l −T s,k )− i−1 X p=j p p,k . (1.20) Now, we have P i p=1 P m∈Jp p i,m = 1. Consider a j < i and a k∈ J j for i≥ 2. It follows that P i p=1,p6=j P m∈Jp p i,m + P l∈J j ,l6=k p i,l = 1−p i,k . Substituting for p i,m and p i,l in terms of p i,k from (1.18), (1.19) and (1.20), we get p i,k in (1.4). Now, for an l∈ J i for any i≥ 1, using (1.16) and (1.20) and substituting for p i,k and p i,m (respectively) in terms of p i,l in P i−1 j=1 P m∈J j p i,m + P k∈J i ,k6=l p i,k = 1−p i,l , we get p i,l in (1.3). We now derive the equilibrium arrival distributions for each population to each serving queue. First, recall that the cost function for population i at queue k ∈ J j ,j < i, is given by C i,k (t) = (α i +β i ) 1 μ k F k (t) +T s,k −α i t∀t∈ [T i−1 ,T i ]. Differentiating this and recalling that at equilibrium the cost is constant over the arrival interval, and that F k (t) = P i−1 p=j p p,k +F i,k (t), we havedF i,k (t) =γ i μ k ∀t∈ [T i−1 ,T i ]. Now, for queuek∈J i , the users arrive over the interval [−T 0,k ,T i ]. Again, differentiating the cost function C i,k (t), we have dF i,k (t) =γ i μ k ∀t∈ [−T 0,k ,T i ]. Finally, we can derive the support of these distributions by backward recursion. Note that for population N,∀k∈J N μ k (T N −T s,k ) =p N,k . Substituting for p N,k from (1.16), we get T N = N + P N j=1 P k∈J j μ k T s,k / P N j=1 P k∈J j μ k . Next, at equilibrium, we must have C i,k (T i ) = C i,k (T i−1 )∀k∈ J j ,j < i, from which we getT i−1 =T i − p i,k γ i μ k . Note that we need to usej <i in order to obtain the recursive definition of T i−1 , since C i,k (t) < C i,l (t) on [T i−1 ,−T 0,l ], for l∈ J i ; that is, there are no arrivals from population i at queue l∈J i on this sub-interval. Finally, population i users arrive at queue k∈J i in the interval [−T 0,l ,T i ]. Thus, at equilibrium we must have C i,k (−T 0,k ) =C i,k (T i ), from which we obtain−T 0,k =T i − p i,k γ i μ k . 168 The proof of uniqueness follows that of Theorem 1.1 and we omit it for brevity. Proof: (Lemma 1.4) From Theorem 1.3 we have T l (k) = l μk + τ 2 (k− 1), for k∈ (1,K]. Differentiating with respect to k we obtain ∂T l (k) ∂k =− l μk 2 + τ 2 , which yields a critical point k ∗ = q 2l/μτ (only the positive value is feasible). Further, the second derivative yields ∂ 2 T l (k) ∂k 2 = 2l μk 3 > 0 ∀k∈ (1,K]. Thus, T l (k) is convex for real k and achieves its minimum at K ∗ l = [k ∗ ]. Proof: (Theorem 1.4) Substituting, for any queue l = 1,··· ,K, μ l =μ and T s,l =τ(l− 1) (i.e., the queue l starts service at time τ(l− 1)), in (1.5) and (1.6), using Lemma 1.4 and taking K l ≈ q 2l/μτ, the expression for η follows after some elementary algebra and is omitted for brevity. To see thatη is upper-bounded by 2, first considerN = 3. The expression forJ opt reduces to J opt = β 1 τ 2 4 3 s 2 μτ + μτ 12 s 2 μτ − 1 ! + μ 2 β 2 τ s 2τ μ √ 2− 1 12 + 4 3μτ (2 √ 2− 1) ! − τ μ ! + μ 2 β 3 τ s 2τ μ √ 3− √ 2 12 + 4 3μτ (3 √ 3− 2 √ 2) ! − τ μ ! . J eq is simply τ 2 1− 1 γ 3 ! P 3 i=1 α π(i) + τ 2 P 3 i=1 β π(i) ≡ τ 2 1− 1 γ 3 ! P 3 i=1 α i + τ 2 P 3 i=1 β i . Using these expressions, we evaluate 2J opt −J eq = τ 2 3 X i=1 (α i ) 1 γ 3 − 1 ! + β 1 τ 2 4 3 s 2 μτ + μτ 12 s 2 μτ − 3 2 ! +β 2 μτ s 2τ μ √ 2− 1 12 + 4 3μτ (2 √ 2− 1) ! − 3τ 2 ! + β 3 μτ s 2τ μ √ 3− √ 2 12 + 4 3μτ (3 √ 3− 2 √ 2) ! − 3τ 2 ! . The first term on the right hand side is > 0, since γ 3 < 1 and α i ≥ 0 for all i. The terms afterβ 2 andβ 3 can easily be verified to be non-negative. The only term left to consider is the 169 one after β 1 . Denote δ = 4 3 q 2 μτ + μτ 12 q 2 μτ − 3 2 . Multiplying and dividing by q 2/μτ we have δ = q μτ 2 4 3 2 μτ + 1 6 − 3 2 q 2 μτ . Letx := q 2/μτ. Then, it can be seen thatδx = ( 4 3 x 2 − 3 2 x+ 1 6 ). Suppose that δx< 0. That is, (after factoring the LHS) (8x− 1)(x− 1)< 0. This implies either (8x− 1) > 0 and (x− 1) < 0, which contradicts the fact that x = q 2/μτ > 1 when μτ < 1; or (8x−1)< 0 and (x−1)> 0 which is impossible. Therefore, it cannot be the case that (8x− 1)(x− 1)< 0, thus proving that 2≥J eq /J opt . It can also be checked (after some tedious algebra) that the terms after β l , for l > 3, are larger than those after β 3 and so it ispossibletousethesameargumentforanarbitrarynumberofarrivingpopulations,N. Proof: (Theorem 1.5) For bottleneck nodeb 1 , the waiting time for an arriving user at time t is W b 1 (t) = X b 1 (t) +I b 1 (t), where X b 1 (t) = F (t) μ b 1 −t and I b 1 (t) = sup −T 0 ≤s≤t (−X b 1 (s)) + is the cumulative idle time of bottleneck node 1. Let T b 1 := inf{t > 0|I b 1 (t) > 0}, be the first time that bottleneck queue 1 idles. Recall that T is the time of the very last arrival to the network. Clearly T b 1 ≤ T, otherwise any user arriving at T could reduce its waiting cost by arriving at timeT b 1 , thus violating the equilibrium condition. Suppose thatT b 1 =T. From the definition of I b 1 , it follows that X b 1 (T b 1 ) ≤ 0, so that 1/μ b 1 ≤ T b 1 = T and W b 1 (T b 1 ) = 0. Fluid leaves bottleneck node b 1 at rate μ b 1 > μ b 2 . This implies that there is a non-zero waiting time at bottleneck node b 2 given by W b 2 (τ b 1 ) = ( μ b 1 μ b 2 − 1)τ b 1 , where τ b 1 = W b 1 (t) +t. For a fluid arrival at time T b 1 = T, τ b 1 = W b 1 (T b 1 ) +T b 1 = T b 1 implying that W b 2 (τ b 1 ) = ( μ b 1 μ b 2 − 1)T b 1 > 0 and τ b 2 = μ b 1 μ b 2 T b 1 . Next, for bottleneck node b 3 , we have W b 3 (τ b 2 ) = ( μ b 2 μ b 3 − 1) μ b 1 μ b 2 T b 1 and τ b 3 = μ b 1 μ b 3 T b 1 . Continuing recursively, the waiting time at bottleneck node b k is W b k (τ b k−1 ) = ( μ b k−1 μ b k − 1) μ b 1 μ b k−1 T b 1 , so that W (T b 1 ) = P L i=2 W b i (τ b i−1 ) = μ b 1 μ b L T b 1 − T b 1 > 0. It follows that the cost of entering the network at time T = T b 1 is C(T ) = (α +β)W (T ) +βT = (α +β) μ b 1 μ b L T−αT. Now, suppose the last arrival atT reneges and arrives at time ˜ T = μ b 1 μ b L T 1 . The cost of entering the system is C( ˜ T ) = β ˜ T, as there is no waiting in the network. Clearly, C(T ) = C( ˜ T ) +αT ( μ b 1 μ b L − 1) > C( ˜ T ), where the last 170 inequality follows since μ b 1 > μ b L . Thus, the arrivals cannot be at equilibrium, implying that T b 1 <T. Let T b k = inf{t > 0|I b k (t) > 0} be the first time bottleneck node b k idles. A similar argument as above shows that T b k < T for 2 ≤ k < L− 1, and T b L = T. From the definition of T b k , k = 1,...,L and the fact that μ b 1 >··· > μ b L , it follows easily that T b 1 <T b 2 <···<T b L . Using this fact together with the equilibrium condition that the cost must be constant over the support of the arrival profile, the exact form of the arrival profile follows. We omit the details due to space considerations. The equilibrium arrival distribution is entirely determined by the bottleneck node. It is then easy to see from the results in [27] that the PoA for the tandem network is 2. Proof: (Lemma 1.1) (i) Recall that the cost function is defined path-wise. We define paths by the tuples of nodes that they traverse. Thus, P 1 = ((1, 1), (2, 1)) and P 2 = ((1, 1), (2, 2)) are the paths from node (1, 1) to layer 2, P 3 = ((1, 2), (2, 1)) and P 4 = ((1, 2), (2, 2)) are the paths from node (1, 2) to layer 2. W P 1 (t) := W 1,1 (t) +W 2,1 (τ 1 ) is the sojourn time through pathP 1 andW P 2 (t) =W 1,1 (t)+W 2,2 (τ 1 ) is the sojourn time through pathP 2 . Using the definition of the waiting time from Section 1.1, we have W 1,1 (t) = F 1,1 (t) μ 1,1 −t ! +I 1,1 (t), W 2,1 (t) = 1 μ 2,1 (μ 1,1 p 1,1 +μ 1,2 p 2,1 −μ 2,1 )t+I 2,1 (t) andW 2,2 (t) = 1 μ 2,2 (μ 1,1 p 1,2 +μ 1,2 p 2,2 −μ 2,2 )t+ I 2,2 (t), whereI · (t) is the idle time process of the nodes. Case A:μ 1,1 +μ 1,2 >μ 2,1 +μ 2,2 . By reductio ad absurdum, it is easy to argue that (p 1,1 ,p 2,1 ) is such thatμ 1,1 p 1,1 +μ 1,2 p 2,1 >μ 2,1 and μ 1,1 p 1,2 +μ 1,2 p 2,2 > μ 2,2 at equilibrium. The cost of taking path P j is C P j (t) = (α + β)W P 1 (t) +βt, for j = 1, 2, 3, 4 and at equilibrium it must be the same across all paths. Equating C P 1 (t) =C P 2 (t), we have W P 1 (t) =W P 2 (t). This implies that W 2,1 (τ 1 ) =W 2,2 (τ 1 ), from which p 1,1 =p 2,1 = μ 2,1 μ 2,1 +μ 2,2 , follows. Case B: Suppose (p 1,1 ,p 2,1 ) is chosen such that μ 1,1 p 1,1 + μ 1,2 p 2,1 > μ 2,1 . Using the condition μ 1,1 +μ 1,2 ≤ μ 2,1 +μ 2,2 it follows that μ 1,1 p 1,1 +μ 1,2 p 2,1 < μ 2,2 , implying that node (2, 2) will idle at all times. This choice of routing cannot be an equilibrium as the 171 cost of entering (2, 2) is lower than the cost of entering (2, 1). Thus, any equilibrium routing profile must be such that both queues idle, implying that I 2,1 (t) = I 2,2 (t). It follows that p 1,1 =p 2,1 = μ 2,1 μ 2,1 +μ 2,2 . CaseC:Withoutlossofgenerality, supposeμ 2,2 ≤μ 1,1 +μ 1,2 <μ 2,1 . Clearly, thetotalrate of fluid departing Layer 1 is less than the cumulative service rate on offer at Layer 2, implying that Layer 1 is the bottleneck layer in this network. Regardless of the fraction of fluid that is routed to node (2, 1), it is never overloaded. However, the maximum rate at which fluid can enter node (2, 2) without overloading it isμ 2,2 . This implies thatμ 1,1 p 1,2 +μ 1,2 p 2,2 ≤μ 2,2 bounds the set of feasible equilibrium routing profiles from above. A similar argument holds for the case where μ 1,1 +μ 1,2 <μ 2,2 and μ 1,1 +μ 1,2 ≥μ 2,1 , but the boundary set is given by μ 1,1 p 1,1 +μ 1,2 p 2,1 ≤μ 2,1 . (ii) Consider Case A so that Layer 2 is the bottleneck. At equilibrium, the very last arrival to the system enters such that it faces no waiting time delay on any path. As the nodes in Layer 2 do not idle (from the argument in (i)) it follows that nodes (2, 1) and (2, 2) finish service at the same time T = 1 μ 2,1 +μ 2,2 . From the equilibrium condition, C P 1 (−T 0 ) =C P 1 (T ) implying T 0 = β α T. At equilibrium, the cost function is constant over the support of the arrival profile. Differentiating the cost function, and solving the resulting equation for the arrival density f ∗ 1,1 we obtain f ∗ 1,1 (t) = μ 11 μ 11 +μ 12 γ(μ 2,1 +μ 2,2 ) ∀t∈ [−T 0 ,T ]. Doing the same for the cost of taking paths P 3 and P 4 , it can be shown that f ∗ 1,2 (t) = μ 12 μ 11 +μ 12 γ(μ 2,1 +μ 2,2 ) ∀t∈ [−T 0 ,T ]. It follows that f ∗ (t) = f ∗ 1,1 (t) +f ∗ 1,2 (t) = γ(μ 2,1 +μ 2,2 ) for t∈ [−T 0 ,T ]. Integrating this expression with respect to t, and noting that 1/(T +T 0 ) =γ(μ 2,1 +μ 2,2 ) we obtain F ∗ (t) = t+T 0 T +T 0 , ∀t∈ [−T 0 ,T ]. Similar arguments hold when Case B or C is assumed. The uniqueness of F ∗ follows by the argument used in the proof of Theorem 1.1. (iii) Note that the equilibrium sojourn time through any path through the network is given byW (t) = ( F ∗ (t) μ 2,1 +μ 2,2 −t) = F ∗ 1,1 (t) μ 1,1 μ 1,1 +μ 1,2 μ 2,1 +μ 2,2 −t = F ∗ 1,2 (t) μ 1,2 μ 1,1 +μ 1,2 μ 2,1 +μ 2,2 −t. Thus, thereisaparallelnode queueing network with service rates ˜ μ 1 = μ 1,1 μ 1,1 +μ 1,2 (μ 2,1 +μ 2,2 ) and ˜ μ 2 = μ 1,2 μ 1,1 +μ 1,2 (μ 2,1 +μ 2,2 ), such that the equilibrium sojourn time equals that of the 2× 2 trellis network. 172 (iv) The PoA follows directly from Theorem 1.2, and is equal to 2. Proof: (Theorem 1.6) (i) The proof of cases A’ and B’ follow directly by a simple generalization of the results in Cases A and B in (i) of Lemma 1.1 to a K× 2 trellis and we omit it. Case C’: Since P K k=1 p (l,j),(l+1,k) = 1, the K − 1 dimensional tuple (p (l,j),(l+1,1) ,...,p (l,j),(l+1,K−1) ) determines the routing profile from node (l,j), and there are K such tuples. A feasible routing profile, p, is a K× (K− 1) dimensional vector formed by concatenating the K vectors together. In general, the sum rate at Layer l will lie between the service rates of some two nodes π(k) and π(k + 1) in Layer l + 1, determined by the permutation operatorπ(·). Nodes with indicesπ(j)≤π(k) will not be overloaded so long as hp,μ π(j) i≤μ l+1,π(j) (Case C of (i) in Lemma 1.1 is a simple example). The inequality defines a K− 1 dimensional hyperplane in the K× (K− 1) dimensional space [0, 1] K×(K−1) . Thus, the feasible routing profiles in [0, 1] K×(K−1) are constrained by π(k) (K− 1)-dimensional hyperplanes. Further, since the routing profiles are constrained to be non-zero, the feasible region is a convex polytope. (ii) We proceed by backward recursion. Consider Layer L. If P K k=1 μ L,k > P K k=1 μ L−1,k , then the internal routing follows Case B’ or Case C’ depending on the exact relationship between the individual service rates of the nodes in LayerL and the sum rate of LayerL− 1 . By part (iii) of Lemma 1.1 the equivalent parallel node network at equilibrium has the same service rates as Layer L− 1. On the other hand, if P K k=1 μ L,k < P K k=1 μ L−1,k , then Layer L is a bottleneck to Layer L− 1, and the internal routing follows Case A’. By (iii) of the lemma, it follows that Layer L− 1 has an equivalent layer of nodes at equilibrium, with node service rate ˜ μ L−1,k = μ L−1,k P K k=1 μ L,k P K k=1 μ L−1,k < μ L−1,k at node (L− 1,k). Note that P K k=1 ˜ μ L−1,k = P K k=1 μ L,k . Continuing in this manner up to Layer 1, it can be seen that that the equivalent parallel nodenetworkatequilibriumtothetrellisnetworkhasservicerates ˜ μ 1,k =μ 1,k P K k=1 μ b,k P K k=1 μ 1,k ,where 173 b = arg min{ P K k=1 μ 1,k ,..., P K k=1 μ L,k }. Note that P K k=1 ˜ μ 1,k = P K k=1 μ b,k . The equilibrium arrival profile and external routing profile now follows from (ii) of Lemma 1.1. (iii) follows directly from Theorem 1.2. Proof: (Proposition 1.3) The proof uses the one-stage deviation principle (see Theorem 4.1 of [84]). Consider a path σ := (θ 1 ,...,θ L ) ∈ Π L l=1 L l . Suppose that at time t, the suggested routing profile from Algorithm 1 is p ∗ (t) = (p ∗ l (t),l = 1,...,L− 1), with p ∗ l,θ l ,θ l+1 (t) > 0 for all l = 1,...,L− 1 (so that path σ is used). Suppose that the non-atomic user at time t can improve its cost by using a different routing strategy p(t) := p ∗ 1 (t),..., p ∗ k−1 (t), p k (t), p ∗ k+1 (t),..., p ∗ L−1 (t) . Then, by the essential uniqueness of the Wardrop equilibrium, it follows that p ∗ (t)) cannot be a Wardrop equilibrium (since the cost at any Wardrop equilibrium solution must be the same, and also should minimize the total downstream delay). Thus, it follows by contradiction that the one-stage deviation principle is satisfied and the solution must be subgame perfect. Since t is arbitrary, this suffices to argue that p ∗ is a Wardrop equilibrium. Proof:(Theorem1.7) (i)TheproofofexistencefollowsdirectlyfromTheorem3of[85]byan application of the Fan-Glicksberg fixed point theorem [86] to a best response correspondence, analogous to the mappingG in Section 6.3 of [85], appropriately defined with respect to the cost functions C F,σ . We skip the details due to space constraints. We now argue essential uniqueness of the equilibrium arrival profiles. Recall that any equilibrium arrival must be such that the cost of arriving at the network is minimized in some arrival intervalT . That is, if F ∗ is an equilibrium arrival profile then for any σ C F ∗ ,σ (τ)≤C F ∗ ,σ (t), and any t∈R + Further, any equilibrium arrival profile must be a Wardrop equilibrium (as the cost of taking any path through the network must be the same). Therefore, using the essential uniqueness property of Wardrop equilibria, and the fact that the equilibrium cost must be 174 minimal, we can see that the equilibrium arrival profiles must be essentially unique. Note that this does not establish uniqueness of the equilibria. (ii) Without loss of generality, suppose the last arrival to node (1,i), under the equilibrium arrivalprofileF ∗ (l,i) , entersbeforethelastuserarrivesattheothernodes. Thelastnon-atomic user to arrive at the network should arrive such that she has zero sojourn time through the network. This implies that the Wardrop equilibrium sojourn time on any path originating from node (1,i) should be zero. However, thelast arrival to anyother node inL 1 can improve its cost by choosing to arrive at (1,i) instead, implying that F ∗ cannot be an equilibrium. This is a contradiction, proving the claim. (iii) Now, as argued before in the case of a Trellis network, there exists an equivalent parallel queue network to the Lattice network at equilibrium. Let D ∗ (t) be the equilibrium sojourn time for a non-atomic user entering node (1,i) at time t. Let ˜ μ i (t) denote the service rate of node i in the parallel network. Then, F ∗ would still be an equilibrium arrival profile for the parallel queue network if F ∗ (1,i) (t) ˜ μ (1,i) (t) −t = F ∗ (1,i) (t) μ (1,i) −t + 1 μ (1,i) sup 0≤s≤t (−F 0 (1,i) (s) +μ (1,i) s) + D ∗ (τ (1,i) (t)). Solving this yields ˜ μ (l,i) (t). Proof: (Proposition 1.4) First, we prove that a four Beneš element trellis network as shown in Figure 1.7(a) has an “equivalent” two Beneš element parallel network that has the same equilibrium arrival profile. For brevity, we call this network a “basic sub-net”. Lemma 1.5 Consider a four Beneš element trellis as in Figure 1.7(a), with service rates such that μ i,1 +μ i,2 > μ i,3 +μ i,4 , i = 1, 2, 3, 4. Then, there exists a unique equivalent two Beneš element parallel network with service rates ˇ μ i,1 = μ i,1 , ˇ μ i,2 = μ i,2 , i = 1, 2, and ˇ μ 1,j = μ j,1 R j , and ˇ μ 2,j = μ j,2 R j where R j = μ j,1 +μ j,2 μ j,3 +μ j,4 , and j = 3, 4, that has the same equilibrium arrival profile. 175 1 2 3 4 5 6 i p1,3 p i 1,4 p i 2,4 p i 2,3 1 2 3 4 p1,4 p2,4 p2,3 (a) A four Beneš element network, or “basic sub- net”. (1,1) (1,2) (2,1) (2,2) (3,1) (3,2) p1,3 p i 1,4 p i 2,4 p i 2,3 1 2 3 4 p1,4 p2,4 p2,3 (b) A feedforward 4× 4 Beneš net- work. (5,1) (6,2) (6,1) (7,1) (7,2) (7,3) (7,4) (1,1) (1,2) (1,3) (1,4) (2,1) (2,2) (2,3) (2,4) (3,1) (3,2) (3,3) (3,4) (4,1) (4,2) (4,3) (4,4) (5,1) (5,2) (5,3) (5,4) (c) A feedforward 8×8 Beneš network. Figure 1.7: Simple Beneš networks. Proof: It is easy to see that the sojourn time on path P 1 = ((1, 1), (1, 3), (3, 1), (3, 3)) is W P 1 (t) = F 11 (t) μ 11 −t+ F 11 (t) μ 1,1 ( (μ 1,1 +μ 1,2 )(μ 3,1 +μ 3,2 )(μ 4,1 +μ 4,2 ) μ 3,1 (μ 4,1 +μ 4,2 )(μ 3,4 +μ 3,3 ) +μ 4,1 (μ 3,1 +μ 3,2 )(μ 4,3 +μ 4,4 ) −1). Thus, there exists a node that offers service rate ˇ μ 1,3 , such that for some equivalent routing probabilities ˇ p 1,3 and ˇ p 1,4 ˇ p 1,3 ˇ μ 1,3 = (μ 1,1 +μ 1,2 )(μ 3,1 +μ 3,2 )(μ 4,1 +μ 4,2 ) μ 3,1 (μ 4,1 +μ 4,2 )(μ 3,4 +μ 3,3 ) +μ 4,1 (μ 3,1 +μ 3,2 )(μ 4,3 +μ 4,4 ) = ˇ p 1,4 ˇ μ 1,4 . Using the fact that ˇ p 1,3 + ˇ p 1,4 = 1, and solving the resulting equation on substitution, we obtain ˇ μ 1,3 = μ 1,3 R 3 , and ˇ μ i,4 = μ 1,4 R 4 . A similar argument holds at node 2 in the four Beneš element network. Now, first note that in a 2 n × 2 n Beneš network, between each successive layer, there are 2 n−2 basic sub-nets and that a 2 n × 2 n Beneš network has 2n− 1 layers. Further, for brevity, we call the first 2 n−2 elements in any layer the “top” elements, and next 2 n−1 elements the “bottom” elements. Now, consider the 4× 4 Beneš network in Figure 1.7(b). A 4× 4 Beneš 176 2 n-1 2 n-1 . . . . . . 2 n-1 2 n-1 . . . . . . . . . . (A) (B) (C) . . . Figure 1.8: Reducing a 2 n × 2 n Beneš network to a 2 n−1 parallel Beneš element network. Step A replaces Layer2 2n− 1 and 2n− 2 by an equivalent parallel element network. Using the induction hypothesis, step B reduces the network further to a two layer network. Finally in step C, the network is reduced to a parallel element network. network has 2 basic sub-nets and three layers in the network, and one top and one bottom element. By Lemma 1.5, the last two layers can be replaced by a single layer of parallel elements, at equilibrium. We repurpose the naming convention and denote the equivalent elements as (2, 1) and (2, 2). The equivalent service rates are, ˇ μ 2,i 1 = μ 2,i 1 , ˇ μ 2,i 2 = μ 2,i 2 , ˇ μ 2,i 3 = μ 3,1 i R 3,1 and ˇ μ 2,i 4 = μ 3,2 i R 3,2 , where i = 1, 2. Recall that R 3,i = μ 3,i 1 +μ 3,i 2 μ 3,i 3 +μ 3,i 4 . We now have an equivalent basic sub-net (identical to Figure 1.7(a)). Applying Lemma 1.5 again to this network, we reduce this network to an equivalent two Beneš element parallel network. The service rates now are ˇ μ 1,i 1 =μ 1,i 1 , ˇ μ 1,i 2 =μ 1,i 2 , ˇ μ 1,i 3 = ˇ μ 2,1 i ˇ R 2,1 and ˇ μ 1,i 4 = ˇ μ 2,2 i ˇ R 2,2 , where i = 1, 2, and ˇ R 2,i = ˇ μ 2,i 1 +ˇ μ 2,i 2 ˇ μ 2,i 3 +ˇ μ 2,i 4 . Substitution and simplification of the expression will show that the resulting expression for the equilibrium routing profile within each of the two elements coincides with that in Lemma 1.1. It is easy to check that these expressions coincide with (1.7), (1.8), (1.9) and (1.10), and that (1.11) and (1.12) are not necessary in this case. For larger networks, with n > 1, we use an inductive argument. In step m = 1 of the induction, consider a 2 n × 2 n = 8× 8 Beneš network, where there are 2 basic sub-nets per layer, and 5 layers, as can be seen from Figure 1.7(c). Once again, we work backwards and, using Lemma 1.5, we replace the 2 basic subnets between Layer 4 and Layer 5 by 177 two parallel elements, resulting in an equivalent network that connects the four input Beneš elements to two 4× 4 Beneš networks. For instance, the elements (4, 1), (4, 3), (5, 1), (5, 2) form a basic subnet. Then the odd numbered (even numbered) elements in Layer 4 are connected to the top elements (bottom elements) in Layer 5. Further, if the element index in Layer 4 is less than 2 3−2 = 2 then these elements are connected to queue 1 inside the corresponding element in Layer 5. If the element index is greater than 2, then the elements are connected to queue 2 inside the corresponding element in Layer 5. For instance, node (4, 1) is connected to the queue 1 in element (5, 1), while element (4, 3) is connected to queue 2 in element (5, 1). It follows that the service rates in the equivalent Beneš element (4, 1) are ˇ μ 4,1 1 = μ 4,1 1 , ˇ μ 4,1 2 = μ 4,1 2 and ˇ μ 4,1 3 = μ 5,1 1 R 5,1 , ˇ μ 4,1 4 = μ 5,2 1 R 5,2 . For Beneš element (4, 2), we have ˇ μ 4,2 1 =μ 4,2 1 , ˇ μ 4,2 2 =μ 4,2 2 and ˇ μ 4,2 3 = μ 5,3 1 R 5,3 , ˇ μ 4,2 4 = μ 5,4 1 R 5,4 . The equivalent Beneš element (4, 3) has service rates ˇ μ 4,3 1 = μ 4,3 1 , ˇ μ 4,3 2 = μ 4,3 2 and ˇ μ 4,3 3 = μ 5,1 2 R 5,1 , ˇ μ 4,3 4 = μ 5,2 1 R 5,2 . Finally, for element (4, 4) we have, ˇ μ 4,4 1 =μ 4,4 1 , ˇ μ 4,4 2 =μ 4,4 2 and ˇ μ 4,4 3 = μ 5,3 2 R 5,3 , ˇ μ 4,4 4 = μ 5,4 1 R 5,4 . Notice that substituting for the indices in (1.9), (1.10), (1.11) and (1.12) will produce the same expressions. From the argument above, there exists a two Beneš element parallel network equivalent to the 4× 4 Beneš networks. Let these equivalent elements be numbered (2,i) i = 1, 2, 3, 4, and service rates are ˇ μ 2,i 1 =μ 2,i 1 , ˇ μ 2,i 2 =μ 2,i 2 and ˇ μ 2,i 3 = μ 3,i 1 ˇ R 3,i , ˇ μ 2,i 4 = μ 3,i+1 1 ˇ R 3,i+1 , ifi = 1, 3. Ifi = 2, 4, the service rates are ˇ μ 2,i 1 =μ 2,i 1 , ˇ μ 2,i 2 =μ 2,i 2 and ˇ μ 2,i 3 = μ 3,i 2 ˇ R 3,i , ˇ μ 2,i 4 = μ 3,i+1 2 ˇ R 3,i+1 . The 8× 8 network has been reduced to a two layer Beneš element network, with 4 elements in each layer. From Figure 1.7(c) it can be seen that there are two basic subnets between Layer 1 and Layer 2, and invoking Lemma 1.5 it can be seen that there is an equivalent two Beneš element network to these basic subnets, and the 8×8 network has been reduced to a 4 parallel Beneš element network, and service rates are ˇ μ 1,i 1 = μ 1,i 1 , ˇ μ 1,i 2 = μ 1,i 2 for all i = 1, 2, 3, 4. For i = 1 ˇ μ 1,1 3 = ˇ μ 2,1 1 ˇ R 2,1 , ˇ μ 1,1 4 = ˇ μ 2,3 1 ˇ R 2,3 . Fori = 2, ˇ μ 1,2 3 = ˇ μ 2,1 2 ˇ R 2,1 , ˇ μ 1,2 4 = ˇ μ 2,3 2 ˇ R 2,3 . Fori = 3, ˇ μ 1,3 3 = μ 2,2 1 ˇ R 2,2 , ˇ μ 1,3 4 = μ 2,4 1 ˇ R 2,4 . And fori = 4, ˇ μ 1,4 3 = μ 2,2 2 ˇ R 2,2 , ˇ μ 1,4 4 = μ 2,4 2 ˇ R 2,4 . It is easy to check that these expressions satisfy (1.7) and (1.8). 178 Now, assume that the induction hypothesis holds for step n− 1, so that there exists an equivalent 2 n−2 element parallel Beneš element network, at equilibrium. In step n, consider a 2 n × 2 n network (as shown in Figure 1.8), and focus on the nodes in Layer 2n− 2. Recall that there are 2 n−2 basic subnets between Layer 2n−2 and 2n−1. Elementl = 1, 2,..., 2 n−1 in Layer 2n− 2 is connected to elements mod (2l− 1, 2 n ) and mod (2l, 2 n−1 ), with the ackowledgement that elementl = 2 n−1 is connected to elements 2 n−1 (= mod (2 n −1, 2 n−1 )) and 2 n−1 . Top (bottom, resp.) elements are connected to node 1 (node 2, resp.) inside the corresponding elements in Layer 2n− 1. Using Lemma 1.5, we can replace each of the 2 n−2 basic subnets by 2 parallel Beneš elements at equilibrium. This results in a network where the 2 n−1 input Beneš elements are connected to two 2 n−1 × 2 n−1 Beneš networks (see reduction-step (A) of Figure 1.8). By the induction hypothesis, there exists an equivalent 2 n−2 Beneš element parallel network for each of the two 2 n−1 Beneš networks, with service rate given by (1.7), (1.8), (1.9), (1.10), (1.11) and (1.12) (with an appropriate change of index). In reduction-step (B) of Figure 1.8, the induction hypothesis can be used to replace the 2 n−1 × 2 n−1 networks by a parallel element network. Finally in reduction-step (C) of Figure 1.8, using Lemma 1.5 on the 2 n−2 basic subnets between the input layer (Layer 1) and the equivalent parallel Beneš element network from the previous reduction-step, we obtain an equivalent 2 n−1 Beneš element parallel network at equilibrium, and the service rates are recursively defined by (1.7), (1.8), (1.9), (1.10), (1.11) and (1.12). Proof: (Theorem 1.8) (i) & (ii) From Proposition 1.4 there exists an equivalent parallel Beneš element network (with 2 n−1 elements) such that it emulates the equilibrium sojourn time through the 2 n × 2 n Beneš network. The service rates of the queues in element (1,i) of the parallel element network are (μ 1,i 1 ,μ 1,i 2 , ˇ μ 1,i 3 , ˇ μ 1,i 4 ). The service rates for the equivalent parallel queue network now follow from (the last statement of) Lemma 1.1. By assumption, 179 μ k,i 1 +μ k,i 2 >μ k,i 3 +μ k,i 4 for all elements (k,i), so that the expressions in (1.13) follow directly. The equilibrium arrival profile can now be determined by invoking Theorem 1.1. (iii) By Corollary 1.1 it follows that the PoA is 2. 180 Conclusions Classical queueing models have focused exclusively on modeling infinite population sources. Asdemonstratedinthisdissertation, therearemanysystemsthatcanandshouldbemodeled as being driven by finite population sources. In the first thesis of this dissertation, we developed a new framework for studying finite source queueing models in a large population asymptotic setting. In particular, we first developed the Δ (i) /GI/1 queueing model - which appears to be the most natural model of a finite population queueing system. We show that the queue length (number-in-system) and workload processes can be approximated by fluid and diffusion processes that emerge as the population size tends to infinity. Next, we extended this study to a much broader class of queueing models that we term Transitory Queueing Models. In particular, we showed that disparate models of traffic - including a generalization of the Δ (i) model, a conditioned renewal model and a scheduled arrival model - satisfy what we term as Population Acceleration limits. Next, we study a network of Δ (i) /GI/1 queues, with Markov routing. We show that while the fluid approximation is relatively easy to obtain, the diffusion approximation is not. Indeed, we are only able to comprehensivelyprovethediffusionapproximation for atwo-nodetandemqueueing network. This leaves significant space for future work (more on this below). In the second thesis, we take a novel approach to studying arrival and routing behavior in queueing networks. Classically, traffic models and routing are assumed as part of the model specification. However, in many systems users arriving at queueing networks are strategic in their choices and react to the available resources. For example, users arriving at a restaurant 181 or a concert queue would choose a time to arrive such that their waiting time (or a function there of) is minimized. We study this type of strategic behavior more generally in the context of generalized Jackson queueing networks as an arrival and routing game. We identify the equilibrium arrival and routing profiles in a "mean field" setting (i.e., in the limit of a large population) for specific network topolgies. However, the network topology is crucial, and we can only provide an algorithm for computing the equilibrium in general networks. The work in this thesis opens several avenues for future work. In the context of the single server Δ (i) /GI/1 queue, the large deviations behavior is almost certainly atypical of standard results for large deviations in queueing networks. Results of this nature would be crucial for understanding capacity provisioning over short time horizons. Second, we have only scratched the surface as far as approximations for finite population systems go. Thus, far we have only focused on fluid and diffusion approximations. One could also study other approximations (such as perturbation expansions for the queue length distribution) that would not scale the number of arrivals. For queueing networks, it would be interesting to extend the current analysis to multi-class networks, while also working out the mathematical difficulties in establishing approximation for general single-class networks. On the game theoretic front, we have mostly focused on single class networks, it is also of interest to study multi-class networks and networks where customers have preferences (something that we have completely ignored for now). Finally, it is also crucial to study mechanisms for influencing arrival and routing behavior of customers, in order to operationalize the results of this thesis. 182 Reference List [1] A. K. Erlang. The theory of probabilities and telephone conversations. Nyt Tidsskrift for Matematik B, 20(33-39):16, 1909. [2] J.F.C.Kingman. ThefirstErlangcentury-andthenext. Queueing Syst., 63(1-4):3–12, 2009. [3] D. Gross and C.M. Harris. Fundamentals of Queueing Theory. Wiley, 1998. [4] D. L. Iglehart and W. Whitt. Multiple channel queues in heavy traffic, I. Adv. Appl. Probab., 2:150–177, 1970. [5] J.M. Harrison. Brownian Motion and Stochastic Flow Systems. John Wiley and Sons, 1985. [6] P. W. Glynn. Diffusion approximations. In Handbooks in Operations Research and Management Science, volume 2, pages 145–198. Elsevier, 1990. [7] L. Brown, N. Gans, A. Mandelbaum, A. Sakov, H. Shen, S. Zeltyn, and L. Zhao. Statistical analysis of a telephone call center: A queueing-science perspective. J. Amer. Statist. Assoc., 100(469):36–50, 2005. [8] S.-H. Kim and W. Whitt. Are call center and hospital arrivals well modeled by nonho- mogeneous poisson processes? Unpublished, 2013. [9] S.-H. Kim and W. Whitt. Choosing arrival process models for service systems: tests of a nonhomogeneous poisson process. Naval Res. Logist., 2013. Accepted for publication. [10] G.F. Newell. Queues with time-dependent arrival rates I: The transition through satu- ration. J. Appl. Probab., pages 436–451, 1968. [11] G.F. Newell. Queues with time-dependent arrival rates II: The maximum queue and the return to equilibrium. J. Appl. Probab., pages 579–590, 1968. [12] G.F. Newell. Queues with time-dependent arrival rates III: A mild rush hour. J. Appl. Probab., pages 591–606, 1968. [13] G.F. Newell. Applications of Queueing Theory. Chapman and Hall Ltd., 2 edition, 1982. 183 [14] J. B. Keller. Time-dependent queues. SIAM Rev., pages 401–412, 1982. [15] M. H. Rothkopf and S. S. Oren. A closure approximation for the nonstationary M/M/s queue. Management Sci., 25(6):522–534, 1979. [16] W. A. Massey. Non-Stationary Queues. PhD thesis, Stanford University, 1981. [17] W. A. Massey and W. Whitt. Uniform acceleration expansions for markov chains with time-varying rates. Ann. Appl. Probab., 8(4):1130–1155, 1998. [18] W.A. Massey. Asymptotic analysis of the time dependent M/M/1 queue. Math. Oper. Res., pages 305–327, 1985. [19] A. Mandelbaum and W.A. Massey. Strong approximations for time-dependent queues. Math. Oper. Res., 20(1), 1995. [20] M. I. Reiman. Open queueing networks in heavy traffic. Math. Oper. Res., 9(3):441–458, 1984. [21] Avi Mandelbaum, William A Massey, and Martin I Reiman. Strong approximations for markovian service networks. Queueing Syst., 30(1-2):149–201, 1998. [22] A. Mandelbaum and K. Ramanan. Directional derivatives of oblique reflection maps. Math. Oper. Res., 35(3):527, 2010. [23] D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes: Elementary Theory and Methods, volume 1. Springer, 2003. [24] Donald L Iglehart and Ward Whitt. Multiple channel queues in heavy traffic. ii: Sequences, networks, and batches. Adv. Appl. Probab., 2(2):355–369, 1970. [25] J.F.C. Kingman. The single server queue in heavy traffic. In Math. Proc. Cambridge Phil. Soc., volume 57(4), pages 902–904. Cambridge Univ Press, 1961. [26] S. Juneja and R. Jain. The concert/cafeteria queueing problem: a game of arrivals. In Proc. Fourth Intl. ICST Conf. on Performance Evaluation Methodologies and Tools, pages 1–6. EAI, 2009. [27] R. Jain, S. Juneja, and N. Shimkin. The concert queueing game: To wait or to be late. Discrete Event Dyn. Syst., 21(1):103–134, 2011. [28] P. Naor. The regulation of queue size by levying tolls. Econometrica: journal of the Econometric Society, 37(1):15–24, 1969. [29] H. Mendelson and S. Whang. Optimal incentive-compatible priority pricing for the M/M/1 queue. Operations Research, pages 870–883, 1990. 184 [30] R. Hassin and M. Haviv. To Queue or not to Queue. Kluwer Academic Publishers, 2003. [31] P. Dube and R. Jain. Bertrand equilibria and efficiency in markets for congestible network services. Automatica, January 2014. [32] A. Glazer and R. Hassin. ?/m/1: On the equilibrium distribution of customer arrivals. European J. Oper. Res., 1983. [33] M.A. Lariviere and J.A. Van Mieghem. Strategically seeking service: How competi- tion can generate poisson arrivals. Manufacturing & Service Operations Management, 6(1):23–40, 2004. [34] R. Hassin and Y. Kleiner. Equilibrium and optimal arrival patterns to a server with opening and closing times. IIE Tran., 43(3):164–175, 2010. [35] R. Lindsey. Existence, uniqueness, and trip cost function properties of user equilibrium in the bottleneck model with multiple user classes. Transportation science, 38(3):293, 2004. [36] W. Whitt. The pointwise stationary approximation for M t /M t /s queues is asymptoti- cally correct as the rates increase. Management Sci., 37(3):307–314, 1991. [37] W. A. Massey. The analysis of queues with time-varying rates for telecommunication models. Telecommun. Syst., 21(2-4):173–204, 2002. [38] D.P. Gaver, J.P. Lehorsky, and M. Perlas. Service systems with transitory demand. In Logistics, volume 1. 1975. [39] G. Louchard. Large finite population queueing systems. The single-server model. Stochastic Proc. Appl., 53(1):117 – 145, 1994. [40] R. Durrett. Probability: Theory and Examples. Cambridge University Press, 4 edition, 2010. [41] H. Chen and D.D. Yao. Fundamentals of Queueing Networks: Performance, asymp- totics, and optimization. Springer, 2001. [42] P. Billingsley. Convergence of Probability Measures. Wiley & Sons, 1968. [43] W. Whitt. Stochastic Process Limits. Springer, 2001. [44] A.V. Skorokhod. Limit theorems for stochastic processes. Theory Probab. Appl., 1(3), 1956. [45] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer, 1991. [46] W. Whitt. Internet Supplement To Stochastic Process Limits. Springer, 2001. 185 [47] J.L.Pomarede. AUnified Approach viaGraphs toSkorohod’sTopologies onthe Function Space D. PhD thesis, Yale University, 1976. [48] A.A. Puhalskii and J.E. Reed. On many-server queues in heavy traffic. Ann. Appl. Probab., 20(1):129–195, 2010. [49] H. Honnappa and R. Jain. Strategic arrivals into queueing networks: The network concert queueing game. Third revision submitted, 2013. [50] D. Aldous. Brownian excursions, critical random graphs and the multiplicative coales- cent. Ann. Probab., pages 812–854, 1997. [51] R. Durrett. Random Graph Dynamics, volume 20. Cambridge University Press, 2007. [52] Anders Martin-Löf. The final size of a nearly critical epidemic, and the first passage time of a wiener process to a parabolic barrier. J. Appl. Probab., 35(3):671–682, 1998. [53] R. Van der Hofstad, A.J.E.M. Janssen, and J.S.H. van Leeuwaarden. Critical epi- demics, randomgraphs, andbrownianmotionwithaparabolicdrift. Adv. Appl. Probab., 42(4):1187–1206, 2010. [54] W. Rudin. Real and Complex Analysis. McGraw-Hill, 2006. [55] M. G. Hahn. Central limit theorems in D[0,1]. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 44(2):89–101, 1978. [56] L. E. Dubins and D. A. Freedman. Random distribution functions. In Proc. Fifth Berkeley Symp. Math. Statist., volume 2(1), pages 183–214. Univ. Calif. Press, 1967. [57] D.R. Cox and W. L. Smith. Queues. John Wiley, 1961. [58] J. A. Wellner. A Glivenko-Cantelli theorem for empirical measures of independent but non-identically distributed random variables. Stochastic Process. Appl., 11(3):309–312, 1981. [59] G. R. Shorack. The weighted empirical process of row independent random variables with arbitrary distribution functions. Stat. Neerl., 33(4):169–189, 1979. [60] U. Liberman. An order statistic characterization of the poisson renewal process. J. Appl. Probab., pages 717–722, 1985. [61] P.D. Feigin. On the characterization of point processes with the order statistic property. J. Appl. Probab., pages 297–304, 1979. [62] D.L. McLeish. Dependent central limit theorems and invariance principles. Ann. Probab., pages 620–628, 1974. 186 [63] N.C. Weber. A martingale approach to central limit theorems for exchangeable random variables. J. Appl. Probab., pages 662–673, 1980. [64] V. F. Araman and P. W. Glynn. Fractional brownian motion with H < 1/2 as a limit of scheduled traffic. J. Appl. Probab., 49(3):710–718, 2012. [65] A.N. Borodin and P. Salminen. Handbook of Brownian motion. Birkhäuser Verlag, 1996. [66] H. Honnappa, R. Jain, and A.R. Ward. The Δ (i) /GI/1 Queueing Model, and its Fluid and Diffusion Limits. submitted, 2013. [67] G. R. Shorack and J. A. Wellner. Empirical Processes with Applications to Statistics, volume 59. SIAM, 2009. [68] J. Komlós, P. Major, and G. Tusnády. An approximation of partial sums of independent RV’s, and the sample DF II. Probab. Theory Related Fields, 34(1):33–58, 1976. [69] P. W. Glynn. Strong approximations in queueing theory. In Asymptotic Methods in Probability and Statistics, pages 133–150. Elsevier Science, 1998. [70] T. M. Liggett. Weak convergence of conditioned sums of independent random vectors. Trans. Amer. Math. Soc., 152(1):195–213, 1970. [71] G.R. Mendieta. Two hyperfinite constructions of the brownian bridge. Stoch. Anal. Appl., 7(1):75–88, 1989. [72] H. Chernoff and H. Teicher. A central limit theorem for sums of interchangeable random variables. Ann. Math. Statist., 29(1):118–130, 1958. [73] W. Rudin. Principles of Mathematical Analysis, volume 3. McGraw-Hill New York, 1964. [74] G. B. Folland. Real Analysis: Modern Techniques and their Applications, volume 2. Wiley New York, 1984. [75] J. M. Harrison and M. I. Reiman. Reflected brownian motion on an orthant. Ann. Probab., pages 302–308, 1981. [76] H. Chen and A. Mandelbaum. Discrete flow networks: bottleneck analysis and fluid approximations. Math. Oper. Res., 16(2):408–446, 1991. [77] H. Chen and A. Mandelbaum. Stochastic discrete flow networks: Diffusion approxima- tions and bottlenecks. Ann.Probab., pages 1463–1519, 1991. [78] H. Honnappa, R. Jain, and A.R. Ward. On Transitory Queueing. submitted, 2013. [79] N.G. Duffield, W.A. Massey, and W. Whitt. A nonstationary offered-load model for packet networks. Telecomm. Syst., 16(3-4):271–296, 2001. 187 [80] Y.LiuandW.Whitt. Anetworkoftime-varyingmany-serverfluidqueueswithcustomer abandonment. Oper. Res., 59(4):835–846, 2011. [81] H. Chen and W. Whitt. Diffusion approximations for open queueing networks with service interruptions. Queueing Syst., 13(4):335–359, 1993. [82] S. Adlakha and R. Johari. Mean field equilibrium in dynamic games with strategic complementarities. Submitted, 2010. [83] N. Nisan, T. Roughgarden, E. Tardos, and V. V Vazirani. Algorithmic game theory. Cambridge University Press, 2007. [84] D. Fudenberg and J. Tirole. Game Theory. MIT Press, 1991. [85] M Ali Khan and Yeneng Sun. Non-cooperative games with many players. Handbook of Game Theory with Economic Applications, 3:1761–1808, 2002. [86] Irving L Glicksberg. A further generalization of the kakutani fixed point theorem, with application to nash equilibrium points. Proceedings of the American Mathematical Society, 3(1):170–174, 1952. 188

Abstract (if available)

Abstract Stochastic network theory, and queueing theory in particular, is the bedrock for the analysis and control of resource constrained systems. Such systems are manifest in our world: in healthcare delivery, shared computing, communications and transportation systems, system operators observe high demand for services necessitating queue management. 'Classical' queueing theory has largely focused on the analysis of stationary and ergodic models. However, most real world resource allocation systems exhibit time-dependent arrival and service. Further, many systems operate only on a finite time horizon, or system operators are interested in the 'small-time' or transient behavior of a queueing system. In this dissertation, we initiate the development of models of such 'transitory' queueing systems. Our first contribution is the introduction of several disparate models of multiple server transitory queues. We develop fluid and diffusion approximations, using a mathematical technique called 'Population Acceleration'. Next, we extend this analysis to generalized Jackson networks. The diffusion approximations are completely unlike the conventional heavy-traffic diffusion approximations. Our second major contribution is the development of game theoretic models of traffic and routing in generalized Jackson networks. Almost all queueing models assume exogeneous arrivals, routing and service. However, in many situations, like early morning commutes, users are strategic in when they decide to join a service system and which route to take, so that they minimize their sojourn time. We identify the Nash equilibrium traffic and routing profile when users are strategic.

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Asset Metadata

Creator Honnappa, Harsha (author)

Core Title Strategic and transitory models of queueing systems

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 10/01/2014

Defense Date 07/07/2014

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag applied probability,empirical process theory,game theory,OAI-PMH Harvest,queueing theory,stochastic process limits

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Jain, Rahul (committee chair), Ward, Amy R. (committee chair), Krishnamachari, Bhaskar (committee member), Neely, Michael J. (committee member)

Creator Email honnappa@gmail.com,honnappa@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-487938

Unique identifier UC11287015

Identifier etd-HonnappaHa-2955.pdf (filename),usctheses-c3-487938 (legacy record id)

Legacy Identifier etd-HonnappaHa-2955.pdf

Dmrecord 487938

Document Type Dissertation

Format application/pdf (imt)

Rights Honnappa, Harsha

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA