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Novel techniques for analysis and control of traffic flow in urban traffic networks
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Novel techniques for analysis and control of traffic flow in urban traffic networks
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Novel Techniques for Analysis and Control of Traffic Flow in Urban Traffic Networks by Seyedpouyan Hosseinialiabad 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 (Civil Engineering) December 2019 Dissertation Committee Prof. Ketan Savla (Chair) Prof. James E. Moore Prof. Petros Ioannou Abstract Traffic congestion has become a major issue in urban society. Performance evaluation of transportation network and utilizing new technologies to improve the existing level of service is crucial. While the majority of traffic signals in arterial networks in the United States operate under fixed-time control, rapid advance- ments in traffic sensing technology have made it possible to use real-time traffic information in road traffic control. This has opened up the possibility of replac- ing traditional fixed-time traffic signal controllers with adaptive signal controllers. Such adaptive controllers can respond to the network condition in real-time, utilize the network capacity in a better manner and improve the mobility and resilience of urban traffic network. Therefore, it is of great interest to have an accurate understanding of existing level of service of transportation system and then design new controllers to improve its performance. In the first part of this dissertation, an ON/OFF traffic flow model is presented for signalized arterial network. A delay differential equation framework is provided to directly simulate queue length dynamics under fixed-time or adaptive control. Existence and uniqueness of the solution to the proposed model for traffic flow dynamics is established for piecewise constant external inflow and capacity func- tions. Additionally, it is shown that if the external inflow and capacity functions 1 are periodic and satisfy a stability condition, there exists a globally attractive peri- odic orbit. In the next step, a novel iterative procedure is provided to compute this periodic orbit with arbitrary accuracy. A periodic trajectory is iteratively updated for every link based on updates to a specific time instant when its queue length transitions from being zero to being positive. The update for a given link is based on the periodic trajectories computed in the previous iteration for its upstream links. The resulting iterates are shown to converge uniformly monotonically to the desired periodic orbit. The proposed computational framework can be uti- lized for health monitoring of existing urban traffic system and is a novel tool for performance evaluation of fixed-time controller. Moreover, it is shown that there is a good consistency between the queue lengths computed using the proposed framework and the output of a complex microscopic traffic simulator. The second part of this dissertation is dedicated to designing decentralized adaptive signal controllers. Motivated by the increasing ubiquity of traffic data and improvements in sensing technologies, adaptive control techniques are pro- posed that update the key elements of traffic signal plan based on real-time mea- surements. The focus is on adapting the well-known Proportionally Fair (PF) controller from the communication networks literature to determine green time allocations for signalized arterial network and providing theoretical results on per- formance analysis of this controller. While most of studies on adaptive signal controllers are focused on designing centralized controllers which requires solving large optimization problems, PF controller is completely decentralized and can be easily implemented with low computational cost, using local information. In addition to that, PF controller has a very minimal nature and requires no knowl- edge of external arrival rates, turning ratios and saturation flow capacities. First, analysis of PF controller is provided under uninterrupted or an averaged traffic 2 flow model. It is proven that under stylized phase architecture the controller is maximally stabilizing, meaning whenever the arrival rates are feasible the result- ing traffic network dynamics admit a globally asymptotically stable equilibrium. The results rely on the use of some entropy-like Lyapunov functions previously considered in the context of communication networks. Next, the PF controller is adapted to the proposed ON/OFF traffic flow model, where green splits are updated at most once per cycle. Specifying the dependence of such controller on queue lengths measurement and the stability analysis becomes more challenging under the new accurate model. These challenges are addressed by proposing an output feedback PF controller that does not require queue length measurement. This is also interesting from the practical perspective, since real-time queue length estimation is known to be difficult and challenging in realistic setups. We identify the scenarios under which the proposed controller is maximally stabilizing and present analytical guarantees. Lastly, performance of PF controller is evaluated in a microscopic traffic simu- lator. The simulation results for a downtown Los Angeles sub-network show that the proposed output feedback PF controller has a good performance in terms of network throughput. Moreover, performance of PF controller is compared with that of other recently studied decentralized controller, Max-Pressure (MP). The results of simulations suggest that PF controllers outperforms MP controllers in terms of throughput and average delay, in spite of its minimalist nature. 3 Acknowledgement I would like to express the greatest gratitude to my advisor, Professor Ketan Savla. Over the past five years, I have learned a lot from him. He has taught me how to develop a hardworking and persistent personality. His expertise was invaluableinidentifyingchallengingandinterestingresearchproblemsandhisdeep knowledge of the field has been a great help in my research. I would like to thank all the great professors at USC who I have taken course with, especially Professor Alexander Sawchuk and Professor Solomon Golomb. I must express my gratitude to Professor James Moore and Professor Petros Ioannou for their guidance and for being a member of my dissertation committee. I would also like to thank Professor Erik Johnson who initiated my interest in USC and highly encouraged me to apply for the PhD program when I reached out to him during my senior year back in undergrad. I have my special thanks for Jennifer Gerson, who has been very helpful and supportive throughout my PhD years and always responded to my questions promptly. I would also like to thank Professor Henry Koffman, it was a pleasure working with him as a teaching assistant for two years and he has taught me many life lessons. I would like to extend my gratitude to my dear relatives who live here in Los Angeles area. They have been really supportive from my first day here and I feel really lucky to have them. I would also like to thank all my friends at USC, 4 especially all my officemates. I shared all my tough and happy moments of my PhD years with them. I have my deep gratitude to my family for their continuous love and support. I am forever indebted to my beloved parents for giving me the opportunities and experiences that have made me who I am. They have always encouraged me to explore new directions in life and seek my own destiny. I am grateful to my brothers, Erfan and Arman, for always being there for me. This journey would not have been possible if not for them, and I dedicate this milestone to my family. Last but not least, I would like to thank National Science Foundation (NSF) and United States Department of Transportation (USDOT) for supporting my research in part. 5 Contents Abstract 1 Acknowledgement 4 List of Figures 8 1 Introduction 11 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Research Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Background and Related Works 16 2.1 Traffic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Traffic Signal Control . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Performance Analysis of Fixed-Time Controller 25 3.1 Traffic Flow Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Existence of Solution to (3.2a) . . . . . . . . . . . . . . . . . . . . . 31 3.3 Periodic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 Steady State Computation for an Isolated Link . . . . . . . . . . . 36 3.6 Steady State Computation For a Network . . . . . . . . . . . . . . 43 3.7 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.7.1 MATLAB Simulations . . . . . . . . . . . . . . . . . . . . . 45 3.7.2 Comparison with Microsimulations . . . . . . . . . . . . . . 48 4 Proportionally Fair Controller (PF) under Averaged Model 56 4.1 Averaged Traffic Flow Model . . . . . . . . . . . . . . . . . . . . . . 57 4.2 PF Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6 5 Output Feedback PF Controller under ON/OFF Model 69 5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6 Evaluation of Controllers in a Microscopic Traffic Simulator 81 6.1 Model and Signal Controllers . . . . . . . . . . . . . . . . . . . . . 82 6.1.1 Proportionally Fair (PF) Controller . . . . . . . . . . . . . 85 6.1.2 Max-Pressure (MP) Controller . . . . . . . . . . . . . . . . . 86 6.2 Case Study 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.2.1 Model Setup in PTV VISSIM . . . . . . . . . . . . . . . . . 89 6.2.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 93 6.3 Case Study 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.3.1 Model Setup and Controller Implementation . . . . . . . . . 100 6.3.2 Simulation Results: Throughput Evaluation . . . . . . . . . 102 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7 Conclusions and Future Work 105 Reference List 109 A Additional Results for Chapter 3 116 A.1 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . 116 A.2 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . 117 A.3 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 A.4 Proof of Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 A.5 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . 119 A.6 Technical Corollary . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 B Additional Results for Chapter 4 123 B.1 Additional Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 C Additional Results for Chapter 5 125 C.1 Proof of Lemma 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7 List of Figures 3.1 Illustration of a capacity function. . . . . . . . . . . . . . . . . . . . 33 3.2 Illustration of transition points, and negative/positive sets. In this case, M w = 3 and M b = 2. Subscript i is not shown for brevity. . . 37 3.3 Illustration of the procedure in (3.10) to computeW α i . . . . . . . . 41 3.4 Graph topology of the network used in the simulations. . . . . . . . 46 3.5 Evolution of RMSE between (a)x (k) andx ∗ , and (b)z (k) andz ∗ for a few representative links. . . . . . . . . . . . . . . . . . . . . . . . 48 3.6 (top)x (k) i fromafewrepresentativeiterationsand(bottom)x ∗ i , both for i = 17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.7 The Los Angeles downtown sub-network used in the simulations: (a) graph topology (b) aerial view in PTV VISSIM. . . . . . . . . . 50 3.8 Illustration of movements and lanes on link 44 at intersection num- ber 7 in the network shown in Figure 6.8. . . . . . . . . . . . . . . . 51 3.9 Evolution of RMSE between (a)x (k) andx ∗ , and (b)z (k) andz ∗ for a few representative links. . . . . . . . . . . . . . . . . . . . . . . . 53 3.10 (top)x (k) i fromafewrepresentativeiterationsand(bottom)x ∗ i , both for i = 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 8 3.11 ComparisonofqueuelengthobtainedfromAlgorithm2andVISSIM simulations for a few representative links (a) i = 20, (b) i = 15 and (c) i = 22, in the network shown in Figure 6.8. . . . . . . . . . . . 55 4.1 (a) Network of four signalized intersections used in the simulations. (b) Evolution of queue lengths under proposed controller with only single phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Illustration of a four phase architecture at an intersection, where the incoming lanes are numbered counterclockwise from 1 to 12. . . 67 4.3 Evolution of queue lengths under PF policy for multiphase case. . . 68 5.1 Extraction of ˜ G in a real traffic signal setup using stop line detectors in (a) undersaturated and (b) oversaturated condition. . . . . . . . 72 5.2 (a) Graph topology of the network used in the simulations. (b) Sum of all queue lengths vs. time. . . . . . . . . . . . . . . . . 80 6.1 (a) Illustration of a movement, lane and a link at a sample inter- section. Link number 20 contains two lanes, and each lane supports multiple movements. (b) Phase architecture at a sample intersection. 82 6.2 TheLosAngelesdowntownsub-networkusedinthesimulations: (a) schematic representation, with the solid disks showing the approx- imate location of loop detector sensors from which we have access to offline traffic count data; (b) aerial view. . . . . . . . . . . . . . . 91 6.3 Overview of adaptive control implementation in PTV VISSIM using COM interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.4 Comparison of the running average of travel time for heavy traffic under PF and MP controllers using (a) Zero offset and (b) Non-zero offset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 9 6.5 Comparison of (a) Running average of travel time and (b) Total number of vehicles in the network in case of incidents under PF and MP3 controllers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.6 Illustrationofphaseshiftinqueuelengthtimeseriesonlinknumbers 1 and 44, which are adjacent to each other (see Figure 6.2(a)), under (a) PF and (b) MP2 controllers under zero offset. . . . . . . . . . . 99 6.7 Illustrationofphaseshiftinqueuelengthtimeseriesonlinknumbers 1 and 44, which are adjacent to each other (see Figure 6.2(a)), under (a) PF and (b) MP2 controllers under non-zero offset . . . . . . . . 99 6.8 The Los Angeles sub-network used in the simulations: (a) graph topology (b) aerial view in PTV VISSIM. . . . . . . . . . . . . . . . 101 C.1 Link queue lengths over η consecutive cycles: link 1 (top) and link 2 (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 10 Chapter 1 Introduction 1.1 Motivation Traffic congestion has been a major problem in modern urban areas. The growing population and increasing travel demand have aggravated congestion of transportation systems causing significant economic loss to our society. Recent surveys show that the average United States commuter spends about 59 hours waitingattheredlightsandloses$1,400idlingawaygasperyear[30]. Eventhough about two-third of miles driven in United States are signalized roadways, most of the recent studies are focused on improving the traffic congestion on freeways. United States’ road network has more than 311,000 traffic signals majority of which operate under fixed-time control [16]. Meaning, the parameters of traffic signals are fixed and are calculated based on offline traffic counts through surveys. Performance evaluation of the existing traffic signals is significantly important to quantify the level congestion and major bottlenecks of the system. On the other hand, new techniques are required to improve the performance of traffic signals and ensure that the existing capacity of the network is not being underutilized. This could postpone or eliminate the need for construction of addi- tionalroadwaysandaddinglanestotheexistingroadwaynetwork. Recentdevelop- ments in communication techniques and sensing technologies have made it possible to re-tune traffic signal controllers based on real-time information and make the traffic signal smarter. In fact, it is hard to think of future smart cities without 11 smart intersections. Infrastructure-based sensors, including loop detectors and traffic cameras provide real-time traffic information such as traffic volume, speed and occupancy of roadway sections. Cell-phone data can be used to directly mea- sure vehicle’s location, speed and acceleration. The ubiquitous availability of traf- fic data provides the opportunity of replacing traditional fixed-time traffic signal controllers with adaptive controllers ( also known as dynamic signal controllers). These controllers response to traffic variations in real-time and are valuable in the sense of network resilience to traffic incidents and extreme conditions. This dissertation presents a set of theoretical results and practical techniques for traffic flow modeling, performance evaluation and control of signalized arte- rial networks. While majority of works in transportation literature only rely on microsimulation studies and lack theoretical results, this work provides rigorous theoretical guarantees for stability and throughput optimality of proposed con- trollers. It uses techniques from communication network, adapts those to trans- portation network and utilizes tools form control theory to establish analytical results in the context of signalized arterial urban networks. In what follows, the main contributions and organization of this thesis is presented. 1.2 Research Overview The first step in analytical study of transportation systems is modeling traffic flow and network dynamics. Modeling traffic flow dynamics for signalized arterial networks has to strike a tradeoff between the ability to capture variations induced by alternating red/green phases and computational complexity of the resulting framework for the purpose of performance evaluation and control synthesis. 12 Chapter 3 of this dissertation presents a macroscopic ON/OFF model for traf- fic flow which captures inter-cycle queue length behavior. It proposes a delay differential equation framework to directly simulate queue length dynamics under fixed-time or adaptive control, by establishing that it has a unique solution as long as the external inflow and capacity functions are piecewise constant. Moreover, under additional periodicity and stability condition, a recently proposed technique is adapted to establish existence of a globally attractive periodic orbit in our set- ting. One of the most novel contribution of this thesis is a procedure to explicitly calculatethisgloballyattractiveperiodicorbit. Infact, thiswasnotedasanimpor- tant open problem in previous works, due to its usefulness in directly quantifying relevant performance metrics for a given fixed-time traffic signal plan. Chapter 3 presents a procedure to compute this periodic orbit with arbitrary accuracy. A periodic trajectory is iteratively updated for every link based on updates to a specific time instant when its queue length transitions from being zero to being positive. This update for a given link is based on the periodic trajectories com- puted in the previous iteration for upstream links. The resulting iterates are shown to converge uniformly monotonically to the desired periodic orbit. Illustrative sim- ulations, including comparison with steady-state queue lengths from a microscopic traffic simulator, are also included. The second part of this dissertation is dedicated to designing decentralized con- trollers for traffic signals. In particular, we are interested in adapting controllers from the field of communications networks to transportation setup and showing theoretical guarantees for performance of such algorithms in signalized arterial net- works. Thefocusisonadaptingthewell-knownProportionallyFair(PF)controller from the communication networks literature to determine green time allocations 13 for signalized arterial network. PF controller has multiple unique features that make it worthy of studying. In addition to being decentralized, PF controller has a minimal nature as it does not require any information about parameters of transportation network such as external inflows, saturation flow capacities of links and turn ratios at the intersections. This feature makes the PF controller robust to disruptions and non-recurrent traffic patterns caused by accidents or road clo- sures. Chapter 4 is dedicated to analysis of PF controllers under uninterrupted or an average traffic flow models. The main contribution is proving that these policies are maximally stabilizing in the sense that, for the largest feasible set of external inflows to the network, they drive the traffic network dynamics to glob- ally asymptotically stable equilibria. Utilizing entropy-like Lyapunov functions is inspired by results in packet-switched stochastic queueing networks. The use of these techniques in the context of traffic signal control in urban traffic networks is a novel contribution. While using average traffic flow models simplifies the analysis, it masks the dependence on offsets and cycle length. Chapter 5 provides analysis of PF con- troller under the more accurate ON/OFF model proposed in chapter 3. While such models have been used in multiple simulation studies, rigorous theoretical analysis of adaptive controllers under such models is missing in the literature. Specifying the dependence of such controller on measurement of queue lengths in the last cycle in order to enable computational tractability of control synthesis as well as facilitating the stability analysis is not straightforward. We address this challenge by proposing an output feedback PF controller that does not require direct queue length measurement. Instead, as an input, it uses the fraction of green time over which the link has a non-zero queue length. In chapter 5, we demonstrate that the input parameter can be readily extracted using presence stop line detectors, which 14 are widely available on urban intersections due to their use in actuated traffic con- trollers. This is a very interesting feature from the practical point of view, since accurate queue length estimation using information from loop detectors is chal- lenging and it becomes even more difficult when the links are congested. In terms of analytical guarantees, it is shown that control synthesis under our proposed adaptation is convex and the resulting output feedback controller is maximally stabilizing for an isolated intersection with a periodic demand. In the next step, we plan to consider more complex scenarios (i.e. general network setup) and pro- vide theoretical guarantees for stability. Finally, the performance of PF controllers is evaluated in a microsimulation environment. Chapter 6 presents the description of microscopic models along with the results of simulations. Performance of the proposed output feedback PF con- troller is studied for a downtown Los Angeles sub-network coded in a microscopic traffic simulator, PTV VISSIM. Moreover, the Max-Pressure (MP) controller, a recently studied traffic signal controller derived from the field of communications networks is implemented in the microsimulator. The PF and MP controllers are implemented in a cycle-based manner where the green splits are updated once every cycle based on local queue measurements at the end of cycle. Also, a modi- fied version of MP is formulated where the controller only requires aggregate queue length of each link in oppose to the existing variants of MP which require individ- ual queue length of each movement and is less practical in realistic traffic signal setup. An extensive set of simulations is conducted to compare performance of PF and MP controllers with respect to different network performance metrics, such as throughput, average travel time and average queue length. 15 Chapter 2 Background and Related Works In this chapter, a brief review of the literature related to this work is presented. We start by reviewing the existing literature on different models for traffic flow and performance evaluation tools for urban traffic systems. Then, the related works on traffic signal control are discussed. 2.1 Traffic Models Analysis of traffic flow and performance evaluation of traffic systems are very important and have been studied in previous works. Different traffic models have been proposed to model the traffic flow in urban traffic networks. In general, traffic models can be divided into two main categories: microscopic and macroscopic. Microscopic models characterize the traffic flow at a vehicle level. Meaning, different behaviors of every vehicle such as car-following, lateral and lane changing behavior, merging behavior, etc., are captured in a detailed manner. Several works have been done on modeling car following behavior of human drivers, using first or second order differential equation [66, 37]. In fact, advanced traffic micro-simulator softwares, e.g., PTVVISSIM[2,17], Aimsun[7]andCORSIM[1]aredesignedbased on such microscopic models. Using these traffic microsimulator softwares requires purchasing expensive licenses. Also, it is usually quite time consuming to create a realistic and well-calibrated models in these softwares and it requires advanced 16 trainings. Ingeneral, microscopictrafficsimulationsarecomputationallyexpensive and are more suitable for analysis of small scale traffic networks. On the other hand, macroscopic models provide description of the traffic flow at a coarser level using aggregated quantities such as traffic flow, density or queue length. Hence, these models are more appropriate for modeling large scale trans- portation networks. While macroscopic simulations are less expensive, capturing all the physical constraints of transportation network and an accurate traffic flow dynamics could be challenging. One such physical constraint is maintaining posi- tivity of queue lengths. Among the well-know macroscopic models are hydrodynamic models, e.g., the LWR (Lighthill-Whitham-Richards) model [36, 58] and Link Transmission Models (LTM) [75, 28]. LTM uses the fundamentals of kinetic wave theory to evaluate the aggregate flow through any point at a given time. This model is based on Newell’s simplified theory of kinetic waves [49]. Particularly, it considers a trian- gular fundamental diagram, where the flow monotonically increases with density up to a point called critical density, after that flow starts to decrease as density increases. Store-and-forward models, e.g., see [54], approximate the traffic dynam- ics by replacing a time-varying outflow due to alternating green and red phase on a link with an equivalent average outflow. Such models have been used for optimal green splits control, e.g., see [14, 4]. Continuous-time versions of these models have also been used for green time control, e.g., in [60, 50]. In part of this thesis, we use this model for theoretical analysis of a specific type of adaptive signal controller. However, this model does not capture the effect of offsets and cycle lengths. These limitations are overcome by discrete-event models, which have been utilized for optimal control synthesis for isolated signalized intersections in some cases, e.g., see [69, 24]. 17 Recently, [47] proposed and analyzed a model, which captures offset and cycle times in the same spirit as discrete-event models. Particularly, in [47], a fixed-time controllerisconsidered, whereeverylinkisendowedwithagivencapacityfunction, that specifies the maximum possible outflow from a link as a function of time. In order to maintain non-negativity of queue lengths, the outflow from every link is constrained to be equal to the capacity function if the queue length is positive, and equal to the minimum of cumulative inflow and capacity function otherwise. The inter-link travel times are modeled by constant (i.e., independent of queue length) parameters, e.g., δ ji for travel time from link j to i. In spite of the resulting dynamics being discontinuous, it is shown in [47] that the traffic dynamics admits a unique queue length trajectory if the parameters δ ji are bounded away from zero. The proof relies on showing desired properties on contiguous time intervals of length equal to the minimum among all δ ji . The proposed model in [47] does not capture dependency of inter-link travel times on queue lengths, and it also does not incorporate spillbacks. In regards to performance evaluation, queue length and delay at intersections are among the quantities of interest used for quantifying the congestion level in signalized arterial networks. In most of the existing works, the queue length or delay is analyzed at the steady state [44, 25, 47]. However, to the best of our knowledge, a procedure to calculate the steady state queue length directly without having to go through the transient states is missing in the literature. In fact, this has been noted as an important “outstanding open problem” in [47], due to its usefulness in directly quantifying network performance metrics under a given fixed- time control. In this dissertation, a novel solution to this problem is presented. 18 2.2 Traffic Signal Control Traffic signal control has been studied in several previous works. An overview of traffic signal control problem and practices is provided in [54, 53, 63]. The key elements of traffic signal plan to be controlled are cycle length, offset, and green splits. [73] was one of the first studies on signal timing optimization, where the optimal parameters of traffic signal are found in order to minimize the delay at an isolated intersection. Classical strategies consist of using extensive surveys to obtain network parameters, which are then used to design traffic signal timings, which are either fixed, e.g., see [59], or constantly re-tuned as in SCOOT, e.g., see [9]. Technological advancements have made it possible to control the traffic lights in a centralized manner, e.g., by coordinating multiple traffic signals along a corridor. One example of such centralized approach is designing the cycle lengths and off- sets of traffic signals such that traffic can move smoothly along a main corridor. In fact, majority of previous works in traffic signal control are focused on centralized techniques. Centralizedadaptivesignalcontrollersforoversaturatednetworkshave been proposed in [39, 5, 64]. Genetic algorithm has been used in multiple works to solve large scale optimization problems for traffic signal control, e.g., see [5, 10, 65]. [56] proposed an ant colony optimization for solving traffic signal coordination and compared it to simple genetic algorithms for a network under over-saturation con- dition. In [46], the network-level traffic signal timing optimization problem was formulatedasamixed-integernon-linearprogram(MINLP)andsolutiontechnique was proposed. Recently, [42] proposed a centralized algorithm for optimizing green splits for a network with unknown demand and capacity, where the proposed iter- ative signal control is guaranteed to converge to an optimal solution. Also, offset optimization has been the focus of many of proposed centralized controllers in the 19 literature. One of the earliest studies is [18], in which a mixed-integer program- ming approach is used for one-way or two-way corridors. Other offline techniques include [57] for a corridor setup, where the backward progression technique is used to avoid long queues on side streets and [26] for a two-way corridor using archived high-resolution traffic signal data. Recently, in [12, 6], a sinusoidal approxima- tion of queue length trajectories is used to facilitate solution to offset optimization problem by semi-definite programming. Ontheotherhand, advancementsinsensingtechnologieshavemadeloopdetec- tors and traffic cameras available at arterial intersections, motivating employment of online optimization techniques. Loop detectors are the most common type of detection used in signalized arterial intersections in United States. The two main categories of loop detector are presence detectors and passage detectors. Volume, occupancy and average speed are the main quantities measured by loop detec- tors. One limitation of using loop detectors is that they are point-detectors and do not provide vehicle-level measurements. Lately, video based detection have attracted significant attention. While video detection has been shown to be capa- ble of collecting information about vehicle counts, lane occupancy, vehicle presence and speed, its accuracy at nights or during the presence of long queues has been questioned [8]. Several techniques have been proposed on retuning the traffic signal timing using the measurements from traffic sensors. On the commercial side, SCOOT[27], SCAT[62] and OPAC[19] are the well-known ones which have the ability to opti- mize green splits, cycle length and offset simultaneously. SCOOT uses the loop detectors to find the cyclic flow profiles (CFP) and estimate the queue, e.g., see [35, 15]. Based on the predicted CFP, the optimizer decides whether the offset of the intersection should remain the same, or change by 4 seconds increments. In 20 SCATS [23], the optimization strategy is based on degree of saturation of incoming links at the intersection. It optimizes the cycle time, green splits and offset on a cycle by cycle basis. In OPAC[19], the controller at each intersection decides on its signal plan to optimize a performance measure of its upstream queues, without taking into account the effect of timing plans on the downstream queues. Even though these techniques are practical and have been shown to improve the per- formance of traffic signals in simulation environments, no analytical guarantees is provided on their throughput optimality and stability properties. Decentralized traffic signal controllers have attracted significant attention recently [33, 43]. While centralized techniques are suitable for network-level opti- mization, they can become intractable and computationally expensive as the net- work size grows or duration of study increases. Decentralized techniques, on the other hand, are mostly scalable and can reduce the computational complexity of the problem. OneofthemostprominentamongdecentralizedalgorithmsistheMax-Pressure (MP) controller, e.g., see [70, 31]. This controller is inspired by back-pressure algorithm [67, 48], a well-known routing algorithm in communications networks. Adaption of this controller to transportation setup has been proposed in [70, 74]. Theoretical guarantees on its throughput optimality and network stability under this controller have been provided in [70, 33]. Max-Pressure controller determines the green time allocations at each intersection based on local queue length mea- surements at upstream and downstream links. While these controllers and other variants proposed recently are fully decentralized, they require explicit information or online estimation of turning rates and saturation flow capacities for their imple- mentation. In [22], a variant of Max-Pressure is proposed where the controller only requires aggregate queue length estimations and does not have a knowledge 21 of turning ratios at intersections. It is shown that the controller has good sta- bility properties, however, is no longer maximally stabilizing. Recently, studies have been done on estimating the turning rations using the information from loop detectors [45, 11]. These techniques can be utilized to provide accurate routing informationfor Max-Pressure controller. Recently, distributedadaptivesignal con- trol algorithms that rely on the estimation of turn ratios at short time scales have also been proposed, e.g., see [34]. Another widely known feedback controller in communication networks liter- ature is Proportionally Fair (PF) controller [29]. The stability analysis of PF controllers in the context of communication networks has been studied in previous works [40, 71]. The adaptive controller proposed and studied in this dissertation is inspired by PF controllers. Application of PF controller in transportation setup was first introduced in previous works [60, 61]. These controllers have attracted great interest due to their unique features. PF controllers are completely decen- tralized, in that the green allocation of the traffic signals at an intersection are chosen as a function of the queue length of the incoming links to that intersec- tion only and do not require any knowledge of network parameters such as arrival rates and saturation flow capacities. Unlike Max-Pressure, PF controllers do not need information about routing and turning ratios. In spite of such a minimalist nature, under stylized settings, PF controllers are known to be throughput opti- mal just as Max-Pressure controllers. The adapting PF controller to traffic signal control has been also studied in [32]. Along the lines of results presented in this dissertation, recently [51], studied PF controller in arterial urban networks and presented theoretical results on throughput optimality of this controller under for average traffic flow models. While proposed controller computes the cycle lengths and green splits simultaneously in every cycle, the analytical result remains limited 22 to the disjoint phase setup, where a link is activated in one and only one phase. To the best of our knowledge, the existing theoretical results on performance anal- ysis of PF controllers are derived for an average traffic flow models. While such models simplify the analysis, it masks the dependence on offsets and cycle length. On the other hand, there has been recent interest in rigorous analysis of the more accurate ON/OFF models, which capture inter-cycle queue length behavior, e.g., analysis of fixed-time control under such model was presented in [47]. These mod- els are more realistic and more compatible with practical hardware constrains on traffic signal systems, hence they are more complex and the theoretical analysis of the controllers becomes more challenging. In this dissertation, we adapt the PF controller to such ON/OFF models and provide theoretical guarantees on its performance analysis. Note that, most of available decentralized techniques require queue length mea- surements as the input to the algorithm. While many intersections are equipped with loop detectors or video detection tools, none of these technologies are capable of directly measuring the queue lengths or number of stationary vehicles at the stop line [55]. Vehicle counts from both advance and stop line detectors can be used for rough queue length estimation by considering the arrival and departure flow. However, accurate estimation is not possible when the links are congested and vehicles are stationary beyond the location of advance detector. Recently, [38] proposed a method for real-time queue length estimation for the case that queue lengthsarelongerthanthedistancebetweenthedetectorandintersectionstopline. On the other hand, while the video detection or cell-phone data can be utilized to improve the queue length estimation, these technologies are not widespread yet. Installation and maintenance cost of video detection systems is relatively high. Moreover, video detectors can be easily blocked by environmental objects and are 23 not capable of detecting long queues due to their small range. The advantage of the output feedback PF controller presented in this dissertation is that it does not require direct knowledge of queue length. Our results suggest that, at least for an isolated intersection, queue length estimation is not necessarily required to design a throughput optimal controller. The input to our proposed controller can be readily measured using presence information collected by stop line detectors and the intermediate step of estimating the queue length is completely circumvented. Our microsimulation studies suggest that the controller has a good performance in terms of throughput for general network setup. To the best of our knowledge, such minimal decentralized controllers with provable stability properties is missing in traffic control literature. 24 Chapter 3 Performance Analysis of Fixed-Time Controller In this chapter, we provide a computational framework to directly simulate queue length dynamics under fixed-time or adaptive control. Particularly, under our model, one obtains queue length trajectories as solution to delay differential equations, where link outflows are obtained from the provably unique solution to a linear program. For given queue lengths, this linear program solves for maximum cumulative outflow from all links subject to constraints imposed by the link capac- ity functions, and subject to maintaining non-negativity of queue lengths. Exis- tence and uniqueness of the solution to delay differential equations is established for piecewise constant external inflow and capacity functions. The existence and uniqueness result also extends to adaptive control policies, as long as the resulting capacity functions remain piecewise constant. This would happen, e.g., if traffic signal control parameters (green time, cycle length, and offsets) at every intersec- tion are updated once per cycle. The piecewise constant assumption is practically justified because a common model for a capacity function is that it is equal to the saturated capacity during the green phase and zero otherwise, and external inflows can be modeled as a sequence of rectangular pulses representing arriving vehicle platoons. The key idea in the proof is that, under constant inflow and capacity, the set of links with zero queue lengths is monotonically non-decreasing, which implies overall finite discontinuities over any given time interval under the 25 piecewise constant assumption. If, additionally, the external inflow and capacity functions are periodic and satisfy a stability condition, then there exists a globally attractive periodic orbit. This result and its proof follows the same structure as in [47], but is adapted to the proposed modeling framework. One consequence of this adaptation is that we work with the ` 1 norm, instead of the sup norm in [47], for continuity arguments in our proofs. Next, we propose a novel procedure to explicitly calculate the globally attrac- tive periodic orbit. The proposed procedure is a useful performance evaluation tool for a given fixed-time signal timing plan. In fact, this was noted as an important “outstanding open problem" in [47], due to its usefulness in directly quantifying relevant performance metrics for a given fixed-time control policy. We provide an iterative procedure to compute the steady state periodic orbit. A periodic tra- jectory is iteratively updated for every link based on updates to a specific time instant when its queue length transitions from being zero to being positive. This update for a given link is based on the periodic trajectories computed in the pre- vious iteration for upstream links. The resulting iterates are shown to converge uniformly monotonically to the desired periodic orbit. Theoutlineofthischapterisasfollows. Section3.1containstheproposeddelay differential equation framework to simulate queue length dynamics. Section 3.5 providesthe(non-iterative)frameworktocomputetheperiodicorbitforanisolated link. This forms the basis for an iterative procedure to compute periodic orbits for a network in Section 3.6 where we also establish uniform monotonic convergence of theiteratestothedesiredperiodicorbit. Section5.3presentsillustrativesimulation results. The proofs for most of the technical results are collected in the Appendix. 26 We introduce the key concepts and notations to be used throughout this chap- ter. R,R ≥0 ,R >0 ,R ≤0 andR <0 will stand for real, non-negative real, strictly pos- itive real, non-positive real, and strictly negative real, respectively, andN denotes the set of natural numbers. For x∈R, we let [x] + = max{x, 0} denote the non- negative part of x. A function f : X ( R→ R n is called piece-wise constant if it has only finitely many pieces, i.e., X can be partitioned into a finite number of contiguous right-open sets over each of which f is constant. The road network topology is described by a directed multi-graphG = (V,E) with no self-loops, whereV is the set of intersections andE is the set of directed links. 3.1 Traffic Flow Dynamics The network state at time t is described by the vector of queue lengths, x(t)∈ R E + corresponding to the number of stationary vehicles, and the history of relevant past departures from the links, β(t), which quantifies the number of vehicles traveling in between links. The quantity β(t) shall be described formally soon. Let c i : R ≥0 → R ≥0 and λ i : R ≥0 → R ≥0 be saturated flow capacity and external inflow functions, respectively, for link i∈E. Let the matrix R ∈ R E×E ≥0 denote the routing of flow, e.g., R ji denotes the fraction of flow departing link j that gets routed to link i. Naturally R ji = 0 if link i is not immediately downstream to link j. We shall assume that R is sub-stochastic, i.e., all of its entries are non-negative, all the row sums are upper bounded by 1, and there is at least one row whose row sum is strictly less than one. We further assume the following on the connectivity ofG. Assumption 1. (i)G is weakly connected, i.e., for every i,j∈E, there exists a directed path inE from i to j, or from j to i. 27 (ii) For every i∈E, either the sum of entries of the i-th row in R is strictly less than one, or there exists a directed path from i to at least one link j such that the entries of the j-th row in R is strictly less than one. Remark 1. The weak connectivity aspect of Assumption 1 is without loss of gen- erality: ifG is not weakly connected, then our analysis applies to each connected component ofG, as long as each of these connected components satisfies (ii) in Assumption 1. Indeed, part (ii) of Assumption 1 implies that, for every vehicle arriving into the network, either it is possible for the vehicle to depart directly from the arrival link, or there exists a directed path to an another link from which the vehicle can depart the network. Formally, part (ii) of Assumption 1 implies that the spectral radius of R, and hence also of R T , is strictly less than one. In particular, this guarantees that I−R T is invertible. We now describe a model for traffic flow dynamics. The queue length dynamics is described by a standard mass balance equation: for t≥ 0, ˙ x i (t) =λ i (t) + X j∈E R ji z j (t−δ ji )−z i (x(t),t), i∈E (3.1) where z i (x(t),t) denotes the outflow from link i at time t. In (3.2a), δ ji ≥ 0 is the model parameter corresponding to travel time from link j to i 1 , and z i (t−δ ji ) is a concise notation for z i (x(t−δ ji ),t−δ ji ). Remark 2. We emphasize that, since the dependency of inter-link travel times on queue length is not explicitly modeled, strictly speaking, δ ji are to be interpreted as 1 Specifically, this is the time to travel from the head of queue on link j to the tail of queue on link i. This is to be distinguished from total travel time, which is the sum of inter-link travel times and the time spent waiting in the queues en route. 28 model parameters inspired by inter-link travel times, rather than being actual inter- link travel times. We shall assume this distinction to be implicit in this paper, and shall not point it out explicitly at all times. In accordance with δ ji being simply model parameter, we allow it to be ignored, i.e., set to be zero, on some or all links. An advantage afforded by the flexibility to ignore δ ji in our model is described in Remark 3. It would be convenient to rewrite the queue length dynamics as: for t≥ 0, ˙ x i (t) = ˜ λ i (t) + X j∈E i R ji z j (x(t),t)−z i (x(t),t), i∈E (3.2a) where ˜ λ i (t) :=λ i (t) + X j∈E\E i R ji z j (t−δ ji ), i∈E (3.3) is the net inflow to linki due to external arrivals and arrivals due to vehicles from upstream which were traveling until t, and E i :={j∈E|R ji > 0 & δ ji = 0} (3.4) is the set of links upstream ofi for whichδ ji = 0 is a resonable approximation. Let ¯ δ j := max{δ ji :i∈E, R ji > 0} be the maximum among all travel time parameters from link j to its downstream links. We let β(t) :={z j (s) :s∈ [t− ¯ δ j ,t)} j∈E (3.2b) 29 be the history of relevant past departures 2 , and kβ(t)k 1 := P i∈E P j∈E R ji R t t−δ ji z j (s)ds be the number of vehicles traveling in between links at time t. 3 Finally, let x(t) := {x i (t)} i∈E , z(x(t),t) ≡ z(t) := {z i (x(t),t)} i∈E , λ(t) := {λ i (t)} i∈E , and c(t) :={c i (t)} i∈E denote the collection of corresponding quantities over all links. (3.2a)-(3.2b) collectively describe the evolution of (x(t),β(t)) start- ing from initial condition (x(0),β(0)). We propose link outflowsz(x(t),t) fort≥ 0 be obtained as solution to the following linear program: maximize z∈R E 1 T z subject to z i ≤c i (t), i∈E z i ≤ ˜ λ i (t) + X j∈E i R ji z j , if i∈I(x) (3.2c) where I(x) :={i∈E|x i = 0} is the set of links with no stationary vehicles. (3.2c) computes the maximum cumulative outflow in the network, subject to twoconstraints. Thefirstoneimposeslink-wisecapacityconstraint, andthesecond one imposes the constraint that, for a link with zero queue length, its outflow is no greater than its inflow. The second constraint is to ensure non-negativity of queue lengths. The well-posedness of our proposed method for computing link outflows, i.e., uniqueness of solution to (3.2c) is established in the next section. Thereafter, we establish existence and uniqueness of the solution to our traffic flow model in (3.2a)-(3.2b)-(3.2c), which we shall collectively refer to as (3.2). 2 If ¯ δ j = 0 for some linkj, then the departure history from such a link is not included in β(t). 3 It is easy to verify that this definition ofkβ(t)k 1 satisfies all the properties of a norm. 30 In order to present our results on existence and uniqueness concisely, we introduce a couple of more notations. Let ¯ δ := max (j,i)∈E×E:R ji >0 δ ji and δ := min (j,i)∈E×E:R ji >0 δ ji be the, respectively, maximum and minimum among all inter- link travel time parameters. Remark 3. In Remark 2, we emphasize that our model allows the inter-link travel time parameters δ ji to be zero. Since a solution to (3.2) is obtained by concatenat- ing its solutions over intervals of length δ, the complexity for obtaining solution increases with decreasing δ. In our model, we can mitigate potentially huge com- plexity due to small δ by ignoring δ ji with small values, so that the complexity is then related to the minimum of the remainingδ ji . Since the model in [47] does not allow δ ji = 0, it does not offer such a flexibility. 3.2 Existence of Solution to (3.2a) The proof of the next result is provided in the Appendix. Proposition 1. Given (x(t),β(t)), λ(t), and c(t), (3.2c) has a unique solution. Moreover, the optimal solution satisfies z i (x,t) = c i (t) i∈E\I(x) min n c i (t), ˜ λ i (t) + P j∈E i R ji z j (x,t) o i∈I(x) (3.5) Proposition 1 implies thatz(x(t),t) in (3.1), or equivalently in (3.2) and (3.3), is well-defined. With regards to (3.5), indeed for i ∈ I(x), z i (x,t) = ˜ λ i (t) + P j∈E i R ji z j (x,t), except possibly at time instants when there is a change inI(x). It is rather straightforward to see that (3.2) admits a unique solution in between 31 such changes. The frequency of such changes in general depends on λ(t), c(t), and the initial condition β(0). We bound the frequency of changes, and thereby establish existence and uniqueness of the solution to (3.2) for all t≥ 0, under the following practical assumption. Assumption 2.{λ i : [0,T ]→R E ≥0 } i∈E ,{c i : [0,T ]→R E ≥0 } i∈E , and{z i : [− ¯ δ, 0]→ R E ≥0 } i∈E are all piece-wise constant. The proof of the next result is provided in Appendix. Proposition2. Letλ(t),c(t) and the initial condition (x(0),β(0)) satisfy Assump- tion 2. Then, there exists a unique solution (x(t),β(t))≥ 0 for all t≥ 0, to (3.2). Remark 4. 1. Unlike [47], existence of a unique solution to (3.2) does not require δ to be strictly greater than zero. However, this comes at the expense of piecewise constant assumption. 2. Assumption 2 is practically justified because a common model for a capacity function is such that it is equal to the saturated capacity during the green phase and zero otherwise, and external inflows, as well as past departures before t = 0 can be modeled as a sequence of rectangular pulses modeling vehicle platoons. Proposition 2 holds true also when the capacity function is state-dependent (referred to as adaptive traffic signal control), but piecewise constant. For exam- ple, let the capacity function c i (t) be equal to c max i if t∈ [θ i ,θ i +g i (0)]∪ [T + θ i ,T +θ i +g i (1)]∪..., and equal to zero otherwise, where θ i ∈ [0,T ] is the offset, and{g i (0),g i (1),...} is a sequence of green times. Such green times can be deter- mined as a function of queue lengths. One such simple proportional rule, when 32 Figure 3.1: Illustration of a capacity function. the capacity functions for all the incoming links at every intersection are mutually exclusive, is: g i (k) = kx i (k− 1 :k)k 1 P j kx j (k− 1 :k)k 1 , k = 1, 2,... The summation in the denominator is over all links incoming to the intersection to whichi is incident, andkx i (k− 1 :k)k 1 := R kT (k−1)T x i (t)dt is proportional to the average queue length during the k-th cycle on link i. 3.3 Periodic Solution It is straightforward to see that the solution to (3.2) can be equivalently describedintermsof (x(t),z(t)). Therefore, weshalluse (x(t),β(t))and (x(t),z(t)) interchangeably to refer to the solution to (3.2). We now develop a result anal- ogous to the one in [47] on the existence of a globally attractive periodic orbit (x ∗ (t),z ∗ (t)), under the following periodicity assumption. 33 Assumption 3. The external inflow functions{λ i (t)} i∈E and capacity functions {c i (t)} i∈E are all periodic with the same period T > 0. 4 Let ¯ λ i := 1 T Z T 0 λ i (t)dt, ¯ c i := 1 T Z T 0 c i (t)dt, i∈E (3.6) be the external inflow and capacity functions averaged over one period. Let ¯ c = {¯ c i : i∈E} and ¯ λ ={ ¯ λ i : i∈E} denote the collection of external inflow and capacity functions, respectively, for all links. The following stability condition will be one of the sufficient conditions for establishing periodicity of (x(t),z(t)) at steady state. Definition1 (StabilityCondition). There exists> 0 such that [I−R T ]¯ c> ¯ λ+1. Theorem 1. Let λ(t), c(t) and the initial condition (x(0),β(0)) satisfy Assump- tions 2 and 3, and the stability condition in Definition 1. Then, there exists a unique periodic state trajectory (x ∗ ,z ∗ ) with period T for (3.2), to which every trajectory converges. The structure of the proof of Theorem 1 follows closely along the lines of [47], with differences due to the combination of dynamics in (3.2a), and the fact that we allow δ = 0. We start with a few necessary technical lemmas. The proof of these technical results are provided in Appendix A. Lemma 1. Suppose x 0 ≤ x 0 0 , λ(t)≤ λ 0 (t) and c(t) = c 0 (t) for all t≥ 0, and {z(t),t∈ [− ¯ δ, 0)}≤{z 0 (t),t∈ [− ¯ δ, 0)}. If{λ(t),λ 0 (t),c(t),c 0 (t)} are all piecewise constant, then the corresponding solutions to (3.2) satisfy x(t)≤x 0 (t) and z(t)≤ z 0 (t) for all t≥ 0. Proof. Provided in Appendix A. 4 As noted in [47], requiring the period to be the same is without loss of generality. 34 Lemma 2. If λ(t), c(t), and (x(0),β(0)) satisfy Assumption 2, and if the stability condition in Definition 1 holds true, then the solution x(t) to (3.2) is bounded. Proof. Provided in Appendix A. We next state an important result on contraction and a global attractivity property of (3.2a). Proposition 3. Let the conditions in Proposition 2 hold true. If (x(t),β(t)) and (˜ x(t), ˜ β(t)) denote the trajectories starting from (x 0 ,β 0 ) and (˜ x 0 , ˜ β 0 ) respectively, then kx(t)− ˜ x(t)k 1 +kβ(t)− ˜ β(t)k 1 ≤kx 0 − ˜ x 0 k 1 +kβ 0 − ˜ β 0 k 1 (3.7) Moreover, if the stability condition in Definition 1 is satisfied, then lim t→∞ kx(t)− ˜ x(t)k 1 = 0, lim t→∞ kβ(t)− ˜ β(t)k 1 = 0 (3.8) Proof. See Appendix A. We can now finish the proof of Theorem 1 as follows. Consider the trajectory starting from (x(0),β(0)) = (0, 0). In particular, consider the sequence of following points on this trajectory: {(x(nT ),β(nT ))} ∞ n=0 . Monotonicity (Lemma 1) and boundedness (Lemma 2) implies that this sequence converges, say to (x ∗ ,β ∗ ). We now establish that the trajectory starting from such a point is periodic. This, together with global attractivity implied by (3.8), then establishes Theorem 1, i.e., every trajectory converges to the periodic trajectory starting from (x ∗ ,β ∗ ). IfF (x((n−1)T ),β((n−1)T )) = (x(nT ),β(nT ))denotestheassociatedPoincare map, then the desired periodicity is equivalent to showingF (x ∗ ,β ∗ ) = (x ∗ ,β ∗ ), i.e., (x ∗ (T ),β ∗ (T )) = (x ∗ ,β ∗ ), i.e., lim n→∞ (x((n + 1)T ),β((n + 1)T )) = (x ∗ ,β ∗ ), i.e., 35 lim n→∞ F (x(nT ),β(nT )) = (x ∗ ,β ∗ ). A sufficient condition for this is continuity of F, which follows from (3.7). 3.4 Problem Statement While one can use (3.2) to obtain the steady state (x ∗ ,z ∗ ) by direct simulations, inthischapter, ourobjectiveistodevelopanalternateframeworktoobtain (x ∗ ,z ∗ ) without having to run lengthy simulations. 3.5 Steady State Computation for an Isolated Link Let y i (t) be the cumulative inflow into link i∈E. Referring to (3.2a), this quantity is given by y i (t) := λ i (t) + P j∈E R ji z j (t−δ ji ). For an isolated link i, y i (t) = λ i (t). It is easy to see that x ∗ i (t)≡ 0 if y i (t)≤ c i (t) for all t∈ [0,T ). In order to avoid such trivialities, we assume that the set{t∈ [0,T )|y i (t)>c i (t)} has non-zero measure. The key in our approach is a procedure to easily compute x ∗ i (s) for some s∈ [0,T ). Thereafter,x ∗ i (t) for allt∈ [0,T ) can be easily obtained by simulating (3.2) over a time interval of length T. The natural candidates for such a s∈ [0,T ) are the time instants when the queue length x ∗ i transitions between zero and positive values. We now provide a detailed procedure to compute such a transition point. We implicitly assume throughout this chapter that Assumption 2 and the stability condition in Definition 1 holds true. Definition 2 (Transition Points). Let{α 1 i ,...,α L i } be the time instants in [0,T ) when x ∗ i transitions from being zero to being positive. 36 Figure 3.2 illustrates the transition points for a sample scenario. 0 T/2 T Time 0 10 20 30 40 50 60 70 80 90 (t) c(t) x(t) 1 2 B 2 W 3 W 1 W 2 B 1 Figure 3.2: Illustration of transition points, and negative/positive sets. In this case, M w = 3 and M b = 2. Subscript i is not shown for brevity. Remark 5. (i) Under the stability condition in Definition 1, L≥ 1, as also noted in [47, Theorem 2]. (ii) For a given y i (t) and c i (t), Theorem 1 implies uniqueness of the resulting (x ∗ i ,z ∗ i ), and hence of{α 1 i ,...,α L i }. (iii) As noted earlier, the knowledge ofx ∗ i (s) at any single time instants∈ [0,T ) is sufficient to determine x ∗ i (t) over the entire period [0,T ). Indeed, x ∗ (s) = 0 if s∈{α 1 i ,...,α L i }. By construction such a x ∗ i corresponds to a periodic orbit for (3.2). Once x ∗ i is computed, inspired by Proposition 1 and remarks immediately following it, let z ∗ i be given by: z ∗ i (t) = y i (t) x ∗ i (t) = 0 c i (t) x ∗ i (t)> 0 (3.9) 37 Periodicity of x ∗ i , y i (t) and c i (t) imply that z ∗ i (t) in (3.9) is periodic, i.e., (x ∗ ,z ∗ ) is a periodic orbit. The uniqueness result in Theorem 1 implies that this is indeed the desired object to be computed. The time instant s referenced in Remark 5 (iii), for whose computation we now provide a procedure, is α L i . We need the notion of negative and positive sets, defined next. Definition 3 (Negative and Positive Sets). Let{B 1 i ,...,B M b i i } be contiguous sub- sets of [0,T ) of non-zero size in which y i (t) < c i (t), and let{W 1 i ,...,W M w i i } be contiguous subsets of [0,T ) of non-zero size in which y i (t)>c i (t). Remark 6. (i) Since the set{t∈ [0,T )|y i (t)>c i (t)} is assumed to have non- zero measure, under the stability condition in Definition 1, we have M b i ≥ 1 and M w i ≥ 1. (ii) The sets{B 1 i ,...,B M b i i } and{W 1 i ,...,W M w i i } do not necessarily form a par- tition of [0,T ). Specifically, they exclude sets where y i (t) =c i (t). Illustration of negative and positive sets are included in Figure 3.2. In prepa- ration for the next result, let B k i = [b k i , ¯ b k i ], k∈{1,...,M b i } and W k i = [w k i , ¯ w k i ], k∈{1,...,M w i } be closures of B k i and W k i , respectively. Proposition 4. Consider a link i with inflow function y i (t) and capacity func- tion c i (t), both periodic with period T. Let the transition points and posi- tive/negative sets be given by Definitions 2 and 3 respectively. Then, there exists a strictly increasing q α :{1,...,L}→{1,...,M w i } such that α ` i = w qα(`) i for all `∈{1,...,L}. Proof. We drop the subscript i for brevity in notation. The strictly increasing property of q α , if it exists, is straightforward; we provide a proof for existence. 38 For a given `∈{1,...,L− 1}, we let γ ` ∈ (α ` ,α `+1 ) denote the time instant in between α ` and α `+1 when the queue length transitions from being positive to being zero. Similarly, we let γ L ∈ (α L ,T ) be the time instant in between α L and T when the queue length transitions from being positive to zero if it exists, or else we let γ L = T. We also let γ 0 ∈ (0,α 1 ) be the time instant in between 0 and α 1 when the queue length transitions from being positive to zero if it exists, or else we let γ 0 = 0. Assume, by contradiction, that there exists ` ∈ {1,...,L} such that α ` / ∈ {w 1 ,...,w M w }. Let a 1 := max n a∈{1,...,M w }|w a <α ` o if it exists, and is equal to zero otherwise. Similarly, let a 2 := max n a∈{1,...,M b }|b a ≤α ` o if it exists, and is equal to zero otherwise. Since a 1 and a 2 can not both be equal to zero, we have a 1 6= a 2 . Therefore, consider the following cases, where we use the convention that w 0 = 0 =b 0 : 1. w a 1 <b a 2 : From the definition ofa 1 , we have (i)α ` ∈ [b a 2 ,w a 1 +1 ) ifa 1 <M w , or (ii) α ` ∈ [b a 2 ,T ] otherwise. In case (i), ∃ > 0 such that α ` + < min{w a 1 +1 ,γ ` }, implying y(t)−z(t) =y(t)−c(t)≤ 0 for all t∈ [α ` ,α ` +]. Similar argument holds true for case (ii). Therefore, x(α ` +) = x(α ` ) + R α ` + α ` (y(t)−c(t))dt≤x(α ` ) = 0 which is in contradiction to x(α ` +)> 0, since α ` +∈ α ` ,γ ` . 39 2. b a 2 <w a 1 : The definitions ofa 1 anda 2 imply thatα ` ∈ (w a 1 , ¯ w a 1 ]. Therefore, ∃ > 0 such that α ` − > max{w a 1 ,γ `−1 }, which implies that y(t) > c(t) for all t∈ [α ` −,α ` ]. Therefore, x(α ` ) =x(α ` −) + R α ` α ` − (y(t)−z(t))dt = R α ` α ` − (y(t)−z(t))dt> R α ` α ` − (c(t)−z(t))dt≥ 0, which contradicts x(α ` ) = 0. This establishes the proposition. Proposition 4 narrows down our search for α L i . We now sharpen this result to the point where it readily yields α L i . In prepartion for this result, we need a few more definitions. For s 1 ,s 2 ∈ [0,T ], let C i (s 1 ,s 2 ) := Z s 2 s 1 c i (t)dt, Y i (s 1 ,s 2 ) := Z s 2 s 1 y i (t)dt LetW α i :={w r 1 i ,...,w rm i } be such that r 1 = 1, and, for j∈{2,...,m}, r j = argmin ind∈{r j−1 +1,...,M w i } ∃p∈{1,...,M b i } s.t. ¯ b p i ∈ [w ind−1 i ,w ind i ] & Y i (w r j−1 i , ¯ b p i )≤C i (w r j−1 i , ¯ b p i ) if ¯ b p i <w ind i , or Y i (w r j−1 i , ¯ b p i )<C i (w r j−1 i , ¯ b p i ) if ¯ b p i =w ind i (3.10) wherem is implicitly defined by the value ofr j where the set over which argmin is taken in (3.10) is empty. In words, (3.10) implies that, forj = 2,...,m,r j is the index of the next positive set before which there exists a negative set over which the solution to (3.2), assuming x(w r j−1 i ) = 0, hits zero. The “or" in the second line of (3.10) is to ensure that the time instant when the trajectory hits zero does not coincide with w ind , which is a candidate for α ` i for some `∈{1,...,L} (cf. Proposition 5). See Figure 3.3 for an illustration. Specifically, the figure illustrates how (3.10) determines w r 3 to be equal to w 5 , given w r 2 = w 3 . Subscript i is not shown in this figure for brevity. 40 t W B W B W . . . 4 3 w b 3 r 5 4 w w b t) , w C( t) , w Y( 2 2 r r 2 r 3 w w Figure 3.3: Illustration of the procedure in (3.10) to computeW α i . Clearly,W α i ⊆{w 1 i ,...,w M w i i }, which from Proposition 4 is known to contain {α 1 i ,...,α L i }. ThenextresultshowsthatinfactthelastLentriesofW α i correspond to{α 1 i ,...,α L i }. Proposition 5. Consider a link i with inflow function y i (t) and capacity function c i (t), both periodic with periodT, and the corresponding setW α i defined via (3.10). Then{α 1 i ,...,α L i }⊆W α i , and, in particular, α ` i =w rm+`−L i , `∈{1,...,L} (3.11) Proof. We drop subscripti for brevity in notation. Assume that there exists a`∈ {1,...,L} such thatα ` =w qα(`) / ∈{w r 1 ,...,w rm }. Let ˆ w be the largest element in {w r 1 ,...,w rm } such that ˆ w<α ` . Sincer 1 = 1 (by definition), taking into account Proposition 4, ˆ w≥ w r 1 is well-defined. Recall the definition of γ `−1 ∈ (α `−1 ,α ` ) from the proof of Proposition 4, and in particular that α ` is the w k immediately afterγ `−1 . Ifγ `−1 < ˆ w, thenα ` ≤ ˆ w, givingacontradiction. Therefore, ˆ w<γ `−1 < α ` . Itiseasytoseethatγ `−1 ∈ (b ζ , ¯ b ζ ]forsomeζ∈{1,...,M b }. Therefore,x( ˆ w)+ 41 Y ( ˆ w, ¯ b ζ )−C( ˆ w, ¯ b ζ ) =x( ˆ w) +Y ( ˆ w,γ `−1 )−C( ˆ w,γ `−1 ) +Y (γ `−1 , ¯ b ζ )−C(γ `−1 , ¯ b ζ ) = Y (γ `−1 , ¯ b ζ )−C(γ `−1 , ¯ b ζ )≤ 0. This in turn would giveY ( ˆ w, ¯ b ζ )−C( ˆ w, ¯ b ζ )≤−x( ˆ w)≤ 0. 5 However, referring to (3.10), this would imply q α (`) ∈ {r 1 ,...,r m }, giving a contradiction. This proves the first claim in the proposition. In order to prove (3.11), observe that if α ` =w r j for some `∈{1,...,L− 1}, then (3.10) implies α `+1 = w r j+1 . If we assume that q α (L) = r m 1 with m 1 < m, then (3.10) implies thatw r m 1 +1 >α L is a point where the queue length transitions frombeingzerotobeingpositive, givingacontradiction. Therefore,α L =w rm . SinceL is not known, (3.11) can not be used to compute allα ` i ,`∈{1,...,L}. However, (3.11) readily gives α L i =w rm i (3.12) from whichx ∗ (t),t∈ [0,T ], can then be computed as explained in Remark 5 (iii). In order to execute this last step, it is more convenient to use: x ∗ i (0) = h Y i (α L i ,T )−C i (α L i ,T ) i + (3.13) which is obtained by integrating (3.2) from α L i to T, and recalling (3.10) for the definition of r m . The entire procedure is summarized in Algorithm 1. Let the relationship between (x ∗ i ,z ∗ i ) and (y i ,c i ), as determined by Algorithm 1, be denoted byx ∗ i =F x (y i ,c i ) andz ∗ i =F z (y i ,c i ) respectively. These notations will be used in extending the procedure to compute steady state for the network. 5 In order to minimize technicalities, we only cover the first case separated by “or" in (3.10); when the second case holds, we would have γ `−1 ∈ (b ζ , ¯ b ζ ) giving strict inequality. 42 Algorithm 1: Computation of (x ∗ i ,z ∗ i ) for isolated link i input : T- periodic inflow function λ i (t) and periodic capacity function c i (t) 1 initialization: y i (t) =λ i (t), t∈ [0,T ]; 2 compute α L i from (3.12) and x ∗ i (0) from (3.13); 3 compute x ∗ i (t), t∈ [0,T ], by simulation of (3.2) with initial condition x ∗ i (0); compute z ∗ i (t), t∈ [0,T ], from (3.9); 3.6 Steady State Computation For a Network Algorithm 2 formally describes the steps to compute steady-state for a general network. The number of iterations in the while loop in Algorithm 2 is determined by a termination criterion. While one could explicitly specify the number of iter- ations for termination criterion, a better criterion can be formulated as follows. For i∈E, let ¯ z ∗ i := 1 T R T 0 z ∗ i (t) be the average outflow from link i at steady-state. Integrating (3.2) over [0,T ] at steady state, we get that 0 = ¯ λ =R T ¯ z ∗ − ¯ z ∗ , where we use notation from (3.6). This then gives ¯ z ∗ = (I−R T ) −1 ¯ λ (cf. Remark 1 for invertibility ofI−R T ). Therefore, considering monotonicity of the iterates z (k) of Algorithm 2 as established in Proposition 6, and letting ¯ z (k) i := 1 T R T 0 z (k) i (t)dt, a termination criterion could be max i∈E ¯ z ∗ i − ¯ z k i ≤, for a specified > 0. Algorithm 2: Computation of (x ∗ i ,z ∗ i ), i∈E input : periodic inflow functions λ i (t) and periodic capacity functions c i (t), i∈E 1 initialization: k = 1; y (1) i (t) =λ i (t), t∈ [0,T ], for all i∈E; 2 while termination criterion is not met do 3 for all i∈E: 4 computex (k) i =F x (y (k) i ,c i ) andz (k) i =F z (y (k) i ,c i ) from Algorithm 1 ; 5 compute y k+1 i (t) =λ i (t) + P j∈E R ji z (k) i (t−δ ji ), i∈E ; 6 k =k + 1; 7 end 43 Proposition 6. Consider a network with T-periodic external inflows λ(t) and T- periodic capacity functionsc(t). The link outflows computed by Algorithm 2 satisfy the following for all k: z (k+1) (t)≥z (k) (t) and x (k+1) (t)≥x (k) (t) for all t∈ [0,T ] Proof. We prove by induction. Algorithm 2 implies that, for all i ∈ E, y (2) i (t) = λ i (t) + P j R ji z (1) j (t− δ ji ) ≥ λ i (t) = y (1) i (t). Therefore, Corollary 1 (in Appendix A.6) implies that x (2) (t)≥ x (1) (t) and z (2) (t)≥ z (1) (t), and hence z (2) (t−δ)≥z (1) (t−δ) for all i∈E, t∈ [0,T ], δ≥ 0. Assume that z (k) (t−δ)≥ z (k−1) (t−δ) for all k = 2,..., ¯ k. Since y ( ¯ k+1) i (t) = λ i (t)+ P j R ji z ( ¯ k) j (t−δ ji )≥λ i (t)+ P j R ji z ( ¯ k−1) j (t−δ ji ) =y ( ¯ k) i (t), Corollary1 implies that x ( ¯ k+1) i (t)≥ x ( ¯ k) i (t) and z ( ¯ k+1) i (t−δ)≥ z ( ¯ k) i (t−δ) for all i∈E, t∈ [0,T ], δ≥ 0. z (k) i (t)≤c i (t)forallk. Anupperboundonx (k) canbeshownalongsimilarlines as Lemma 2 (in Chapter 3). Combining this with monotonicity from Proposition 6 implies that (x (k) ,z (k) ) converges to (ˆ x, ˆ z). Periodicity of (x (k) ,z (k) ) for every k implies periodicity of (ˆ x, ˆ z). It is easy to see from the construction of Algorithm 2 that, for every iteration k: ˙ x (k) i (t) =λ i (t) + P j∈E R ji z (k−1) j (t−δ ji )−z (k) i (t) for all i∈E. Therefore, for any t≥ 0: 0 =x (k) i (t+T )−x (k) i (t) = Z t+T t λ i (s) + X j∈E R ji z (k−1) j (s−δ ji )−z (k) i (s) ds, i∈E 44 where the first equality follows from periodicity ofx (k) by construction. Therefore, taking the limit as k→ +∞, we get that, for all t≥ 0: 0 = ˆ x i (t +T )− ˆ x i (t) = Z t+T t λ i (s) + X j∈E R ji ˆ z j (s−δ ji )− ˆ z i (s) ds, i∈E This implies that (ˆ x, ˆ z) is a periodic orbit for (3.2a). The uniqueness result in Theorem 1 then implies that it is indeed the object to be computed. Remark 7. Algorithm 2 naturally lends itself to a distributed implementation: during an iteration, all the links independently update their respective (x (k) i ,z (k) i ), andat theend ofthe iteration, each linktransmits itsupdatedz (k) i toits immediately downstream links. 3.7 Simulations In this section, we report simulation results in two parts. In Section 3.7.1, we illustrate consistency between steady-state computations from Algorithm 2 and the steady-state obtained from direct simulations in MATLAB (using (3.2)), on a synthetic network. In Section 3.7.2, we report comparison between steady-state computations from Algorithm 2 with the output from PTV VISSIM, a well-known microscopic traffic simulator, for a sub-network in downtown Los Angeles. 3.7.1 MATLAB Simulations The graph topology of the network used for simulations is shown in Figure 5.2. All intersections have common cycle time of T = 20. The external inflows are constant λ i (t)≡λ i , i∈E, and the values are provided in Table 3.1. The capacity functions are of the form: c i (t) = c max i if t∈ [θ i ,θ i +g i ] and zero otherwise. The 45 Figure 3.4: Graph topology of the network used in the simulations. values of c max i , θ i and g i , which can be interpreted as saturation capacity, offset and green split are given in Table 3.2. Link ID (i) 1 2 3 4 5 6 7 8 9 10 11 12 λ i 1.70 2.27 4.35 3.11 9.23 4.30 1.84 9.04 9.79 4.38 1.11 2.58 Link ID (i) 13 14 15 16 17 18 19 20 21 22 23 24 λ i 4.08 5.94 2.62 6.02 7.11 2.21 1.17 2.96 3.18 4.24 5.07 0.85 Table 3.1: External inflow values. Link ID (i) 1 2 3 4 5 6 7 8 9 10 11 12 c max i 47.81 20.34 147.74 212.15 363.33 1192.03 362.82 67.30 706.05 69.93 142.51 114.89 θ i 16.02 5.24 0.58 1.49 18.57 3.47 14.60 5.05 9.77 7.01 15.88 4.72 g i 5.47 18.22 6.42 3.56 6.15 1.77 1.27 10.96 1.78 13.57 8.14 8.92 Link ID (i) 13 14 15 16 17 18 19 20 21 22 23 24 c max i 48.06 154.75 134.76 279.76 98.60 131.25 94.35 98.10 398.25 94.87 107.44 176.15 θ i 11.56 4.02 11.57 13.65 4.74 9.45 9.17 2.09 15.94 19.69 17.20 18.05 g i 10.53 5.43 8.12 4.92 11.19 5.15 8.02 7.67 2.11 11.87 11.97 6.68 Table 3.2: Parameters of link capacity functions. 46 These values ofλ(t) andc(t) satisfy the stability condition in Definition 1. The routing matrix is chosen to be: R = 0 0.44 0.23 0 0 0 0.23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.33 0 0 0 0.57 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.34 0 0 0 0 0.56 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.19 0.5 0 0 0 0 0.21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.34 0 0 0 0 0.35 0 0 0 0.21 0 0 0 0 0 0 0 0 0 0.05 0 0 0 0.85 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.06 0 0 0 0.27 0.3 0 0 0 0.27 0 0 0 0 0 0 0 0 0.08 0.55 0 0 0 0.27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.41 0 0 0 0.26 0 0 0 0 0.23 0 0 0 0 0 0 0 0 0.38 0 0 0 0 0.52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.2 0 0 0 0.24 0.13 0 0 0 0.33 0 0 0 0 0 0 0 0 0 0 0 0 0.08 0 0 0 0 0.33 0 0 0 0.49 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.32 0 0 0 0.22 0 0 0 0 0.36 0 0 0 0 0 0 0 0 0 0 0 0 0.12 0 0 0 0.43 0.12 0 0 0 0.23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.57 0 0 0 0 0.33 0 0 0 0 0 0 0 0 0.84 0 0 0 0 0.02 0 0 0 0.04 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.15 0 0 0 0.39 0.36 0 0 0 0 0 0 0 0 0.23 0 0 0 0.29 0.03 0 0 0 0.34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.19 0 0 0 0.71 0 0 0 0 0 0 0 0 0 0.22 0 0 0 0.35 0 0 0 0 0.33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.28 0 0 0 0.32 0.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.54 0 0 0 0 0.36 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.42 0 0 0 0.48 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.19 0 0 0 0.28 0.43 0 While the entries of R are chosen arbitrarily, the sum of entries on each row is 0.9 < 1. This combined with the fact that the network shown in Figure 5.2 is weakly connected, Assumption 1 is satisfied. The inter-link travel time parameters are all approximated as zero, i.e., ¯ δ = 0. Figure 3.5 shows the root mean squared error (RMSE) between the (x ∗ ,z ∗ ) obtained by direct simulation of (3.2), with initial condition x i (0) = 10 for all i∈E, over a sufficiently long time horizon, and the output (x (k) ,z (k) ) of Algo- rithm 2 during various iterations. The RMSE between x (k) i and x ∗ i is defined as r R T t=0 (x (k) i (t)−x ∗ i (t)) 2 dt T . The RMSE between z (k) i and z ∗ i is defined similarly. The monotonically decreasing RMSE in Figure 3.5 (a) and (b) illustrates the monotonic convergence of the iterates of Algorithm 2 to the desired periodic orbit (x ∗ ,z ∗ ). 47 0 10 20 30 40 50 60 70 0 100 200 300 400 500 600 700 800 900 1000 iterations RMSE link 3 link 4 link 5 link 19 link 20 link 21 0 20 40 60 80 0 50 100 150 200 iterations RMSE link3 link4 link5 link19 link20 link21 (a) (b) Figure 3.5: Evolution of RMSE between (a) x (k) and x ∗ , and (b) z (k) and z ∗ for a few representative links. Figure 3.6 illustrates uniform monotone convergence of x (k) i to x ∗ i , as stated in Proposition 6, for a sample link. 3.7.2 Comparison with Microsimulations In this section, we report comparison between steady-state computations from Algorithm 2 with the output from PTV VISSIM for the downtown Los Angeles sub-network shown in Figure 6.8. All the intersections have common cycle time of T = 90 second. Referring to the notations in Chapter 3, the values of offsets and green times for various capacity functions were obtained from Los Angeles Department of Transportation (LADOT) signal timing sheets, and are reported in Table 3.3. The values of saturation flow capacities, denoted as c max in Table 3.3, are based on the values commonly reported in the literature, e.g., [52]: 1800 vehi- cle/hour/lane for through movement, 1600 vehicle/hour/lane for right-turn and left-turn movements, 1200 vehicle/hour/lane for lanes that are shared between through movement and left/right turn and 960 vehicle/hour/lane for permissive 48 0 T/2 T 0 100 200 300 400 time x(t) k=5 k=10 k=70 0 T/2 T 0 100 200 300 400 time x(t) Direct simulation Figure 3.6: (top) x (k) i from a few representative iterations and (bottom) x ∗ i , both for i = 17. left turns. For each link, the total saturated flow capacity is computed by adding up the capacities of all the lanes associated with it; For example, as shown in Figure 6.1, link number 44 contains two lanes: one lane supports through+right movements, and the second lane supports through+left movements. Therefore, the saturation capacity of link 44 isc max 4 = 1200+1200 = 2400 veh/hour.The resulting values for all the links are reported in Table 3.3. The external inflows λ(t)≡ λ are taken to be non-zero only on the boundary links. These values, which are estimated from loop detector data during weekday PM peak hour (between 4pm to 6pm) from May 1 to May 31, 2013, are reported in Table 6.3. For every link, the turn ratios are chosen to be proportional to the number of lanes dedicated to each movement. For example, for link 44 (cf. Figure 3.8), the ratios are 0.5, 0.25 and 0.25 for through movement, right turn and left turn, respectively. Asaresult, thesumofentriesofrowsofR associatedwithlinkswhich 49 (a) (b) Figure 3.7: The Los Angeles downtown sub-network used in the simulations: (a) graph topology (b) aerial view in PTV VISSIM. 50 Figure 3.8: Illustration of movements and lanes on link 44 at intersection number 7 in the network shown in Figure 6.8. have downstream exit links, shown in dashed arrow in Figure 6.8, is strictly less than one. An example is link 8. On the other hand, for links with no downstream exit links, e.g., link 44, the entries of the corresponding row in R add up to be equal to one. Combining this with the fact that the network shown in Figure 6.8 (a) is weakly connected, Assumption 1 is satisfied in this case. Moreover, matrix R and values in Tables 3.3 and 6.3 satisfy the stability condition in Definition 1. The link travel time parameters ¯ δ i , i∈E, are constant and taken to be equal to free-flow travel time, i.e., the link length divided by the link speed limit. These values are presented in Table 3.3. Figure 3.9 shows RMSE between the (x ∗ ,z ∗ ) obtained by direct simulation of (3.2), with initial condition x i (0) = 10 for all i∈E, over a sufficiently long time horizon, and the output (x (k) ,z (k) ) of Algorithm 2 during various iterations. The monotonically decreasing RMSE in Figure 3.9 (a) and (b) illustrates the monotonic convergence of the iterates of Algorithm 2 to the desired periodic orbit (x ∗ ,z ∗ ). Figure 3.10 illustrates uniform monotone convergence of x (k) i to x ∗ i , as stated in Proposition 6, for a sample link. 51 Link ID (i) 1 2 3 4 5 6 7 8 c max i (veh/hour) 7860 5100 4560 3960 3960 3000 4560 8500 θ i (sec) 88 77 77 73 73 55 38 50 g i (sec) 52 43 43 48 48 48 53 36 ¯ δ i (sec) 8 8 8 8 8 8 9 9 Link ID (i) 9 10 11 12 13 14 15 16 c max i (veh/hour) 6600 3000 2400 7000 4560 6600 6400 3000 θ i (sec) 1 31 31 50 16 38 65 63 g i (sec) 44 44 40 49 51 53 59 52 ¯ δ i (sec) 10 11 11 11 16 16 11 11 Link ID (i) 17 18 19 20 21 22 23 24 c max i (veh/hour) 3000 4800 3360 6600 4200 4260 3000 6300 θ i (sec) 31 64 30 16 64 87 63 84 g i (sec) 44 54 44 51 44 39 52 39 ¯ δ i (sec) 11 11 11 11 12 10 10 9 Link ID (i) 25 26 27 28 29 30 31 32 c max i (veh/hour) 2400 3000 3000 2400 2400 3000 7860 6600 θ i (sec) 76 18 18 36 36 35 88 30 g i (sec) 44 45 45 51 51 49 52 45 ¯ δ i (sec) 9 9 8 8 9 9 9 10 Link ID (i) 33 34 35 36 37 38 39 40 c max i (veh/hour) 2400 8500 4560 5200 3000 4260 7560 6060 θ i (sec) 31 13 55 11 35 87 30 76 g i (sec) 40 40 48 39 49 39 44 44 ¯ δ i (sec) 10 10 10 8 5 22 23 23 Link ID (i) 41 42 43 44 45 46 47 48 c max i (veh/hour) 6200 1500 5200 2400 3100 3000 2400 3000 θ i (sec) 1 47 47 77 67 34 27 30 g i (sec) 36 44 44 44 37 29 36 34 ¯ δ i (sec) 7 7 8 8 7 7 8 8 Table 3.3: Parameters of link capacity functions and link travel time. Link ID (i) 31 32 33 34 35 36 37 38 39 40 λ i (veh/hour) 910 1271 270 755 573 414 694 186 1323 827 Table 3.4: External inflow on links located on the boundary of the network. We further compare the queue length obtained from Algorithm 2 with micro- scopic traffic simulations in PTV VISSIM run for a 2-hour scenario starting 52 10 20 30 40 50 60 70 0 2 4 6 8 iterations RMSE link1 link6 link12 link14 link19 link24 10 20 30 40 50 60 70 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 iterations RMSE link1 link6 link12 link14 link19 link24 (a) (b) Figure 3.9: Evolution of RMSE between (a) x (k) and x ∗ , and (b) z (k) and z ∗ for a few representative links. from zero initial condition. Figure 3.11 compares the queue length from the last 20 cycles in VISSIM simulations, and steady-state computations from Algo- rithm 2, for a few representative links. Here, “queue length" for link i is equal to x ∗ i (t) + P j∈E R ji R 0 δ ji z ∗ j (t−s)ds, i.e., it corresponds to the number of vehicles on link i which are stationary as well as in transit from upstream. For the queue lengths from VISSIM, the figure plots the mean queue length (obtained from last 20 cycles), as well as one standard deviation represented by the error bars. In spite of the fact that the dynamical model in (3.2) is a coarse approximation of the microscopic traffic dynamics, e.g., (3.2) neglects lane changing and car follow- ing behavior of vehicles, does not model dependency of link travel times on queue lengths, and utilizes a simplified abstraction of capacity function in the form of a rectangular pulse, the plots in Figure 3.11 show reasonable match between the steady-state corresponding to the two models. The mismatch, especially the shift inFigure3.9(b),couldbeattributedtotheaforementionedfeaturesthatweignored in our model. 53 0 T/2 T 0 5 10 time x(t) k=10 k=30 k=70 0 T/2 T 0 5 10 time x(t) Direct simulation Figure 3.10: (top) x (k) i from a few representative iterations and (bottom) x ∗ i , both for i = 22. We plan to extend the model and its analysis to capture spillbacks and depen- dency of inter-link travel time on queue length. While the well-posedness of the dynamical model proposed in this thesis extends to a reasonable class of adaptive control policies, extensions of the steady-state analysis and computation remains to be done in future work. 54 0 10 20 30 40 50 60 70 80 90 Time (sec) 0 5 10 15 20 Queue length (veh) VISSIM simulation Algorithm 2 (a) 0 10 20 30 40 50 60 70 80 90 Time (sec) 0 5 10 15 20 Queue length (veh) VISSIM simulation Algorithm 2 (b) 0 10 20 30 40 50 60 70 80 90 Time (sec) 0 5 10 15 20 Queue length (veh) VISSIM simulation Algorithm 2 (c) Figure 3.11: Comparison of queue length obtained from Algorithm 2 and VISSIM simulations for a few representative links (a) i = 20, (b) i = 15 and (c) i = 22, in the network shown in Figure 6.8. 55 Chapter 4 Proportionally Fair Controller (PF) under Averaged Model The rest of this dissertation is dedicated to designing decentralized adaptive controllers for traffic signals. The focus is on adapting the well-known Proportion- ally Fair (PF) controller from the communication networks literature to determine green time allocations for signalized arterial network and providing theoretical results on performance analysis of this controller. Particularly, in this chapter, analysis of PF controller is provided under unin- terrupted or averaged traffic flow models. The main novelty of this study with respect to [60, 61] is the use of entropy-like Lyapunov functions in the stability analysis, asopposedtomonotonicityandl 1 -contractionargumentsemployedthere. While the results in [60, 61] we were restricted to acyclic networks, our proposed approach proves stronger results in that it allows us to deal with cyclic networks. Our use of entropy-like Lyapunov functions is inspired by results in packet- switched stochastic queuing networks, see e.g., the recent work [72]. In particular, some of the technical results in the proofs rely on adaptations of arguments devel- oped in the context of proportional fairness bandwidth allocation problems [41]. The use of these techniques in the context of traffic signal control in urban traffic networks is a novel contribution to our knowledge. The rest of this chapter is organized as follows. In Section 4.1, we describe the averaged traffic flow model for signalized arterial networks. In Section 4.2, we 56 introduce PF controller for a continuous-time averaged model. In Section 4.3 we prove that, in the special case when only single lanes can be activated at every intersection, such green light policies drive the system to a globally asymptoti- cally stable equilibrium. In Section 4.4 the proposed PF controller is simulated in MATLAB and simulation results for the case when multiple lane can be activated simultaneously are also presented. Appendix B contains a few technical lemmas used in this chapter. 4.1 Averaged Traffic Flow Model We refer to Section 3.1 for key parameters and notations. The saturation flow capacity is noted as C i in this chapter (while it was shown as c max i in chapter 3). Conservation of mass implies that P j∈E R ij ≤ 1 for all i∈E, the quantity 1− P j∈E R ij ≥ 0 representing the fraction of flow out of lanei that leaves the network directly. Throughout this chapter, we make the following assumption on the network topologyG, the routing matrix R, and the arrival vector λ. Assumption 4. The routing matrix R is adapted to the topologyG = (V,E). Moreover, for every lane i∈E, (i) there exists some k∈E such that P j∈E R kj < 1 and (R l ) ik > 0 for some l≥ 0; (ii) there exists some h∈E such that λ h > 0 and (R l ) hi > 0 for some l≥ 0. Part (i) of Assumption 4 states that from every lane i∈E it is possible to reach some lane k∈E with P j∈E R kj < 1 by a length-l path i =j 0 ,j 1 ,...,j l =k such that R j s−1 ,js > 0 for all 1≤ s≤ l. Physically, this ensures that from every lane there exists a path to an exit of the network. Mathematically, this implies 57 that the spectral radius of R (hence, of its transpose R T ) is strictly smaller than 1: in particular, this ensures that the matrix (I−R T ) is invertible with inverse nonnegative inverse (I−R T ) −1 = P k≥0 (R T ) k . On the other hand, part (ii) of Assumption 4 states that every lane i ∈ E is reachable by some lane k∈E with positive external inflow λ k > 0 by a length-l pathk =j 0 ,j 1 ,...,j l =i such thatR j s−1 ,js > 0 for all 1≤s≤l. In particular, this implies that all the entries of the vector (I−R T ) −1 λ = P m≥0 (R T ) m λ are strictly positive. In order to complete the description of the signalized arterial network, we intro- duce the notion of phases. Phases are subsets of lanes that can be given green light simultaneously. We thus identify every phase with a binary vector p∈{0, 1} E whose i-th entry p i equals 1 if lane i is activated during phase p and 0 otherwise. The set of all phases at intersection v∈V will be denoted by Ψ v ⊆{0, 1} E . The continuous-time dynamics with state vector x(t)∈R E + is considered for the network as follows: ˙ x i =λ i + X j∈E R ji z j (x)−z i (x), ∀i∈E. (4.1) In averaged traffic flow model, the outflow term is defined as z i (x) = C i h i (x), where C i is the saturation flow capacity of lane i and the term h i (x) represents the total fraction of time that lane i is given green light whenx i > 0. This can be expressed as h i (x) = X p∈Ψ v γ p (x)p i , if x i > 0, (4.2) where γ p (x) represents the fraction of time that phase p is activated. Here, γ(x) is a adaptive green light policy. The domain of γ isR E + , while its range is the simplexS of probability vectors over the set of phases Ψ. In other 58 terms, for all network states x∈R E + , γ(x) is a vector with nonnegative entries γ p (x) indexed by the phases p∈ Ψ, such that P p∈Ψ vγ p (x) = 1. Remark 8. The continuous green light policies in this section can be interpreted as the time-averaged green light duration. Often traffic signal policies are designed in a cycle-based setting, we use such models in our analysis in chapter 5 and in the simulation studies presented in chapter 6. Remark 9. Equation (4.2) characterizes the value ofh i (x) only whenx i > 0. The case when x i = 0 has to be treated specifically in order to guarantee the physically obvious requirement that the dynamical system (4.1) keeps the nonnegative orthant R E + invariant. In the special case when there are only single phases, this issue is easily dealt with for the specific green light policies considered in this report, as shown in Section 4.3. For multiphases, the issue is more delicate, as discussed more in detail in Section 4.4. The ON/OFF model we proposed in Chapter 3 handles such cases. In fact, we address this issue by solving the linear program 3.2c. This is comprehensively discussed in Section 3.1. If an equilibrium x ∗ of the dynamical system (4.1) exists with all positive entries, it must satisfy 0 =λ i + X j∈E R ji C j h j (x ∗ )−C i h i (x ∗ ), which can be compactly written as λ + (R T −I)diagCh(x ∗ ) = 0, or h(x ∗ ) =diagC −1 z, z := (I−R T ) −1 λ. (4.3) 59 This argument implies the following result. Proposition 7 (Necessary condition for stability). LetG = (V,E) be a traffic network topology, R a routing matrix adapted toG, C∈R E capacity vector, and λ∈R E + an arrival vector such that Assumption 4 is satisfied. Let Ψ⊆{0, 1} E be a nonempty set of phases,S the simplex of probability vectors over Ψ, andγ :R E + → S a green light policy. If the dynamical system (4.1) admits an equilibriumx ∗ with all positive entries, then it must hold that diagC −1 z∈ conv(Ψ), (4.4) where z = (I−R T ) −1 λ. (4.5) Proof. Ifx ∗ is an equilibrium of (4.1) whose entries are all positive, then it follows from (4.2) and (4.3) that diagC −1 z = X p∈Ψ γ p (x)p, i.e., diagC −1 z is a convex combination of phases. In this chapter, we focus on the case where diagC −1 z ∈ int(conv(Ψ)) and study green time policies that admit globally asymptotically stable equilibria. The objective is to show that the controller is maximally stabilizing. Meaning, for any feasible choice of arrival vector the queue lengths on all lanes remain bounded. We will consider sets of phases that model local constraints among the incoming lanes in each intersection v∈V. Specifically, observe that the set of lanes can be partitioned asE =∪ v∈V E v , whereE v stands for the set of lanes incoming junction v. We then have the following assumption. 60 Assumption 5. The set of phases Ψ = Q v∈V Ψ v , where Ψ v ⊆{0, 1} Ev is the local set of phases at junction v∈V. Moreover, each local of set of phases Ψ v contains the all-zero phase 0∈ Ψ v . The Assumption 5 ensures that there are no joint constraints among the green lights that can be activated simultaneously at the different intersections. 4.2 PF Controller We focus on PF policiesγ(x) that can be written as the concatenation of local policies γ (v) (x (v) ) of the following form γ (v) (x (v) )∈ argmax γ∈Sv X i∈E − v x i log( X p∈Ψv γ p p i ) +κ v logγ 0 , (4.6) where x (v) = (x i ) i∈E − v is the vector of queue lengths on the incoming lanes to intersection v∈V,S v is the simplex of probability vectors over Ψ v and κ v > 0 is the zero phase weight. The zero phase is introduced to capture the fact that under normal traffic demands, a fraction of the possible utilization is used for phase shifts and all-red phase. Moreover, it is straightforward to see that when each phase activates at most one lane, the given green time to a lane is proportional to the queue length asso- ciated with that lane. Specifically, we assume that the local set of phases at every intersection is Ψ v ={p∈{0, 1} E − v : X i∈E − v p i ≤ 1}, ∀v∈V. (4.7) In what follows, we express the controller in closed form for this specific case. 61 Lemma 3. LetG = (V,E) be a traffic network topology and Ψ = Q v∈V Ψ v a set of phases satisfying (4.7). Then, for every junction v∈V, and every strictly positive local state vector x (v) , the maximizing green light policy satisfies h (v) i (x (v) ) = x i P j∈E − v x j +κ v , ∀i∈E − v (4.8) Using the expression above, h (v) can be extended by continuity to x (v) ∈R E − v + . Proof. Let us identify the set of lanesE − v into junctionv with the integers 1,...,k, where k :=|E − v |. Then, for the set of phases given by (4.7), the maximization problem in (4.6) reduces to maximize X 1≤i≤k x i log(γ i ) +κ v log(γ 0 ) subject to X 0≤i≤k γ i = 1, and γ i ≥ 0 for 0≤i≤k. Let μ be the Lagrange multiplier associated to the equality constraint. The Lagrangian of the relaxed problem without nonnegativity constraints is then f(γ,μ) = X 1≤i≤k x i log(γ i ) + κ v log(γ 0 ) + μ X 0≤i≤k γ i − 1 . The zero gradient conditions ∂f ∂γ i = x i γ i +μ = 0, ∀1≤i≤k, ∂f ∂γ 0 = κ v γ 0 +μ = 0, ∂f ∂μ = X 0≤i≤k γ i − 1 = 0, 62 then give that γ i =− x i μ = x i P j∈E − v x j +κ v ≥ 0, ∀1≤i≤k. Since the objective function is concave, this is the maximizing green light policy. 4.3 Stability Analysis In this section we focus on the special case of phase sets that do not allow for multiphases, i.e., where every phase can prescribe green light to at most one lane incoming to a junction. In this case, the necessary condition for stability (4.4) takes the form z i ≥ 0, ∀i∈E, X i∈E − v z i C i < 1, ∀v∈V. Moreover, the green light policy can be expressed explicitly shown in 4.8. Using this explicit expression allows one to prove the following stability result. Theorem 2. LetG = (V,E) be a traffic network with capacity vectorC∈R E ,R a routing matrix, and λ∈R E + an arrival vector such that Assumption 4 is satisfied. Let Ψ = Q v∈V Ψ v . Then the dynamical system (4.1), with green light policies given by (4.8), satisfying X i∈E − v z i C i < 1, ∀v∈V, (4.9) admits a globally asymptotically stable equilibrium x ∗ , where x (v)∗ =κ v I− z i C i i∈E − v 1 T ! −1 z i C i i∈E − v 63 for all v∈V. Proof. Using the explicit expression for the maximizing green light policy given in Lemma 3, together with the stability condition (4.9) for invertiablity of (I− ( z i C i ) i∈E − v 1 T ), yields the expression for the limit queue lengths. To prove that x ∗ is globally asymptotically stable, first observe that x(t)≥ 0 for all t≥ 0, since when x i = 0 for i∈E, h i (x) = 0 and ˙ x i ≥ 0. Introduce the function V :R E + →R as V (x) = X i∈E x i log C i h i (x) z i ! + X v∈V κ v log h (v) 0 (x) h (v) 0 (x ∗ ) , (4.10) where, with a slight abuse of notation, h (v) 0 (x) = 1− X i∈E − v h i (x). We will now prove that V (x) is a Lyapunov function. • Negative drift ˙ V (x)< 0 for all x6=x ∗ : By use of Lemma 6 (see Appendix B), it holds that dV dt = X i∈E ∂V ∂x i dx i dt = X i∈E X j∈E R ji h j (x)C j − h i (x)C i + λ i log h i (x)C i z i ! . Introduce u i = log h i (x)C i z i , then the time derivative can be written as dV dt = X i∈E ∂V ∂x i dx i dt = X i∈E X j∈E R ji z j e u j −z i e u i +λ i u i , 64 or equivalently in matrix form as dV dt =u T (λ− (I−R T )(diagze u )). From here, Lemma 7 ensures that the Lyapunov function has negative drift for all u6= 0, when u = 0, it holds that h i (x) = a i C i which is the equilibrium. • V (x ∗ ) = 0 and V (x)> 0: Sinceh i (x ∗ ) = a i C i for alli∈E at equilibrium, it follows thatV (x ∗ ) = 0. Moreover, if x6=x ∗ , then V (x) = X i∈E x i log C i h i (x) z i ! + X v∈V κ v log h v 0 (x) h v 0 (x ∗ ) ! > X i∈E x i log C i h i (x ∗ ) z i ! + X v∈V κ v log h v 0 (x ∗ ) h v 0 (x ∗ ) ! = 0, where the strict inequality follows from the suboptimality of the strictly concave optimization problem forx> 0. Ifx i = 0 for a subset ˜ E⊆E, the above inequality is still strict, due to the fact that for every node v∈V it holds that X i∈E − v \ ˜ Ev h i (x ∗ ) +h v 0 (x ∗ )< X i∈E − v \ ˜ Ev h i (x) +h v 0 (x). • V (x) radially unbounded: Due to the stability condition (4.9), it is possible to choose an > 0, such that ˜ h i = a i C i + and ˜ h (v) 0 = 1− P i∈E − v a i C i −n where n =|E − v |. ˜ h (v) is a feasible but 65 not optimal solution to the maximization problem stated in (4.8). Hence, due to suboptimality it holds that V (x) > X i∈E x i log 1 + C i z i + X v∈V κ n log ˜ h (v) 0 h (v) 0 (x ∗ ) → ∞, when|x|→∞. 4.4 Simulation Results The PF control policy presented in (4.8) is implemented into simulations in MATLAB withκ = 2.5 for a network of four intersections, as shown in Figure 4.1 (a). Each intersection has 12 incoming lanes. The network parameters are as follows. Turn ratios are assumed to be 0.17 for left turning, 0.33 for through movement and 0.5 for right turning. The lane saturation flow capacities are chosen to be symmetrical throughout the network and are specified to be 1.5 for the left lane, 1.6 for the middle lane and 1.7 for the right lane. The evolution of lane queue lenghts is presented in Figure 4.1 (b). In the this simulation we use the single phase policy proposed in Section 4.3. External arrival rate λ i is considered to be 0.35 for all lanes located on external roads of the network. We assume that the arrival rate is zero for internal lanes of network. How the dynamics evolves is shown in Figure 4.1 (b). We further consider the case with multiphases, where each phase can activate more than one lanes. We introduce a set of four disjoint phases as illustrated in Figure 4.2. The simulation parameters are chosen as follows. Turning ratios are 66 500 1,000 1,500 5 10 Time Density Time (sec) Queue Length (a) (b) Figure 4.1: (a) Network of four signalized intersections used in the simulations. (b) Evolution of queue lengths under proposed controller with only single phases. 1 12 Figure 4.2: Illustration of a four phase architecture at an intersection, where the incoming lanes are numbered counterclockwise from 1 to 12. assumed to be 0.15 for left turning, 0.4 for through movement and 0.25 for right turning. External arrival rate λ i is considered to be 0.3 for all links located on external roads of the network and is 0.1 for all links located on internal roads. The lane flow capacities are symmetrical throughout the network and are specified to be 1.5 for the left lane, 1.6 for the middle lane and 1.7 for the right lane. At each intersection the set of phases, Ψ v , contains four phases p (1) ,p (2) ,p (3) and p (4) as following: 67 Time (sec) Queue Length 0 100 200 300 400 0 1 2 3 4 5 Time Density Figure 4.3: Evolution of queue lengths under PF policy for multiphase case. p (1) = 1 0 0 0 0 0 1 0 0 0 0 0 , p (2) = 0 1 1 0 0 0 0 1 1 0 0 0 , p (3) = 0 0 0 1 0 0 0 0 0 1 0 0 , p (4) = 0 0 0 0 1 1 0 0 0 0 1 1 The evolution of queue length on each lane under PF control policy is presented in Figure 4.3. In this case the simulation results show that even if each phase activates several lanes simoultaneously, the dynamical system seems to remain stable. We plan to study the stability of this case and provide formal theoretical guarantees for stability in future work. 68 Chapter 5 Output Feedback PF Controller under ON/OFF Model In chapter 4, we presented stability analysis of Proportionally Fair (PF) con- troller under an average traffic flow model. While using such models simplify the analysis, it masks the dependence on offsets and cycle length. Hence, specifying the dependence of such controller on queue lengths measurement (e.g., instanta- neous queue at the end of cycle vs. averaged queue over the last cycle) becomes challenging when implementing the controller in a real traffic signal setup. One such choice is using queue length measurement at the end of cycle, for which we present extensive simulation studies in chapter 6. Formal theoretical analysis of PF controller under ON/OFF model for such choice of queue length measure- ment turned out to be difficult and is remain to be done. However, our work in this chapter suggest that an output feedback PF control is amenable to rigor- ous theoretical analysis under such models. This is also interesting from practical point of view because an output feedback controller does not require direct queue length measurements, which makes it more compatible with practical constrains due to existing challenges in real-time queue length estimation. The proposed PF controller calculates the green time for the next cycle based on event based measurements from previous cycle. We provide theoretical guarantee for through- put optimality of the new PF controller under the more accurate ON/OFF model (presented in chapter 3). Recently, [47] used an ON/OFF model for analysis of 69 fixed-time control. However, to the best of our knowledge, stability analysis of adaptive traffic controllers under such models is missing in the literature. The proposed output feedback PF controller is minimal, in the sense that it does not require any knowledge of arrival rates, turning ratios and saturation capacities. In particular, the only measurement required to calculate the green splits for the next cycle is the total fraction of green time over which the queue length is nonzero during each cycle. This type of measurement can be done using widely available stop line detectors that detect vehicle presence at incoming links to each intersection. The rest of this chapter is organized as follows. First, we present the problem formulation and introduce the new output feedback PF controller as the solution to a convex optimization problem. Then, we provide stability analysis of PF controller for a isolated intersection with time-varying arrival rates and prove that the controller is maximally stabilizing. Lastly, we present simulation results for a general network setup that suggest the controller is stabilizing and motivates rigorous theoretical analysis of network case for our future work. 5.1 Problem Formulation Let there existT 1 > 0 andT 2 > 0 such that the external arrival ratesλ 1 (t) and λ 2 (t) satisfy the following for all t≥ 0, and for some > 0: Z t+T i t λ i (s)ds≤ ¯ λ i T i , i = 1, 2 ¯ λ 1 c max 1 + ¯ λ 2 c max 2 + = 1 (5.1) Note the distinction between T i , i = 1, 2, and T, the period of the signal controller (i.e. cycle length). LetT be the least common multiple (lcm) of T 1 ,T 2 , 70 and T, with η = T T , η 1 = T T 1 , and η 2 = T T 2 . Also, in the rest of this chapter, we drop the superscript in c max i , i = 1, 2, for brevity in notation. Remark 10. For the regular periodic case where the arrival rates and the capacity functions are all T-periodic, i.e., if T =T 1 =T 2 , then η =η 1 =η 2 = 1. Let G min i > 0 be the minimum green time for link i, which are assumed to satisfy: G min i ≤ ¯ λ i T c i , i∈{1, 2} (5.2) In particular, in conjunction with (5.1), (5.2) implies that P i G min i <T. Fork = 0, 1,..., thek-th cycle refers to the time period [kT, (k + 1)T ]. x i [k] is the queue length on thei-th link at the beginning ofk-th cycle. G i [k] is the green time on the i-th link during the k-th cycle. Consider the following PF controller. Let ˜ G i [k] be the maximum of G min i and thepartofG i [k]duringwhichx i [k]ispositive. ThePFcontroller{G 1 [k+1],G 2 [k+ 1]} is then defined to be the unique solution to the following convex optimization problem: max G≥G min ˜ G 1 [k] logG 1 + ˜ G 2 [k] logG 2 s.t. G 1 +G 2 =T (5.3) Note that, unlike the PF controller in chapter 4, the proposed PF is an output feedback controller and does not require queue length measurements. Loop detec- tors and traffic cameras do not measure the queue length directly and accurate real-time queue length estimation is challenging in traffic intersections [55]. How- ever, the value of ˜ G i [k] used here, can be readily measured by stop line detectors which are used in actuated traffic signal systems and are widely available at traffic intersections. Stop line detectors can detect lane occupancy and vehicle presence 71 2 [ ] ̃ 2 [ ] = ∑ 3 = 1 1 2 3 2 [ ] 1 2 3 ̃ 2 [ ] = ∑ 3 = 1 2 [ ] 1 2 3 ̃ 2 [ ] = ∑ 4 = 1 4 (a) (b) Figure 5.1: Extraction of ˜ G in a real traffic signal setup using stop line detectors in (a) undersaturated and (b) oversaturated condition. on incoming links of intersections. Figure 5.1, demonstrates the green time alloca- tions and the presence data from a stop line detector at a two-phase intersection, modeled in a microsimulator. As shown in this figure, the value of ˜ G 2 [k] is the part of G 2 [k] during which the corresponding detector has detected presence of vehicles. In this figure, τ i is a time span over which vehicles presence is detected continuously. We use a threshold of 1 second for τ i ’s, meaning if τ i < 1 it is not contributing to ˜ G. Figure 5.1(b), demonstrates the extraction of ˜ G when the link is oversaturated. It is easy to see that ˜ G 2 [k] approachesG 2 [k] as the link becomes fully saturated. 5.2 Stability Analysis We say that U i [k] (respectively, S i [k]) is true if link i is undersaturated during the k-the cycle (respectively, G i [k]> ¯ λ i T c i ). We shall say that S δ i [k], δ> 0, is true, if G i [k]≥T ¯ λ i c i +δ . 72 For any k∈{0, 1,...}, the following trivial upper bound holds true for change in queue length: x i [k +η]−x i [k]≤ ¯ λ i ηT, i = 1, 2 (5.4) The following set of basic results will prove to be useful for the main result. Lemma 4. The following are true for every λ 1 and λ 2 satisfying (5.1), and for every k = 1, 2,...: (a)¬U i [k] =⇒ G i [k + 1]≥G i [k], i = 1, 2 (b) U 1 [k], G 1 [k]≥G 1 [k +η], and S /2 1 [κ] for all κ∈ [k,k +η] =⇒ U 1 [k + ] and ˜ G 1 [k + ]≤ ¯ λ 1 T c 1 for some k + ∈ [k + 1,k +η] (c) U 2 [k] and S /2 2 [κ] for all κ∈ [k + 1,k +η] =⇒ U 2 [k + ] and ˜ G 2 [k + ]≤ ¯ λ 2 T c 2 for some k + ∈ [k + 1,k +η] Proof of Lemma 4 is presented in Appendix C. The next result establishes that, starting from any initial condition, every link becomes undersaturated in a finite number of cycles. Proposition 8. Let the necessary condition in (5.1) hold. If¬S /2 i [0], then U i [k] for some k≤K :=K 0 +K 1 +K 2 , where K 0 := 2x −i [0] ηc −i T , K 1 := log G min i T ( ¯ λ i /c i +/2) ! / log(1−/2) , K 2 := x i [0]/η + ¯ λ i T (K 0 +K 1 ) c i T/2 (5.5) 73 Proof. i = 1: Assume by contradiction that¬U 1 [k] for all k≤ηK. We first show that S /2 1 [ ˜ kη] for some ˜ k≤K 0 +K 1 (5.6) (5.4) implies that x 1 [ ˜ kη]≤x 1 [0] + ¯ λ 1 T ˜ kη 1 ≤x 1 [0] + ¯ λ 1 Tη(K 0 +K 1 ) (5.7) Moreover, Lemma 4(a) implies that S /2 1 [k] for all k∈ [η ˜ k,η( ˜ k +K 2 )]. Combining this with our assumption that¬U 1 [k] for all k∈ [η ˜ k,η( ˜ k +K 2 )] gives x 1 [η( ˜ k +K 2 )]≤x 1 [η ˜ k] +ηK 2 ¯ λ 1 T− ¯ λ 1 T−c 1 T/2 ≤x 1 [0] + ¯ λ 1 Tη(K 0 +K 1 )−c 1 T 2 ηK 2 ≤ 0 where the last inequality follows from the definition of K 2 in (5.5). Therefore, we have a contradiction to¬U 1 [k] for all k≤ηK. All that remains is to prove (5.6), for which it is sufficient to show¬S /2 2 [ ˜ kη] for some ˜ k≤K 0 +K 1 . Assume by contradiction that S /2 2 [k], and hence¬S /2 1 [k], for allk≤η(K 0 +K 1 ). Therefore,U 2 [ ˆ k] for some ˆ k≤ηK 0 . Following Lemma 4(c), let 74 k + be the maximum value in [ ˆ k + 1, ˆ k +η] such thatU 2 [k + ] and ˜ G 2 [k + ]≤ ¯ λ 2 T/c 2 . Therefore, under the PF controller: G 1 [ ˆ k +η + 1]≥G 1 [k + + 1] = max ( G min 1 , ˜ G 1 [k + ] ˜ G 1 [k + ] + ˜ G 2 [k + ] T ) ≥ max{G min 1 ,G 1 [k + ]} G 1 [k + ] + ¯ λ 2 T/c 2 T ≥ max{G min 1 ,G 1 [k + ]} 1−/2 ≥ max n G min 1 ,G 1 [ ˆ k + 1] o 1−/2 where the second inequality follows from¬S /2 1 [k + ] and (5.1). Repeating this procedure, and noting that G 1 [k] is non-decreasing in k, we get G 1 [ ˆ k + 1 +ηK 1 ]≥ max n G min 1 ,G 1 [ ˆ k + 1] o (1−/2) K 1 ≥ G min 1 (1−/2) K 1 ≥T ¯ λ 1 c 1 + 2 ! wherethelastinequalityfollowsfromthedefinitionofK 1 in (5.5). Thiscontradicts our assumption that¬S /2 1 [η(K 0 +K 1 )]. i = 2: Assume by contradiction that¬U 2 [k] for all k≤ηK. We first show that S /2 2 [ ˜ kη] for some ˜ k≤K 0 +K 1 (5.8) (5.4) implies that x 2 [ ˜ kη]≤x 2 [0] + ¯ λ 2 T ˜ kη 2 ≤x 2 [0] + ¯ λ 2 Tη(K 0 +K 1 ) (5.9) 75 Moreover, Lemma 4(a) implies that S /2 2 [k] for all k∈ [η ˜ k,η( ˜ k +K 2 )]. Combining this with our assumption that¬U 2 [k] for all k∈ [η ˜ k,η( ˜ k +K 2 )] gives x 2 [η( ˜ k +K 2 )]≤x 2 [η ˜ k] +ηK 2 ¯ λ 2 T− ¯ λ 2 T−c 2 T/2 ≤x 2 [0] + ¯ λ 2 Tη(K 0 +K 1 )−c 2 T 2 ηK 2 ≤ 0 where the last inequality follows from the definition of K 2 in (5.5). Therefore, we have a contradiction to¬U 2 [k] for all k≤ηK. All that remains is to prove (5.8), for which it is sufficient to show¬S /2 1 [ ˜ kη] for some ˜ k≤K 0 +K 1 . Assume by contradiction that S /2 1 [k], and hence¬S /2 2 [k], for allk≤η(K 0 +K 1 ). Therefore,U 1 [ ˆ k] for some ˆ k≤ηK 0 . Following Lemma 4(b) (recallG 1 [ ˆ k]≥...≥G 1 [ ˆ k +η]), letk + be the maximum value in [ ˆ k +1, ˆ k +η] such that U 1 [k + ] and ˜ G 1 [k + ]≤ ¯ λ 1 T/c 1 . Therefore, under the PF controller: G 2 [ ˆ k +η + 1]≥G 2 [k + + 1] = max ( G min 2 , ˜ G 2 [k + ] ˜ G 1 [k + ] + ˜ G 2 [k + ] T ) ≥ max{G min 2 ,G 2 [k + ]} ¯ λ 1 T/c 1 +G 2 [k + ] T ≥ max{G min 2 ,G 2 [k + ]} 1−/2 ≥ max n G min 2 ,G 2 [ ˆ k + 1] o 1−/2 where the second inequality follows from¬S /2 2 [k + ] and (5.1). Repeating this procedure, and noting that G 2 [k] is non-decreasing in k, we get G 2 [ ˆ k + 1 +ηK 1 ]≥ max n G min 2 ,G 2 [ ˆ k + 1] o (1−/2) K 1 ≥ G min 2 (1−/2) K 1 ≥T ¯ λ 2 c 2 + 2 ! wherethelastinequalityfollowsfromthedefinitionofK 1 in (5.5). Thiscontradicts our assumption that¬S /2 2 [η(K 0 +K 1 )]. 76 The above proposition straightforwardly leads to the following main result. Theorem 3. Let the necessary condition in (5.1) hold. Then the queue lengths are upper bounded as: x i [k]≤ max{x i [0],η ¯ λ i T} +η(K 0 +K 1 ) ¯ λ i T, ∀k = 0, 1,..., i = 1, 2 (5.10) where K 0 and K 1 are as defined in (5.5). Proof. The proof of Proposition (8) implies that, before linki becomes undersatu- rated for the first time, its queue length is non-decreasing for at most η(K 0 +K 1 ) cycles. Since the increment in queue length over η consecutive cycles is at most ¯ λ i ηT (cf. (5.4)), (5.10) is then satisfied before i becomes undersaturated for the first time. The queue length at the end ofη cycles after it becomes undersaturated is again upper bounded byη ¯ λ i T. One can now apply the previous logic with initial conditionx i [0]≤η ¯ λ i T to verify that (5.10) holds true in between the cycles when i becomes undersaturated. In particular, Theorem 3 implies that the output feedback PF controller in (5.3) is maximally stabilizing. 77 5.3 Simulation Results In thissection, we providenumericalsimulationsto evaluate theperformance of proposed output feedback PF controller on a grid network. We present numerical examples which motivates future theoretical analysis for a general network setup. The output feedback PF controller is implemented in MATLAB using the traffic model in (3.1)-(3.2c). The graph topology of the network used for simulations is shown in Figure 5.2 (a). The network has 4 nodes and 8 links. All intersections have common cycle length of T = 90 seconds. The external inflows are constant λ i (t)≡ λ i , i∈E, and ranging from 11.59 to 47.88 veh/sec. The capacity functions are of the form: c i (t) =c max i if link i is given green and zero otherwise. The values of c max i , which can be interpreted as saturation flow capacity, are given in Table 5.1. These values of λ(t) and c(t) satisfy the stability condition. However, the values of λ i are chosen to be close to the boundary of stability region. While the entries of R are chosen arbitrarily, the sum of entries on each row is 0.9 < 1, meaning 0.1 of vehicles coming to each node leave the network. The entries of routing matrix R are chosen as following: R = 0 0.59 0.31 0 0 0 0 0 0.44 0 0 0 0 0 0.46 0 0 0 0 0.85 0 0.05 0 0 0 0.76 0.14 0 0 0 0 0.21 0 0 0 0.43 0 0.47 0 0 0 0 0 0 0.28 0 0 0.62 0 0 0 0 0.41 0 0 0.49 0.13 0 0 0 0 0 0.77 0 We run the simulations for 40 cycles, where the simulation starts with a set of arbitrary green time values and non-zero initial queue lengths ranging from 170 to 78 Link ID (i) 1 2 3 4 5 6 7 8 c max i 2994.80 1199.20 643.54 438.83 584.12 246.88 5095.10 913.23 Table 5.1: Saturation flow capacities (veh/sec). 200. The proposed controller updates the values of green time at the end of every cycle. As shown in Figure 5.2 (b), the sum of all queue lengths in the network stays bounded and the network is stabilized. Note that the controller is fully decentralized and has no knowledge of external inflows, saturation flow capacities and turn ratios, yet it is stabilizing. This motivates rigorous analysis of general network setup for our future work. In section 6.3, we study the performance of proposed output feedback PF on a sub-network in downtown Los Angeles. We evaluate the throughput optimality of the controller through extensive microsimulations, in a close to real world scenario. 79 2 1 5 6 8 7 3 4 (a) 0 500 1000 1500 2000 2500 3000 3500 Time 0 1 2 3 4 5 6 7 8 9 Sum of Link-wise Queue Lengths 10 4 (b) Figure 5.2: (a) Graph topology of the network used in the simulations. (b) Sum of all queue lengths vs. time. 80 Chapter 6 Evaluation of Controllers in a Microscopic Traffic Simulator In this chapter, we evaluate the performance of different variants of Propor- tionally Fair (PF) controller, studied in previous chapters, in a microscopic traffic simulation software. We start by presenting a cycle-based version of PF policy that is readily implementable in real traffic network setup. We, also, study another well-known decentralized adaptive controller, Max-Pressure (MP), and compare its performance with that of PF controller under different scenarios. We formu- late a modified version of MP where the controller only requires aggregate queue lengthofeachlinkinopposetotheexistingvariantsofMPwhichrequireindividual queue length of each movement and is less practical on realistic traffic signal setup. Lastly, performance of our proposed output feedback PF controller is studied for a downtown Los Angeles sub-network. Our simulation studies are developed in PTV VISSIM, a well-known micro- scopic traffic simulator, on different sub-networks in downtown Los Angeles. We use offline data collected from loop detector sensors installed on major surface arterials in downtown Los Angeles, and relevant information from signal timing charts procured from LADOT for these intersections to calibrate our simulations. The controllers are implemented using a combination of C++, MATLAB and cvx, and they interface with VISSIM through its COM interface. 81 6.1 Model and Signal Controllers In this section, we illustrate some aspects of our model using a sub-network from downtown Los Angeles, which we also use in one of our simulation studies. This sub-network is shown in Figure 6.2. Weremarkthatinthischapter,ourpurposeistorepresenttrafficdynamicsonly to provide context for traffic signal control design. In particular, the controllers presented in this chapter will be evaluated in a microscopic traffic simulator, and notbysimulatingthedynamicsthatwepresenthere. Thecyclelengthandoffsetat every intersection is assumed to be fixed. Their values are described at appropriate places in this chapter. We describe dynamics for one sample intersection, referring to Figure 6.1 to illustrate the key concepts. I 4 ! Phase&1! I 4 Phase 2 (a) (b) Figure 6.1: (a) Illustration of a movement, lane and a link at a sample intersection. Link number 20 contains two lanes, and each lane supports multiple movements. (b) Phase architecture at a sample intersection. We further adapt the traffic flow model presented in Chapter 3 to the setup that uses the queue length for each movement. A movement at the intersection is denoted by (i,j) corresponding to an admissible maneuver from the upstream 82 link i into the downstream link j at the intersection. Every link i is divided into multiple lanes. A lane can support multiple movements, and the same movement can be made possible through multiple lanes. Every movement (i,j) is associated with a saturation flow capacity, denoted by C ij and a turning ratio R ij denoting the fraction of flow entering linki from upstream and from external to the network, that intends to perform the movement (i,j) and hence the corresponding traffic will queue up in any of the lanes on link i supporting that movement. The saturation flow capacities for a movement are cumulative over all the lanes which facilitate that movement. The R ij are also referred to as turning ratios, and naturally satisfy P j R ij ≤ 1, where the residual 1− P j R ij represents the fraction of flow that departs the network from link i. Let the set of phases at the intersection be fixed, and denoted by{φ 1 ,...,φ m }. Each phase is associated with, possibly multiple, non-conflicting movements that become active when the phase is given green. At most one phase is given green at any time. We assume that the sub-networks used in the simulations in this chapter satisfies the following: (P1) Every movement belongs to one and only one phase. While we impose (P1) for simplicity in this chapter, it is not difficult to extend the descriptions of the controllers to the settings where (P1) is not satisfied. In that case we have: (P2) All the movements associated with a given lane are activated in the same phase. (P2’) All the movements associated with a given link are activated in the same phase. It is straightforward to see that (P2’) implies (P2). (P2) is expected to be true for every network. The sub-network used in our simulations satisfies (P2’). 83 Let x ij [t] be the number of vehicles waiting in queue to perform the (i,j) movement at the beginning of thet-th cycle, and letλ i [t] be the number of vehicles entering linki from outside the network during thet-th cycle. At the beginning of every cycle, the traffic signal controller updates allocation of green time to every phase. Let h φ [t] be the fraction of cycle length T allocated to phase φ during the t-th cycle. The dynamics is then given by: x ij [t + 1] =x ij [t] +R ij λ i [t] + X ` z `i [t] ! −z ij [t] (6.1) wherez ij [t]isthemaximumnumberofvehiclesthatcanexecute (i,j)movement during the t-th cycle, i.e., z ij [t] =C ij h φ [t]T, where φ is the phase under which (i,j) is activated. We implicitly assume that, if the right hand side of (6.1) is negative, then it is reset to zero. This would correspond to emptying of the corresponding queue. In this chapter, we consider decentralized dynamic traffic signal controllers that measure the queue lengths in the immediate vicinity of the intersection at the beginningofeachcycletoupdatethegreentimeallocations. Ideally, manyexisting decentralized controllers, some of which we describe in the next section, require queue length measurements for individual movements. This may be impractical, especially when lanes can support multiple movements. Fortunately, the following features, which are satisfied by real networks, allow us to implement our proposed PF controllers using only queue length measurements for individual lanes. These features also allow intuitive extensions of existing MP controllers which we use for comparison purposes. 84 In what follows, we first propose a cycle-based version of Proportionally Fair controller, describe extensions of Max-Pressure controllers and compare their per- formance in a microsimulator. Then, we study the performance of our proposed output feedback PF controller. 6.1.1 Proportionally Fair (PF) Controller Referring to the formulation of PF controller presented in Chapter 4, its imple- mentation requires, for every phase, aggregate queue length over all the movements associated with the phase. This is the same as the total queue length over all the lanes (or links) activated by a phase under (P2). Let x φ [t] be the aggregate queue length associated with phase φ at the beginning of the t-th cycle. The green time allocation under PF controller h PF [t] is equal to the value of γ corresponding to the optimal solution of the following convex optimization problem: maximize m X r=1 x φr [t] logγ r subject to γ r ≥ 0, r = 1,...,m m X r=1 γ r = 1 (6.2) When (P1) is satisfied, the solution to (6.2) can be written in closed form as follows: h PF φr [t] = x φr [t] P m s=1 x φs [t] , r = 1,...,m (6.3) This means that the green time allocated to a phaseφ r under PF policy is propor- tional to the aggregate queue length associated with φ r . We choose to write (6.2) in its current form because it can be readily adapted to settings where (P1) is not 85 satisfied, e.g., see our recent work [50] for details. In fact, the control implemen- tation in PTV VISSIM described in Figure 6.3 implements the PF controller for the general setting, for which the optimization in eq. (6.2) is convex, and can be solved in real-time using available software tools such as cvx [13, 20]. We reiterate that, the PF controller is also inspired by similar controllers from communication networks. There are several attractive features of the PF con- troller from the perspective of implementation, including requiring queue length information only at an aggregate level, and more importantly not requiring infor- mation about turning ratios, saturation flow capacities and external inflows, and yet possessing the throughput maximizing property. 6.1.2 Max-Pressure (MP) Controller The MP controller involves adding up pressures associated with movements constituting a phase, and then allocating green times as a function of the pressure associated with different phases. The pressure associated with a movement (i,j) at the beginning of the t-th cycle is computed as p ij [t] =C ij x ij [t]− X ` R j` x j` [t] ! , (6.4) where the summation in ` is over all the movements from the downstream link j. The pressure associated with a phaseφ is the sum of the pressures associated with the movements constituting the phase: p φ [t] = X (i,j)∈φ p ij [t]. (6.5) 86 Finally, the green time allocation under MP controller is given by: h MP φr [t] = exp(ηp φr [t]) P m s=1 exp(ηp φs [t]) r = 1,...,m, (6.6) where η > 0 is a parameter to be tuned. The controller given by (6.4), (6.5) and (6.6) is inspired primarily by scheduling algorithms for communication networks. It was originally proposed for communication networks in [68] and extended to traffic signal control in [70]. Since then various extensions have been proposed, e.g., see [33]. Since it is impractical to have queue length measurements for every movement, as would be required to implement (6.4), we use the following approximation to relate queue lengths for movements and links: x ij [t] =R ij x i [t] Thelinkqueuelengthscanbereadilymeasured,e.g.,inthetrafficsimulatorthatwe use for our simulation studies. Under the approximation, the pressure associated with a movement can be expressed in terms of link queue lengths as: p ij [t] =C ij R ij x i [t]− X ` R 2 j` x j [t] ! . (6.7) We will denote the MP controller given by (6.7), (6.5) and (6.6) as h MP1 . The following approximation has also been proposed, e.g., in [22], to compute the pressure associated with a movement in terms of link queue lengths as: p ij [t] =C ij (x i [t]−x j [t]). (6.8) 87 We will denote the MP controller given by (6.8), (6.5) and (6.6) ash MP2 . The main advantage of (6.8) is that it does not require information about turning ratios. Anotherapproachtocomputepressureusingaggregatelinkqueuelength, under (P2’), is proposed in [33] as: p φ [t] = X i: (i,j)∈φ C i x i [t]− X j R ij x j [t] , (6.9) where C i := P j C ij is the aggregated saturation flow capacity of all movements associated with link i. We will denote the MP controller given by (6.9) and (6.6) as h MP3 . A few other variants of MP controllers have been proposed recently, including explicit consideration for lane capacities, e.g., see [31, 21], and replacing turning ratios with their online estimates, e.g., see [33]. One can also write an approximation similar to (6.7) to compute pressure for a movement in terms of lane queue length measurements. Such an approximation will be relevant, e.g., when the network satisfies the weaker property of (P2). We conclude this subsection by noting that, while several variants of MP con- trollers exist in the literature, their available descriptions do not give sufficient details for implementation, especially when the network contains lanes support- ing multiple movements, and when the network satisfies (P2) but not (P2’). We believe that the framework that we provide above, which is absent from the lit- erature, allows to distinguish between the applicability of various MP controllers in terms of availability of granularity of queue length information required for implementation. Moreover, existing descriptions of MP controllers do not clarify how the pres- sure is computed for a movement (e.g., the through movement on link 10 in Fig- ure 6.2 (a)) that exits the network, specifically if the downstream queue (link 54 in 88 Figure 6.2 (a)) faces any stoppage before exiting the network. In this study, for the purpose of computation of pressure only, we assume that the downstream queue length is zero for such movements. For example, for the pressure computation of the through movement on link 10, we assume zero queue length on link 54. While MP controllers have attracted considerable attention recently, the PF controllers have received very little attention. Note that, unlike the MP policies, the PF policy does not take into account downstream queue lengths in its compu- tations. An implication of this feature is that, under a PF policy, a phase might get green light even if downstream links are saturated, and therefore vehicles on the lanes constituting the phase are not able to discharge. This might lead to under- utilization of the discharge rate of intersections under PF policies. However, our simulations suggest that, in spite of such under-utilizations, the PF policy gives better performance than existing variants of MP policies. 6.2 Case Study 1 We perform a comparison study between PF and MP controllers in a micro- scopic traffic simulator. We consider two scenarios in our simulations: one cor- responding to zero offset and the other corresponding to existing non-zero offsets obtained from LADOT signal timing charts. We start by providing a detailed description on how to create the model in the microsimulator software. 6.2.1 Model Setup in PTV VISSIM The road network shown in Figure 6.2(a) is coded in PTV VISSIM while respecting appropriate geometric proportions and lane archiectures, using Google 89 Maps and signal timing charts procured from LADOT. The vehicle type is set to be ’Car’ for all the vehicles in the network. For simplicity, we do not model pedestrians in our simulations. In order to run the simulations in VISSIM, it is essential to define the external arrival rates at vehicle input points. We enter the traffic volume in vehicles per hour for each external link. Then, in order to define the turning ratios at each intersection we use the vehicle routing decision in VISSIM. For each turning movement we set the ’RelFlow’ to be the fraction of the traffic flow that intends to perform the movement. These values are defined once in the beginning of the simulation, however we are able to change them during the simulation through COM interface due to different simulation scenarios. The traffic lights are defined as signal heads at the end of each incoming lane at every intersection. Phase architecture (See Figure 6.1 (b)) and cycle length at each intersection is obtained from LADOT signal timing charts and is defined in ’Signal program’ of each intersection in VISSIM. We use ’Conflict areas’ in order to model the permissive left turns. We used arterial loop detector data collected from May 9 to May 31, 2013 at the locations shown in Figure 6.2 (a) to extract the arrival rates. The data included volume, speed and occupancy, recorded once every 5 minutes. External arrival rate for each incoming link is set to the mean value between 5-7 pm. The resulting arrival rates are as follows (all in veh / hour): λ 17 = 231, λ 4 = 156, λ 26 = 45, λ 31 = 30, λ 30 = 79, λ 20 = 240, (6.10) 90 44 50 49 48 46 53 54 52 55 1 9 10 20 8 12 11 19 17 16 4 2 7 33 14 30 29 27 28 25 26 22 34 31 5 3 W. Olympic Blvd. W. 11 th St. W. 12 th St. W. Pico Blvd. S. Hope St. S. Hope St. S. Flower St. S. Flower St. I 1 I 2 I 4 I 6 I 8 I 7 I 5 I 3 (a) (b) Figure 6.2: The Los Angeles downtown sub-network used in the simulations: (a) schematic representation, with the solid disks showing the approximate location of loop detector sensors from which we have access to offline traffic count data; (b) aerial view. 91 λ 44 ,λ 49 ,λ 52 ,λ 53 were missing in the data set and they all are set to 200 veh/hour in our simulations. The saturated flow capacities are measured offline in VISSIM for every move- ment. The saturation flow capacity is 2200 vehicle/hour/lane for through move- ment and 1800 vehicle/hour/lane for right-turn and left-turn movements. These values are relatively close to the standard values reported in HCM 2010 [3]. Turn ratios are assumed to be equally distributed between the admissible movements at incoming link of each intersection. The values of cycle length for each intersection are obtained from LADOT signal timing charts and are all 90 seconds. We consider two scenarios for the simulations: zero offset and non-zero offset. For the non-zero offset scenario, we use the offsets from LADOT timing chart. The signal offsets are 77, 73, 47, 77, 15, 27, 64, and 36 seconds for intersections I1 to I8 respectively. TheMPandPFcontrollersareimplementedasexternalsignalcontrollersusing a combination of C++ and MATLAB through VISSIM COM interface. All the simulation runs were started with an empty network, and, referring to (6.6), we used η = 0.1 in all our MP controller implementations. Every intersection has two phases: one corresponding to north south movements, and the other one for the east west movements, as shown in Figure 6.1(b) for a sample intersection. The overall architecture for interfacing our control implementation with VISSIM simulator is shown in Figure 6.3. At the beginning of every cycle, the Visual Studio block extracts queue lengths associated with various lanes. This information is then passed on to the MAT- LAB+cvx block to compute the corresponding green time allocation as per MP1, MP2, MP3 or PF controllers. The computed green times are then sent back to VISSIM through its COM interface. 92 Queue length, Average travel /me, Throughput analysis PTV VISSIM Computa/on of green /me Op/miza/on backbone Read queue length and vehicle aAributes from VISSIM Figure 6.3: Overview of adaptive control implementation in PTV VISSIM using COM interface. 6.2.2 Simulation Results We compared MP1, MP2, MP3 and PF controllers with respect to network throughput, average travel time and queue lengths. A detailed description of each of these performance criteria, and the corresponding comparison results are described next. Comparison of Throughput The network throughput is described in terms of feasible external inflows. A set of external inflows λ is said to be feasible for a given network under given turning ratiosβ, saturated flow capacitiesC, and a control policy if the fraction of vehicles spilling outside the boundary of the network is negligible. Since there are 10 entry points in our network, the set of feasible external inflows is a 10 dimensional set. 93 In general, it is hard to compute this set exactly. We first describe a simple procedure to compute an outer approximation to this set. Let z(λ,β,C) be the corresponding vector of steady state flows under fixed-time controllers associated with various movements. Then a necessary condition for the external inflow λ to be feasible is that the corresponding steady state flow satisfies the following at every intersection. m X r=1 max (i,j)∈φr z ij C ij ≤ 1, (6.11) i.e., the sum of the critical flow ratio associated with different phases, at every intersection, is less than the ratio of green light time to total cycle time which is approximated by 1 here. Such a condition is standard for fixed time controllers. It is reasonable to assume that these theoretical estimates provide an outer approx- imation of the capacity region under practical constraints including, finite queue length capacity, imperfect observation by loop detectors, the transient effects in saturated flow at the start and end of a green phase, effect of lane changing and lateral behaviors, etc. We compared the outer approximation with simulations in VISSIM for various control policies. For every control policy, we start with nominal external inflows in (6.10), and then increase one component until the fraction of vehicles spilling outside the network at each of the 10 entry points is less than 5 %. The maxi- mum value is reported in Table 6.1 for each of the three control policies for a few representative components. Each row gives the maximum feasible external inflow, in vehicles per hour, at the corresponding entry point of the network, when the arrival rates at all other entry points are fixed at their nominal values given by (6.10). These results suggest that in both zero offset and non-zero offset scenarios, the throughput of PF controller is comparable to the MP1 controller proposed in 94 this study, and significantly better than the MP2 and MP3 controllers proposed in the literature. PF MP1 MP2 MP3 λ max i Upper bound Zero offset Non-zero offset Zero offset Non-zero offset Zero offset Non-zero offset Zero offset Non-zero offset i=4 3140 2500-2600 2100-2200 2300-2400 2400-2500 2000-2100 1200-1300 1900-2000 1300-1400 i=26 4310 2600-2700 2800-2900 2500-2600 2500-2600 1500-1600 1400-1500 1600-1700 1400-1500 i=30 4510 2800-2900 2800-2900 2900-3000 2800-2900 1900-2000 1600-1700 1800-1900 1800-1900 Table 6.1: Comparison of the upper bound on network throughput capacity as given by the outer approximation in (6.11), and the empirical throughput values found through simulation studies under PF, MP1, MP2 and MP3 controllers. Comparison of Average Travel Time We computed the travel time under the three control policies, for external inflow equal to 4 times the nominal values given in (6.10). This external inflow was first checked to be feasible, and was chosen so as to simulate heavy traffic. We use the output vehicle record data from VISSIM to calculate the average travel time. The output file from VISSIM gives, for every simulation instant, the time spent by every vehicle currently present in the network up to that time instant. This data is used to compute the running average of the travel time for all the vehicles, including the ones which have departed the network. Figure 6.4 shows the average travel time based on 5 stochastic simulation runs in VISSIM. The confidence intervals of the results are shown by the error bars. Each error bar represents one standard deviation on either side of the mean. Figure 6.4 shows that, in non-zero offset scenario, the average travel time under PF is consistently better than the MP2 and MP3 controllers. Under the zero offset scenario, the performance of PF controllers is comparable to its MP counterparts in spite of its minimalistic nature. 95 0 1000 2000 3000 4000 5000 6000 20 40 60 80 100 120 140 Simulation Time (sec) Running Average of Travel Time (sec) MP2 PF 0 1000 2000 3000 4000 5000 6000 20 40 60 80 100 120 140 Simulation Time (sec) Running Average of Travel Time (sec) MP3 PF (a) 0 1000 2000 3000 4000 5000 6000 20 40 60 80 100 120 140 Simulation Time (sec) Running Average of Travel Time (sec) MP2 PF 0 1000 2000 3000 4000 5000 6000 20 40 60 80 100 120 140 Simulation Time (sec) Running Average of Travel Time (sec) MP3 PF (b) Figure 6.4: Comparison of the running average of travel time for heavy traffic under PF and MP controllers using (a) Zero offset and (b) Non-zero offset. Comparison of Travel Time during Incidents In the next step, we compared travel time during incidents under PF and MP controllers. The simulation parameters are kept unchanged. Meaning, we know that the external inflow was feasible, and was chosen so as to simulate heavy traffic. However, at time t = 5400, we model accidents on link 14 and link 19. These two links remain fully blocked during the rest of simulation and the turning ratios at intersectionI4 andI8 are changed, such that the coming vehicles avoid the blocked links. Figure 6.5 shows the average travel time of the network and total number of 96 vehicles in the network under PF and MP3 controllers. The results indicate that, the PF controller adapts to the new traffic diversion pattern and outperforms the MP counterpart. 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 20 40 60 80 100 120 140 160 180 Simulation Time (sec) Running Average of Travel Time (sec) MP3 PF 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10000 150 200 250 300 350 400 450 500 550 Time (sec) Total Number of Vehicles in the Network MP3 PF (a) (b) Figure6.5: Comparisonof(a)Runningaverageoftraveltimeand(b)Totalnumber of vehicles in the network in case of incidents under PF and MP3 controllers. Comparison of Steady State Queue Lengths Table 6.2 shows that for zero offset scenario, with some exceptions, steady state queue lengths are lower under PF in comparison to MP controllers. The links which show the reverse trend (e.g., link 10) are typically the ones whose dominant movementhasadownstreamlinkexitingthenetwork. Ontheotherhand, e.g., link 1 also has a movement (right turning) which exits the network, but this movement is not dominant because it shares only one lane with through movement, whereas the through movement has 3 dedicated lanes, and is not exiting. For the non- zero offset scenario, the queue lengths under PF policy are comparable to the ones under MP1 and are lower than MP2 and MP3 on most links. We recall that, for the purpose of computation of pressure in our implementation of MP controllers, we assume zero queue length on the exit lengths, and hence the links 97 with dominant movement exiting the network has a relatively high pressure in comparison to the other links at that intersection whose dominant movements do not exit the network. This feature highlights the boundary effect on the queue lengths under MP controllers. However, the PF controllers do not exhibit such a boundary effect because the green time allocation is determined based on queue length measurements only on links at that intersection. Link No. 1 2 3 4 7 9 10 11 12 19 20 28 33 44 46 48 49 52 53 PF 5.11 21.38 5.55 4.66 7.32 5.43 15.46 6.43 15.03 2.85 6.15 4.84 3.26 3.26 2.68 5.66 5.61 3.19 14.45 MP1 12.47 25.18 9.78 6.33 3.09 9.09 6.31 9.10 6.19 4.82 10.48 2.03 5.06 7.20 5.58 7.08 6.48 9.41 10.45 MP2 12.69 18.15 11.30 6.73 8.53 9.21 6.27 11.10 5.01 4.17 10.21 3.44 5.64 7.07 10.34 9.91 7.44 8.79 9.29 MP3 10.95 29.25 8.88 6.21 7.59 10.83 5.99 9.31 4.33 6.44 9.78 2.33 4.73 6.67 13.61 9.43 7.77 8.35 10.12 (a) Link No. 1 2 3 4 7 9 10 11 12 19 20 28 33 44 46 48 49 52 53 PF 11.27 16.07 10.03 4.48 5.45 8.20 5.41 9.66 4.41 3.72 10.87 3.57 6.12 7.70 11.33 5.67 5.23 18.95 7.71 MP1 11.85 17.12 10.97 4.79 5.54 8.48 6.37 8.06 3.45 4.31 11.31 3.22 5.90 9.08 9.70 5.20 5.06 10.27 7.77 MP2 13.44 19.67 9.89 6.06 4.86 9.95 5.63 8.94 4.57 2.74 10.30 3.11 6.53 6.91 12.65 9.97 7.73 8.29 10.79 MP3 9.96 13.44 11.05 9.27 9 11.42 5.03 10.73 5.35 5.56 10.53 2.78 5.21 6.05 10.85 10.79 7.46 10.99 10.36 (b) Table 6.2: Comparison of steady state average queue lengths on representative links under various control policies using (a) Zero offset (b) Non-zero offset. Refer toFigure6.2(a)forlinknumberidentifiers. Forbrevity, queuelengthsarereported onlyforthoselinkswhichshowsignificantdifferenceacrossthefourcontrolpolicies. The feature of the MP controllers to allocate green time based on comparison between upstream and downstream queue lengths gives rise to phase shift in the queue length profiles between adjacent links of the network, as illustrated in Fig- ure 6.6 and 6.7. Indeed, this phase shift builds the necessary pressure to extract green time for the upstream link (link no. 44 in Figure 6.6). The queue lengths under PF controller, however are decoupled, and do not show a phase shift. Fig- ure 6.7 shows that the phase shift behavior changes in non-zero offset scenario compare to the zero offset scenario. However, even in non-zero offset scenario, the MP controller shows larger phase shift in comparison to PF controller. This phase shift might be responsible for higher average queue lengths under MP controllers in comparison to PF controller. 98 4000 4100 4200 4300 4400 4500 0 5 10 15 20 25 30 35 Time (sec) Queue length (veh) Link No. 1 Link No. 44 4000 4100 4200 4300 4400 4500 0 5 10 15 20 25 30 35 Time (sec) Queue length (veh) Link No. 1 Link No. 44 (a) (b) Figure 6.6: Illustration of phase shift in queue length time series on link numbers 1 and 44, which are adjacent to each other (see Figure 6.2(a)), under (a) PF and (b) MP2 controllers under zero offset. 4000 4100 4200 4300 4400 4500 4600 0 5 10 15 20 25 30 Time (sec) Queue length (veh) Link No. 1 Link No. 44 4000 4100 4200 4300 4400 4500 4600 0 5 10 15 20 25 30 Time (sec) Queue length (veh) Link No. 1 Link No. 44 (a) (b) Figure 6.7: Illustration of phase shift in queue length time series on link numbers 1 and 44, which are adjacent to each other (see Figure 6.2(a)), under (a) PF and (b) MP2 controllers under non-zero offset 6.3 Case Study 2 In this study, we evaluate the throughput performance of output feedback PF controller (see chapter 5) for a general network setup. The goal is to empirically evaluate in a real world scenario how close is the resulting stability region to the 99 outer bound obtained in (6.11). To that aim, we study an arterial grid network. We consider downtown Los Angeles arterial sub-network consisting of 16 signalized intersections and 48 links, which is adjacent to Interstate 10 and State Route 110. The topology of this network is shown in Figure 6.8(a). In the next subsection, we describe the simulation setup in detail. 6.3.1 Model Setup and Controller Implementation The network is coded in VISSIM while respecting appropriate geometric pro- portions (see Figure 6.8(b)), using Google Maps. In general, the simulation setup is similar to the one in the previous case study. We used arterial loop detector data collected from May 1 to May 31, 2013 at the input points of the network to extract the arrival rates (see Figure 6.8a). The dataset includes volume, speed and occu- pancy, recorded once every 5 minutes. External arrival rate for each incoming link is set to the value of vollume averaged over the weekday’s afternoon peak (4:00 PM-6:00 PM). The resulting external inflows on entry links of the network are shown in Table 6.3. Turning ratios at every intersection are defined using ‘Vehicle Routing Decision’ in VISSIM. The values are defined once at the beginning of the simulation, and kept constant thereafter. Link ID (i) 31 32 33 34 35 36 37 38 39 40 λ i 910 1271 270 755 573 414 694 186 1323 827 Table 6.3: External inflow on entry links of the network (veh/hour). Cycle length, offset and phase architecture at each intersection are obtained from LADOT signal timing charts and are input through ‘Signal program’ for each intersection. Permitted left turns are coded using the ‘Conflict areas’. Based on the LADOT signal timing chart, all the intersections have the common cycle length of 90 seconds 100 (a) (b) Figure 6.8: The Los Angeles sub-network used in the simulations: (a) graph topol- ogy (b) aerial view in PTV VISSIM. 101 We refer to Figure 6.3 for an illustration of the implementation of controller in VISSIM. The output feedback PF controller is implemented through VISSIM COM interface. As shown in Figure 5.1, stop line detectors record presence of a vehicle at the stop line of each intersections. The Control Implementation block checks the information from stop line detectors every second. This information is used to calculate the values of ˜ G and green splits. These values are fed back to VISSIM simulation. All the simulation scenarios start from an empty network as an initial condition. 6.3.2 Simulation Results: Throughput Evaluation Since there are 10 entry points in our network, the set of feasible external inflows is a 10 dimensional set. One can design a simple procedure to compute an outer approximation to this set, inspired by discussion around (6.11). We compared the outer approximation with empirical values from VISSIM microsimulations for output feedback PF controllers. In order to compute the throughput in VISSIM, we start with external inflows in Table 6.3, except we increase the arrival rate for one component until the fraction of vehicles spilling outside the network at each of the 10 entry points is less than 5 %. The maximum value (λ max i ) is reported in Table 6.4 for output feedback PF controller for a few representative components of external inflow vector. The theoretical upper bound corresponds to the outer approximation. We calculated the empirical values of throughput for entry links 36 and 40, located on east and west boundary of the network, respectively (see Figure 6.8(a)). For these two representative links, our simulation results show that the gap between the theoretical upper bound and empirical values of throughput is within 20-30 %. 102 We believe that the gap can be attributed to the assumptions used in com- putation of the outer approximation. In calculating the outer approximation the microscopic behavior of traffic flow, e.g. lane changing and car following behavior, has been ignored. However, these behaviors are fully captured in microsimula- tor and could be responsible for the gap that we observe between empirical and theoretical values. While calculating the empirical throughput values in VISSIM, it is essential to carefully calibrate the model to avoid nonrealistic congestions. For instance, it is crucial that the "Vehicle Routing Decision" and its parameters are coded properly, such that the distance between the routing decision marker and the downstream link is large enough to allow the vehicles to change lanes. Otherwise, unrealistic interference and queue back up will arise that can result in unreasonably low empirical values for throughput. Link ID Upper bound (veh/hr) λ max i (veh/hr) gap (%) i=36 3500 2600 25.7 i=40 3200 2300 28.1 Table 6.4: Comparison of outer approximation and empirical throughput from microsimulations. 6.4 Conclusions In this chapter, we studied two classes of decentralized adaptive traffic control policies: proportionally fair (PF) and max-pressure (MP) in microscopic traffic environment. Of these, several variants of MP controller exist in the literature, whereas the PF controller does not seem to have received much attention. We present a unified framework to distinguish between several variants of MP con- trollersintermsofthegranularityofreal-timequeuelengthmeasurementsrequired 103 for their implementation. PF controller does not require information about turn- ing ratios, saturation flow capacities and downstream queue length in comparison to the majority of its MP counterparts. In spite of this minimalist nature, in our microscopic simulation studies on a Los Angeles downtown sub-network in PTV VISSIM, PF controllers outperforms MP controllers in terms of throughput and average delay time in non-zero offset scenario. In the zero offset scenario, the performance of PF controller is comparable to its MP counterparts. The better performance of PF controller reported in this paper motivates thorough investiga- tion of its performance with respect to MP controllers in other scenarios. We plan to incorporate the effect of downstream queue length in PF controller. Moreover, we evaluated the performance of proposed output feedback PF con- troller in microscopic simulations. The simulations results suggest that the con- troller performs well in terms of network throughput, despite the fact that it does not require direct queue length measurements. We plan to compare the perfor- mance of this controller with that of other controllers that use similar detector information such as Ring-Barrier Controllers (RBC). 104 Chapter 7 Conclusions and Future Work In this dissertation, we proposed novel techniques for analysis and control of traffic flow in urban arterial networks. In the first part, we proposed a novel macro- scopic model for signalized arterial network that captures the inter-cycle behavior of queue length. In particular, we proposed a delay differential equation frame- work to simulate queue length dynamics for signalized arterial networks under fixed-time or adaptive control. We established the existence of a unique solution when the external inflow and capacity functions are piecewise constant. More- over, under periodicity and stability conditions, existence of a globally attractive periodic orbit is established for fixed-time control. An iterative procedure is also providedtocomputethisperiodicorbitwitharbitraryaccuracywithoutdirectsim- ulations. Illustrativesimulationsarepresentedthatshowgoodconsistencybetween the output of proposed iterative procedure and steady-state queue lengths from a microscopic traffic simulator. Collectively, these results provide useful computa- tional tools to evaluate the performance of signalized arterial networks for a given traffic signal timing plan. Next, we studied decentralized control algorithms for traffic signal control. Specifically, we focused on adapting the well-known Proportionally Fair (PF) con- troller from the communication networks literature to determine green time alloca- tions for signalized arterial network and provided theoretical results on its perfor- mance analysis. In addition to being completely decentralized, the unique feature 105 of PF controller is that it does not require any information about network param- eters such as external arrival rates, saturation capacity and turning ratios. First, we studied the stability of PF controller under uninterrupted traffic flow models. Our main theoretical result shows that, in the case when only single phases are allowed, the resulting traffic network dynamics admit a globally asymptotically stable equilibrium, provided that the arrival rates belong to the interior of stabil- ity region. In other words, it is shown that the PF controller is throughput optimal in spite of its minimal nature. These results rely on using some entropy-like Lya- punov functions previously considered in the context of communication queuing networks. In the next step, we provided analysis of PF controller under the more accurate ON/OFF model proposed in this dissertation. It turned out that choos- ing the right dependency of controller on queue length measurement is nontrivial under the ON/OFF model. An output feedback PF controller, which calculates the green splits based on real-time event-driven measurements in every cycle, is proposed to enable computational tractability of control synthesis and facilitate the stability analysis. In particular, it is shown that control synthesis under our proposed adaptation is maximally stabilizing for an isolated intersection with time varying arrival rates. Unlike other variants of PF controllers, the output feedback PF controller does not require queue length measurement. This unique feature of ouroutputfeedbackcontrollermakesiteasiertoimplementinrealistictrafficsignal setup, where the input the controller is readily extracted from the presence data from stop line detectors, without having to estimate the queue length in real-time. We studied performance of PF controllers in a microscopic simulation environ- ment to evaluate its performance under a more realistic and accurate traffic flow dynamics. We compared the PF controller with Max-Pressure (MP) controller, 106 another recently studied decentralized controller inspired by controllers in commu- nication networks. While several variants of MP controller exist in the literature, the PF controller does not seem to have received much attention. We presented a unified framework to distinguish between several variants of MP controllers in terms of the granularity of real-time queue length measurements required for their implementation. Unlike PF controller, most variants of MP controllers require information about turning ratios, saturation flow capacities and downstream queue length. In spite of this minimalist nature, in our microscopic simulation studies on a Los Angeles downtown sub-network in PTV VISSIM, PF controllers out- performs MP controllers in terms of throughput and average delay time. While these decentralized algorithms inspired by similar algorithms from communica- tion networks, give higher throughput than fixed-time controller, the well-known throughput optimality results from communication networks do not appear to hold in micro-simulation environments. This is possibly because there is not yet a def- inite adaptation of PF to address spillback and to integrate offset optimization, and the traffic flow model does not account for effect of lane change and lateral behavior on link flow capacity. Hence, the upper bound estimates for throughput from communication networks are optimistic for urban traffic networks. There are several interesting directions for future work. On the analysis side, we plan to extend the traffic flow model presented in chapter 3 and its analysis to account for finite queue capacity and capture spillbacks. This requires modeling inter-link travel time as a function of queue length. Also, extending the compu- tational framework proposed in chapter 3 to capture spillbacks, is an immediate next step due to its usefulness in performance evaluation of given fixed-time signal plan in oversaturated condition. 107 Furthermore, we plan to to extend the theoretical analysis from chapter 5 to a more general setup, i.e. general network. The suggested framework in this thesis opens up the possibility of adapting other class of scheduling algorithms, such as MP controllers to ON/OFF traffic flow models. Hence, one immediate direction of future work is to develop the necessary analytical tools for establishing maximally stabilizing property of such adaptations. 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PhD thesis, Katholieke Universiteit Leuven, 2007. 115 Appendix A Additional Results for Chapter 3 A.1 Proof of Proposition 1 The feasible set for (3.2c) is non-empty (z = 0 is always feasible) and compact. Therefore, there exists at least one solution, say ˆ z, to (3.2c). We first note that if z and ˜ z satisfy the constraints in (3.2c), then so does z max defined by z max i = max{z i , ˜ z i } for all i ∈ E. This is because z ≤ c(t) and ˜ z ≤ c(t) implies that z max ≤ c(t), and therefore the first inequality in (3.2c) is trivially satisfied by z max . With respect to the second inequality, fix some i ∈ I, and let z max i = z i (without loss of generality). Then, z max i = z i ≤ ˜ λ i + P j∈E i :z max j =z j R ji z j + P j∈E i :z max j =˜ z j R ji z j ≤ ˜ λ i + P j∈E i :z max j =z j R ji z max j + P j∈E i :z max j =˜ z j R ji z max j = ˜ λ i + P j∈E i R ji z max j , where the first inequality follows from (3.2c) and the second one follows from the definition of z max . This argument is used to prove that ˆ z is unique as follows. Let z and ˜ z be two optimal solutions. Since z6= ˜ z, there exist i,j∈E such that z max i > z i and z max j > ˜ z j . Therefore, 1 T z max >1 T z =1 T ˜ z, contradicting optimality of z and ˜ z. In order to prove the first part of (3.5), let ˆ z i < c i (t) for some i∈E\I(x). Then, a small increase in ˆ z i will trivially maintain feasibility of the first set of constraints in (3.2c), and also maintains feasibility with respect to the second set of constraints because it only affects the right hand side which increases with increase in ˆ z i . However, increasing ˆ z i strictly increases the objective, thereby contradicting optimality of ˆ z. 116 With regards to the second part of (3.5), if c i (t)< ˜ λ i (t) + P j∈E i R ji ˆ z j for some i∈I(x), then the proof follows along the same lines as the first part of (3.5). Allowing ˆ z i < ˜ λ i (t) + P j∈E i R ji ˆ z j <c i (t) for somei∈I(x) leads to a contradiction for similar reason. A.2 Proof of Proposition 2 Once the solution (x(t),β(t)) to (3.2) is proven to exist and be unique, its non-negativity follows from the constraint on z(x,t) in (3.2c). Our approach to showing the existence and uniqueness of solution to (3.2) is to show it on contigu- ous intervals [0,4), [4, 24),.... 4 > 0 is chosen to be (the greatest) common divisor of: (i) time instants in [− ¯ δ, 0] corresponding to switch in values ofz(t); (ii) time instants in [0,T ] corresponding to switch in values of λ(t) and c(t); and (iii) {δ ji } j,i∈E . Under Assumption 2, such a4 > 0 exists if, e.g., the three types of quantities are all rational numbers. The next result establishes the required existence and uniqueness of solution to (3.2) over [0,4), along with an important input-output property. Lemma 5. If ˜ λ : [0,4)→ R E ≥0 is piece-wise constant and non-increasing, and c : [0,4)→ R E ≥0 is constant, then, for any x(0)∈ R E ≥0 , there exists a unique solution x : [0,4)→ R E ≥0 to (3.2). Moreover, z : [0,4)→ R E ≥0 is piece-wise constant and non-increasing. Proof. (3.2c) and Proposition 1 imply that z(x,t) remains constant over a time interval if so do ˜ λ(t),c(t) andI(x). Let (τ 1 ,τ 2 ,..., )∈ (0,4) be the finite number of time instants corresponding to changes in the value of ˜ λ(t). Since ˜ λ(t) and c(t) are constant over [0,τ 1 ), z(x,t) will remain constant at least until say at t s ∈ [0,τ 1 ) when there is possibly a change inI(x). This also implies the existence 117 and uniqueness of solution to (3.2) over [0,t s ). Moreover, since z(x,t) is constant over [0,t s ), if x i (t = 0 + ) = 0 for some i∈E, then x i (t)≡ 0 over [0,t s ]. Thus, a change in the setI(x) at t s could only involve its expansion. Therefore, (3.2c) implies that z(x(t s ),t s )≤z(x(t − s ),t − s ). Continuing along these lines,I(x) is non- contracting over [0,τ 1 ). Since ˜ λ(τ 1 ) < ˜ λ(τ − 1 ) by assumption, (3.2c) implies that I(x(τ − 1 ))(I(x(τ 1 )), and hence also z(x(τ 1 ),τ 1 )≤z(x(τ − 1 ),τ − 1 ). Collecting these facts together implies thatI(x) is non-contracting and z(x,t) is non-increasing over [0,4). Combining this with the fact thatI(x) can take at most 2 E distinct values, implies that the total number of changes inI(x) over [0,4) are finite. Concatenating the unique solutions to (5) from between changes inI(x) gives the lemma. Since z(t) is non-increasing and piece-wise constant in each of the intervals [− ¯ δ,− ¯ δ +4),..., [0,4) (cf. Lemma 5 and the assumption in Proposition 2), this implies that ˜ λ(t) is non-increasing and piece-wise constant over [4, 24). One can then use Lemma 5 to show existence and uniqueness of solution over [4, 24). Recursive application of the procedure then proves Proposition 2. A.3 Proof of Lemma 1 It suffices to show the result for a small time interval starting from zero. More- over, it is sufficient to show that x(t)≤ x 0 (t) in this interval, since this implies I(x 0 (t))⊆I(x(t)), and hence z(t)≤ z 0 (t) along the same lines as the proof of Lemma 5. We shall prove this for each component ofx(t) andx 0 (t) independently. Fix a component i∈E. 1. If 0 < x 0,i ≤ x 0 0,i , then, recalling (3.4), ˙ x i (t = 0) = λ i (0) + P j∈E\E i R ji z j (−δ ji )+ P j∈E i R ji z i (t = 0)−c i (0)≤λ 0 i (0)+ P j∈E\E i R ji z 0 j (−δ ji )+ 118 P j∈E i R ji z 0 j (t = 0)−c 0 i (t) = ˙ x 0 i , where we have used the fact that x 0 ≤ x 0 0 impliesz(0)≤z 0 (0). Hencex(t)≤x 0 (t) for small time interval starting from zero. 2. Now consider the case when 0 =x 0,i ≤x 0 0,i . If x i (t)≡ 0 for a small interval starting from zero, then trivially x i (t)≤x 0 i (t) over that interval. Otherwise, the proof follows along the same lines as Case 1. A.4 Proof of Lemma 2 (3.2a) can be rewritten as ˙ x i = λ i (t) + P j∈E R ji z j (t)− z i (t) +4 i (t), where 4 i (t) = P j∈E R ji (z j (t−δ ji )−z j (t)). Therefore, R t s 4 i (r)dr = P j∈E R ji R s s−δ ji z j (r)− R t t−δ ji z j (r) dr. Sinceδ ji ≤ ¯ δ andz j (r)≤c j (r) is bounded, it follows that| R t s 4(r)dr|≤d1 for some constant d> 0. Suppose x i (t 0 )>NT ¯ c i for some constant N > d T and t 0 ≥ 0, where > 0 is from Definition 1. Since x i (t +T )−x i (t)≥−T ¯ c i , we have x i (t)> 0 for t 0 ≤t≤t 0 +NT. Therefore, x i (t 0 +NT )−x i (t 0 )≤NT ¯ λ i +NT X j∈E R ji ¯ c j −NT ¯ c i +d≤−NT +d< 0 where we use the stability condition from Definition 1 in the second inequality. This is sufficient to show that x i (t) is bounded, since the queue length increments per cycle are upper bounded. A.5 Proof of Proposition 3 Let ¯ x 0,i := max{x 0,i , ˜ x 0,i } andx 0,i := min{x 0,i , ˜ x 0,i }, for alli∈E. Let ¯ β 0 andβ 0 be defined similarly. Therefore,kx 0 − ˜ x 0 k 1 = P i∈E |x 0,i − ˜ x 0,i | = P i∈E (¯ x 0,i −x 0,i ) = k¯ x 0 −x 0 k 1 . Similarly,kβ 0 − ˜ β 0 k 1 =k ¯ β 0 −β 0 k 1 . Let (¯ x(t), ¯ β(t)) and (x(t),β(t)) 119 be the trajectories starting from (¯ x 0 , ¯ β 0 ) and (x 0 ,β 0 ) respectively. Lemma 1 then implies that x(t)≤ x(t)≤ ¯ x(t), x(t)≤ ˜ x(t)≤ ¯ x(t), β(t)≤ β(t)≤ ¯ β(t), and β(t)≤ ˜ β(t)≤ ¯ β(t), which then implies thatkx(t)− ˜ x(t)k 1 ≤k¯ x(t)−x(t)k 1 and kβ(t)− ˜ β(t)k 1 ≤k ¯ β(t)−β(t)k 1 . Therefore, it suffices to show k¯ x(t)−x(t)k 1 +k ¯ β(t)−β(t)k 1 ≤k¯ x 0 −x 0 k 1 +k ¯ β 0 −β 0 k 1 (A.1) We show (A.1) using an intuitive argument similar to the one used in [47, Lemma 2]. Color the vehicles in the initial state (i.e., (¯ x 0 , ¯ β 0 ) and (x 0 ,β 0 )) red, and all the vehicles arriving after that as black. Therefore, the right hand side in (A.1) represents the excess red vehicles initially in the system with the larger initial condition. Subsequently, in each queue, there will be black and red vehicles. Change the service discipline in each queue so that all black vehicles are served ahead of every red vehicle. This has two impli- cations. First, since the service times for red vehicles in each queue are the same in each of the two systems, every red vehicle common to both the sys- tems receives identical service in the two systems. That is, more red vehicles depart the system starting from (¯ x 0 , ¯ β 0 ) than in the system starting from (x 0 ,β 0 ), 120 i.e., P i∈E (¯ x 0,i + P j∈E R ji R 0 −δ ji ¯ z i (s)ds)− P i∈E ¯ x red i (t) + P j∈E R ji R t t−δ ji ¯ z i (s)ds ≥ P i∈E (x 0,i + P j∈E R ji R t t−δ ji z i (s)ds)− P i∈E x red i (t) + P j∈E R ji R t t−δ ji z red i (s)ds , i.e., X i∈E ¯ x red i (t)−x red i (t) + X i∈E X j∈E R ji Z t t−δ ji ¯ z red i (s)−z red i (s) ds≤ X i∈E ¯ x 0,i −x 0,i + X i∈E X j∈E R ji Z 0 −δ ji (¯ z i (s)−z i (s)) ds i.e., k¯ x red (t)−x red (t)k 1 +k ¯ β red (t)−β red (t)k 1 ≤ k¯ x 0 −x 0 k 1 +k ¯ β 0 −β 0 k 1 (A.2) Second, service of black vehicles is unaffected by red vehicles in both the sys- tems. Therefore, the number of black vehicles in each queue, and in particular, the total number of black vehicles in the entire network for both the systems are the same at any time. This combined with (A.2) gives (A.1). In order to prove (3.8), let (ˆ x(t), ˆ β(t)) be the trajectory starting from the ini- tial condition (0, 0). Sincekx(t)− ˜ x(t)k 1 ≤kx(t)− ˆ x(t)k 1 +k˜ x(t)− ˆ x(t)k 1 and kβ(t)− ˜ β(t)k 1 ≤kβ(t)− ˆ β(t)k 1 +k ˜ β(t)− ˆ β(t)k 1 , it suffices to prove (3.8) for (˜ x(0), ˜ β(0)) = (ˆ x(0), ˆ β(0)) = (0, 0). Using the red vehicle terminology from before, kx(t)−˜ x(t)k 1 +kβ(t)− ˜ β(t)k 1 thendenotesthenumberofredvehiclesin (x(t),β(t)). Stability condition implies that all red vehicles eventually leave the network, i.e., lim t→∞ kx(t)− ˜ x(t)k 1 +kβ(t)− ˜ β(t)k 1 = 0. A.6 Technical Corollary The following corollary to Lemma 1 and Proposition 3 is used in Section 3.6. 121 Corollary 1. Consider an isolated link i with T-periodic capacity function c i (t). Lety i (t) andy 0 i (t) beT-periodic inflow functions, both satisfying the stability condi- tion in Definition 1, and y i (t)≤y 0 i (t) for all t∈ [0,T ]. If the corresponding steady stateT-periodic queue lengths arex ∗ i (t) andx ∗0 i (t) respectively, and the steady state T-periodic link outflows are z ∗ i (t) and z ∗0 i (t) respectively, then x ∗ i (t)≤ x ∗0 i (t) and z ∗ i (t)≤z ∗0 i (t) for all t∈ [0,T ]. Proof. Let (x(t),z(t)) and (x 0 (t),z 0 (t)) be the system trajectories for the two systems, both starting from initial condition (x 0 ,β 0 ). Lemma 1 implies that x 0 (t)≥x(t) and z 0 (t)≥z(t) for all t≥ 0. On the other hand, Theorem 1 implies that (x(t),z(t)) and (x 0 (t),z 0 (t)) converge to T-periodic trajectories (x ∗ (t),z ∗ (t)) and (x ∗0 (t),z ∗0 (t)) respectively. Combining these facts gives the desired result. 122 Appendix B Additional Results for Chapter 4 B.1 Additional Lemmas Lemma 6. Let V (x) be as in (4.10), then for all x> 0, ∂V (x) ∂x i = log C i h i (x) z i ! . Proof. Let x ξ be a vector, such that x ξ i =x i +ξ and x ξ j =x j , j6=i. Then V (x ξ )−V (x) = X i∈E x ξ i log C i h i (x ξ ) z i ! + X v∈V κ v log h (v) 0 (x ξ ) h (v) 0 (x ∗ ) − X i∈E x i log C i h i (x) z i ! + X v∈V κ v log h (v) 0 (x) h (v) 0 (x ∗ ) ≥ X i∈E x ξ i log C i h i (x) z i ! + X v∈V κ v log h (v) 0 (x) h (v) 0 (x ∗ ) − X i∈E x i log C i h i (x) z i ! + X v∈V κ v log h (v) 0 (x) h (v) 0 (x ∗ ) =ξ log C i h i (x) z i ! , 123 where the inequality follows from the suboptimality of the solution. In the same manner, we have that V (x ξ )−V (x)≤ξ log C i h i (x ξ ) z i ! . The combination of the two inequalities yields log C i h i (x) z i ! ≤ 1 ξ V (x ξ )−V (x) ≤ log C i h i (x ξ ) z i ! . Letting ξ→ 0 proves the lemma. Lemma 7. Let R be a routing matrix and λ an arrival vector satisfying Assump- tion 4. Then, for a∈R E + satisfying the relation, z = (I−R T ) −1 λ, and for any u∈R E + it holds that u T (λ− (I−R T )(diagze u ))≤ 0, with equality if and only if u = 0. Proof. See [41, Lemma 7]. 124 Appendix C Additional Results for Chapter 5 C.1 Proof of Lemma 4 We introduce a couple of notations for this proof. For t 2 ≥ t 1 ≥ 0, ˜ G i (t 1 : t 2 ) is the sum of length of time intervals in [t 1 ,t 2 ] when x i > 0. z i (t) is the outflow from link i at time t. (a) The G i [k] = G min i case is trivial. Therefore, let us consider G i [k] > G min i . ¬U i [k] implies that ˜ G i [k] =G i [k]. Referring to (5.3), it is sufficient to show that G i logG i + ˜ G −i logG −i >G i log (G i −δ) + ˜ G −i log (G −i +δ) (C.1) for all feasible δ> 0, where we drop the argument k for brevity in notation. It is easy to see that [ ˜ G i ˜ G i + ˜ G −i , ˜ G −i ˜ G i + ˜ G −i ] is the solution to the optimization problem in 5.3. Considering that ˜ G i =G i , one can write the following for all feasible δ> 0: ˜ G i log G i G i + ˜ G −i + ˜ G −i log ˜ G −i G i + ˜ G −i > ˜ G i log G i +δ G i + ˜ G −i + ˜ G −i log ˜ G −i −δ G i + ˜ G −i 125 then, it is easy to see that, G i G i + ˜ G −i log G i G i + ˜ G −i + ˜ G −i G i + ˜ G −i log ˜ G −i G i + ˜ G −i > G i G i + ˜ G −i log G i −δ G i + ˜ G −i + ˜ G −i G i + ˜ G −i log ˜ G −i +δ G i + ˜ G −i i.e., G i logG i + ˜ G −i log ˜ G −i >G i log(G i −δ) + ˜ G −i log( ˜ G −i +δ) i.e., G i log G i G i −δ + ˜ G −i log ˜ G −i ˜ G −i +δ > 0 G −i ≥ ˜ G −i implies G −i G −i +δ ≥ ˜ G −i ˜ G −i +δ , which then gives G i log G i G i −δ + ˜ G −i log G −i G −i +δ > 0, which is the same as (C.1). (b) It is sufficient to show that ˜ G 1 [k + ] ≤ ¯ λ 1 T c 1 for some k + ∈ [k + 1,k +η]. Combined withS /2 1 [k + ], this implies ˜ G 1 [k + ]<G 1 [k + ], which impliesU 1 [k + ]. Let t 0 be the time instant when x 1 = 0 the last time during the green phase of the k-th cycle, and let τ := kT +G 1 [k]−t 0 ≥ 0; see Fig C.1 (top) for illustration. x 1 (t 0 ) = 0 implies: 0≤x 1 (t 0 +ηT )−x 1 (t 0 ) = Z t 0 +ηT t 0 λ 1 (s)ds− Z t 0 +ηT t 0 z 1 (s)ds = Z t 0 +η 1 T 1 t 0 λ 1 (s)ds− Z t 0 +ηT t 0 z 1 (s)ds ≤η 1 ¯ λ 1 T 1 − Z t 0 +ηT t 0 z 1 (s)ds =η ¯ λ 1 T− Z t 0 +ηT t 0 z 1 (s)ds (C.2) 126 Figure C.1: Link queue lengths over η consecutive cycles: link 1 (top) and link 2 (bottom). The last term in (C.2) is the total outflow from link 1 during [t 0 ,t 0 +ηT ], and is lower bounded by the total outflow during times whenx 1 > 0 in [t 0 ,t 0 +ηT ]: Z t 0 +ηT t 0 z 1 (s)ds≥c 1 τ + ˜ G 1 ((k +η)T :t 0 +ηT ) +c 1 k+η−1 X κ=k+1 ˜ G 1 [κ] (C.3) Since G 1 [k] ≥ G 1 [k + η], the duration of the remaining green period (if any) after t 0 + ηT is upper bounded by τ. Therefore ˜ G 1 [k + η] ≤ τ + ˜ G 1 ((k +η)T :t 0 +ηT ). Using this with (C.3) and plugging into (C.2) gives 0≤ η ¯ λ 1 T−c 1 P k+η κ=k+1 ˜ G 1 [κ], which implies that there exists at least one k + ∈ [k + 1,k +η] such that ˜ G 1 [k + ]≤T ¯ λ 1 c 1 . 127 (c) It is sufficient to show that ˜ G 2 [k + ] ≤ ¯ λ 2 T c 2 for some k + ∈ [k + 1,k +η]. Combined withS /2 2 [k + ], this implies ˜ G 2 [k + ]<G 2 [k + ], which impliesU 2 [k + ]. Let t 0 be the time instant when x 2 = 0 for the last time in the green phase of the k-th cycle, and let τ := (k + 1)T−t 0 ≥ 0; see Fig C.1 (bottom) for illustration. x 2 (t 0 ) = 0 implies: 0≤x 2 (t 0 +ηT )−x 2 (t 0 ) = Z t 0 +ηT t 0 λ 2 (s)ds− Z t 0 +ηT t 0 z 2 (s)ds = Z t 0 +η 2 T 2 t 0 λ 2 (s)ds− Z t 0 +ηT t 0 z 2 (s)ds ≤η 2 ¯ λ 2 T 2 − Z t 0 +ηT t 0 z 2 (s)ds =η ¯ λ 2 T− Z t 0 +ηT t 0 z 2 (s)ds (C.4) The last term in (C.4) is the total outflow from link 2 during [t 0 ,t 0 +ηT ], and is lower bounded by the total outflow during times whenx 2 > 0 in [t 0 ,t 0 +ηT ]: Z t 0 +ηT t 0 z 2 (s)ds≥c 2 τ + ˜ G 2 ((k +η)T +G 1 [k +η] :t 0 +ηT ) +c 2 k+η−1 X κ=k+1 ˜ G 2 [κ] (C.5) Note that ˜ G 2 [k +η] = ˜ G 2 ((k +η)T +G 1 [k +η] :t 0 +ηT ) + ˜ G 2 (t 0 +ηT : (k +η + 1)T ) ≤ ˜ G 2 ((k +η)T +G 1 [k +η] :t 0 +ηT ) +τ Using this with (C.5) and plugging into (C.4) gives 0 ≤ η ¯ λ 2 T − c 2 P k+η κ=k+1 ˜ G 2 [κ], which implies that there exists at least onek + ∈ [k+1,k+η] such that ˜ G 2 [k + ]≤T ¯ λ 2 c 2 . 128
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Novel techniques for analysis and control of traffic flow in urban traffic networks
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