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Essays on dynamic control, queueing and pricing

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Essays on dynamic control, queueing and pricing

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Content ESSAYS ON DYNAMIC CONTROL, QUEUEING AND PRICING by Vasiliki Kostami 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 (BUSINESS ADMINISTRATION) August 2010 Copyright 2010 Vasiliki Kostami Acknowledgements I would like to foremost express my gratitude to my supervisor and chair of my thesis committee, professor Amy Ward, for her continuous guidance and inspiration over the years of my doctoral work. I am also heartily grateful to professor Raj Rajagopalan for advising me and encouraging me especially during the last year of my PhD. Without their persistent help, this dissertation would not have been possible. I would also like to express my appreciation to professor Sheldon Ross, the outside member of my committee. I am especially grateful to my brother Dimitris for the emotional support and caring he provided during these five years. I would also like to thank Akis for being tolerant and helping me get through the difficult times. I wish to thank my friend Konstantinos Papakonstantinou for his technical assistance with part of the numerical work. I offer my regards to all of those who supported me in any respect during the completion of the project. Lastly, and most importantly, I wish to thank my parents and my family for the loving and supportive environment. ii Table of Contents Acknowledgements ii List of Tables v List of Figures vi Abstract viii Chapter 1: Introduction 1 Chapter 2: Managing Service Systems with an Offline Waiting Option and Cus- tomer Abandonment 4 2.1 Introduction 4 2.2 Model Formulation 10 2.2.1 System Equations 13 2.3 Revenue Optimization 15 2.3.1 Approximating Problem 16 2.3.2 Solution to Approximating Problem 19 2.4 Waiting Time Quotation 25 2.5 Concluding Remarks 29 Chapter 3: Speed Quality Tradeoffs in a Dynamic Model 30 3.1 Introduction 30 3.2 Literature Review 34 3.3 The Single Period Model 37 3.4 The Multi-Period Model: Constant Price 41 3.5 The Multi-Period Model: Constant Service Speed 48 3.6 The General Multi-Period Model 53 3.7 General Queueing Systems 56 3.8 Conclusions 61 iii Chapter 4: Analysis and Comparison of Inventory Systems: Dynamic vs Static Policies 63 4.1 Introduction 63 4.2 Model Formulation 68 4.3 The Optimal Static Policy 73 4.4 The Optimal Dynamic Policy 78 4.5 Model Comparison 80 4.6 Decision on the Production 85 4.6.1 The Optimal Policy with Static Pricing 85 4.6.2 The Optimal Policy with Dynamic Pricing 88 4.6.3 Model Comparison 91 4.7 Convex Leadtime Cost Functions 92 4.8 Conclusions and Future Research 92 References 96 Appendices 102 A Managing Service Systems with an Offline Waiting Option & Customer Abandonment: Technical Appendix 102 B Managing Service Systems with an Offline Waiting Option & Customer Abandonment: Companion Note 124 C Speed Quality Tradeoffs in a Dynamic Model: Technical Appendix 135 D Analysis and Comparison of Inventory Systems with Dynamic vs Static Policies: Technical Appendix 160 iv List of Tables 2.1 Comparison of Approximated and Simulated Cost to a Simulation [Poisson arrivals with rate 100 per hour, deterministic service with mean 0.01 hours (¹=100), and parameters° = 1,c=40,h=10,r =5, andw I =w O . ] 20 2.2 Comparison of Approximated and Simulated Cost for Overloaded Systems to a Simulation [Poisson arrivals with rate¸ per hour, deterministic service with mean 0.01 hours (¹ = 100), and parameters ° = 1, c = 40, h = 5, r =10 andw I =w O .] 24 2.3 Comparison of Approximated Expected Number of Customers in Inline and Offline queue and Number of Abandonments at Optimal Capacity Allo- cation to a Simulation [Poisson arrivals with rate¸ per hour, deterministic service with mean 0:01 (¹ = 100), and parameters ° = 1, c = 40, h = 5, r =10 andw I =w O .] 24 2.4 Comparison of Actual and Quoted Wait Times for Inline and Offline Queues to a Simulation [Poisson arrivals with rate¸ per hour, deterministic service with mean 0.01 hours (¹ = 100), and parameters ° = 1, ® = 0:5 and w I =w O =1.] 26 2.5 Value of² Such ThatW O (t)¡²<W O (t) for 95%, 98%, and 99% of Cus- tomers Joining Offline Queue [Simulation has Poisson arrivals with rate 100 per hour, deterministic service with mean 0.01 hours (¹ = 100), aban- donment rate° =0:01, andw I =w O =1.] 28 3.1 Sensitivity Analysis for the parameters in the Single Period Model when ¤> ^ ¹. 42 v List of Figures 2.1 The model 10 2.2 Solution to (2.13), ~ ® ? as a function of µ and w I w O for c = 40;° = 1;r = 5;h=7, and¾ =1. 21 2.3 Solution to (2.13), ~ ® ? as a function ofc andr forw I =w O = 1;° = 1;µ = 0:5;h=1, and¾ =1. 22 2.4 Solution to (2.13), ~ ® ? as a function ofc andh forw I =w O = 1;° = 1;µ = 1;r =2, and¾ =1. 22 2.5 Histogram Difference Between Waiting Time Quotation and Actual Wait- ing Time [® =0:5 in a simulation having Poisson arrivals with rate 100 per hour, deterministic service with mean 0:01, and parameters ° = 0.01, and w I =w O =1.] 27 3.1 Service Speed Behavior when price is constant and ¹ 1 < ^ ¹ (¤ 1 = 6, ® = 0:5,° =0:4, and ^ ¹=1:5). 45 3.2 Service Speed Behavior when price is constant and ¹ 1 > ^ ¹ (¤ 1 = 9, ® = 0:5, and ^ ¹=1). 46 3.3 Service Speed Behavior when price is constant and ¹ 1 < ^ ¹ (¤ 1 = 6, ® = 0:5,± =0:6, and ^ ¹=1:5). 47 3.4 Service Speed Behavior with ‘normal’ speed when price is constant and ¹ 1 < ^ ¹ (¤ 1 =6,® =0:5,° =0:4 and± =0:6). 49 3.5 Utilization with Quality Sensitivity. (¤ 1 =6,® =0:5, and° =0:4). 53 4.1 The expected infinite horizon discounted revenue v 0 as a function of the static price p 0 , for the parameter values ¤ = 6, v = 120, ¯ = 4, ± = 0:1, B =100,¹=1,c=10 andh(x)=0 forx2f¡B;¡B+1;:::g. 76 vi 4.2 The maximum expected infinite horizon discounted revenue v ? 0 as a func- tion of ¯, for the parameter values ¤ = 6, v = 120, ± = 0:1, B = 100, ¹=1,c=10 andh(x)=0 forx2f¡B;¡B+1;:::g. 77 4.3 The optimal dynamic pricing policyp ? and value functionv ? d for the param- eter values¤ = 6,v = 120,¯ = 4,± = 0:1,B = 100,¹ = 1,c = 10 and h(x)=0 forx2f¡B;¡B+1;:::g. 81 4.4 v ? d (solid line) andv ? 0 (dotted line) as a function of the starting statex (¤= 6, v = 120, ¯ = 4, ± = 0:1, B = 100, c = 10 and h(x) = 0 for x 2 f¡B;¡B+1;:::g) 82 4.5 The effect of¯ (¤ = 6,¹ = 1,v = 120,¯ = 4,± = 0:1,B = 100,c = 10 andh(x)=0 forx2f¡B;¡B+1;:::g) 83 4.6 v ? d (solid line) andv ? 0 (dotted line) as a function of the starting statex when there is control over the production process (¤ = 6, v = 120, ¯ = 4, ± =0:1,B =100,c=10 andh(x)=0 forx2f¡B;¡B+1;:::g) 89 4.7 Optimal Production Decision: w ? d (solid line) andw ? 0 (dotted line) as a func- tion of the statex (¤ = 6,v = 120,¯ = 4,± = 0:1,B = 100,c = 10 and h(x)=0 forx2f¡B;¡B+1;:::g) 90 4.8 (¤ = 6, ± = 0:1, B = 100, ¹ = 3, c = 10 and h(x) = 0 for x 2 f¡B;¡B+1;:::g) 93 vii Abstract This thesis focuses on the analysis of stochastic models that frequently arise in the manage- ment of service and manufacturing operations. Determining the waiting time, the routing choice or proposing a dynamic pricing policy in stochastic systems are of great importance in practice but can be challenging. Waiting is an integral part of the service experience and customers joining a line to acquire a product can rarely estimate their waiting time so any relevant information is appreciated by them. But sometimes disclosing delay information to the customers may not benefit the provider. In other settings, a manager has to trade off speed and quality. Speeding up a service or manufacturing operation may reduce conges- tion or increase production but at the expense of quality and customer satisfaction. In my dissertation, I focus on modeling these phenomena and obtaining significant managerial insights on how firms can improve the service experience and maximize their revenue. My main findings should be of interest to managers; in laymen’s terms, I show how to estimate wait times in a complex queueing system, how to choose the optimal price and speed to deliver a good product and I provide insights on when firms should reveal more information to customers. viii Chapter 1 Introduction The first topic of my dissertation (Chapter 2) studies a queueing system that offers cus- tomers the choice of either waiting in line or going offline and returning at a dynamically determined future time. The best known example is the FASTPASS system at Disneyland. To operate such a system, the service provider must make an upfront decision on how to allocate service capacity between the two lines. Then, during system operation, she must provide estimates of the waiting times for both lines to each arriving customer. The esti- mation of offline waiting times is complicated by the fact that some offline customers do not return for service at their appointed time. I show that when demand is large and service is fast, for any fixed-capacity allocation decision, the two-dimensional process tracking the number of customers waiting in line and offline collapses to one dimension, and this one- dimensional limit process is characterized as a reflected diffusion with linear drift. The analytic tractability of this one-dimensional limit process allows solving for the capacity allocation that minimizes the average cost when there are costs associated with customer abandonments and queueing. Finally, I show that in this limit regime, a simple scheme, based on Little’s Law, to dynamically estimate in line and offline wait times, is effective. In the second essay of my dissertation (Chapter 3), I consider an important tradeoff organizations face in many environments between the speed of processing and the quality of the product (or service) delivered. The speed at which some services are performed can be varied with a resultant impact on quality. Working faster may result in greater output and less delay but may result in lower quality and dissatisfied customers. In this work, we consider dynamic models in a monopoly setting to explore this trade-off between quality 1 and speed. We also allow the price of the product or service to be a lever in determining demand levels. In a single-period setting, we find that greater sensitivity to quality among customers may result in increased or decreased profits depending on the speed. In the dynamic model, we capture the impact of speed on quality through a change in market potential as a function of the firm’s actual service speed relative to a typical or “normal” speed. We obtain several interesting results and insights. First, in scenarios where a firm may not be able to change prices but can adjust the speed, we find that the firm starts at a speed that may be faster or slower than typical speeds but gradually approaches the “normal” speed over time. In scenarios where speed may be difficult to change over time (e.g. some automated production lines) but price can be changed, we show that the optimal price charged is such that the demand rate remains constant over time, even though the price and market potential are changing. Furthermore, we characterize the conditions under which a firm may work at a slower or faster speed than “normal”. We also characterize the behavior of price and speed in settings where both can be changed over time. Interestingly, we find that firms typically start at a slower speed than normal and increase the speed, price, and demand over time. In the last essay in my dissertation (Chapter 4), I consider a firm that faces the follow- ing pricing and leadtime disclosure decision for a given product, and allows orders for that product to be backlogged. The first option is to offer the same price to all arriving cus- tomers, and disclose no leadtime information. Then, customers know only the steady-state expected leadtime. The second option is to vary the price dynamically, as the system state changes, and to also disclose leadtime information dynamically. This allows the firm to offer customers that face a long leadtime a price discount, and to also price lower when there are many products in inventory. Our objective is to determine which pricing and lead- time disclosure policy attains higher maximum infinite horizon discounted revenue. To do 2 this, we formulate and solve a Markov decision problem for each situation, and then com- pare the two solutions. The system state is the inventory position, and is the order backlog when it is positive and the on-hand inventory when it is negative. We show that in general there is a critical level such that when the initial inventory is above this level, static pricing and no leadtime disclosure yields higher revenue, and when the initial inventory is below this level, dynamic pricing and leadtime disclosure yields higher revenue. We also identify when this critical level is negative infinity; that is, when the maximum revenue for the static policy equals or exceeds the maximum revenue for the dynamic policy in all initial states. We further show that there is the same structure when the system manager also has control over the production process, except that the critical level appears to always be finite. 3 Chapter 2 Managing Service Systems with an Offline Waiting Option and Customer Abandonment 2.1 Introduction An inherent part of the service experience that customers dislike is waiting. In deference to the fact that waiting influences customer evaluation of service (Taylor 1994), service providers aim to minimize wait times. However, it is generally economically infeasible to eliminate waiting. Hence, it is important to manage customers’ perceptions of their wait (see, for example, Maister 1985, Katz et al. 1991, Bitran et al (2008), and to realize that different mechanisms for managing the customer perception of wait time produce different customer reactions (Munichor and Rafaeli 2007). One factor that influences the psychological cost of waiting is whether the customer physically waits in a line, or is offline, and free to engage in other activities. In practice, we observe many different implementations of the offline idea. For example, many restaurants give their patrons wireless devices that signal when a table becomes available. In call cen- ters, the idea of giving customers a call-back option was studied by Armony and Maglaras (2004a, b). Cruises and all-inclusive resorts often allow customers to wander while they wait for space to become available in a desired activity. Student healthcare clinics may offer 4 noncritical drop-in patients who face a long delay in seeing a doctor or nurse the option of returning later in the day. Perhaps the best known real-life example of an offline queue is the FASTPASS R ° sys- tem in Disneyland. For the most popular rides in Disneyland, visitors have a choice. They can either wait in a line or obtain a FASTPASS. The FASTPASS specifies a time at which the visitor can take the ride, making it possible for the customer to visit other parts of the park instead of waiting in a line. The FASTPASS also benefits Disneyland because offline customers may spend money on food or entertainment while they wander around the park. Hence, the offline queue benefits both Disneyland and its customers. The question that then arises is why Disneyland, or any other service provider, does not offer only offline queueing. One compelling reason to maintain an inline queue in addition to an offline queue is that some customers who join the offline queue become consumed in other activities and do not return at their appointed time for service. So the inline queue ensures capacity is not wasted. Also, customers joining the inline queue generally do not leave, and there may be costs other than having idle capacity associated with customers who leave the line. For example, in the amusement park setting, abandoning customers that do not experience certain rides may be foregoing an important element of the park’s value proposition, and thus be less likely to return (eliminating a future revenue source). Finally, customer preference for an inline or an offline wait may change according to the required amount of waiting associated with each option. One convenient implementation of offline queueing is having a reservation system. However, for very popular services, reservations tend to fill quickly. This may be accept- able for a restaurant anxious to maintain an image of exclusivity, but it is unacceptable for many service providers. In particular, in an all-inclusive service setting, such as an amuse- ment park, where customers pay a fixed price for access to a number of different attractions, customers expect to be able to visit any attraction of their choosing throughout the course 5 of a day. In fact, Disneyland attempted to implement a reservation system in the mid-1990s but found that early-arriving guests would quickly book all available reservation capacity on all their most popular rides. Guests arriving after 11am were denied the reservation option (Dickson et al. 2005). Thus, it is important to investigate service models in which customers can choose between inline and offline queueing at the time of their arrival. In our model, the cus- tomers are homogeneous, have linear delay costs that depend on whether the wait is inline or offline, and join the queue that minimizes the cost of waiting. To operate such a system, the service provider must: 1. Make an upfront static decision on how to allocate capacity between the inline and the offline queue, and 2. Provide arriving customers with waiting time estimates in real time for both the inline and offline queue. In some settings, such as a restaurant, where the server is able to communicate with cus- tomers, incorrect waiting time estimates can be corrected. However, in other settings, such as Disneyland or any other amusement park, where communication with offline customers is prohibitively difficult, accurately estimating waiting times is essential. Our objective is to allocate the capacity between the two queues to minimize the aver- age cost, when there are costs associated with customer abandonments and inline queueing and an assumed revenue per customer in the offline queue. The upfront static capacity allocation decision is motivated by the amusement park setting, in which seats on each ride are allocated in predetermined proportions to the inline and the offline queue. We fur- ther dynamically derive wait time estimates that depend on that allocation decision using a simple scheme based on Little’s Law. The difficulty inherent in making such estimates accurately is complicated by the presence of customers in the offline queue who may aban- don, and it is not a priori clear that a simple scheme can work. 6 The capacity-allocation problem is intractable. However, we can solve it explicitly in a heavy-traffic asymptotic regime in which demand is large and close to the service rate, meaning service times are short. Then, capacity utilization is near 100%. The capacity- allocation problem becomes tractable because there is a reduction in problem dimension- ality: the two-dimensional process tracking the number of customers waiting inline and offline collapses to one dimension. Most amusement parks have hundreds of customers arriving per hour for rides popu- lar that and last only minutes. Furthermore, almost every departing train has customers in every available seat. Hence, our heavy traffic analysis is directly applicable to this setting, provided the demand is close to–but not grossly exceed–the service rate. Eventually, the system departs from the heavy traffic regime, where diffusion approximations are appro- priate, and moves to an overloaded regime, where a fluid analysis such as in (Whitt, 2006) becomes relevant. To understand where our analysis and conclusions break down, we pro- vide numerics showing that the performance of our approximations remain accurate for arrival rates that exceed the service rate by as much as 20% and degrades thereafter. The remainder of the chapter is organized as follows. We first review some relevant literature. In Section 2.2, we present our basic model formulation, which is a single-server queue with general interarrival and service times and both inline and offline waiting. In Section 2.3, we solve for the capacity allocation that minimizes average cost as demand becomes large and service fast and demonstrate the accuracy of our solution through sim- ulation. Section 2.4 validates that our wait time estimates are correct as demand becomes large and service fast. Section 2.5 presents concluding remarks. The proofs of all our results can be found in Appendix A. We provide the details of how to extend our model and results to a setting that more closely resembles an amuse- ment park ride (specifically, to a setting in which all customers are served in batches at deterministically spaced intervals) in Appendix B. 7 Literature Review The service model we analyze is novel because it combines the features of customer aban- donment and customer choice. To do this, we have considered a simple scenario in which the customers are homogeneous, the abandonment distribution is given exogeneously, and the delay costs are linear and depend on whether the wait is inline or offline. Previous work that focuses on one of these two features exclusively incorporates heterogeneous cus- tomers. Specifically, Mandelbaum and Shimkin (2000) propose a model in which customer abandonment times are determined by each customer optimizing individual utility function, which balances waiting costs against perceived service benefits, and show that the abandon- ment distribution emerges as an equilibrium point for the model. The models in Armony and Maglaras (2004a) (2004b) do not have customer abandonments, but instead focus on how to manage a system in which customers can choose between the equivalent of inline and offline service, where the offline service is guaranteed to be completed within a max- imum delay. Both of the aforementioned papers are motivated by call center applications and so are multiserver models. In relation to the queueing literature, our model is a variant of a join-the-shorter queue model. The traditional join-the-shorter queue model that is well studied in the queueing literature has no customer abandonment. For this model, under the assumption of expo- nential interarrival and service times, the exact solution for the generating function of the stationary distribution of the number of customers in each queue is known, both in the case in which the two service rates are identical (Flatto and McLean (1977)) and when they are not (Adan et al (1991)). Results for a join-the-shorter-queue model with general interar- rival and service times must rely on an asymptotic analysis, and the heavy traffic analysis is given in Reiman (1984) (along with results on several other models that show state space collapse in a heavy traffic asymptotic regime). 8 Accurate wait prediction is well studied in the service operations and queueing lit- erature. This is because accurate wait prediction improves customer satisfaction. Whitt (1999) shows how to exploit state information, such as the number of customers ahead of the current customer, to dynamically predict the customer waiting time distribution in a multiserver model that can include customer abandonments. Our analysis is rougher in the sense that we provide only a point estimate. However, the point estimate is enough for our purposes because in our asymptotic regime the waiting time quotes we provide to customers coincide with the waiting times customers actually experience. In particular, our waiting time quotation policy is asymptotically compliant in the sense of Plambeck et al (2001). Although the work of Puhalskii (1994) suggests such a result, the presence of customer abandonments complicates the analysis. Finally, our model considers a single-server system operating in isolation. In an amuse- ment park, as discussed by Ahmadi (1997), there is a larger issue of how to manage capac- ity and visitor flow throughout the park. It would be interesting to extend Parlakt¨ urk and Kumar’s (2004) model to investigate how the presence of inline and offline queues and abandonments affects customer routing decisions when there is more than one service sta- tion. In general, a complete analytic analysis that incorporates an arbitrary number of service stations with inline and offline queueing and abandonments appears intractable; however, the model formulation in this essay could be used as input into a simulation model such as that developed in Mielke et al (1998). This would allow for the investigation of further questions of interest from an economics standpoint, such as the one discussed in Oi (1971): Should the amusement park owner set a two-part tariff in which there is one lump sum admission fee into the park and then separate fees per ride? 9 2.2 Model Formulation We begin our analysis with a single-server system in which each arriving customer chooses between waiting for service in a line or going offline and returning for service at a specified future time point, as shown in Figure 2.1. The service discipline is head-of-the-line gener- alized processor sharing. Specifically, when there are customers waiting in both lines, the server processes the customers in the inline queue at rate¹® and those in the offline queue at rate¹(1¡®), for¹ > 0 and®2 [0;1]. We assume each customer in the offline queue may become distracted by other activities while offline and abandon. To model these possi- ble abandonments, we use a nonhomogeneous Poisson process that has rate°[Q O (t)¡1] + at time t, for ° > 0, when the number of customers from the offline queue in the system, including any customer in service, isQ O (t). Inline Queue Offline Queue server abandonment rate γ arrival rate λ service rate = μ α 1-α Figure 2.1: The model Customers choose which queue to join based on an estimation of the waiting times in the inline and offline queues provided by the server. Note that the server must provide wait time estimations, because customers cannot make their own. This is because customers cannot observe the offline queue themselves, and, in many settings, such as amusement 10 parks, also cannot observe the inline queue. The waiting time estimations we propose are oriented to the situation in which accurate estimation is most important–when demand is large and there is little leftover capacity, meaning there are many customers in both queues. For the inline queue, we estimate the wait time as the expected time to serve all the customers, given that the server is splitting the effort between both queues, so that the wait time estimate at timet is W I (t)´ Q I (t) ¹® ; where Q I (t) represents the number of customers from the inline queue in the system, including any customer in service. The parallel waiting time estimation for the offline queue cannot be based onQ O (t), because the server does not see an offline customer aban- don. In particular, the server only realizes the abandonment has occurred after the fact, when the customer fails to return for service at the designated time. So an abandonment cannot be recognized as such by the server until the designated time to receive service has passed. We let O(t) ¸ Q O (t) denote the number of customers in the offline queue recognized by the server, and propose W O (t)´ O(t) ¹(1¡®) as the waiting time estimation for the offline queue. In general, we expect that the waiting time quoteW O (t) will be too high. This is due both to customers who have abandoned the offline queue that the server has not yet seen and customers who are currently in the offline queue but will eventually abandon and not receive service. However, we will show that such an overestimation is small when demand is large and service is fast. In that case, the arrival and service rates are large compared to the abandonment rate, because the abandonment rate is held fixed. Under these conditions, 11 even though the queue sizes are large, the waiting times are much smaller than the mean abandonment times. Hence, the simple wait time estimation suffices. We assume customers are homogeneous in their waiting time costs. Let w I > 0 and w O > 0 be the waiting costs per hour for the inline and the offline queues, respectively. A customer arriving at the system at timet minimizes the cost of waiting by joining the inline queue if w I W I (t)·w O W O (t) and by joining the offline queue otherwise. We choose the division of server effort ® to minimize infinite horizon average cost. There is a cost, c > 0, associated with any customer who abandons who may represent a refund for a service not rendered. There is a holding cost,h>0 per customer, in the inline queue that can be used to penalize the server for the customer’s inconvenience. There is a revenue generated, r > 0 per customer, in the offline queue that can be used to quantify the value of being free to engage in other activities. We assumer < c° so that the offline queue is costly. In the amusement park setting, the costs c and h represent an expected future revenue loss from the customer being less likely to return at a later date and pay another park entrance fee. The parameter r is actually revenue generated per customer while he wanders around the park, because those customers may purchase food and spend money on entertainment. The total cost aftert hours is C(®;t)´cN µZ t 0 °[Q O (s)¡1] + ds ¶ + Z t 0 hQ I (s)ds¡ Z t 0 rQ O (s)ds; (2.1) whereN is a standard Poisson process. We put the® into the notation explicitly to empha- size the dependence of the infinite horizon average cost on it. Let C(®)´ lim t!1 1 t C(®;t): 12 Our objective is min ®2[0;1] C(®): (2.2) 2.2.1 System Equations Before considering how to solve the optimization problem (2.2), we specify the detailed evolution equations for each queue. Letfu i ;i ¸ 1g be an i.i.d. sequence of nonnegative, mean 1 random variables having finite variance ¾ 2 A . Letfv O i ;i ¸ 1g andfv I i ;i ¸ 1g be independent i.i.d. sequences of nonnegative, mean 1 random variables having the same distribution and finite variance¾ 2 S . The renewal processes A(t) ´ maxfi¸0: i X j=1 u j ·¸tg S I (t) ´ maxfi¸0: i X j=1 v I j ·¹tg S O (t) ´ maxfi¸0: i X j=1 v O j ·¹tg represent respectively the cumulative number of arrivals to the system in [0;t] and the cumulative number of departures from the inline and offline queues after the server has devoted t hours to the queue working at rate ¹. Then the evolution equations for Q I and Q O are Q I (t) ´ A(t) X i=1 1fw I W I (t i ¡)·w O W O (t i ¡)g¡S I (T I (t)) (2.3) Q O (t) ´ A(t) X i=1 1fw I W I (t i ¡)>w O W O (t i ¡)g¡N µZ t 0 °[Q O (s)¡1] + ds ¶ ¡S O (T O (t)); (2.4) 13 where T I (t) ´ Z t 0 ®1fQ I (s)>0g ®+(1¡®)1fQ O (s)>0g ds (2.5) T O (t) ´ Z t 0 (1¡®)1fQ O (s)>0g ®1fQ I (s)>0g+1¡® ds: (2.6) Note that d dt T I (t) and d dt T O (t) provide the percentage of effort the server allocates to the inline and offline queues, respectively, at time t. When Q I (t) > 0 and Q O (t) > 0, d dt T I (t) = ® and d dt T O (t) = (1¡®). Otherwise, if either Q I (t) = 0 or Q O (t) = 0, but Q I (t)+Q O (t) > 0, then d dt T O (t) = 1 or d dt T I (t) = 1 accordingly, so that the non-empty queue receives full server effort. Define Q ´ Q I +Q O to be the process tracking the total number of customers in the system. The server must work whenever customers are present, and so I(t)´ Z t 0 1fQ(s)=0gds (2.7) is the cumulative server idle time. Then T I (t)+T O (t)+I(t)=t (2.8) Z 1 0 Q(t)dI(t)=0: (2.9) We have made the simplifying assumption that the customers in the offline queue who are served are all present at the service facility when the server is ready to serve them. This is legitimate because we will show that the waiting time estimates we provide are arbitrar- ily close to the true waiting times experienced by customers in our regime of interest, when demand is large and service is fast (see Theorem 2.2 in Section 2.4). For example, when waiting times are around one hour, it suffices to ask customers to return to the service facil- ity five minutes before the estimated time at which their service will begin and to assume 14 that serviced customers return at this requested time (see the results of our simulation study in Section 2.4). Note that it is difficult to specify the process O exactly. However, if we let W O (t) represent the actual waiting time a customer arriving to the offline queue at time t would experience, we can bound the processO as follows Q O (t)·O(t)·Q O (t) + N µZ t 0 °[Q O (s)¡1] + ds ¶ ¡ N Ã Z [ t¡sup 0·s·t W O (s) ] + 0 °[Q O (s)¡1] + ds ! : (2.10) The lower bound is obvious. To see the upper bound, realize that all customers who have arrived at the offline queue by time t will have either reached the server or abandoned by time t +W O (t). Hence, all customers who have arrived at the offline queue by time £ t¡sup 0·s·t W O (s) ¤ + will have either reached the server or have abandoned by time · t¡ sup 0·s·t W O (s) ¸ + +W O Ã · t¡ sup 0·s·t W O (s) ¸ + ! ·t: Therefore, the server knows at time t all the customers arriving prior to time £ t¡sup 0·s·t W O (s) ¤ + who have abandoned, and the upper bound onO in (2.10) follows. 2.3 Revenue Optimization The capacity-allocation problem (2.2) cannot be solved with an exact analysis. Even in the case of exponential interarrival and service times, gaining insight from solving the Markov decision problem is difficult. Fortunately, we can solve an approximating problem that becomes accurate in our regime of interest when demand is large and service is fast. In Subsection 2.3.1 we derive the approximating problem, and in Subsection 2.3.2, we solve the approximating problem and verify the accuracy of the solution via simulation. 15 2.3.1 Approximating Problem The key to developing a tractable approximating problem for (2.2) is to show that the two-dimensional queue-length process can be described by the following one-dimensional diffusion process. Let ~ X be a Brownian motion having drift µ ´ ¸¡¹ p ¸ and variance ¾ 2 ´ ¾ 2 A +¾ 2 S . Given ~ X, define the regulated Ornstein-Uhlenbeck process on[0;1) ~ Q(t)´ ~ X(t)¡° (1¡®)w I ®w O +(1¡®)w I Z t 0 ~ Q(s)ds+ ~ I(t)¸0; (2.11) for ~ I a nondecreasing process having ~ I(0)=0 and R 1 0 ~ Q(t)d ~ I(t)=0. Consider a system in which the arrival rate ¸ becomes large and the service rate is defined as an increasing function of¸. The abandonment rate°, the server-sharing constant ®, and the waiting costs w I and w O all remain constant. Our convention is to superscript any process or quantity associated with the system having arrival rate¸ by¸. Theorem 2.1. Consider a system having arrival rate¸ and service rate¹(¸)´ ¸¡ p ¸µ for someµ2<. (i) For anyT >0,sup 0·t·T 1 p ¸ ¯ ¯ w I ® Q ¸ I (t)¡ w O 1¡® Q ¸ O (t) ¯ ¯ !0, in probability, as¸!1. (ii) For ³ ~ Q; ~ I ´ defined by (2.11) in which ~ X is a Brownian motion with infinitesimal driftµ and infinitesimal variance¾ 2 , ³ Q ¸ p ¸ ; I ¸ p ¸ ´ ) ³ ~ Q; ~ I ´ , as¸!1. The following corollary to Theorem 2.1 allows us to state a tractable approximating problem for the original capacity-allocation problem (2.2). 16 Corollary 2.1. Consider a system having arrival rate¸ and service rate¹(¸)´¸¡ p ¸µ for someµ2<. As¸!1, Q ¸ I p ¸ ) ®w O (1¡®)w I +®w O ~ Q Q ¸ O p ¸ ) (1¡®)w I ®w O +(1¡®)w I ~ Q N ³ R ¢ 0 ° £ Q ¸ O (s)¡1 ¤ + ds ´ p ¸ ) ° (1¡®)w I ®w O +(1¡®)w I Z ¢ 0 ~ Q(s)ds: Specifically, Corollary 2.1 suggests that for the total cost up to timet,C(®;t), defined as in (2.1), C(®;t) p ¸ ¼ · (c°¡r) (1¡®)w I ®w O +(1¡®)w I +h ®w O ®w O +(1¡®)w I ¸Z t 0 ~ Q(s)ds: Hence, for large t, letting the random variable ~ Q(1) have the steady-state distribution of the process ~ Q in (2.11), and noting thatP ³ lim t!1 t ¡1 R t 0 ~ Q(s)ds!E h ~ Q(1) i´ = 1, it follows from the definition ofC(®) that 1 p ¸ C(®)¼ · (c°¡r) (1¡®)w I ®w O +(1¡®)w I +h ®w O ®w O +(1¡®)w I ¸ E h ~ Q(1) i : Proposition 18.3 in Browne and Whitt (1995) shows that forÁ and© the density and cumu- lative distribution functions, respectively, of a standard normal random variable and ·´° (1¡®)w I ®w O +(1¡®)w I ; the steady-state mean of the process ~ Q is E h ~ Q(1) i = µ · + ¾ p 2· Á ³ ¡µ ¾ q 2 · ´ 1¡© ³ ¡µ ¾ q 2 · ´: (2.12) 17 Therefore, defining ~ C(®)´ · (c°¡r)(1¡®)w I ®w O +(1¡®)w I + h®w O ®w O +(1¡®)w I ¸ 0 B @ µ · + ¾ p 2· Á ³ ¡µ ¾ q 2 · ´ 1¡© ³ ¡µ ¾ q 2 · ´ 1 C A ; it follows that the problem min ®2[0;1] ~ C(®) (2.13) approximates the original capacity allocation problem in (2.2). In particular, min ®2[0;1] C(®)¼ p ¸ min ®2[0;1] ~ C(®); Letting® ? and ~ ® ? denote the respective capacity allocations that result in the minimum cost in (2.2) and (2.13), we expect that® ? ¼ ~ ® ? . It is interesting to compare the optimization problem in (2.13) to the solution for the case when there is no abandonment. In this case, the inline queue is costly, and the offline queue provides revenue, so it is clear that it is optimum to have only an offline queue. We can also see this in the analysis by adapting the objective function in (2.13) to the case when there is no abandonment, as follows. Similar to the setting in Reiman (1984) (the difference being that his setting has two servers with equal service rates instead of a single server with processor-sharing), Theorem 2.1 holds, except that the process ~ Q is a reflected Brownian motion with drift µ and variance ¾ 2 . When µ < 0, the steady-state mean of ~ Q is¾ 2 =jµj. (See, for example, equation (12) in Section 5.6 in Harrison (1985).) Hence, the objective (2.13) becomes min ®2[0;1] µ ¡r (1¡®)w I ®w O +(1¡®)w I +h ®w O ®w O +(1¡®)w I ¶ ¾ 2 2jµj : 18 The minimum occurs at® =0, so that having only an offline queue is optimum. In general, the solution to (2.13) has ~ ® ? 2 [0;1]. Hence the presence of customer abandonments provides the cost trade-off between inline and offline queueing that makes maintaining both an inline and an offline queue desirable. 2.3.2 Solution to Approximating Problem The optimization problem in (2.13) minimizes a continuous function over a bounded region and so is always solvable numerically. It is easily analytically tractable when µ = 0, and, in Subsection 2.3.2, we present the closed form expression for ~ ® ? . In Subsection 2.3.2, we solve (2.13) numerically to understand the effect of the cost parameters r, c, h, w I , and w O and the capacity parameterµ on ~ ® ? . In both subsections, we present simulation results that validate determining the optimum capacity allocation for the original problem (2.2) by solving the approximating problem (2.13). Case: µ =0. For intuition, we solve (2.13) in the case that there is exact balance between the arrival and service rates so thatµ =0. Then (2.13) becomes min ®2[0;1] f(®); (2.14) where f(®)´ ¾ p ¼ p ° µ ® 1¡® w O w I +1 ¶ ¡1=2 µ c°¡r+h w O w I ® 1¡® ¶ : The functionf has first derivative f 0 (®)= ¾ 2 p ¼ p ° w O w I 1 (1¡®) 3 µ ® 1¡® w O w I +1 ¶ ¡ 3 2 0 @ ® ³ c°¡r¡2h+h w O w I ´ +2h¡(c°¡r) 1 A : 19 ® Simulated Cost(C(®)) Approximated Cost ³ p ¸f(®) ´ Error 0.0 19.006 19.747 3.90% 0.1 19.639 19.328 1.59% 0.2 19.401 18.923 2.46% 0.3 18.733 18.544 1.01% 0.4 18.461 18.209 1.36% 0.5 18.793 17.952 4.48% 0.6 17.264 17.841 3.34% 0.7 18.188 18.026 0.89% 0.8 18.082 18.923 4.65% 0.9 24.031 22.302 7.20% Table 2.1: Comparison of Approximated and Simulated Cost to a Simulation [Poisson arrivals with rate 100 per hour, deterministic service with mean 0.01 hours (¹ = 100), and parameters° = 1,c=40,h=10,r =5, andw I =w O . ] The solution ~ ® ? = c°¡r¡2h c°¡r¡2h+h w O w I is valid when2h< (c°¡r) and has ~ ® ? 2 (0;1). This is becausef 0 (~ ® ? ) = 0,f 00 (~ ® ? ) > 0, f 0 (0) < 0,f(®)!1 as®" 1, and sof(~ ® ? ) < f(0) andf(~ ® ? ) < lim ®"1 f(®). Note that when there are no holding costs for the inline queue (h = 0), ~ ® ? = 1, and so it is optimum to only maintain an inline queue, which matches intuition. Otherwise, in the case that 2h¸ (c°¡r), it follows thatf 0 (®)¸ 0 for all®2 [0;1]. Then the minimum achievable cost occurs at ~ ® ? =0, so having only an offline queue is optimum. Recall that Corollary 2.1 suggests thatC(®)¼ p ¸f(®). Table 2.1 shows via simulation that the error in this approximation is low (less than 10%) when the system arrival and service rates are 100 or more. (The approximation error decreases as the mean interarrival and service times become shorter, consistent with Corollary 2.1; however, we do not show these simulation results because of space considerations.) The error sizes in Table 2.1 are indicative of the error sizes in the approximations suggested by Corollary 2.1 for both the expected queue lengths and the total number of customer abandonments. 20 All simulation runs shown in Table 2.1, and in every table in this chapter, are run long enough to generate 10,000,000 arrivals. This ensures that the system has settled into its steady state. Case: µ6=0. In the case that the system is either overloaded or underloaded, we can solve (2.13) numer- ically. Figure 2.2 shows that for any values of w I and w O , there exist µ;µ 2 < such that whenµ2 (µ;µ) it is optimal to maintain both an inline and an offline queue (~ ® ? 2 (0;1)). Otherwise, whenµ = 2(µ;µ), maintaining only one queue (~ ® ? =0 or ~ ® ? =1) is optimal. This is representative of the behavior we find in general, regardless of the specific parameter values of c, °, r, h, and c. In particular, as µ becomes small, meaning the ser- vice capacity is exceeding the arrival rate by more and more, having only an inline queue, ~ ® ? = 1, eliminates all abandonment costs and produces a very small inline holding cost, because waiting times are negligible. Otherwise, asµ becomes larger, ~ C(®)¼ · (c°¡r) (1¡®)w I ®w O +(1¡®)w I +h ®w O ®w O +(1¡®)w I ¸ µ · ; 0 2 4 6 8 10 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 θ wI/wO Figure 2.2: Solution to (2.13), ~ ® ? as a function ofµ and w I w O forc=40;° =1;r =5;h=7, and¾ =1. 21 and the right-hand side is minimized at® =0. Note thatµ=· is the steady-state mean of an unregulated Ornstein-Uhlenbeck process that has the same infinitesimal mean and variance as ~ Q in (2.11). The term µ=· in the preceding display is reflective of the fact that the idleness process in a very heavily loaded system rarely increases; in particular, the process ~ Q behaves similarly to the unregulated process having ~ I(t)=0 for allt¸0. We also observe that Figure 2.2 is consistent with the solution for ~ ® ? in Subsection 2.3.2 whenµ =0. In particular, ~ ® ? increases as the ratiow I =w O increases. 0 20 40 60 80 100 0 5 10 0 0.2 0.4 0.6 0.8 1 c r Figure 2.3: Solution to (2.13), ~ ® ? as a function of c and r for w I =w O = 1;° = 1;µ = 0:5;h=1, and¾ =1. 0 50 100 150 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 h c Figure 2.4: Solution to (2.13), ~ ® ? as a function of c and h for w I =w O = 1;° = 1;µ = 1;r =2, and¾ =1. 22 When abandonment costs are low, because customers in the offline queue generate revenue, we expect costs to be minimized by maintaining only an offline queue. Figures 2.3 and 2.3 confirm this intuition. In particular, when c is low relative to either r or h, ~ ® ? =0, and asc becomes high relative tor orh, ~ ® ? becomes positive and increases. Corollary 2.1 suggests that for an unbalanced system(µ6=0) min ®2[0;1] C(®)¼ p ¸ min ®2[0;1] ~ C(®) when the arrival rate is within order p ¸ of the service rate. Tables 2.2 and 2.3 confirm that this is the case. In particular, our approximation has less than 10% relative error for arrival rates that are as high as 120 per hour when the service rate is 100 per hour. We focus on the case that the arrival rate exceeds the service rate because this is when our approximations are most relevant; otherwise, when the service rate exceeds the arrival rate, the wait times are small (as can be seen in Table 2.4), so accurate approximation is not as important. Note that to find® ? we simulated the total cost for various values of®(® =0;0:1;0:2;:::;0:9) and chose the minimum cost value. In all cases, the minimum cost value was achieved at exactly the ~ ® ? predicted by our approximation (2.13). Also note that for the given values of¸ and¹, the value ofµ is specified as in Corollary 2.1; i.e.,µ =¸ ¡1=2 (¸¡¹). We note that Tables 2.2 and 2.3 also show that our proposed cost approximation breaks down as the arrival rate increases far past the service rate, by more than 20%. This is not surprising, because the system is moving out of a heavy traffic regime and into an overloaded regime, where a fluid analysis becomes relevant. 23 ¸ ¸¡¹ ¹ ® ¤ Approx. Cost ( p ¸ ~ C(~ ® ? )) Simulated Cost (C(® ? )) Error 150 50% 0.0 122.47 149.98 18.34% 140 40% 0.0 101.42 120.08 15.54% 130 30% 0.0 78.944 90.019 12.30% 120 20% 0.0 55.076 60.266 8.61% 110 10% 0.0 32.346 33.529 3.53% 109 9% 0.1 30.316 31.735 4.47% 108 8% 0.2 28.288 29.557 4.29% 107 7% 0.3 26.266 27.184 3.38% 106 6% 0.4 24.255 25.193 3.72% 105 5% 0.5 22.269 23.861 6.67% 104 4% 0.5 20.278 20.839 2.69% 103 3% 0.6 18.297 19.064 4.03% 102 2% 0.7 16.374 17.258 5.12% 101 1% 0.7 14.504 15.108 4.00% 100 0% 0.8 12.616 13.622 7.39% Table 2.2: Comparison of Approximated and Simulated Cost for Overloaded Systems to a Simulation [Poisson arrivals with rate ¸ per hour, deterministic service with mean 0.01 hours (¹=100), and parameters° = 1,c=40,h=5,r =10 andw I =w O .] E[Inline Queue-length] E[Offline Queue-length] # of Abandonments ¸ ® ¤ Approx. Error Approx. Error Approx. Error 150 0.0 0 N/A 40.825 18.28% 4082480 22.51% 140 0.0 0 N/A 33.806 15.58% 3380620 18.23% 130 0.0 0 N/A 26.315 12.36% 2631450 13.99% 120 0.0 0 N/A 18.359 8.69% 1835850 9.64% 110 0.0 0 N/A 10.782 3.38% 1078180 6.16% 109 0.1 1.1024 4.87% 9.9218 4.70% 992176 3.94% 108 0.2 2.2631 0.73% 9.0523 4.67% 905231 3.10% 107 0.3 3.5021 0.97% 8.1715 3.62% 817154 3.19% 106 0.4 4.8510 5.73% 7.2765 3.49% 727649 2.30% 105 0.5 6.3624 8.83% 6.3625 2.12% 636255 0.56% 104 0.5 5.7936 5.01% 5.7936 2.26% 579364 1.62% 103 0.6 7.3186 6.62% 4.8791 3.33% 487907 0.45% 102 0.7 9.1696 8.94% 3.9298 3.86% 392983 1.70% 101 0.7 8.1221 2.39% 3.4809 4.63% 348089 3.66% 100 0.8 10.093 13.69% 2.5231 7.51% 252313 3.91% Table 2.3: Comparison of Approximated Expected Number of Customers in Inline and Offline queue and Number of Abandonments at Optimal Capacity Allocation to a Simula- tion [Poisson arrivals with rate¸ per hour, deterministic service with mean0:01 (¹=100), and parameters° = 1,c=40,h=5,r =10 andw I =w O .] 24 2.4 Waiting Time Quotation We claimed in Section 2.2 that the waiting time estimations we proposed for the inline and offline queues at timet, W I (t)= Q I (t) ¹® andW O (t)= O(t) ¹(1¡®) ; were very close to the waiting time a customer joining either queue at timet would expe- rience in our parameter regime of interest, when arrival and service rates are close and large compared to the abandonment rate. This is not surprising for the inline queue. How- ever, this is not obvious for the offline queue, because some customers in the offline queue may abandon, and the process O bounded in (2.10) includes customers who have already abandoned the offline queue but of whom the server is not yet aware. Our next theorem shows thatW I (t) is very close to the actual waiting time a customer joining the inline queue at time t would experience, which we denote by W I (t), and that W O (t) is very close to the actual waiting time a customer joining the offline queue at time t would experience, which we denote byW O (t). As in Theorem 2.1, we consider a system in which the arrival rate ¸ becomes large and the service rate is defined as an increasing function of¸, and we superscript any process or quantity associated with the system having arrival rate¸ by¸. Theorem 2.2. Consider a system having arrival rate¸ and service rate¹(¸)´ ¸¡ p ¸µ for someµ2<. For anyT >0, as¸!1 sup 0·t·T p ¸ ¯ ¯ W ¸ I (t)¡W ¸ I (t) ¯ ¯ !0 and sup 0·t·T p ¸ ¯ ¯ W ¸ O (t)¡W ¸ O (t) ¯ ¯ !0; in probability: Theorem 2.2 shows that our waiting time quotations are very accurate when the arrival rate¸ is large and within p ¸ of the service rate. As in Section 2.3.2 when considering the 25 accuracy of our proposed cost function approximation, we would also like to understand how our proposed waiting time quotations perform as the arrival rate increases past the service rate. To do this, in Table 2.4 we simulate a system having fixed capacity allocation and for every arriving customer, we record the actual wait time and the wait time quote in minutes. We then report the average actual inline and offline wait times and the average absolute difference between the actual and quoted wait times in minutes. We expect that the inline wait time quotes will be very accurate, even outside the parameter regime stated in Theorem 2.2. Because the service times are deterministic, eliminating an inherent system variability, the only reasons the inline wait time quotes should not match actual wait times would be because of the residual service times and a possibly empty offline queue. We include them in Table 2.4 as a benchmark for comparison purposes. Inline Queue Offline Queue ¸ Avg. Wait Avg. Abs. Diff. Avg. Wait Avg. Abs. Diff. P(abandon) 150 61.57 0.0133 41.65 18.5623 0.50 140 49.38 0.0223 35.30 12.8790 0.44 130 37.37 0.0216 28.30 8.0390 0.38 120 25.23 0.0197 20.27 4.0955 0.29 110 13.50 0.0170 11.39 1.4332 0.17 100 5.96 0.0401 4.99 0.3477 0.08 Table 2.4: Comparison of Actual and Quoted Wait Times for Inline and Offline Queues to a Simulation [Poisson arrivals with rate ¸ per hour, deterministic service with mean 0.01 hours (¹=100), and parameters° =1,® =0:5 andw I =w O =1.] We conclude that the offline wait time quotes are very accurate for parameters that sat- isfy the conditions of Theorem 2.2. (When the arrival rate is less than the service rate, the wait times in both queues are very small, so accurate wait time quotation is not so impor- tant.) It is also true that the accuracy of the offline wait quotes decreases monotonically as the arrival rate increases past the service rate. This is because the percentage of customers abandoning the offline queue increases monotonically as the arrival rate increases past the service rate. 26 0 200000 400000 600000 800000 1000000 1200000 1400000 -1.2 -0.9 -0.6 -0.3 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 w a i t i n g t i m e q u o t a t i o n e r r o r (i n m i n) # o f o b s e r v a t i o n s o u t o f 5, 0 0 0, 0 0 0 o f f l i n e a r r i v a l s Figure 2.5: Histogram Difference Between Waiting Time Quotation and Actual Waiting Time [® =0:5 in a simulation having Poisson arrivals with rate 100 per hour, deterministic service with mean0:01, and parameters° = 0.01, andw I =w O =1.] Recall that the system evolution equations in (2.3)-(2.9) make the simplifying assump- tion that the customers in the offline queue who are served are all present at the service facility when the server is ready to serve them. How can we ensure that this is indeed the case? Theorem 2.2 suggests that for a system in which the arrival and service rates are close and large compared to the abandonment rate, we can simply ask customers to return a little before their estimated service time. Figure 2.5 quantifies the meaning of “a little” for one particular example in which the arrival and service rates are 100 customers per hour, the average offline wait time is 47.22 minutes, and the probability a customer abandons is 0.82%, meaning° = 0:01. (Note that we have changed the abandonment rate from that in the previous paragraph so that the average wait time in the offline queue will be more than 5 minutes.) The largest wait time quotation error we see over 5,000,000 customer arrivals to the offline queue is 10 minutes; therefore, if we have customers return to the service facility 10 minutes before their estimated service time, we would ensure that all served customers are present at the service facility when the server is ready to serve them. 27 In general, the amount of time before the estimated offline wait time that customers must return to the service facility to ensure their presence when it is desired varies accord- ing to the system parameters and the average offline waiting time. Suppose we ask a cus- tomer who chooses to join the offline queue at timet to return to the service facility at time W O (t)¡². Table 2.5 shows what the value of ² must be to ensure that 95%, 98%, and 99% of the customers choosing to join the offline queue are present at the service facility when desired. For example, in a system having arrival and service rates of 100 customers per hour, abandonment rate ° = 0:01, and ® = 0 (so there is only an offline queue), asking customers to return 1.24 minutes before their estimated service time ensures 99% of the customers are present when required. This 1.24 minutes is a negligible amount of “padding” when the average offline wait time is 34.72 minutes. Of course, the system evolution equations in (2.3)-(2.9) are not exactly modeling what happens for the small per- centage of customers for which we grossly err, i.e., for those for whichW O (t)¡²>W O (t). Are these customers effectively abandoned? Are they absorbed into the offline queue when they appear? In either case, when their percentage is small enough, their effect on the overall system behavior is negligible, so our model is representative of the overall system behavior. ² in min to achieve Simulated avg. offline wait ® 95% 98% 99% (in min) 0.0 0.51 0.10 1.24 34.72 0.1 0.76 1.08 1.41 35.87 0.2 0.62 1.40 1.79 38.92 0.3 1.06 1.44 1.81 39.51 0.4 1.34 1.76 2.18 43.11 0.5 1.67 2.15 3.11 47.22 0.6 2.13 3.26 3.83 56.26 0.7 2.89 3.65 5.16 60.56 0.8 5.34 6.61 7.89 80.91 0.9 7.81 12.41 17.01 105.73 Table 2.5: Value of² Such ThatW O (t)¡²<W O (t) for 95%, 98%, and 99% of Customers Joining Offline Queue [Simulation has Poisson arrivals with rate 100 per hour, deterministic service with mean 0.01 hours (¹=100), abandonment rate° =0:01, andw I =w O =1.] 28 2.5 Concluding Remarks In this chapter, we analyze a single-server system with two waiting modes: inline and offline. Customers have linear delay costs and pick the mode with the smaller delay cost based on their waiting time quote. The customers who join the offline queue may aban- don. We show that when demand is large and service is fast, the two-dimensional pro- cess tracking the number of customers waiting in line and offline can be described by a one-dimensional reflected diffusion with linear drift. The analytic tractability of this limit process allows us to provide an approximation of the capacity allocation that minimizes the average cost. Moreover, we can accurately predict the waiting time of any new arrival using a simple scheme based on Little’s Law, despite the abandonments that may occur in the offline queue. We demonstrate the accuracy of our approximations via simulation. We end by noting that our results continue to hold in a setting that more closely models an amusement park ride, in which customers are served in batches at discrete time points. For the details of this setting, we refer the interested reader to Appendix B. 29 Chapter 3 Speed Quality Tradeoffs in a Dynamic Model 3.1 Introduction Organizations face numerous tradeoffs in managing their operations. A key tradeoff faced by organizations in both for-profit and non-profit sectors is the one between speed and qual- ity. Consider a restaurant which attempts to provide high quality service to customers by pampering them and paying careful attention to their every need. This will result in highly satisfied customers who keep returning to the restaurant even if it charges high prices. On the other hand, this may result in slower service and longer waits for other customers. Some customers may not want to wait and may leave the restaurant. On the other hand, rushing the customers to provide quick service may result in poor quality and unhappy customers who may never return. Similar tradeoffs occur in other service environments such as health- care, call centers, etc. Such a tradeoff is not limited to service environments. Workers at a manufacturing plant may work faster to produce more output but this may result in lower quality which is not detected until the customer uses the product. Thus, as firms compete for customers, they have to compete on the multiple dimensions of speed, quality and price. A firm that develops a high quality reputation may be able to charge higher prices. Thus, the tradeoff is not just between quality and speed, but along all the three dimensions. Healthcare is an important sector where this tradeoff exists. For instance, doctors offices and urgent care facilities need to deliver a high level of quality to their patients by providing 30 adequate attention to each of them. Spending less time with a patient can lead to poor out- comes in a clinical sense as well as in terms of patient satisfaction. In a study of endoscopy procedures, Cohen (2008) points out that increased emphasis on doing more procedures can result in more errors due to factors such as not following standard processes, work- ing faster to finish the procedure and fatigue. Kc and Terwiesch (2009) show the impact of increased demand on service rate or speed and the resultant negative impact on patient quality. Therefore, a hospital that takes better care of its patients by spending more time with them will gain a better reputation and attract more patients. In a study based on over 20 years of data, researchers at the University of California, San Diego, found that hospi- tal pharmacists committed more prescription errors during periods of increased workload which resulted in significantly higher fatalities (Croasmun 2005). Flynn and Barker (2000) emphasize that workload can be an important contributor to medication errors. They point out that several studies showed that increased workload resulted in prescription errors and that one study showed that“the risk of error increases when the pharmacist fills more than 10-12 prescriptions per half-hour”. Oliva and Sterman (2001) and Anand, Pac and Veer- araghavan (2009) provide several other examples of such tradeoffs in the service industry and Lovejoy and Sethuraman (2000) identify similar issues in manufacturing. Call centers represent another environment where speed and quality of service needs to be balanced carefully. What is the most important metric in a call center: speed of answer, quality of the call, customer satisfaction or all of them? Responding to the customer’s needs and satisfying them by answering their queries in detail is critical for the success of any call center. On the other hand, an agent at a call center cannot spend an inordinate amount of time with customers because this will reduce productivity and increase wait times for other customers. Ren and Zhou (2008) and Hasija et al. (2008) mention this tradeoff in the call center environment. 31 In the manufacturing context, consider Coverking (www.coverking.com) a privately owned firm that designs and manufactures custom car covers, seat covers, etc. Their value proposition is in producing custom, make-to-order products designed to fit the particular vehicle with a short turnaround time. They are able to charge a higher price due to the cus- tomization and high quality. When demand is high causing in high congestion levels, this results in long lead times and an inability to meet specified lead times. Many make-to-order manufacturing companies want to maintain a high quality level and at the same time stick to the promised delivery time. Speeding up the process to meet demand can result in lower quality, poorer reputation and lower market share and prices. Toyota recently admitted that they had to delay several products in order to address quality issues. In his testimony, the Toyota CEO said that the reason why a number of quality control issues cropped up last year was due to the company’s rapid growth. (http://www.leftlanenews.com/toyota-ceo- rapid-growth-hurts-quality.html) In this essay, we consider the key tradeoff between the speed of processing and quality of the product (service) delivered but we also consider price as a lever to control demand. As described in the previous examples, high quality service usually implies longer time spent with the customer but this may result in happier customers. In some contexts, the customer does not observe the service speed (e.g. auto repair or manufacturing) but faster speed may result in a lower quality product which negatively impacts customer satisfaction. Happier customers often implies loyal customers and so the firm’s future demand potential may be higher and we model this aspect. The actual demand the firm realizes in the future, however, also depends on the price charged. Slower service speeds may have a negative effect in that they can result in higher congestion levels and waiting times. While we consider these wait times in our model, we focus on a different and perhaps more important dimension of quality which is the customer experience with the service or product. 32 In our model, the firm optimizes its profit by modifying the speed of processing and the price of the product. The expected profit consists of the revenue less costs due to congestion and costs associated with poor quality or unhappy customers. We analyze this model in both single period and multi period scenarios. In the single period model, we model the cost of lower quality by including a term in the profit function that reflects this cost as an immediate outcome of working at a speed that is faster than the “normal” speed. This normal speed can be thought of as a base processing time as in Lovejoy and Sethuraman (2000) or a benchmark service rate or speed of processing as discussed in Anand, Pac and Veeraraghavan (2009) – we discuss this aspect of the model in greater detail in section 3.3. In the multi-period model, in any period, the speed of the process relative to the normal speed determines the quality of the output which impacts the demand potential in the next period. The actual demand realized in a period is a function of the price charged. Thus, the two levers the firm has to optimize profits in each period are the price charged and the speed of processing. Given the tradeoffs discussed, we address the following questions. If the firm has the flexibility, will it work faster or slower than the normal speed? How would the speed and price change over time? Will the speed change monotonically over time or will it fluctuate above and below the normal speed? Will prices decrease or increase over time and will they be higher or lower than that charged by a monopolist when this tradeoff does not exist? Will utilization levels be lower or higher and how will utilization levels change over time? The main results of this essay are as follows. First, in the single period formulation, we show that the service provider will slow down but will increase the price charged when customers care more about the quality of the product offered. On the other hand, when customers are more sensitive to waiting, she will increase the speed of the service and the optimal price will be higher. Second, in the multi-period model when price is constant over time but endogenous, we show that independent of the starting speed which can be slow 33 or fast compared to the ‘normal’, the firm will speed up or slow down over time until it reaches the ‘normal’ speed and it is stable thereafter. Third, we study the scenario where speed is endogenous but cannot be changed over time and prices can be varied over time to control demand and thus, the system congestion. Interestingly, we find that the demand rate remains constant over time and so does the congestion level. Also, depending on the parameters, the firm may work at a slower or faster speed than the ‘normal’ speed. Finally, we consider the case where both prices and service speed can vary over time. Interestingly, in most scenarios, price increases over time but despite this, demand also increases over time. This is because the market potential increases at a fast enough rate over time which in turn is due to the fact that the firm typically starts at a slow speed which is lower than the ‘normal’ speed and gradually increases the speed over time. In this chapter, we will use the terms service and product interchangeably because as discussed above their role in the model is the same. We will also refer to the speed of the service as the service rate interchangeably. The rest of the chapter is organized as follows. In Section 3.2, we discuss the existing literature and highlight our contribution relative to the literature. Our single model formu- lation and its main insights are presented in section 3.3. In Section 3.4, we discuss the multi-period model in contexts where price is constant over time and in Section 3.5, we discuss situations where the service speed cannot change over time. The general multi- period model is presented in Section 3.6. In Section 3.7, we extend some of the results under a G/G/1 framework and finally in Section 3.8, we summarize the main contribution and propose further directions for research. 3.2 Literature Review There is limited literature in Operations Management that studies the speed quality interac- tion. Lovejoy and Sethuraman (2000) is among the earliest papers to consider the tradeoff 34 between quality and speed using a M/M/1 queuing model in a specific manufacturing con- text – rushing jobs to meet scheduled deadlines which can reduce production yield and increase defects. They consider many details such as inspection, labor and material costs, etc. that we do not consider, but, unlike us, they do not consider price and dynamic trade- offs. In the same vein, Lu et al. (2009) consider the fact that the time spent with a product determines the probability of a good quality product but our focus is very different. Hasija et al. (2009) discuss models of speed up behavior and explain why the staffing level cannot determine the actual service speed that can vary. There exists some additional literature that studies the tradeoff between quality and congestion but these papers consider waiting time or service level as the dimension of quality, while we focus on a different dimension of quality which is the customer experience with the service or product. Some examples of this literature are Allon and Federgruen (2007) which considers competition between firms in a service setting where the demand rate depends on the price and the service level and Gans (2002), like Png and Reitman (1994), which measures quality in terms of the delay but uses the idea of a threshold in the service speed that determines the quality level. In a novel model, Hopp, Iravani and Yuen (2007) allow workers to adjust time spent with the customer depending on current workload and the value or revenue from the customer is an increasing function of the time spent. We do not assume this to be the case and our approach is more appropriate for services where the price charged is based on a specific service provided and not a function of the time spent. Veeraraghavan and Debo (2009) present counterintuitive findings to explain why longer queues may imply better service in some environments. The work closest to ours has been done by Anand, Pac and Veeraraghavan (2009). In their work, they focus on customer-intensive services and model explicitly the dependence of service quality on service duration to provide the equilibrium behavior of customers and optimal quality-speed tradeoff in both single and multiple-server queuing settings. While 35 we obtain results similar to theirs in a single period setting, our main focus is on a dynamic setting wherein the demand potential is a function of the speed in the previous period. In a dynamic setting, we show that price increases over time and still demand rate increases, in services that require more time with the customer similar to a result in their ‘high customer- intensive’ case. Our main contribution lies in characterizing the optimal dynamic policy with respect to the service speed and price in three different scenarios and providing new insights based on these results. To our knowledge, there is no previous literature that has addressed such a dynamic tradeoff between quality and speed. The productivity, congestion and quality tradeoff often arises in the call center litera- ture, although quality in this literature is often a function of wait time. Firms that decide to outsource their call centers have to decide on a contract that ensures a certain service level along with low congestion levels. Hasija et al. (2008) discuss contracts where penalties are imposed when agreements on the waiting times or service levels are not satisfied. In Ren and Zhou (2008), service quality depends on the staffing level; longer time with the cus- tomer often leads to call resolution and thus to high service quality, an assumption similar to ours. There are a few recent empirical papers in the operations literature that explore the speed quality tradeoff. As mentioned earlier, Kc and Terwiesch (2009) show that overwork- ing is associated with quality reduction in hospital operations, a situation also discussed by Needleman et al. (2002). They also point out that hospitals accelerate the service rate in the short term in response to an increased load. Ren and Wang discuss an empirical study of the relationship of the service quality and the patient volume in hospitals. Oliva and Sterman (2001) provide some anecdotal evidence and show, using simulation, that working overtime may lead to quality issues. 36 3.3 The Single Period Model We consider a monopolist firm selling a product or service to homogenous consumers in a market with a demand potential¤. The firm charges a pricep for its product. We consider a typical downward sloping demand function given by ¸=¤¡®p; where¸ is the demand rate and® is the price sensitivity of the customers. So, the effective demand rate ¸ is a function of the demand potential ¤ and the price charged p. In this section, we consider a single-period setting and model it as an M/M/1 queueing system with a First Come First Served queue discipline similar to the approach in Anand et al (2009) and Ren and Zhou (2008). Later in Section 3.7, we will discuss how some of our results can be extended to a more general queueing system. Customers arrive to the system according to a Poisson process with demand rate ¤. Other than price, which controls the demand rate, the firm also determines the speed of service (rate¹) or service times; service times are assumed to be independent exponential. To ensure stability in the system, we need¸<¹. An important characteristic of our model is the modeling of service speed and its impact on customer satisfaction. In many environments, the quality of the product and customer satisfaction is a function of the service speed or service time. Customers consider a ser- vice to be of ‘normal’ quality when the service provider has spent an adequate amount of time. We assume that the quality of service increases when the service time increases. For instance, if a professor spends more time going through exams and provides detailed com- ments, this takes more time but the students feel they are getting a higher quality service. Assuming that our customers are homogeneous in their perception of the quality of the product, we assume that there is a service speed that we will consider as ‘normal’. When 37 the provider works at this level, the service has the expected quality level. The ‘normal’ speed could be a benchmark time based on industry standards or could be based on cus- tomer expectations. In some environments, the customer may be a part of the process (for example, a hair salon) and in such cases, the customer has an expectation of a certain ser- vice time or speed and this would be the normal speed. In other environments, the customer may not be part of the process or even observe the process but the customer observes the outcome of the process (e.g. auto service). In such cases, we assume that the outcome depends on the processing speed. For example, many fast-food restaurants, call centers and manufacturing firms have standard times associated with specific tasks or certain ser- vices. Pharmacies have standard times associated with filling a prescription and auto repair shops have standard times for typical services. Anand et al. (2009) use a similar idea of a benchmark service time or service rate and Lovejoy and Sethuraman (2000) use a similar idea and refer to it as “base” time for processing a unit. If the provider works faster than the ‘normal’ speed, then the service has lower quality and the company hurts its reputation. On the contrary, if the speed is lower than ‘normal’, the customers are happier and the firm gains from this goodwill. We denote ^ ¹ as the ‘nor- mal’ speed of service and± as the importance of quality to the customers and in turn, to the firm’s profit. Therefore, the loss or gain to the firm due to the speed at which it operates relative to the normal speed is given by ±¸(¹¡ ^ ¹): We include the term¸ in the expression so that the loss or gain is a function of the number of customers (demand). For simplicity we consider the case where the cost (gain) is linearly dependent on the difference of the actual service speed from the ‘normal’ speed. Also, we could replace (¹¡ ^ ¹) by (¹¡ ^ ¹)=^ ¹ to represent the percentage impact of service speed but this will not change our results. Note that we have not included an explicit cost for 38 increasing the speed. This is because we want to focus on the tradeoff between speed and quality and so the cost of increasing the speed is lower quality. However, including a cost for increasing the speed will not change the insights derived here. We have also not placed any upper or lower limits on the speed to keep the exposition uncluttered but these can be incorporated. In reality, the speed clearly cannot be increased to any desired level and at no cost. We emphasize that we do this only to focus on the speed-quality tradeoff and to keep the exposition simple. When the service provider decides to work at a slow speed, depending on the demand rate, the congestion in the system could be high and the waiting time long. As is conven- tional in the literature (Anand et al 2009, Ren and Zhou 2008), we assume that the provider incurs a congestion cost. If° denotes a customer’s sensitivity to waiting or the congestion cost per customer, then the firm will incur a total cost which depends on the demand rate and is given by °¸ 1 ¹¡¸ ; Note here that we consider the total sojourn time in the system. This is consistent with the existing literature (Anand et al. 2009). The system’s total expected profit will consist of the revenue generated from the cus- tomers paying pricep for the service less the congestion cost and the costs (gains) associ- ated with quality due to unsatisfied (satisfied) customers. We denote the service provider ’s expected profit asR and so, R(¹;p)=p¸¡° ¸ ¹¡¸ ¡±¸(¹¡ ^ ¹): 39 The service provider’s objective is to maximize her expected profit by choosing the pricep and the service speed¹. The objective function of the service provider can be equivalently written as R(¹;p)=p(¤¡®p)¡° ¤¡®p ¹¡¤+®p ¡±(¤¡®p)(¹¡ ^ ¹): We solve the service provider’s profit optimization problem and in the next theorem, we state the optimal pricing decision and speed of service. Theorem 3.1. Consider the single period model. 1. The profit function is concave. 2. The profit maximizing price chargedp ¤ and service rate¹ ¤ for the system will be p ¤ = ¤+2®¤±¡®±^ ¹+2® p °± 2®(1+®±) (3.1) and ¹ ¤ = ¤+®±^ ¹+2 p ° ± 2(1+®±) : (3.2) At optimality, the expected profit of the service provider will be R ¤ = ¡ ¤+®±^ ¹¡2® p °± ¢ 2 4®(1+®±) : (3.3) A further examination of the optimal price and service speed shows that when cus- tomers do not value the quality of the product as much (smaller±), the firm speeds up the service process and this allows for larger demand, so customers will pay less for the prod- uct. The opposite happens when customers are sensitive to product quality. In this case, the company slows down the processing and produces higher quality but charges higher prices. 40 Interestingly, when a firm works at a speed that is slower than normal speed, an increase in± results in an increase in profits. In such a scenario, higher quality sensitivity results in higher prices, lower speed and higher profits thus benefiting the firm. This is similar to the more customer-intensive or high-value services as described in Anand et al. (2009) where they reach similar conclusions. It is also reminiscent of Chase and Tansik (1983) where they differentiate the services depending on the customer contact required. They recom- mend that high customer contact services should maximize the quality offered, whereas low customer contact services should focus on efficiency and maximize productivity. If congestion cost is high, i.e. customers consider the time waiting in line to be crucial, the service provider has to increase the service speed in order to limit the unhappy cus- tomers. In the same vein, the firm can increase the price, thus decreasing the demand rate and the congestion level. The effects of any changes in price sensitivity and the demand potential are obvious. When the market size ¤ increases, the service provider gains more flexibility in all her decisions. Thus, we expect her to increase the price and the service speed and the expected profit will be larger. A higher price sensitivity from the customers will obligate the service provider to decrease the price and expect a lower profit. We summarize the impact of the parameters on the optimal price and service speed decisions in Table 3.1 and we show these results analytically in the Appendix. 3.4 The Multi-Period Model: Constant Price In this section, we consider the multi-period problem. We begin by formulating a multi- period model where the price is a decision made at the beginning of the horizon and is the same in all the periods. In later sections, we consider scenarios where price may vary over time but speed is constant or both can vary over time. In some service environments, prices are not varied frequently due to regulatory or other reasons. This situation is not uncommon for instance in hospitals or pharmacies. 41 speed price demand profit with respect to± # " # #/" with respect to° " " # # with respect to® # # " # with respect to¤ " " " " with respect to ^ ¹ " # " " Table 3.1: Sensitivity Analysis for the parameters in the Single Period Model when¤> ^ ¹. Similar to the single-period model formulation, there is a market or demand potential ¤ j at the beginning of each periodj (= 1;:::;N +1), where¤ 1 is exogenously specified. The market potential can increase or decrease from one period to another depending on the quality of the product offered in the last period. In particular, customers take into consideration the speed of the service offered in one period; if the speed was slower than what they were expecting, they will be happy, the firm’s goodwill increases and the market potential will increase. But if the service was not good, because it was faster than the expected ‘normal’ speed, then the potential customers in the next period will be fewer. We define the demand potential recursively as follows, ¤ j =¤ j¡1 ¡±¸ j¡1 (¹ j¡1 ¡ ^ ¹) j =2;:::;N +1 where ± is the parameter that reflects the sensitivity to quality. Note that ± has a slightly different interpretation than in the single-period case. Here, it controls the gain or loss in market potential and does not directly impact profits. We have assumed that the change in market potential is deterministic. In reality, the change in market potential may be stochas- tic and this could be captured by making± a random parameter. Also, as discussed earlier, ¸ j¡1 has been included in the term±¸ j¡1 (¹ j¡1 ¡ ^ ¹) to capture the fact that the change in 42 market potential is proportional to the current set of customers (whose satisfaction deter- mines the change in market potential). The realized demand rate ¸ j in each period j will depend on the price p and the cus- tomers’ price sensitivity®, that is ¸ j =¤ j ¡®p; j =1;:::;N: The total expected profit R will consist of the revenue acquired from the customers served less the costs due to congestion levels and is given by, R= N X i=1 · p¸ i ¡° ¸ i ¹ i ¡¸ i ¸ +µ¤ N+1 = N X i=1 · p(¤ i ¡®p)¡° ¤ i ¡®p ¹ i ¡¤ i +®p ¸ +µ¤ N+1 ; where° denotes the congestion cost per customer. The last termµ¤ N+1 has been included to incorporate an appropriate ending condition and µ represents the salvage value of the market potential at the end of the horizon. If this term is not included, note that ¹ N can be increased with no penalty but this would result in a negative ¤ N+1 . We also do not have any explicit cost for increasing the speed but there is an implicit cost captured through the reduction in market potential if the speed is higher than normal speed. We ignore dis- counting of profits to keep the exposition less cluttered but this feature can be incorporated easily. In a dynamic setting, the tradeoffs are interesting since the firm has to consider the impact of current decisions on price and speed on future profits. Increasing the speed now will decrease the current period congestion costs but will decrease future demand potential and profits. The price will be higher than that charged by a monopolist optimizing current period revenues (and the corresponding demand will be lower). This is due to both the impact of high demand on current period congestion costs as well as the negative impact of 43 high current demand and the resultant high speed on future market potential and, therefore, future profits. It is straightforward to show that the profit function is concave in the multi-period prob- lem using the fact that the single-period profit function is concave (proved in section 3.3) and the multi-period profit function is a sum of concave functions. Hence, there exists a unique set of ¹ i and p values that maximizes the profit. Next, we explore the behavior of the optimal speed over time. Theorem 3.2. If ¹ ¤ i < ^ ¹ and N is large, then the optimal service speed ¹ ¤ i is increasing over time and approaches the value of ^ ¹. Furthermore, service speed increases faster than demand rate,i.e. ¹ ¤ i ¡¹ ¤ i¡1 ¸¸ ¤ i ¡¸ ¤ i¡1 ; i=2;:::;N: If ¹ ¤ i > ^ ¹ and N is large, then the optimal service speed ¹ ¤ i is decreasing over time and approaches the value of ^ ¹. The optimal pricep ¤ will satisfy ¤ 1 ® ( 1+ N X i=2 i¡1 Y k=1 [1¡±(¹ ¤ k ¡ ^ ¹)] ) +p ¤ N X i=2 i¡1 Y k=1 [1¡±(¹ ¤ k ¡ ^ ¹)]¡ °[1¡±(2¹ ¤ 1 ¡ ^ ¹)] ±(¹ ¤ 1 ¡¤ 1 +®p ¤ ) 2 =0: The result suggests that the firm over time gradually reaches the normal speed ^ ¹. Sup- pose the firm starts at a speed slower than ^ ¹ then the market potential increases but this will result in increased demand and congestion costs. The firm cannot change its price and so cannot use it as a lever to lower the demand. So, the congestion levels increase but this is more than compensated for by the increased revenue. However, as the firm’s speed gets closer to the normal speed, the increase in market potential diminishes and speed increases at a slower rate. WhenN is large, the service speed will increase over time until it reaches the normal speed. IfN is small, depending onµ the service speed may fluctuate around the ‘normal’ speed before eventually converging to it. 44 0.90 1.00 1.10 1.20 1.30 1.40 1.50 μ δ=0.3 δ=0.5 δ=0.7 δ=0.9 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1 2 3 4 5 6 7 8 μ i δ=0.3 δ=0.5 δ=0.7 δ=0.9 Figure 3.1: Service Speed Behavior when price is constant and¹ 1 < ^ ¹ (¤ 1 = 6,® = 0:5, ° =0:4, and ^ ¹=1:5). Note that the firm can also start at a higher speed than normal but it will lose market potential over time and the optimal speed will gradually edge closer to the normal speed over time. Whether the firm starts at a speed faster or slower than the normal speed depends on the problem parameters. Numerical analysis suggests that the firm is more likely to start at a higher speed than normal when ^ ¹ is small and ± is small. If normal speed is low, then the firm cannot satisfy much demand and so it starts at a high speed even though this means lower quality and loss of market potential. But the low demand also implies high prices and this compensates to an extent for the loss of market potential and demand over time. Similarly, if± is high, customers are more sensitive to the difference in actual versus normal speed and so the firm is less likely to try and work at a faster speed than normal. The optimal price is substantially higher than what a single-period monopolist would charge to maximize revenues and the demand is correspondingly smaller. Interestingly, we find that even if the normal speed is lower than the speed required to meet a single-period monopolist’s demand, the firm will start at a slower speed than normal if± is not too low. 45 1.25 1.45 1.65 1.85 2.05 μ δ=0.2 δ=0.4 δ-0.6 δ=0.8 0.85 1.05 1.25 1.45 1.65 1.85 2.05 1 2 3 4 5 6 7 8 μ i δ=0.2 δ=0.4 δ-0.6 δ=0.8 Figure 3.2: Service Speed Behavior when price is constant and¹ 1 > ^ ¹ (¤ 1 = 9,® = 0:5, and ^ ¹=1). We performed some numerical studies to understand the behavior of price, speed and profits as a function of the parameters±;° and ^ ¹: Consider the scenario where¹ ¤ i < ^ ¹: As quality or speed sensitivity± increases, speed or service rate is lower in the initial periods but is higher in later periods (see Figure 1). Demand is lower when speed is lower and prices are correspondingly higher. The converse behavior occurs if ¹ ¤ i > ^ ¹ (See Figure 3.2). The impact on profits is interesting - if the firm works at a speed higher than normal, then profit decreases if customers are more sensitive to speed and quality. But if the firm works at a speed lower than normal (and reaches the normal speed over time), then higher customer sensitivity is actually beneficial in that profit increases with ±. The rationale is as follows. When ± is large, quality or speed sensitivity is high and so the firm starts at a slower speed (and charges higher prices) and thus maximizes the increase in market potential and reaps the rewards in later periods. A further increase in quality sensitivity benefits the firm in terms of increased market potential and the resultant higher profits. On the other hand, if± is small, the firm starts at a higher speed and charges lower prices. 46 1.1 1.2 1.3 1.4 1.5 μ γ=0.3 γ=0.5 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1 2 3 4 5 6 7 8 μ i γ=0.3 γ=0.5 γ=0.7 γ=0.9 Figure 3.3: Service Speed Behavior when price is constant and¹ 1 < ^ ¹ (¤ 1 = 6,® = 0:5, ± =0:6, and ^ ¹=1:5). The impact of° is as expected because it implies higher congestion costs and so speed is higher (see Figure 3.3), demand is lower and profits are lower. Consider the impact of an increase in ^ ¹. One would expect that an increase in normal speed would imply that the firm can work faster and so demand will be higher and prices will be lower. But this is not always the case. If the optimal speed is less than normal speed, then an increase in normal speed ^ ¹ actually results in an increase in price. The firm surprisingly starts out working at a lower speed when ^ ¹ increases and the corresponding demand is also lower and prices higher. However, over time, the speed increases at a faster rate and reaches the normal speed as can be seen in Figure 3.4. When ^ ¹ is higher, the firm can satisfy more demand and so it is willing to start at a lower speed and exploit the increase in market potential and corresponding higher demand in the future that this allows. This results in higher overall profits. Consider a firm providing call center services for technical support of PCs where cus- tomers may expect a lot of hand-holding and support. In such an environment, ^ ¹ is likely 47 to be low since the support person has to spend a lot of time with the customer to pro- vide a reasonable quality of service. Whether the firm starts at a higher (lower) speed and decreases (increases) speed over time depends on the parameters and in particular on ±. If the quality sensitivity is high, the firm will start at a slow speed, charge a higher price and build its reputation and market potential over time. Our results suggest that in such a scenario, higher quality or speed sensitivity can result in higher profits. Numerical studies also provide some insights into the rate of convergence of ¹ ¤ i to ^ ¹ over time. When ^ ¹ is large, the firm is more likely to start at a speed below that threshold; further, when customers’ sensitivity to quality ± is high, the lower is the starting service speed ¹ ¤ 1 . One may anticipate that if customers care more about the quality, the service provider will try to maintain a low service speed for a longer time to satisfy her customers. But the service speed actually reaches the ‘normal’ speed faster when the quality sensitivity is higher. This is because the high± combined with “better” service (¹ ¤ i < ^ ¹) attracts more customers, increasing the future demand, and the firm has to adapt to the increasing demand by speeding up the process and so the ‘normal’ speed is reached faster. On the other hand, when^ ¹ is low, numerical results show that we are likely to start from a higher service speed; in particular when quality sensitivity± is low and customers value the service time less, it is optimal to start from a relatively high service speed and gradually reach the ‘normal’ speed (Figure 3.2). 3.5 The Multi-Period Model: Constant Service Speed Now we study the case where the price can be varied over time but the speed is difficult to change. This setting is appropriate for scenarios where the process may be automated and the speed is therefore difficult to change. This is, for instance, true in manufacturing environments where the machine speed cannot be changed easily over time. However, the firm may have the flexibility to change prices which helps modulate demand and congestion 48 0.90 1.30 1.70 2.10 μ 0.10 0.50 0.90 1.30 1.70 2.10 1 2 3 4 5 6 7 8 μ i `normal' speed = 0.9 `normal' speed = 1.3 `normal' speed = 1.7 `normal' speed = 2.1 Figure 3.4: Service Speed Behavior with ‘normal’ speed when price is constant and¹ 1 < ^ ¹ (¤ 1 =6,® =0:5,° =0:4 and± =0:6). levels. This scenario may also be applicable in service environments where the process is more automated, e.g. call centers where a customer interacts primarily with an automated system. Even if the service is not automated, the process speed may be solely a function of the process layout and design which cannot be changed easily. We consider an N period model in which the firm decides on the price p i to quote every period, whereas the speed ¹ is a one-time decision made at the beginning of the horizon. At the beginning of each periodi, there is a demand potential¤ i that depends on the quality of the service provided in the previous period as before, that is ¤ i =¤ i¡1 ¡±(¹¡ ^ ¹)(¤ i¡1 ¡®p i¡1 ) 49 The demand function is ¸ i = ¤ i ¡®p i in period i and ® represents the price sensitivity of the customers. The formulation is similar to Section 3.4 in other respects. The total expected profitR will consist of the revenue less congestion costs and so R= N X i=1 · p i (¤ i ¡®p i )¡° ¤ i ¡®p i ¹¡¤ i +®p i ¸ +µ¤ N+1 We now explore the behavior of prices and demand over time. The next theorem states that prices vary in such a manner that the demand rate remains constant over time. Theorem 3.3. The demand rate remains constant in all the periods, i.e. ¸ 1 = ::: =¸ N =¸ ¤ : The intuition behind the result is as follows. Suppose the optimal speed is less than the normal speed. Then, the market potential will increase over time and suppose demand also increases over time. Then, congestion will increase since speed cannot be changed. The only way the firm can offset this increase in congestion cost is to increase price so that revenue increases but the increase in price implies lowering demand to the point where demand essentially does not change over time. Thus, demand and utilization do not change over time and neither does congestion cost. But demand potential increases over time (since ¹ ¤ < ^ ¹) and so prices increase too, resulting in higher revenues over time. This scenario illustrates a situation where the firm is able to leverage the slower speed by increasing prices and revenues over time. One of the interesting consequences of this result is that utilization and therefore congestion is constant over time. Next, we explore the optimal price and the conditions under which the firm will choose to work at a slower or faster speed than normal. 50 Theorem 3.4. The optimal pricing policy will be p ¤ i = ¹ ¤ ° (¹ ¤ ¡¸ ¤ ) 2 + ¸ ¤ ® +±µ(¹ ¤ ¡ ^ ¹)+ (N¡i)¸ ¤ ±(¹ ¤ ¡ ^ ¹) ® where (¸ ¤ ;¹ ¤ ) is the solution to ¤ 1 ¡2¸ ¤ ¡ ®¹ ¤ ° (¹ ¤ ¡¸ ¤ ) 2 ¡®±µ(¹ ¤ ¡ ^ ¹)¡(N¡1)¸ ¤ ±(¹ ¤ ¡ ^ ¹) = 0 N X i=1 [¸ ¤ +(®µ+(N¡i)¸ ¤ )±(¹ ¤ ¡ ^ ¹)]A i¡1 (¸ ¤ ;¹ ¤ )+®µA N (¸ ¤ ;¹ ¤ ) + ®°N¸ ¤ ±(¹ ¤ ¡¸ ¤ ) 2 = 0 where A i¡1 (¸ ¤ ;¹ ¤ )=¡(i¡1)[1¡±(¹ ¤ ¡ ^ ¹)] i¡2 ¤ 1 + P i¡1 j=1 [1¡±(¹ ¤ ¡ ^ ¹)] i¡j¡2 [1¡(i¡j)±(¹ ¤ ¡ ^ ¹)] 2 4 ®°¹ ¤ (¹ ¤ ¡¸ ¤ ) 2 +¸ ¤ +±(¹ ¤ ¡ ^ ¹)(µ®+(N¡j)¸ ¤ ) 3 5 Moreover, suppose that ^ ^ ¸ and ^ ^ ¹ solve the following system of equations: ¤ 1 ¡2 ^ ^ ¸¡ ® ^ ^ ¹° ( ^ ^ ¹¡ ^ ^ ¸) 2 =0 ^ ^ ¹= 2(¤ 1 ¡2 ^ ^ ¸) ± ³ ^ ^ ¸(N¡1)+2®µ ´: Then 1. If ^ ¹= ^ ^ ¹, then¹ ¤ = ^ ¹ and prices will remain constant over time. 2. If ^ ¹> ^ ^ ¹, then¹ ¤ < ^ ¹ and prices will increase over time. 51 3. If ^ ¹< ^ ^ ¹, then¹ ¤ > ^ ¹ prices will decrease over time. We find that when normal service speed is high (^ ¹ > ^ ^ ¹), the firm is more likely to work at a slower speed than normal (¹ ¤ < ^ ¹) and vice versa. Of course, the value of ^ ^ ¹ which determines when this scenario is likely to occur is a function of many parameters and while we do not have a closed-form solution, we can predict how ^ ^ ¹ varies with the problem parameters. When quality sensitivity increases, ^ ^ ¹ will decrease. This happens because customers care about the quality and so, the firm is more likely to work slower than the ‘normal’ speed devoting more time to each customer. This result can be interpreted as follows. Consider a firm that is providing a service where customers care deeply about the quality of the service. In this case, ± will be high and ^ ^ ¹ is likely to be lower. Suppose the firm has carefully designed the service and trained its workforce so that they can be more productive and hence the normal speed is high. In this case, ^ ¹ is more likely to be greater than ^ ^ ¹ and we have the middle scenario in the result where¹ ¤ < ^ ¹. This allows the firm to exploit its higher productivity by working at a slower speed than normal and thus building a reputation and increasing market potential and prices over time resulting in increasing profits. This result also illustrates the important role that operations can play by developing a process that is efficient enough to provide a buffer for the firm where it can work at a slower speed so as to make the customers happier. On the other hand, consider a firm that has a poor process whereby ^ ¹ is low. This firm may have to work at a faster speed than normal resulting in lower quality service and paying for it by having declining demand potential and profits over time. We also conducted numerical studies to understand the impact of the parameters ±; ° and ^ ¹ on the behavior of speed, utilization, etc. As one would expect, speed and cor- respondingly demand increase as ° or ^ ¹ increases and speed decreases as ± increases. The impact on utilization and queue length is more interesting. As sensitivity to qual- ity and speed (±) increases, one would expect congestion costs to be less important and 52 0.75 0.8 0.85 0.9 utilization 0.65 0.7 0.75 0.8 0.85 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 utilization δ μ<`normal' speed = 1.5 μ>`normal' speed = 0.5 Figure 3.5: Utilization with Quality Sensitivity. (¤ 1 =6,® =0:5, and° =0:4). hence queue length to increase. However, this depends on the scenario in the theorem, i.e. whether the optimal speed is higher or lower than normal speed. When optimal speed is lower than normal speed, the utilization and queue length does increase with ±. However, if the speed is higher than normal, then the utilization actually decreases as ± increases (Figure 3.5). Another interesting finding is that as ^ ¹ increases, one would expect the firm to increase speed and thus reduce utilization but we find the reverse. The utilization and queue length actually increase as ^ ¹ increases because demand increases at a faster rate than speed. While congestion costs are higher, these are more than made up by the increased revenue that comes from the faster growth in market potential due to the higher ^ ¹. 3.6 The General Multi-Period Model We now consider the scenario where both price and speed are allowed to vary over time. Recall from section 3.5 that when speed cannot be changed, the firm cannot take full advan- tage of the increased market potential and allow demand to increase due to the concern about congestion costs. Since the firm can change both price and speed, this gives the firm 53 more flexibility in optimizing its operations and we will explore how this can benefit the firm. The demand potential evolves as follows: ¤ i =¤ i¡1 ¡±(¹ i¡1 ¡ ^ ¹)¸ i¡1 =¤ i¡1 ¡±(¹¡ ^ ¹)(¤ i¡1 ¡®p i¡1 ): The total expected profitR is given by, R= N X i=1 · p i (¤ i ¡®p i )¡° ¤ i ¡®p i ¹ i ¡¤ i +®p i ¸ +µ¤ N+1 : We then have the following result on the evolution of price, speed and demand over time. Theorem 3.5. If¹ ¤ i < ^ ¹, then¹ ¤ i+1 >¹ ¤ i andp ¤ i+1 >p ¤ i and service speed increases faster than demand rate, i.e.(¹ ¤ i+1 ¡¸ ¤ i+1 )¸(¹ ¤ i ¡¸ ¤ i ). If¹ ¤ i > ^ ¹, then¹ ¤ i+1 <¹ ¤ i andp ¤ i+1 <p ¤ i and service speed decreases slower than demand rate, i.e.(¹ ¤ i+1 ¡¸ ¤ i+1 )¸(¹ ¤ i ¡¸ ¤ i ). First consider the scenario where the problem parameters are such that the service speed in the first period is less than the normal speed. We discuss the implications of the result followed by some insights into when this scenario (¹ ¤ 1 < ^ ¹) occurs. It is interesting to see that the service speed is increasing as in the constant price case. What is more interesting is that the price and demand rate are both increasing over time (demand rate behavior is from numerical studies). Thus, the firm increases price over time as we observed in the constant speed scenario. However, unlike in that scenario, it is allowed to change speed and this allows the firm to satisfy a higher demand rate over time. So, the firm exploits the increase in market potential by increasing both price and demand over time. Unlike in the constant speed case, the price increase over time is not large enough to keep the demand constant over time. 54 This result has some interesting implications in practice. Firms that have developed efficient processes are more likely to have higher ^ ¹ and be able to operate at less than nor- mal speed. This gives them greater leverage as they will be able to enhance their reputation and market potential over time. In turn, this means that they are in an enviable position where they can raise both prices and demand over time. Of course, some of these advan- tages will gradually vanish over time as their demand increases, necessitating higher speed which approaches the normal speed and thus diminishing further increases in market poten- tial and profits. But meanwhile they have would have increased their profits and solidified their market position. Consider a specific example such as the “Minute Clinics” which provides routine, low- risk healthcare services for minor illnesses and injuries, wellness tests, etc. and has grown to be the largest provider of retail healthcare in the US with 500 clinics (Scott 2006). Sup- pose they have designed their processes for such services so that they are better designed than the ones in typical doctor offices and urgent care facilities in hospitals. For instance, one area in which they are ahead of many healthcare facilities is in the use of informa- tion technology. They have used electronic kiosks to help patients provide information and proprietary software systems to help nurse practitioners arrive at a quick and accurate diagnosis to speed up the process (Scott 2006). They are likely to have faster than typical speed of service and this may allow their providers to spend more time with patients and provide a better customer experience. This in turn results in a better reputation, increased market potential and all the other benefits discussed earlier. While there are other factors such as healthcare regulation and competition that certainly impact their profits and perfor- mance over time, our model and results provide some insights into this phenomenon. It is important to understand the conditions under which this scenario is likely to occur and we address this issue next. 55 While it is not possible to provide closed-form conditions based on the model parame- ters to describe when the scenario (¹ ¤ i < ^ ¹) occurs, we have some insights based on numer- ical studies. We find that this scenario is most likely when± is high and ^ ¹ is large. These represent scenarios where quality sensitivity is high as this implies that the firm benefits from increasing its market potential. The normal service speed also plays a role. For exam- ple, if the normal service speed is large enough to cover the demand at optimum monopoly prices in a single period, then we find that ¹ ¤ i < ^ ¹ even if ± is small. But if ± is high, then even if ^ ¹ is not very high (for instance, not enough to cover optimum single-period monopoly demand) we find that ¹ ¤ i < ^ ¹ and speed is increasing over time but typically staying below ^ ¹. 3.7 General Queueing Systems In this section, we extend some of the key results in the previous sections to a more general queueing framework, specifically a G/G/1 queueing system. Consistent with the model formulation in earlier sections, we consider the sojourn time in the system in the estimation of congestion costs. For the G/G/1 queue, we have the following standard approximation for the time spent in the system (Marchal 1978), W s = ¸ ¹(¹¡¸) C 2 a +C 2 s 2 + 1 ¹ ; whereC a andC s represent the coefficients of variation of the arrival and service processes respectively. We defineC = C 2 a +C 2 s 2 to simplify the expressions. The Single Period Model: We start first with the single period setting modeled as a G/G/1 queueing system keeping the First Come First Served service discipline. The only difference from Section 3.3 is that 56 the interarrival and service times can now have a general distribution. Customers arrive to the system with rate ¤ and the firm has to decide on the price p and the service speed ¹. The actual demand rate will be ¸=¤¡®p; where ® is the price sensitivity of the customers. The system’s total expected profit will consist of the revenue acquired from the customers for selling the product (service) at price p less the congestion and quality costs (gains). Thus, the modified expected profit will be R(¹;p)=p¸¡° ¸ ¹¡¸ ¡° ¸ 2 (C¡1) ¹(¹¡¸) ¡±¸(¹¡ ^ ¹): The following theorem characterizes the optimal price and service speed that maxi- mizes the firm’s expected profit. Theorem 3.6. The optimal pricing and service speed decisions will be the solution to ®±(2¹ ¤ ¡ ^ ¹)=¤¡2¹ ¤ +2¹ ¤ s °C °(C¡1)+±¹ ¤2 and p ¤ = ¤+®±(2¹ ¤ ¡ ^ ¹) 2® : Although in Theorem 3.6, we do not provide closed form expressions for the optimal price and service speed, one can easily derive the solution of the system for specific values of the parameters. To better understand the impact of this modification on the optimal deci- sions, let us discuss the role of the system variation (C). As expected, a high variation will deteriorate system performance and particularly, the congestion level. Numerical analysis suggests that the firm will increase the price to contain the demand but it will also speed up the service so as to moderate the increasing delay cost. But the increase in speed and price does not compensate for the congestion costs and the firm’s profit declines asC increases. 57 The effects of the delay sensitivity (°), the quality sensitivity (±), the price sensitivity (®) and the demand potential (¤) on the optimal decisions and the maximum profit are similar to those discussed in Section 3.3. The Multi-Period Model We now consider the multi period model under the G/G/1 framework. The results we provide in this section are restricted to the cases of either constant price or constant service speed. Similar to Sections 3.4 and 3.5, the demand potential at the beginning of each period is impacted by the service speed in the previous period. The notation throughout this section is consistent with the notation used earlier. The demand potential is ¤ 1 at the beginning of period 1 and is defined recursively as ¤ j =¤ j¡1 ¡±¸ j¡1 (¹ j¡1 ¡ ^ ¹) j =2;:::;N +1 where the parameter ± denotes the importance of the quality level provided. The realized demand is ¸ j =¤ j ¡®p j j =1;:::;N depending on the pricep j and the price sensitivity®. The firm’s total expected profit is, R= N X i=1 p i (¤ i ¡®p i )¡° ¤ i ¡®p i ¹ i ¡¤ i +®p i ¡° (¤ i ¡®p i ) 2 (C¡1) ¹ i (¹ i ¡¤ i +®p i ) +µ¤ N+1 : Many of the results already discussed under the M/M/1 framework in the multi-period model continue to hold and are stated in the following theorem. Theorem 3.7. Under the G/G/1 framework in the dynamic setting, Constant Price: When the price is constant over time, i.e.,p 1 =p 2 =:::=p N =p, then, (a) if ¹ ¤ i < ^ ¹, the optimal service speed ¹ ¤ i increases over time and approaches the value 58 of ^ ¹, (b) if¹ ¤ i > ^ ¹, the optimal service speed¹ ¤ i decreases over time and approaches the value of ^ ¹. Constant Service Speed: When the service speed is constant over time, i.e., ¹ 1 = ¹ 2 = :::=¹ N =¹, then, (a)The demand rate remains constant in all periods, i.e ¸ 1 =¸ 2 =:::=¸ N =¸ ¤ : (b) The optimal pricing policy will be p ¤ i = ¹ ¤ °C (¹ ¤ ¡¸ ¤ ) 2 + ¸ ¤ ® ¡ °(C¡1) ¹ ¤ ¡±µ(¹ ¤ ¡ ^ ¹)+ (N¡i)¸ ¤ ±(¹ ¤ ¡ ^ ¹) ® where(¸ ¤ ;¹ ¤ ) is the solution of ¤ 1 ¡2¸ ¤ ¡ ®¹ ¤ °C (¹ ¤ ¡¸ ¤ ) 2 + ®°(C¡1) ¹ ¤ ¡®±µ(¹ ¤ ¡ ^ ¹)¡(N¡1)¸ ¤ ±(¹ ¤ ¡ ^ ¹) = 0 (3.4) N X i=1 [¸ ¤ +(®µ+(N¡i)¸ ¤ )±(¹ ¤ ¡ ^ ¹)]B i¡1 (¸ ¤ ;¹ ¤ )+®µB N (¸ ¤ ;¹ ¤ ) + ®°N¸ ¤ ±(¹ ¤ ¡¸ ¤ ) 2 · 1+ (C¡1)¸ ¤ (2¹ ¤ ¡¸ ¤ ) ¹ ¤2 ¸ = 0 (3.5) where B i¡1 (¸ ¤ ;¹ ¤ )=¡(i¡1)[1¡±(¹ ¤ ¡ ^ ¹)] i¡2 ¤ 1 + P i¡1 j=1 [1¡±(¹ ¤ ¡ ^ ¹)] i¡j¡2 [1¡(i¡j)±(¹ ¤ ¡ ^ ¹)] 2 4 ®°¹ ¤ (¹ ¤ ¡¸ ¤ ) 2 +¸ ¤ ¡ ®°(C¡1) ¹ ¤ +±(¹ ¤ ¡ ^ ¹)(µ®+(N¡j)¸ ¤ ) 3 5 Moreover, suppose that ^ ^ ¸ and ^ ^ ¹ solve the following system of equations: 59 ¤ 1 ¡2 ^ ^ ¸¡ ® ^ ^ ¹°C ( ^ ^ ¹¡ ^ ^ ¸) 2 + ®°(C¡1) ^ ^ ¹ =0 ^ ^ ¹= 2(¤ 1 ¡2 ^ ^ ¸) ± ³ ^ ^ ¸(N¡1)+2®µ ´: Then 1. If ^ ¹= ^ ^ ¹, then¹ ¤ = ^ ¹ and prices will remain constant over time. 2. If ^ ¹> ^ ^ ¹, then¹ ¤ < ^ ¹ and prices will increase over time. 3. If ^ ¹< ^ ^ ¹, then¹ ¤ > ^ ¹ prices will decrease over time. We focus on the additional insights that arise due to the impact ofC, i.e. greater varia- tion in interarrival and service times. First, consider the case where price is constant over time and service speed is varied in each period. Our results suggest that independent of the starting speed, it is optimal to vary the speed so that it approaches the normal speed. Numerical analysis shows that when the system variation (C) is higher resulting in higher congestion, the starting service speed¹ ¤ 1 is also higher so as to reduce delays. In particular, when¹ ¤ 1 < ^ ¹ andC is large, we start with a high service speed¹ ¤ 1 and approach the normal speed ^ ¹ in a shorter period or time. On the other hand, when ¹ ¤ 1 > ^ ¹ and C is large, we start with a speed far higher than the normal speed and therefore, it takes longer to reach the normal speed. Recall here thatC affects only the delay costs and has no impact on the evolution of the demand potential. Finally, there is another lever to control the congestion – the firm also increases the price whenC is large in order to decrease the demand rate and congestion. As expected, a high system variation reduces the total profit. Now let us move to the situation where the service speed cannot be changed. Again here, the firm faces high congestion levels when the system variation is high. Numerical 60 results show that for a largerC, it is optimal to start at a higher pricep ¤ 1 and thus lower the demand rate but this changes over time with the optimal price in later periods becoming lower than that for a lower C value. Note that ¹ ¤ is also higher when C is larger and the higher service speed is enough to moderate the congestion levels. However, the total profit declines at higherC values as expected. 3.8 Conclusions We summarize the key results and insights from the chapter and point out opportunities for further research. The essay addressed a fundamental tradeoff between speed and quality faced by organizations when they optimize their operations. We provide several interesting results from the dynamic tradeoffs in a multi-period setting. Speeding up a process enables a firm to meet more demand with less congestion but a faster process can result in poorer quality and this impacts customer satisfaction and future demand potential and profits. Using this basic tradeoff, we find that it is optimal for a firm to keep demand constant over time when the speed has to be kept constant. This speed can be faster or slower than the normal speed depending on how sensitive customers are to speed and quality and other factors. When price has to be kept constant, the firm tends to gradually approach the normal speed over time with the rate of convergence dependent upon congestion costs and quality sensitivity. The results in the essay suggest that firms which design their processes so that the normal speed can be high but they can operate at lower than normal speed, and thus make customers happier, tend to benefit greatly over time. There are several possible directions along which the models in the essay can be extended. First, we assumed that the change in market potential is linear and symmet- ric as a function of the actual speed relative to normal speed and these assumptions can be relaxed. Second, we could extend the model to consider strategic consumers as in Anand et al (2008). This implies that we will have to derive the demand potential in each period 61 as a function of the consumers’ behavior in this period as well as their satisfaction with the service in prior periods. Third, we have assumed that speed in a period impacts the change in demand potential immediately in the next period and in a deterministic manner. These assumptions can be relaxed by assuming that the impact of speed on quality and market potential may be lagged and stochastic. 62 Chapter 4 Analysis and Comparison of Inventory Systems: Dynamic vs Static Policies 4.1 Introduction The ability to price dynamically, that is, to charge different customers different prices for the same product, has great appeal to firms. This pricing flexibility can mitigate the negative effects on profit that stem from demand uncertainty by allowing firms to change prices in response to fluctuating inventory levels and customer waiting times. In more recent years, the growth of online sales has increased the attractiveness of dynamic pricing, because the cost of changing prices on an everyday basis is negligible (Coy 2000). It is also the case that firms in many industries do not price dynamically. This is because customers in general strongly dislike dynamic pricing. For example, Amazon attempted to price dynamically (Wall Street J. 2000), but stopped due to negative customer reaction (Washington Post 2000). Also, Coca-Cola did not have a better outcome when it tried to test “smart” vending machines that adjusted the soda prices according to the outside temperature; later, it denied that it had priced dynamically (Washington Post 2000). Customers are willing to accept dynamic pricing when they perceive it as fair in some sense. One acceptable mechanism through which dynamic pricing can be implemented is to offer customers a price discount in exchange for accepting a long product lead time. This can be attractive to both firms and customers, because when lead times are long, the firm retains customers it otherwise would have lost, and those customers pay less. Another 63 acceptable mechanism is sales promotions, in which the product price is decreased for a limited time. This may be in response to the firm having excess inventory that it would like to disburden, so that leadtime quotations are zero. The following question then arises. Does implementing dynamic pricing jointly with dynamic leadtime quotation produce greater revenue than having no dynamic pricing and no leadtime quotation? The reason it is not clear that the dynamic policy will have greater revenue is that the revenue advantage that having pricing flexibility offers may be offset by the restriction that dynamic leadtime information must always be given. Then, in the dynamic model, there is more flexibility in setting prices, but less flexibility in leadtime information disclosure. The objective of this paper is to formulate and analyze a model that quantifies the difference in revenue between a dynamic pricing and leadtime quotation policy, and a static pricing policy with no dynamic leadtime quotation. Our model is a single product inventory system. We assume Poisson arrivals of customers, and Poisson arrivals of products. There is a cost to obtain each product (that could represent a manufacturing cost or a contracted cost with a supplier), and an inventory holding cost. Customers are served from inventory using the first-come first-served discipline. There is a customer-specific valuation of the product, and customers are sensitive to price and leadtime. The customer’s utility for having the product equals his valuation minus the price, and then minus an assumed leadtime cost. For most of the paper, we assume that the leadtime cost function is linear. A customer will buy the product if his utility is positive. The objective is to maximize expected infinite horizon discounted revenue. We analyze our model both in the case that there is a static price and no dynamic leadtime quotation, and in the case that there is dynamic pricing and dynamic leadtime quotation. In the first case, we assume that customers know the steady-state expected lead- time. This may be because the customers have enough long term experience with the firm 64 in order to have formed their own estimation of the steady-state expected leadtime, or it may be because the firm provides the same steady-state expected leadtime to all its cus- tomers, regardless of the system state. We show that the optimal static price and maximum expected discounted revenue are independent of the starting state when the holding cost function is the zero function and we find their values. In the latter case, the leadtime quo- tation is the expected state-dependent time to receive the product, and we require that the quotation is truthful. Specifically, when there is a backlog in the system, the leadtime quo- tation is the expected time to clear the backlog and when there is inventory, the leadtime quotation is zero. We show that the optimal price is non-increasing in both the on-hand inventory and in the number of orders waiting to be filled under mild assumptions on the holding cost function. Next, we obtain closed-form expressions for the optimal price and maximum expected discounted revenue in the case that holding costs are zero and in the states where there is backlog. We find that there is a critical initial inventory level that determines which policy results in higher maximum expected discounted revenue. Below this level, the dynamic pricing and leadtime quotation policy is preferred, and, above this level, the static pricing policy is preferred. We further show that this result continues to hold in a more general model, in which there is also control over the production decision. We, additionally, identify cases when the static policy has higher maximum expected discounted revenue in all starting states. We complete this introduction with a brief review of the most relevant literature. Then, in Section 4.2, we specify the details of our model. Next we solve for the optimal static price and associated maximum revenue under the assumption that leadtimes are not dis- closed, but that the customers know the expected steady-state leadtime in Section 4.3. We use Markov decision problem methodology to solve for the optimal dynamic pricing pol- icy and associated maximum revenue under the requirement that expected leadtimes are 65 dynamically quoted in Section 4.4. In Section 4.5, we find the critical initial inventory level that determines whether a dynamic or static policy results in higher maximum expected dis- counted revenue, and we perform a sensitivity analysis to show how this critical inventory level depends on other system parameters. In Section 4.6, we show that when there is also a decision on when to produce products, the optimal production policy has a control limit form regardless of whether there are dynamic price and leadtime quotations or one static price and no leadtime quotations. We demonstrate numerically that there is in general a critical inventory level that determines whether a dynamic or static policy results in higher revenue. We consider leadtime cost functions that are not linear in Section 4.7, and demon- strate numerically that a convex leadtime cost function can lead to static policies being strictly dominant; that is, the maximum revenue obtained using a static policy exceeds that obtained using a dynamic policy for all initial inventory positions. Finally, we conclude in Section 4.8, and also discuss some interesting future directions. Literature Review The literature on inventory management is extensive, and the book by Zipkin (2000) illu- minates many of the guiding principles. Here we discuss only the most closely related streams of literature. We begin with papers that have focused on the question of how to price products in an inventory system. Early works on pricing focused on static pricing policies; see for example, Karlin and Carr (1962) and Veinott and Wagner (1965). More recent works have focused on dynamic pricing policies and a thorough review of that literature can be found in Elmaghraby and Keskinocak (2003). The works most closely related to ours that consider dynamic pricing but without a leadtime quotation are these of Thomas (1970), Thowsen (1975), Federgruen and Heching (1999), and Chen and Simchi- Levi (2004). Thomas (1970) studies dynamic price and production decisions but under deterministic demand 66 and no backlogging, assumptions that are later relaxed by Thowsen (1975). Federgruen and Heching (1999), and Chen and Simchi- Levi (2004) both consider pricing policies that depend on the inventory level for different ordering cost functions but, unlike us, they also focus on the optimal inventory policies. There is a separate stream of literature that focuses on leadtime quotation. The papers most relevant to ours are the ones that analyze models in which quoted leadtimes affect the customer arrival process. The first to analyze such a model are Duenyas and Hopp (1995) and Duenyas (1995). Later works include Kapuscinski and Tayur (2007), Maglaras and Van Mieghem (2005), and Keskinocak et al. (2001). The focus in these papers is different from ours because there is a simultaneous leadtime quotation and scheduling, whereas there is no scheduling question in our work. The papers by So and Song (1998), Plambeck (2004), C ¸ elik and Maglaras (2008), Ata and Olsen (2009) and Akan et al.(2009) consider models with joint pricing and leadtime quotation decisions. So and Song (1998) derive a static price and static leadtime quotation that maximizes the average net profit in an M/M/1 queue. In Plambeck (2004) and C ¸ elik and Maglaras (2008), customers are heterogeneous, and the objective is to use prices (static in Plambeck (2004) and dynamic in Maglaras (2008)) and dynamic leadtime quotation in a make-to-order queue to discriminate between customers with different price and leadtimes sensitivities, in order to extract more revenue. Ata and Olsen (2009) and Akan et al.(2009) both consider dynamic pricing and leadtime quotation decisions in a single product inven- tory system, in which there is also a scheduling control. The difference between the two papers is that in one customers are homogeneous and in the other customers are heteroge- neous. All of the papers mentioned in this paragraph except for So and Song (1998) use an asymptotic analysis to motivate their proposed policy whereas we perform an exact MDP analysis. Ours is the only paper to explicitly compare the maximum attainable revenue under a dynamic policy vs a static policy. 67 The idea of understanding the value of sharing information with customers is in the same spirit as the papers by Dobson and Pinker (2006), Guo and Zipkin (2007) and Guo and Zipkin (2008). Dobson and Pinker (2006) show that more information can increase a firm’s profit and improve customers’ experience under certain conditions. Guo and Zipkin (2007) and Guo and Zipkin (2008) show that the decision of sharing certain amounts of information with the customers depends on the leadtime sensitivity of the customers. None of the three papers allows for dynamic pricing. The idea that information disclosure may not be favorable was also discussed in Hassin (1986) in the context of a single server queue, when the question is whether or not to tell customers the queue length. A final related stream of literature encompasses inventory models in which there are production decisions, and we refer the reader to Yano and Gilbert (2002) for a review of this work. The closest work to ours is Chan et al. (2006), which has both pricing and production decisions, but in which leadtime information is not conveyed to customers. 4.2 Model Formulation We consider a single product inventory system. The state of the system at time t, X(t), represents the inventory position; that is, whenX(t) < 0,jX(t)j is the on-hand inventory, and when X(t) ¸ 0, X(t) is the number of unfilled orders. The system state increases by one every time a customer arrives and decides to purchase a product, and decreases by one every time a component is produced. There is a lower bound¡B << 0, that can be arbitrarily large and represents the maximum amount of inventory that can be stored. When X(t)=¡B, no components are produced, so thatX(t)¸¡B for allt>0. Components are produced one unit at a time, and placed in inventory. The amount of time to produce one component follows an exponential distribution with mean 1=¹, and has a per-unit cost c > 0. There is a physical holding cost per unit time of holding items 68 in inventory, and this is given by the function h. We assume h is increasing, convex, and h(0)=0. Prospective customers arrive according to a Poisson process with rate¤, and each cus- tomer has a valuation V for obtaining the product immediately, where V is drawn from a uniform distribution on [0;v]. Every prospective customer has the same cost function ¯(l) for a given expected lead time l, where ¯ is a positive, increasing, continuous, and unbounded function having ¯(0) = 0. Then, an arriving customer that has an expected lead timeL E has expected utility U :=V ¡p¡¯(L E ) (4.1) for a product priced atp> 0, and will purchase the product only ifU ¸0. Let the process A(t) be the thinned Poisson process that tracks the cumulative number of prospective cus- tomers that arrived by timet> 0, and decided to purchase the product. The objective is to maximize expected infinite horizon discounted profit v(x):=E x ·Z 1 0 e ¡±t p(X(t))dA(t)¡ Z 1 0 e ¡±t h ¡ [X(t)] ¡ ¢ dt¡ Z 1 0 e ¡±t cdC(t) ¸ ; (4.2) where E x [¢] := E[¢jX(0)=x] and [x] ¡ := max(¡x;0). The first term in (4.2) is the revenue obtained from selling products, and the second term represents the cost of holding inventory. The third term in (4.2) is the cost of producing components. Note that this term will not be relevant to the optimal policy until Section 4.6, when we generalize the model so that the system manager also can control production. We would like to compare the maximum attainable profit in (4.2) when the pricing policy is restricted to be static, but customers know the expected steady-state lead time, to that when the pricing policy can be dynamic, but it is required that customers are given expected state-dependent lead time information. The two scenarios lead to different utility 69 computations in (4.1), because the L E differs. We first discuss the static case, and then discuss the dynamic case. In the static case, L E (p) is the expected steady-state lead time, and depends on the pricep>0. It follows from (4.1) that when the product price isp 0 , each arriving customer purchases the product if V >V 0 :=p 0 +¯(L E (p 0 )): This computation is the same for all customers, and so the fraction of prospective customers that purchase products is P(V >V 0 )= v¡V 0 v : Note thatV 0 affectsL E , because the steady-state lead time is stochastically increasing in the fraction of prospective customers that purchase products, andL E also affectsV 0 , for a given pricep 0 . So findingV 0 means finding a price under which the system is in equilibrium with respect to the steady-state expected leadtime and the probability that an arriving customer purchases a product. Once V 0 is known, it follows that A = A 0 is a Poisson process with rate ¸ 0 :=¤ µ v¡V 0 v ¶ : The inventory position X(t) is a birth-and-death process with constant birth rate ¸ 0 and constant death rate¹. Then, the revenue function in (4.2) is v 0 (x):=E x ·Z 1 0 e ¡±t p 0 dA 0 (t)¡ Z 1 0 e ¡±t h ¡ [X(t)] ¡ ¢ dt¡ Z 1 0 e ¡±t cdC(t) ¸ : The objective is to find the pricep ? 0 that maximizesv 0 ; i.e., to solve v ? 0 (x)= sup p 0 ¸0 v 0 (x): 70 In the dynamic case, the pricep may depend on the inventory positionx. Specifically, lettingP denote the set of functions that map the integersf¡B;:::;¡2;¡1;0;1;2;:::g into the positive real line< + , an admissible pricing policy has p 2 P. This restriction to deterministic policies that do not vary with time is without loss of generality because there is an upper bound on the price we will set (p · v) and our action space is compact. In particular, the conditions of Theorem 6.2.10 in Puterman 2005 are satisfied and so, there exists a policy p 2 P that maximizes the expected discounted profit. Next, L E is the expected lead time given that the inventory position isx; that is, L E = x + ¹ : Then, it follows from (4.1) that a customer that arrives when the inventory position isx and the price isp(x) purchases the product if V >V(x;p):=p(x)+¯ µ x + ¹ ¶ : LetA = A d be the associated thinned Poisson process that has the state-dependent proba- bility that a prospective customer purchases the product P(V >V(x;p))= [v¡V(x;p)] + v : The inventory positionX(t) is a birth-and-death process with state-dependent birth rate ¸(x;p):=¤ [v¡V(x;p)] + v =¤ h v¡p(x)¡¯ ³ x + ¹ ´i + v 71 and constant death rate¹. The profit function in (4.2) is v d (x):=E x ·Z 1 0 e ¡±t p(X(t))dA d (t)¡ Z 1 0 e ¡±t h ¡ [X(t)] ¡ ¢ dt¡ Z 1 0 e ¡±t cdC(t) ¸ : (4.3) The objective is to find the dynamic pricing policyp ? that maximizesv d ; i.e., to solve v ? d (x)=sup p2P v d (x): It is not clear a priori whether v ? 0 (x) or v ? d (x) will be higher. There is more pricing flexibility when computingv ? d (x), but there is also required dynamic leadtime information disclosure. On the other hand, the pricing is restricted to be a single price when computing v ? 0 (x), but all arriving customers base their purchasing decision on the same steady-state expected leadtime, even when the system is very congested. In general, whenx is large, we expect thatv ? 0 (x)¸ v ? d (x), and whenx is small, thatv ? 0 (x) < v ? d (x). The question is, can we show that there is exactly one state where the inequality reverses, and can we compute that state? Then, can we also determine whetherv ? 0 (x)¸v ? d (x)? The first step in answering those last questions is to findv ? 0 (x) andv ? d (x). We findv ? 0 in Section 4.3 andv ? d (x) in Section 4.4 when the system manager does not have control over the production process. We find that when the holding cost function is the zero function, v ? 0 (x) = v ? 0 is a constant that does not depend on x. Then, to compare v ? 0 and v ? d (x), we need only to determine if and where the functionv ? d (x) crosses the valuev ? 0 , and we do this in Section 4.5. Next, in Section 4.6, we computev ? 0 (x) andv ? d (x) and compare their values, in the case that the system manager does have control over the production process. Then, even when the holding cost function is the zero function, v ? 0 (x) depends on x, and so the comparison betweenv ? 0 (x) andv ? d (x) is not as easy. 72 4.3 The Optimal Static Policy In this section, we find the optimal policy when pricing is static, no dynamic leadtime information is given, and customers know the expected steady-state leadtime. The first step is to solve for the revenue v 0 (x) when the product price is p 0 . The second step is to optimize overp 0 to findv ? 0 (x). The time from when the processX enters statex >¡B to when either a customer or component arrives follows an exponential distribution with parameter¤+¹. It then jumps to statey2fx¡1;x;x+1g when the price isp 0 , so that¸ 0 =¸ 0 (p 0 ), with probabilities q 0 (yjx):= 8 > > > > > > > > > > < > > > > > > > > > > : ¸ 0 ¤+¹ 1¡ ¸ 0 +¹ ¤+¹ ¹ ¤+¹ ify =x+1 ify =x ify =x¡1 : The time from when the process X enters state ¡B to when a customer arrives fol- lows an exponential distribution with parameter ¤. From state ¡B, it jumps to state y2f¡B;¡B+1g with probability q 0 (yj¡B):= 8 > > > < > > > : ¸ 0 ¤ 1¡ ¸ 0 ¤ ify =¡B+1 ify =¡B : Note thatq 0 (yjx) depends onp 0 through¸ 0 , but that this is implicit in the notation. Let ¿ 1 be the time of the first customer or component arrival event when the state is x>¡B, so that¿ 1 has an exponential distribution with parameter¤+¹. By conditioning on the time of the first event, and then conditioning on whether that event is a customer that 73 arrives and purchases a product, a customer that arrives and does not purchase a product, or a component that arrives, it follows that under any static pricep 0 >0, the revenue function v 0 must satisfy v 0 (x) = Z 1 0 E x £ e ¡±t ¡ p 0 dA(t)¡cdC(t)¡h([X(t)] ¡ )dt ¢ j¿ 1 2du ¤ P (¿ 1 2du) = Z 1 0 E x 2 6 6 6 4 ¡ e ¡±u p 0 ¡h([x] ¡ ) R u 0 e ¡±t dt+e ¡±u v 0 (x+1) ¢ q 0 (x+1jx) + ¡ e ¡±u c¡h([x] ¡ ) R u 0 e ¡±t dt+e ¡±u v 0 (x¡1) ¢ q 0 (x¡1jx) + ¡ ¡h([x] ¡ ) R u 0 e ¡±t dt+e ¡±u v 0 (x) ¢ q 0 (xjx) 3 7 7 7 5 ¤ ¤P (¿ 1 2du) = µ p 0 ¸ 0 ¤+¹ ¡c ¹ ¤+¹ ¶ E x £ e ¡±¿ 1 ¤ ¡h([x] ¡ )E x ·Z ¿ 1 0 e ¡±t dt ¸ +E x £ e ¡±¿ 1 ¤ X y2fx¡1;x;x+1g q 0 (yjx)v 0 (y) = p 0 ¸ 0 ¡c¹¡h([x] ¡ ) ±+¤+¹ + ¤+¹ ±+¤+¹ X y2fx¡1;x;x+1g q 0 (yjx)v 0 (y): Similarly, v 0 (¡B)= p 0 ¸ 0 ¡c¹¡h(B) ±+¤ + ¤ ±+¤ X y2f¡B;¡B+1g q 0 (yj¡B)v 0 (y): Equivalently, v 0 (x+1)¡ ±+¸ 0 +¹ ¸ 0 v 0 (x)+ ¹ ¸ 0 v 0 (x¡1)=¡p 0 + c¹+h([x] ¡ ) ¸ 0 ; (4.4) forx2f¡B+1;¡B+2;:::;¡1;0;1;:::g, and v 0 (¡B+1)¡ ±+¸ 0 ¸ 0 v 0 (¡B)=¡p 0 + c¹+h([x] ¡ ) ¸ 0 : (4.5) 74 Note here that since the production process cannot be interrupted, the costc associated with the production of a component has to be paid when in state¡B, even though there is no storage space. When h(x) = 0 for all x 2 f0;1;2;:::g, the equation (4.4) is the same for all state x >¡B, which suggests that the value function is constant. Assuming thatv 0 (x) = v 0 for allx2<, we can solve (4.4) forv 0 to find that v 0 = ¸ 0 p 0 ¡c¹ ± : (4.6) We can double-check that v 0 in (4.6) also satisfies (4.5). This implies that for any price p 0 >0, the expected discounted revenue does not depend on the starting state. Substituting for¸ 0 in (4.6) yields v 0 = 1 ± µ p 0 ¤ v¡p 0 ¡¯(L E (p 0 )) v ¡c¹ ¶ : (4.7) We can solve forL E (p 0 ) when¸ 0 <¹ by finding the steady-state probabilities¼ i ofX. These are the same as those for the uniformized chain, ¼ i := µ ¸ 0 ¹ ¶ i ¼ 0 ; i=f¡B;¡B+1;:::;¡1;0;1;2;:::g where ¼ 0 = µ ¸ 0 ¹ ¶ B µ 1¡ ¸ 0 ¹ ¶ : 75 It follows that L E (p 0 ) = 1 X i=0 i ¹ ¼ i = ¸ 0 ¹ 2 1 ³ 1¡ ¸ 0 ¹ ´ 2 ¼ 0 = µ ¸ 0 ¹ ¶ B+1 1 ¹¡¸ 0 : Substituting for ¸ 0 = ¤ v¡p 0 ¡¯(L E (p 0 )) v provides an equation that L E (p 0 ) must satisfy. A graph of the value function with respect to the static price is shown at Figure 4.1. 0 20 40 60 80 100 120 0 100 200 300 400 500 600 700 800 900 p 0 v 0 Figure 4.1: The expected infinite horizon discounted revenuev 0 as a function of the static price p 0 , for the parameter values ¤ = 6, v = 120, ¯ = 4, ± = 0:1, B = 100, ¹ = 1, c=10 andh(x)=0 forx2f¡B;¡B+1;:::g. The following theorem provides the pricep 0 that maximizes the right-hand side of (4.7) when the leadtime cost function is linear. Theorem 4.1. Assume¯(l)=¯l for anyl¸0, then the optimal static pricep ? 0 satisfies (v¡2p ? 0 )(¹¡¸ 0 )+¯L E (p ? 0 )B¹=0; (4.8) 76 and is p ? 0 =¡ 1 2 ¹ ¤ v¡ 1 2 ¯L E (p ? 0 )+ 3 4 v+ s · 3 4 v¡ 1 2 ¹ ¤ v¡ 1 2 ¯L E (p ? 0 ) ¸ 2 + 1 2 v¯L E (p ? 0 )(¤¡B¹); whereL E (p ? 0 ) is the solution of L E (p ? 0 )= ³ ¤ v¡p ? 0 ¡¯(L E (p ? 0 )) v ´ B+1 ¹ B+1 ³ ¹¡¤ v¡p ? 0 ¡¯(L E (p ? 0 )) v ´: (4.9) Furthermore, v ? 0 = 1 ± µ p ? 0 ¤ v¡p ? 0 ¡¯(L E (p ? 0 )) v ¡c¹ ¶ : We expect that the more sensitive customers are to leadtime; that is the higher¯ is, the lower the effective arrival rate¸ 0 , and, therefore, the lower the maximum revenue. Figure 4.2 provides a numeric illustration of the effect that¯ has on the maximum revenue. 0 2 4 6 8 10 12 14 16 18 20 840 845 850 855 860 865 β v 0 Figure 4.2: The maximum expected infinite horizon discounted revenue v ? 0 as a function of ¯, for the parameter values ¤ = 6, v = 120, ± = 0:1, B = 100, ¹ = 1, c = 10 and h(x)=0 forx2f¡B;¡B+1;:::g. 77 4.4 The Optimal Dynamic Policy In this section, we find the optimal policy when price can change dynamically and the state- dependent expected leadtime is announced. Then, we need to solve for the deterministic pricing policyp2P that maximizesv d (x). The time from when the processX enters statex >¡B to when either a customer or component arrives follows an exponential distribution with parameters¤+¹. It then jumps to statey2fx¡1;x;x+1g when the price isp:=p(x) with probabilities q d (yjx;p):= 8 > > > > > > > > > > < > > > > > > > > > > : ¸(x;p) ¤+¹ 1¡ ¸(x;p)+¹ ¤+¹ ¹ ¤+¹ ify =x+1 ify =x ify =x¡1 : The time the process X remains in state¡B follows an exponential distribution with parameter ¤. From state¡B, it jumps to statef¡B;¡B +1g when the price is p with probability q d (yj¡B;p):= 8 > > > < > > > : ¸(¡B;p) ¤ 1¡ ¸(¡B;p) ¤ ify =¡B+1 ify =¡B : An argument very similar to that in the third paragraph of Section 4.3 shows that under any pricing policyp2P, the revenue functionv d must satisfy 78 v d (x)= p¸(x;p)¡c¹¡h([x] ¡ ) ±+¤+¹ + ¤+¹ ±+¤+¹ X y2fx¡1;x;x+1g q d (yjx;p)v d (y); forx2f¡B+1;¡B+2;:::;¡1;0;1;:::g, and v d (¡B)= p¸(¡B;p)¡c¹¡h(B) ±+¤ + ¤ ±+¤ X y2f¡B;¡B+1g q d (yj¡B;p)v d (y): Hence, a pricing policy that maximizes expected discounted revenue satisfies the optimality equations v ? d (x)=sup p2P 8 < : p¸(x;p)¡c¹¡h([x] ¡ ) ±+¤+¹ + ¤+¹ ±+¤+¹ X y2fx¡1;x;x+1g q d (yjx;p)v ? d (y) 9 = ; ; (4.10) forx2f¡B+1;¡B+2;:::;¡1;0;1;:::g, and v ? d (¡B)=sup p2P 8 < : p¸(¡B;p)¡c¹¡h(B) ±+¤ + ¤ ±+¤ X y2f¡B;¡B+1g q d (yj¡B;p)v ? d (y) 9 = ; : (4.11) Note again that, as explained in the previous section, we still have to pay for the component that may arrive when in state¡B. When the leadtime cost function is linear, we have an explicit expression for the pricing policy and the value function in the positive part of the state space. Theorem 4.2. Assume¯(l)=l forl¸0. The policyp ? satisfies p ? (x)=¡ v ? d (x+1)¡v ? d (x) 2 + v¡¯ x + ¹ 2 : (4.12) 79 Furthermore, whenx>0, v ? d (x) = rx 2 +sx+t (4.13) p ? (x) = 2¤v p ¢+(2¹¡±) ³ ¤¯ ¹ +v±¡v p ¢ ´ 2¤(±+ p ¢) ¡ " ¯ ¹ +v ±¡ p ¢ 2¤ # x; (4.14) where ¢ = ±(±+ 2¤¯ v¹ ); r = ¤¯ ¹ +v(±¡ p ¢) 2¤ >0; s = ¤¯ ¹ (±¡2¹)+ ³ ¤v+ ¤¯ 2¹ +±v¡2¹v ´ (±¡ p ¢) ¤(±+ p ¢) ; t = ¤v 4 ¡c¹+ ¤ 2 (r+s)+ ¤ 4v (r+s) 2 +r¹¡s¹ ± : Although it is not possible to have an explicit expression for the value function in the negative part of the state space, it is possible to know its value forx=¡B. Proposition 4.1. Whenh(x)=0, for large enoughB >> 0, v ? d (¡B)= ¤v 4 ¡c¹ ± andp ? (¡B)= v 2 : 4.5 Model Comparison We have found the maximum expected discounted revenue under two different assumptions regarding pricing and leadtime quotation. In the model analyzed in Section 4.3, there is one static price, and all customers are offered the product at that price. Arriving customers do not receive dynamic information regarding how long they must wait for the product, but they do know the mean waiting time. In the model analyzed in Section 4.4, the system 80 -30 -20 -10 0 10 20 30 0 200 400 600 800 1000 1200 1400 1600 x v d * -30 -20 -10 0 10 20 30 0 10 20 30 40 50 60 70 80 90 x p * (a) The function v d * (b) The function p * Figure 4.3: The optimal dynamic pricing policyp ? and value functionv ? d for the parameter values ¤ = 6, v = 120, ¯ = 4, ± = 0:1, B = 100, ¹ = 1, c = 10 and h(x) = 0 for x2f¡B;¡B+1;:::g. manager quotes a price and leadtime to arriving customers that changes as the system state changes. The leadtime information is the expected wait conditional on the system state, and the price may be raised or lowered to depending on the system state. The question of interest is: does the firm attain higher maximum revenue under a static pricing policy in which state-dependent leadtime information may be kept private, or under a dynamic pricing policy in which state-dependent leadtime information must be disclosed? That is, doesv ? d exceedv ? 0 or doesv ? 0 exceedv ? d ? 81 -30 -20 -10 0 10 20 30 0 200 400 600 800 1000 1200 1400 1600 x v d * x 0 -30 -20 -10 0 10 20 30 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 x v d * X 0 = -B (a) μ = 1 (b) μ = 4 Figure 4.4: v ? d (solid line) andv ? 0 (dotted line) as a function of the starting statex (¤ = 6, v =120,¯ =4,± =0:1,B =100,c=10 andh(x)=0 forx2f¡B;¡B+1;:::g) We first observe that the answer to the question of which ofv ? d orv ? 0 is highest depends in general on the system state. This is because v ? d is decreasing in the system state x, whereasv ? 0 is constant. We next note that v ? 0 ·v ? d (¡B): This is because when ¯ = 0, it follows from Theorem 4.1 that the maximum attainable revenue under a static policy is ¤v 4 ¡c¹ ± ; 82 0 2 4 6 8 10 12 14 16 18 20 -100 0 100 200 300 400 500 β v d * (0) - v 0 * 0 2 4 6 8 10 12 14 16 18 20 -5 0 5 10 15 20 25 β x 0 (a) v d * - v 0 * (b) x 0 (β) Figure 4.5: The effect of¯ (¤ = 6,¹ = 1,v = 120,¯ = 4,± = 0:1,B = 100,c = 10 and h(x)=0 forx2f¡B;¡B+1;:::g) and also that v ? 0 · ¤v 4 ¡c¹ ± : From Proposition 4.1, v ? d (¡B)= ¤v 4 ¡c¹ ± : Hence v ? 0 ·v ? d (¡B); 83 and there exists a critical starting statex 0 for which v ? 0 8 < : <v ? d (x) ifx<x 0 ¸v ? d (x) ifx¸x 0 : The question of interest then becomes: when is x 0 > ¡B and when is x 0 = ¡B. That is, when can the state space be divided into some initial states in which a dynamic policy attains higher maximum revenue and some initial states in which a static policy attains higher maximum revenue, and when is it the case that a static policy attains higher maxi- mum revenue in all initial states? Figure 4.4 findsx 0 by plotting the value functionsv ? 0 andv ? d as a function of the initial statex for a system with no holding costs and parameters¤=6,v =120,¯ =4,± =0:1, B = 100, andc = 10 in for both the case that¹ = 1 and¹ = 4. The statex 0 is the state where the two functions intersect. In general,v ? d will be higher when there is inventory and not too much backlog, because the wait times conditional on the system state will be smaller than the unconditional expected waiting time. This situation is depicted in Figure 4.4 (a). However, when¹ is high, so that the system is not very congested, then the unconditional expected waiting time is very small. Intuitively, that implies that v ? 0 may always equal or exceedv ? d . This other situation is depicted in Figure 4.4 (b). It is also of interest to understand the effect of the parameter ¯. The parameter ¯ measures how sensitive customers are to increases in the leadtime. When ¯ is large, we expect that the system manager will not want to disclose any positive waiting time to his customers, because the price cannot be set low enough to maintain customers having a positive expected utility (as defined in (4.1)) for buying the product. This is illustrated in Figure 4.5(a), which plots the difference in maximum revenuev ? d (0)¡v ? 0 as a function of ¯ for the fixed starting state 0, and in Figure 4.5(b), which plots x 0 as a function of ¯. In both figures, the system has no holding costs and parameters ¤ = 6, v = 120, ± = 0:1, 84 B = 100, c = 10, and ¹ = 1. Consistent with the aforementioned intuition, note that in Figure 4.5(a),v ? 0 exceedsv ? d (0) as¯ becomes large, and in Figure 4.5(b),x 0 decreases as¯ becomes large. 4.6 Decision on the Production In this section, we generalize the model so that the system manager also has control over the production process. First, in Section 4.6.1, we assume pricing is static, no dynamic leadtime information is given, customers know the expected steady-state leadtime, and the system manager can dynamically decide whether or not to shut down the production process. Next, in Section 4.6.2, we assume the price can change dynamically, that the state- dependent expected leadtime is announced, and that the system manager can dynamically decide whether or not to shut down the production process. In both models, we find that the production policy that maximizes the expected discounted revenue is a control limit policy. Finally, in Section 4.6.3, we compare the model with static pricing in Section 4.6.1 and the model with dynamic pricing in Section 4.6.2, in order to answer the question of which model can attain higher revenue. 4.6.1 The Optimal Policy with Static Pricing The functionw 0 :f¡B;:::;¡1;0;1;2;:::g!f0;1g represents the production decision; that is, w 0 (x) = 0 implies that the production process will be shut down and w 0 (x) = 1 85 implies that components are continued to be produced. Then, the transition probabilities are q 0 (yjx;w 0 )= 8 > > > > > > > > > > < > > > > > > > > > > : ¸ 0 ¤+¹ 1¡ ¸ 0 +¹w 0 (x) ¤+¹ ¹ ¤+¹ w 0 (x) ify =x+1 ify =x ify =x¡1 and q 0 (yj¡B)= 8 > > > < > > > : ¸ 0 ¤ 1¡ ¸ 0 ¤ ify =¡B+1 ify =¡B sincew 0 (¡B):=0. For a given static price p 0 ¸ 0, an argument very similar to the third paragraph in Section 4.3 shows thatv 0 satisfies v 0 (x)= max w 0 2f0;1g 8 < : p 0 ¸ 0 ¡c¹w 0 (x)¡h([x] ¡ ) ±+¤+¹ + ¤+¹ ±+¤+¹ X y2fx¡1;x;x+1g q 0 (yjx;w 0 )v 0 (y) 9 = ; (4.15) forx>¡B and v 0 (¡B)= p 0 ¸ 0 ¡c¹¡h(B) ±+¤ + ¤ ±+¤ X y2f¡B;¡B+1g q 0 (yj¡B)v 0 (y): The following theorem shows that for any fixed p 0 > 0, a control limit production policy maximizes the expected discounted revenue when there are no holding costs. 86 Theorem 4.3. Assumeh is the zero function. There exists aS 0 2< ¡ such that w ? 0 (x)= 8 > > > < > > > : 1; 0; whenx>S 0 whenx·S 0 : (4.16) Then, an analysis similar to that used to deriveL E (p 0 ) in Section 4.3 shows that L E (p 0 )= ³ ¤ v¡p 0 ¡¯(L E (p 0 )) v ´ S 0 +1 ¹ S 0 +1 ³ ¹¡¤ v¡p 0 ¡¯(L E (p 0 )) v ´: In contrast to Section 4.3, it is no longer the case that, for a givenp 0 ¸ 0, the expected discounted revenue does not depend on the starting state when there are no holding costs. This can be seen by comparing equation (4.4), which is the same for all states, and equation (4.15), which is not the same for all states. Intuitively, this is because the system manager no longer has to pay a production cost regardless of how much inventory he has, and so not all states have the same cost. Looking ahead in the paper, this observation is demonstrated numerically in Figure 4.6(b). It remains to optimize (4.15) over p 0 ¸ 0 to find the static price that results in the maximum expected infinite horizon discounted revenue. This is difficult because the ¸ 0 that appears in (4.15) depends on L E (p 0 ). However, it is possible to do this through an exhaustive search. In particular, for each possible value ofS 0 ,f¡B;¡B +1;:::;0g, we can compute the maximum static price p ? 0 (S 0 ), and then set S ? 0 so that v 0 (x;p ? 0 (S ? 0 )) ¸ v 0 (x;p ? 0 (S 0 )) for anyS 0 2f¡B;¡B+1;:::;0g. 87 4.6.2 The Optimal Policy with Dynamic Pricing As in Section 4.6.1, we let the function w d : f¡B;:::;¡1;0;1;:::g ! f0;1g represent the production decision. Then, the transition probabilities are q d (yjx;p;w d )= 8 > > > > > > > > > > < > > > > > > > > > > : ¸(x;p) ¤+¹ 1¡ ¸(x;p)+¹w d (x) ¤+¹ ¹ ¤+¹ w d (x) ify =x+1 ify =x ify =x¡1 and q d (yj¡B;p;w d )= 8 > > > < > > > : ¸(x;p) ¤ 1¡ ¸(¡B;p) ¤ ify =¡B+1 ify =¡B : The optimality equations are v ? d (x)= max p¸0;w d 2f0;1g 8 < : p(x)¸(x;p)¡c¹w d (x)¡h([x] ¡ ) ±+¤+¹ + ¤+¹ ±+¤+¹ P y2fx¡1;x;x+1g q d (yjx;p;w d )v ? d (y) 9 = ; (4.17) forx>¡B and v ? d (¡B)=max p¸0 ( p(x)¸(¡B;p)¡c¹¡h(B) ±+¤ + ¤ ±+¤ X y q d (yj¡B;p;w d )v ? d (y) ) : The main theorem of this subsection shows that the production policy that maximizes expected revenue is a control limit policy, and that the dynamic pricing that maximizes expected revenue has the same form as in Theorem 4.2, when there is no decision on production. 88 -30 -20 -10 0 10 20 30 0 200 400 600 800 1000 1200 1400 1600 x v -30 -20 -10 0 10 20 30 900 1000 1100 1200 1300 1400 1500 1600 1700 x v (a) μ = 1 (b) μ = 4 Figure 4.6: v ? d (solid line) and v ? 0 (dotted line) as a function of the starting state x when there is control over the production process (¤ = 6, v = 120, ¯ = 4, ± = 0:1, B = 100, c=10 andh(x)=0 forx2f¡B;¡B+1;:::g) Theorem 4.4. Assumeh is the zero function. There exists a thresholdS d 2< ¡ such that w ? d (x)= 8 > > > < > > > : 1 0 forx¸S d forx<S d (4.18) In addition, the optimal pricing policyp ? has the form p ? (x)=¡ v ? d (x+1)¡v ? d (x) 2 + v¡¯ x + ¹ 2 : 89 -30 -20 -10 0 10 20 30 0 0.2 0.4 0.6 0.8 1 x w -30 -20 -10 0 10 20 30 0 0.2 0.4 0.6 0.8 1 x w S 0 = -57 (a) μ = 1 (b) μ = 4 Figure 4.7: Optimal Production Decision: w ? d (solid line) andw ? 0 (dotted line) as a function of the state x (¤ = 6, v = 120, ¯ = 4, ± = 0:1, B = 100, c = 10 and h(x) = 0 for x2f¡B;¡B+1;:::g) It is also true that parts 1. and 2. of Theorem 4.2 continue to hold; that is, the optimal price is non-increasing in the number of orders and also non-increasing in the on-hand inventory. This statement can be checked following the same reasoning as in the proof of Theorem 4.2, and so we have omitted that argument. We conjecture that the value function is always monotonically decreasing, regardless of whether of not the holding cost function is zero. 90 4.6.3 Model Comparison We observe numerically that when the system manager has control over the production process it may no longer be the case that a static pricing policy with no leadtime disclo- sure can attain higher maximum revenue than a dynamic pricing policy with announced state-dependent expected leadtimes in all initial starting states. Figure 4.6 plots v ? d and v ? 0 for the same parameter values as in Figure 4.4. The difference is that in Figure 4.6 the system manager has control over the production process and in Figure 4.4 he does not. Observe that Figures 4.6(a) and 4.4(a) are very similar, but that Figures 4.6(b) and 4.4(b) are not. In particular, in Figure 4.4(b) the optimal static pricing policy attains higher max- imum revenue than the optimal dynamic pricing policy in all starting states. However, in Figure 4.6(b), the optimal static pricing policy attains higher maximum revenue only in the starting states with large backlog, and the optimal dynamic pricing policy attains higher maximum revenue in the states with positive on-hand inventory or small backlog, exactly as in Figures 4.6(a) and 4.4(a). To understand the reason for this difference, it is instructive to also compare the values S 0 andS d . These values are shown in Figure 4.7. Note thatS 0 <S d in both Figures 4.7(a) and 4.7(b). The implication is that the system manager wants to hold more inventory on average under the optimal static pricing policy than under the optimal dynamic pricing pol- icy. This is because higher average inventory leads to a smaller unconditional expected waiting time, and, in the static pricing model, arriving customers base their purchase deci- sions on the unconditional expected waiting time. The result is that there is more flexibility in the model with dynamic pricing to trade-off the risk of having backlog with the com- ponent production cost. Hence we expect that the model with dynamic pricing can attain higher maximum revenue in the states with positive on-hand inventory or small backlog, for any parameter values. This contrasts with the situation in which the system manager does not have control over the production process. 91 4.7 Convex Leadtime Cost Functions Intuitively, the more sensitivity customers are to longer leadtimes, the more incentive the system manager has to not disclose real-time leadtime information where there is a backlog. It is natural to capture an increasing sensitivity to longer leadtimes by increasing the con- vexity of the leadtime cost function; that is by allowing the first derivative of the leadtime cost function to increase more and more rapidly as the backlog increases. Then we expect that the critical starting state x 0 for which v ? 0 (x) < v ? d (x) if x < x 0 and v ? 0 (x) ¸ v ? d (x) if x¸x 0 decreases as the leadtime cost function becomes more convex. We verify the intuition in the previous paragraph in Figure 4.8. In particular, we assume the leadtime cost function is ¯(l) = 2l n for n = 1;2;3;4;5 and n = 100, and Figure 4.8 plotsv ? 0 andv ? d in each of these cases. We assume that the system manager does not have control over the production process, and that the holding cost function is the zero function. Then, because v ? 0 (x) satisfies (4.4) and (4.5), v ? 0 (x) = v ? 0 for all x 2 <, is constant and does not depend on the starting state x as explained in the fourth paragraph of Section 4.3. We solve forv ? d numerically from the optimality equation (4.10) and (4.11). Note that the equations (4.4), (4.5), (4.10) and (4.11) are valid for any leadtime cost function, even though Sections 4.3 and 4.4 focus on linear leadtime cost functions. Finally, observe that the critical statex 0 decreases in Figure 4.8, asn increases. 4.8 Conclusions and Future Research In this paper, we investigated the trade-off between static and dynamic pricing models for a single product inventory system in which customers dislike waiting. The static pric- ing model does not allow the system manager to change the product price, and also does not require that the system manager disclose real-time expected leadtime information that is conditional on the current system backlog. In the dynamic pricing model, the system 92 -30 -20 -10 0 10 20 30 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 x v d * -15 -10 -5 0 5 10 15 1150 1200 1250 1300 1350 1400 1450 1500 x v d * -15 -10 -5 0 5 10 15 1250 1300 1350 1400 1450 1500 x v d * -15 -10 -5 0 5 10 15 1300 1350 1400 1450 1500 x v d * -25 -20 -15 -10 -5 0 5 10 15 20 25 700 800 900 1000 1100 1200 1300 1400 1500 x v d * -15 -10 -5 0 5 10 15 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 x v d * (l )=2l 2 β(l )=2l (l )=2l 3 (l )=2l 4 (l )=2l 5 (l )=2l 100 Figure 4.8: (¤ = 6,± = 0:1,B = 100,¹ = 3,c = 10 andh(x) = 0 forx2f¡B;¡B + 1;:::g) 93 manager can change the product price in real-time, in response to increased backlog or increased on-hand inventory, but must also announce the expected leadtime conditional on the current system state. There is more pricing flexibility in the dynamic model but also a stringent leadtime disclosure requirement. We show that in general the model with static pricing has higher maximum expected infinite horizon discounted revenue when the system begins with a large backlog, and the model with dynamic pricing has higher maximum rev- enue otherwise. We further show it is sometimes the case that the model with static pricing has higher maximum revenue for all starting states. Finally, we generalize our model so that the system manager also has control over the production process. Then, it appears to always be the case that the model with static pricing has higher maximum revenue when the system begins with a large backlog, and the model with dynamic pricing has higher maximum revenue otherwise. There are several interesting directions for future research. First, we have focused our analysis on linear leadtime sensitivity functions, and have also performed numerical studies when the leadtime sensitivity function is convex. There is value in extending the analysis to include a richer space of leadtime sensitivity functions, in order to understand the effect of the leadtime sensitivity function on the question of whether the model with static pric- ing or the model with dynamic pricing can attain higher maximum revenue. Next, we have assumed that the system manager reports the true expected state-dependent leadtime. However, there is in reality a strategic game that is played between the system manager, that would prefer to quote leadtimes that are shorter than what may be accurate, and the customers, that will react negatively if the leadtimes they experience are longer than the quoted leadtimes. Finally, our model is very simple in the sense that there is one product, and customers are homogeneous in their product valuation distribution, and in their lead- time tolerances. It is of interest to extend the model to allow for multiple products, that may 94 be compliments or substitutes, and to allow for customers that may differ in their product valuations and/ or leadtime tolerances. 95 References –. (2000). Amazon.com Varies Prices of Identical Items for Test. The Wall Street J. Sept. 7, 2000. Adan, I. J., Wessels, J., & Zijm, W. H. M. (1991). Analysis of the asymmetric shortest queue problem. Queueing Systems, 8, 1-58. Ahmadi, R. H. (1997). Managing capacity and flow at theme parks. Operations Research, 45(1), 1-13. Akan, M., Ata, B., & Olsen, T. L. (2009). Congestion-Based Leadtime Quotation and Pricing for Revenue Maximization with Heterogeneous Customers. Working Paper. Allon, G., & Federgruen, A. (2007). Competition in Service Industries. Operations Research, 55(1), 37-55. Anand, K. S., Pac ¸, M. F., & Veeraraghavan, S. K. (2009). Quality-Speed Conundrum: Tradeoffs in Customer-Intensive Services. Working Paper. Armony, M., & Maglaras, C. (2004a). Contact centers with a call-back option and real-time delay information. Operations Research, 52(4), 527-545. Armony, M., & Maglaras, C. (2004b). On customer contact centers with a call-back option: Customer decisions, routing rules, and system design. Operations Research, 52(4), 271-292. Ata, B., & Olsen, T. L. (2009). Near-Optimal Dynamic Lead-Time Quotation and Schedul- ing Under Convex-Concave Customer Delay Costs. Operations Research, 57(3), 753-768. Billingsley, P. (1999). Convergence of Probability Measures. New York: John Wiley & Sons, Inc. (Second Edition) Bitran, G. R., Ferrer, J. C., & Oliveira, P. R. (2008). Managing customers experiences: Perspectives on the temporal aspects of service encounters. Manufacturing & Service Operations Management, 10(1), 61-83. 96 Browne, S., & Whitt, W. (1995). Piecewise-linear diffusion processes. In J. Dshalalow (Ed.), Advances in Queueing: Theory, Methods, and Open Problems (p. 463-480). Boca Raton, Florida: CRC Press. C ¸ elik, S., & Maglaras, C. (2008). Dynamic Pricing and Lead-Time Quotation for a Multi- class Make-to-Order Queue. Management Science, 54(6), 1132-1146. Chan, L. M. A., Simchi-Levi, D., & Swann, J. (2006). Pricing, Production, and Inven- tory Policies for Manufacturing with Stochastic Demand and Discretionary Sales. Manunfacturing and Service Operations Management, 8(2), 149-168. Chen, X., & Simchi-Levi, D. (2004). Coordinating Inventory Control and Pricing Strategies with Random Demand and Fixed Ordering Cost: The Finite Horizon Case. Opera- tions Research, 52(6), 887-896. Cohen, L. B. (2008). Production Pressure in Endoscopy: Balancing Quantity and Quality. Gastroenterology, 135(6), 1842-1844. Coy, P. (2000). The power of smart pricing. Business Week (April 10). Croasmun, J. (2005, January). Overworked Pharmacies Could Be Causing Medication Errors. Ergonomics Today. Dickson, D., Ford, R. C., & Laval, B. (2005). Managing real and virtual wait in hospitality and service organizations. Cornell Hotel and Restaurant Administration Quarterly, 46, 52-68. Dobson, G., & Pinker, J. (2006). The Value of Sharing Lead Time Information. IIE Transactions, 38(3), 171-183. Duenyas, I. (1995). Single Facility Due Date Setting with Multiple Customer Classes. Management Science, 41(4), 608-619. Duenyas, I., & Hopp, W. J. (1995). Quoting Customer Lead Times. Management Science, 41(1), 42-57. Elmaghraby, W., & Keskinocak, P. (2003). Dynamic Pricing in the Presence of Inven- tory Considerations: Research Overview, Current Practices, and Future Directions. Management Science, 49(10), 1287-1309. Federgruen, A., & Heching, A. (1999). Combined Pricing and Inventory Control Under Uncertainty. Operations Research, 47(3), 454-474. Flatto, L., & McLean, H. P. (1977). Two queue in parallel. Communications in Pure and Applied Mathematics, 30, 255-263. 97 Flynn, E. A., & Barker, K. N. (2000). Medication Errors: Causes, Prevention, and Risk Management. (V ol. Chapter 6; M. R. Cohen, Ed.). Sudbury, MA: Jones and Bartlett Publishers. Gans, N. (2002). Customer Loyalty and Supplier Quality Competition. Management Science, 48(2), 207-221. Guo, P., & Zipkin, P. (2007). Analysis and Comparison of Queues with Different Levels of Delay Information. Management Science, 53(6), 962-970. Guo, P., & Zipkin, P. (2008). The effects of information on a queue with balking and phase-type service times. Naval Research Logistics, 55(5), 406-411. Harrison, J. M. (1985). Brownian motion and stochastic flow systems. Malabar, Florida: Krieger. Hasija, S., Pinker, E. J., & Shumsky, R. A. (2008). Call Center Outsourcing Contracts Under Information Asymmetry. Management Science, 54(4), 793-807. Hasija, S., Pinker, E. J., & Shumsky, R. A. (2009). OM Practice–Work Expands to Fill the Time Available: Capacity Estimation and Staffing under Parkinson’s Law. Manufac- turing and Service Operations Management., 12(1), 1-18. Hassin, R. (1986). Customer information in markets with random products quality: The case if queues and balking. Econometrica, 54(5), 1185-1195. Hopp, W. J., Iravani, S. M. R., & Yuen, G. Y . (2007). Operations Systems with Discre- tionary Task Completion. Management Science, 53(1), 61-77. Iglehart, D. L., & Whitt, W. (1970). Multiple channels queues in heavy traffic I. Adv. in Applied Probability, 2, 150-177. Kapuscinski, R., & Tayur, S. (2007). Reliable Due-Date Setting in a Capacitated MTO System with Two Customer Classes. Operations Research, 55(1), 56-74. Karlin, S., & Carr, C. R. (1962). Prices and optimal inventory policy (K. Arrow, S. Karlin, & H. Scarf, Eds.). Stanford, CA: Stanford University Press. Katz, K., Larson, B., & Larson, R. (1991). Prescription for the waiting in line bluse: Entertain, enlighten and engage. Sloan Management Review, Winter, 44-53. Kc, D. S., & Terwiesch, C. (2009). Impact of Workload on Service Time and Patient Safety: An Econometric Analysis of Hospital Operations. Management Science, 55(9), 1486-1498. 98 Keskinocak, P., Ravi, R., & Tayur, S. (2001). Scheduling and Reliable Lead-Time Quota- tion for Orders with Availability Intervals and Lead-Time Sensitive Revenues. Man- agement Science, 47(2), 264-279. Kostami, V ., & Ward, A. R. (2008a). Managing service systems with an offline waiting option and customer abandonment: Companion note. Kostami, V ., & Ward, A. R. (2008b). Managing service systems with an offline waiting option and customer abandonment: Technical appendix. Kruk, L., Lehoczky, J., Ramanan, K., & Shreve, S. (2007). An explicit formula for the skorokhod map on [0,a]. Annals of Probability, 35(5), 1740-1768. Lovejoy, W. S., & Sethuraman, K. (2000). Congestion and Complexity Costs in a Plant with Fixed Resources that Strives to Make Schedule. Manufacturing and Service Operations Management, 2(3), 221-239. Lu, L. X., Mieghem, J. A. V ., & Savaskan, R. C. (2009). Incentives for Quality Through Endogenous Routing. Manufacturing and Service Operations Management, 11(2), 254-273. Maglaras, C., & Mieghem, J. A. V . (2005). Queueing systems with leadtime constraints: A fluid-model approach for admission and sequencing control. European Journal of Operational Research, 167(1), 179-207. Maister, D. (1985). The psychology of waiting in lines. In J. A. Czepiel, M. Solomon, & C. S. Surprenant (Eds.), The service encounter (p. 113-123). Lexington, MA: Lexington Books. Mandelbaum, A., & Shimkin, N. (2000). A model for rational abandonments from invisible queues. Queueing Systems, 36, 141-173. Marchal, W. G. (1978). Some Simpler Bounds on the Mean Queuing Time. Operations Research, 26(6), 1083-1088. Mielke, R., Zahralddin, A., Padam, D., & Mastaglio, T. (1998). Simulation applied to theme park management. In D. J. Medeiros, E. F. Watson, J. S. Carson, & M. S. Manivannan (Eds.), Proceedings of the 1998 Winter Simulation Conference (p. 1199- 1203). Munichor, N., & Rafaeli, A. (2007). Number of apologies? Customer reactions to tele- phone waiting time fillers. Journal of Applied Probability, 92, 511-518. 99 Needleman, J., Buerhaus, P., Mattke, S., Stewart, M., & Zelevinsky, K. (2002). Nurse Staffing Levels and the quality of care in hospitals. The New England Journal of Medicine, 326(22), 1715-1722. Oi, W. I. (1971). A disneyland dilemma: Two-part tariffs for a mickey mouse monopoly. The Quarterly Journal of Economics, 85(1), 77-96. Oliva, R., & Sterman, J. D. (2001). Cutting Corners and Working Overtime: Quality Erosion in the Service Industry. Management Science, 47(7), 894-914. Parlakt¨ urk, A., & Kumar, S. (2004). Self-interested routing in queueing networks. Man- agement Science, 50(7), 949-967. Plambeck, E., Kumar, S., & Harrison, J. M. (2001). Asymptotic optimality of a single server queueing system with constraints on throughput times. Queueing Systems, 39, 23-54. Plambeck, E. L. (2004). Optimal Leadtime Differentiation via Diffusion Approximations. Operations Research, 52(2), 213-228. Png, I. P. L., & Reitman, D. (1994). Service Time Competition. RAND Journal of Eco- nomics,, 25(4), 619-634. Puhalskii, A. (1994). On the invariance principle for the first passage time. Mathematics of Operations Research, 19, 946-954. Puterman, M. (2005). Markov decision processes. Wiley Interscience. Reed, J., & Ward, A. R. (2008). Approximating the GI/GI/1+GI queue with a nonlin- ear drift diffusion: Hazard rate scaling in heavy traffic. Mathematics of Operations Research, 33(3), 606-644. Reiman, M. I. (1984). Some diffusion approximations with state space collapse. In F. Bac- celi & G. Fayolle (Eds.), Modelling and Performance Evaluation Methodology (p. 209-240). Springer-Verlag. Ren, Z. J., & Wang, X. (2009). Should Patients be Steered to High V olume Hospitals? An Empirical Investigation of Hospital V olume and Operations Service Quality. Working Paper. Ren, Z. J., & Zhou, Y . (2008). Call Center Outsourcing: Coordinating Staffing Level and Service Quality. Management Science, 54(2), 369-383. So, K. C., & Song, J.-S. (1999). Price, delivery time guarantees and capacity selection. European Journal of Operations Research, 111(1), 28-49. 100 Streitfeld, D. (Sept 27, 2000). On the Web, Price Tags Blur; What You Pay Could Depend on Who You Are. Washington Post. Taylor, S. (1994). Waiting for service: the relationship between delays and evaluations of service. Journal of Marketing, 58, 56-69. Tatsu Ride Statistics. (2007). (http://en.wikipedia.org/wiki/Tatsu) Thomas, J. (1970). Price-Production Decisions with Deterministic Demand. Management Science, 16(11), 747-750. Thowsen, G. T. (1975). A dynamic nonstationary inventory problem for a price/quantity setting firm. Naval Res. Logist. Quart., 22, 461-476. Veeraraghavan, S., & Debo, L. (2009). Joining Longer Queues: Information Externalities in Queue Choice. Manufacturing and Service Operations Management, 11(4), 543- 562. Veinott, A., & Wagner, H. (1965). Computing Optimal (s,S) Inventory Policies. Manage- ment Science, 11(5), 525552. Whitt, W. (1999). Improving service by informing customers about anticipated delays. Management Science, 45(2), 192-207. Whitt, W. (2002). Stochastic-process limits. New York: Springer. Whitt, W. (2006). Fluid models for multiserver queues with abandonments. Operations Research, 54(1), 37-54. Yano, C. A., & Gilbert, S. M. (n.d.). Coordinated pricing and production/procurement decisions: A review (A. Chakravarty & J. Eliashberg, Eds.). Zipkin, P. (2000). Foundations of inventory management. Burr Ridge, IL: McGraw-Hill,. 101 Appendices A Managing Service Systems with an Offline Wait- ing Option & Customer Abandonment: Technical Appendix In this Technical Appendix we provide the proofs for Theorem 2.1, Corollary 2.1, Theorem 2.2, and Proposition 4.2 stated in Chapter 2 and Appendix B. This requires several Lemmas, which we state upfront, but whose proofs we defer to the end of this Technical Appendix. Recall that we are considering a system in which the arrival rate ¸ becomes large, and the service rate is ¸¡ p ¸µ for some µ 2 <. We superscript any process or quantity associated with the system having arrival rate¸ and service rate¹(¸)=¸¡ p ¸µ by¸. Note that we require the following technicalities. All random variables are defined on a common probability space (;F;P). For each positive integer d, let D be the space of right continuous functions with left limits (RCLL) in< d having time domain [0;1). We endow D with the usual Skorokhod J 1 topology, and let M d denote the Borel ¾-algebra associated with the J 1 topology. All stochastic processes are measurable functions from (;F;P) into(D;M d ) for some appropriate dimensiond. Supposef» n g 1 n=1 is a sequence of stochastic processes. The notation» n ) » means that the probability measures induced by the » n ’s on (D;M d ) converge weakly to the probability measure on (D;M d ) induced by the stochastic process ». Note that we suppress d from the notation unless necessary. We often reference the functional strong law of large numbers, the functional central limit theorem, and the continuous mapping theorem. A convenient reference for these theorems is Billingsley (1999) or Whitt (2002). We use the notatione to denote the identity process e(t) = t for allt¸ 0. We let “a.s.” denote “almost surely” and “u.o.c.” denote “uniformly on compact sets”. 102 It is useful to work with the system processes under law of large numbers (fluid) and central limit theorem (diffusion) scaling. Define the fluid scaled quantities ¹ A ¸ (t) ´ 1 ¸ A ¸ (t)¡t ¹ S ¸ I (t) ´ 1 ¸ S ¸ I (t)¡(1¡ µ p ¸ )t ¹ S ¸ O (t) ´ 1 ¸ S ¸ O (t)¡(1¡ µ p ¸ )t ¹ N ¸ (t) ´ 1 ¸ N(¸t)¡t ¹ Q ¸ I (t) ´ 1 ¸ Q ¸ I (t) ¹ Q ¸ O (t) ´ 1 ¸ Q ¸ O (t) ¹ Q ¸ (t) ´ 1 ¸ Q ¸ (t) ¹ ¿ ¸ (t) ´ 1 ¸ Z t 0 °[Q ¸ O (s)¡1] + ds; 103 and the diffusion scaled quantities ~ A ¸ (t) ´ p ¸( 1 ¸ A ¸ (t)¡t) ~ S ¸ I (t) ´ p ¸( 1 ¸ S ¸ I (t)¡(1¡ µ p ¸ )t) ~ S ¸ O (t) ´ p ¸( 1 ¸ S ¸ O (t)¡(1¡ µ p ¸ )t) ~ N ¸ (t) ´ p ¸( 1 ¸ N(¸t)¡t) ~ Q ¸ I (t) ´ 1 p ¸ Q ¸ I (t) ~ Q ¸ O (t) ´ 1 p ¸ Q ¸ O (t) ~ Q ¸ (t) ´ 1 p ¸ Q ¸ (t) ~ I ¸ (t) ´ p ¸(1¡ µ p ¸ )I ¸ (t) ~ W ¸ I (t) ´ p ¸W ¸ I (t) ~ W ¸ O (t) ´ p ¸W ¸ O (t) ~ W ¸ I ´ p ¸W ¸ I ~ W ¸ O ´ p ¸W ¸ O It is additionally useful to introduce the processesP ¸ I andP ¸ O , which represent the work- load processes in the inline and offline queues respectively. We use the term “workload” to indicate the total processing time of all the customers in the queue that will eventu- ally receive service when all the effort of the server is given exclusively to their queue (® = 1 or 0). Note that the workload process is defined conditionally on future abandon- ments, because the wait time of a customer is not affected by the customers in front of him that abandon. In particular, the actual waiting times a customer arriving to the inline or offline queue at time t would experience, W ¸ I (t) and W ¸ O (t), are exactly P ¸ I (t)=® and P ¸ O (t)=(1¡ ®) when the server works continuously at rate ® on the inline queue and at rate (1¡ ®) on the offline queue. Also define the diffusion-scaled workload processes ~ P ¸ I = p ¸P ¸ I and ~ P ¸ O = p ¸P ¸ O . Next, we will use the following three Lemmas, whose proofs we defer to the end of the appendix. Lemma 4.1. As¸!1, ¡ ¹ Q ¸ ;P ¸ I ;P ¸ O ;¹ ¿ ¸ ;T ¸ I +T ¸ O ;I ¸ ¢ !(0;0;0;0;e;0); a.s., u.o.c.. 104 It is useful to observe that as a consequence of Lemma 4.1 ~ N ¸ ±¿ ¸ )0; (A.1) as ¸ ! 1. This weak convergence follows because the functional central limit theorem establishes that ~ N ¸ weakly converges to a Brownian motion as ¸ ! 1. Since the initial position of the Brownian motion is 0, and¿ ¸ is a non-decreasing process, the random time change theorem establishes (A.1). Lemma 4.2. The sequencef ~ Q ¸ ; ~ I ¸ g is tight inD. Lemma 4.3. Consider a system having arrival rate¸ and service rate¹(¸)´¸¡ p ¸µ for someµ 2<. For ~ Q defined by (2.11) in which ~ X is a Brownian motion with infinitesimal driftµ and infinitesimal variance¾ 2 , p ¸W ¸ I ) w O (1¡®)w I +®w O ~ Q and p ¸W ¸ O ) w I (1¡®)w I +®w O ~ Q; as¸!1: (A.2) Furthermore, for anyT > 0, as¸!1; sup 0·t·T ¯ ¯ ¯ ¯ ¯ ~ W ¸ I (t)¡ ~ P ¸ I (t) ® ¯ ¯ ¯ ¯ ¯ !0 and sup 0·t·T ¯ ¯ ¯ ¯ ¯ ~ W ¸ O (t)¡ ~ P ¸ O (t) 1¡® ¯ ¯ ¯ ¯ ¯ !0; in probability. (A.3) Finally, it is useful to write the stochastic equation for the diffusion ~ Q in (2.11) in terms of the one-sided linearly generalized regulator mapping, whose definition we provide below. Definition 4.1. (The one-sided linearly generalized regulator mapping) Given · a non-negative constant and x 2 D([0;1);<) having x(0) ¸ 0, the one-sided linearly generalized regulator mapping (Á · ;Ã · ):D([0;1);<)7!D([0;1);[0;1)£[0;1)) is defined by (Á · ;Ã · )(x)´(z;l) where (C1) z(t)=x(t)¡· R t 0 z(s)ds+l(t)2[0;1) for allt¸0; (C2) l is nondecreasing,l(0)=0, and R 1 0 z(t)dl(t)=0. Specifically, for ·´° (1¡®)w I ®w O +(1¡®)w I ; 105 it follows that ³ ~ Q; ~ I ´ =(Á · ;Ã · ) ³ ~ X ´ : (A.4) Proposition 4.1 part (i) in Reed and Ward (2008) establishes the existence and uniqueness of the regulator mapping in Definition 4.1 1 , and so the representation (A.4) is equivalent to the representation for ~ Q in (2.11). Proof of Theorem 2.1 Proof of (i) The structure of our proof follows the proof of Theorem 1 in Section 5 in Reiman (1984), which establishes state-space collapse for a join the shorter queue system in heavy traffic with no abandonments. However, more delicate argument is required to handle the cus- tomer abandonments. We need to show that for any²>0, P µ sup 0·t·T ¯ ¯ ¯ ¯ w I ® ~ Q ¸ I (t)¡ w O 1¡® ~ Q ¸ O (t) ¯ ¯ ¯ ¯ >² ¶ !0 as¸!1: (A.5) Fix²>0 and let » ¸ ´ inf ½ t¸0: ¯ ¯ ¯ ¯ w I ® ~ Q ¸ I (t)¡ w O 1¡® ~ Q ¸ O (t) ¯ ¯ ¯ ¯ >² ¾ » ¤ ¸ ´ sup ½ t·» ¸ : ¯ ¯ ¯ ¯ w I ® ~ Q ¸ I (t)¡ w O 1¡® ~ Q ¸ O (t) ¯ ¯ ¯ ¯ · ² 2 ¾ : 1 Actually, the regulator mapping in Definition 4.1 is a specific instance of the more general regulator mapping in Reed and Ward (2008). 106 It will also be useful to define the processes ~ U ¸ 1 (t;s;u;v) ´ ¡ w I ® n ~ S ¸ I (u+®(t¡s))¡ ~ S ¸ I (u) o + w O 1¡® n ~ S ¸ O (v+(1¡®)(t¡s))¡ ~ S ¸ O (v) o ¡ w O 1¡® n ~ A ¸ (t)¡ ~ A ¸ (s) o + ½ µ p ¸ (w I ¡w O )¡ ® 1¡® w O ¡w I ¾ p ¸(t¡s) ~ U ¸ 2 (t;s;u;v) ´ ¡ w O 1¡® n ~ S ¸ O (v+(1¡®)(t¡s))¡ ~ S ¸ O (v) o + w I ® n ~ S ¸ I (u+®(t¡s))¡ ~ S ¸ I (u) o ¡ w I ® n ~ A ¸ (t)¡ ~ A ¸ (s) o + ½ µ p ¸ (w O ¡w I )¡ (1¡®) ® w I ¡w O ¾ p ¸(t¡s): An upper bound for the left-hand-side of (A.5) First assumew I ~ Q ¸ I (» ¤ ¸ )=®>w O ~ Q ¸ O (» ¤ ¸ )=(1¡®). Then, for» ¤ ¸ ·t·» ¸ , all customers join the offline service queue, and so ¯ ¯ ¯ ¯ w I ® ~ Q ¸ I (t)¡ w O 1¡® ~ Q ¸ O (t) ¯ ¯ ¯ ¯ = w I ® ~ Q ¸ I (» ¤ ¸ ¡)¡ w O 1¡® ~ Q ¸ O (» ¤ ¸ ¡)¡ w I ® 1 p ¸ © S ¸ I (T ¸ I (t))¡S ¸ I (T ¸ I (» ¤ ¸ ¡)) ª + w O 1¡® 1 p ¸ £ S ¸ O (T ¸ O (t))¡S ¸ O (T ¸ O (» ¤ ¸ ¡)) ¤ ¡ w O 1¡® h ~ A ¸ (t)¡ ~ A ¸ (» ¤ ¸ ¡)+ p ¸(t¡» ¤ ¸ ) i + w O 1¡® 1 p ¸ · N µZ t 0 ° £ Q ¸ O (s)¡1 ¤ + ds ¶ ¡N µZ » ¤ ¸ ¡ 0 ° £ Q ¸ O (s)¡1 ¤ + ds ¶¸ : (A.6) The inline queue does not become empty during[» ¤ ¸ ;» ¸ ], so that T ¸ I (t)¡T ¸ I (» ¤ ¸ ¡)¸®(t¡» ¤ ¸ ): The offline queue may become empty during[» ¤ ¸ ;» ¸ ], so that T ¸ O (t)¡T ¸ O (» ¤ ¸ ¡)·(1¡®)(t¡» ¤ ¸ ): 107 SinceS ¸ I andS ¸ O are non-decreasing processes, S ¸ I (T ¸ I (t))¡S ¸ I (T ¸ I (» ¤ ¸ ¡)) ¸ S ¸ I (T ¸ I (» ¤ ¸ ¡)+®(t¡» ¤ ¸ ))¡S ¸ I (T ¸ I (» ¤ ¸ ¡)) = p ¸ · ~ S ¸ I ¡ T ¸ I (» ¤ ¸ ¡)+®(t¡» ¤ ¸ ) ¢ ¡ ~ S ¸ I ¡ T ¸ I (» ¤ ¸ ¡) ¢ +® µ 1¡ µ p ¸ ¶ p ¸(t¡» ¤ ¸ ) ¸ ; and S ¸ O (T ¸ O (t))¡S ¸ O (T ¸ O (» ¤ ¸ ¡)) · S ¸ O (T ¸ O (» ¤ ¸ ¡)+(1¡®)(t¡» ¤ ¸ ))¡S ¸ O (T ¸ O (» ¤ ¸ ¡)) = p ¸ " ~ S ¸ O ¡ T ¸ O (» ¤ ¸ ¡)+(1¡®)(t¡» ¤ ¸ ) ¢ ¡ ~ S ¸ O ¡ T ¸ O (» ¤ ¸ ¡) ¢ +(1¡®) ³ 1¡ µ p ¸ ´ p ¸(t¡» ¤ ¸ ) # : The definition of» ¤ ¸ and substitution of the above upper bounds into (A.6) establish ¯ ¯ ¯ ¯ w I ® ~ Q ¸ I (t)¡ w O 1¡® ~ Q ¸ O (t) ¯ ¯ ¯ ¯ · ² 2 + ~ U ¸ 1 (t;» ¤ ¸ ¡;T ¸ I (» ¤ ¸ ¡);T ¸ O (» ¤ ¸ ¡)) + w O 1¡® 1 p ¸ · N µZ t 0 ° £ Q ¸ O (s)¡1 ¤ + ds ¶ ¡N µZ » ¤ ¸ ¡ 0 ° £ Q ¸ O (s)¡1 ¤ + ds ¶¸ : Whenw I ~ Q ¸ I (» ¤ ¸ )=®·w O ~ Q ¸ O (» ¤ ¸ )=(1¡®), a similar argument shows ¯ ¯ ¯ ¯ w O 1¡® ~ Q ¸ O (t)¡ w I ® ~ Q ¸ I (t) ¯ ¯ ¯ ¯ · ² 2 + ~ U ¸ 2 (t;» ¤ ¸ ¡;T ¸ I (» ¤ ¸ ¡);T ¸ O (» ¤ ¸ ¡)) ¡ w O 1¡® 1 p ¸ · N µZ t 0 ° £ Q ¸ O (s)¡1 ¤ + ds ¶ ¡N µZ » ¤ ¸ ¡ 0 ° £ Q ¸ O (s)¡1 ¤ + ds ¶¸ : Also noting the processN is non-negative, we conclude ¯ ¯ ¯ ¯ w I ® ~ Q ¸ I (t)¡ w O 1¡® ~ Q ¸ O (t) ¯ ¯ ¯ ¯ · ² 2 +max n ~ U ¸ 1 (t;» ¤ ¸ ¡;T ¸ I (» ¤ ¸ ¡);T ¸ O (» ¤ ¸ ¡)); ~ U ¸ 2 (t;» ¤ ¸ ¡;T ¸ I (» ¤ ¸ ¡);T ¸ O (» ¤ ¸ ¡) o + w O 1¡® 1 p ¸ µ N µZ t 0 °[Q ¸ O (w)¡1] + dw ¶ ¡N µZ » ¤ ¸ ¡ 0 °[Q ¸ O (w)¡1] + dw ¶¶ : 108 Therefore, the left-hand side of (A.5) can be bounded as follows P µ sup 0·t·T ¯ ¯ ¯ ¯ w I ® ~ Q ¸ I (t)¡ w O 1¡® ~ Q ¸ O (t) ¯ ¯ ¯ ¯ >² ¶ (A.7) · P Ã sup 0·s·t·T sup 0·u;v·s max n ~ U ¸ 1 (t;s;u;v); ~ U ¸ 2 (t;s;u;v) o + w O 1¡® 1 p ¸ ³ N ³ R t 0 °[Q ¸ O (w)¡1] + dw ´ ¡N ³ R s 0 ° £ Q ¸ O (w)¡1 ¤ + dw ´´ > ² 2 ! : Convergence of the right-hand-side of (A.7) to zero Let´ be arbitrarily small. We will show that the right-hand side of (A.7) is less than´ for large enough¸. Define ~ B ¸ I (t) ´ sup 0·s·t;0·u;v·s ¡ w I ® n ~ S ¸ I (u+®(t¡s))¡ ~ S ¸ I (u) o + w O 1¡® n ~ S ¸ O (v+(1¡®)(t¡s))¡ ~ S ¸ O (v) o ¡ w O 1¡® n ~ A ¸ (t)¡ ~ A ¸ (s) o ~ B ¸ O (t) ´ sup 0·s·t;0·u;v·s ¡ w O 1¡® n ~ S ¸ O (v+(1¡®)(t¡s))¡ ~ S ¸ O (v) o + w I ® n ~ S ¸ I (u+®(t¡s))¡ ~ S ¸ I (u) o ¡ w I ® n ~ A ¸ (t)¡ ~ A ¸ (s) o : Also observe that max ³ ~ U ¸ 1 (t;s;u;v); ~ U ¸ 2 (t;s;u;v) ´ · max ³ ~ B ¸ I (t); ~ B ¸ O (t) ´ + p ¸(t¡s)max ( µ p ¸ (w I ¡w O )¡ ® 1¡® w O ¡w I ; µ p ¸ (w O ¡w I )¡ 1¡® ® w I ¡w O ) (because for any constants d 1 , d 2 , d 3 , and d 4 , max(d 1 + d 2 ;d 3 + d 4 ) · max(d 1 ;d 3 ) + max(d 2 ;d 4 )). Furthermore, forw´min(w I ;w O ), max ( µ p ¸ (w I ¡w O )¡ ® 1¡® w O ¡w I ; µ p ¸ (w O ¡w I )¡ 1¡® ® w I ¡w O ) · µ p ¸ (w I +w O )¡w; and so max ³ ~ U ¸ 1 (t;s;u;v); ~ U ¸ 2 (t;s;u;v) ´ · max ³ ~ B ¸ I (t); ~ B ¸ O (t) ´ + p ¸(t¡s) ³ µ p ¸ (w I +w O )¡w ´ : 109 Next, for anyt> 0, 1 p ¸ N µZ t 0 ° £ Q ¸ O (w)¡1 ¤ + dw ¶ = ~ N ¸ ¡ ¿ ¸ (t) ¢ +° Z t 0 £ Q ¸ O (w)¡1 ¤ + p ¸ dw; and so 1 p ¸ µ N µZ t 0 ° £ Q ¸ O (w)¡1 ¤ + dw ¶ ¡N µZ s 0 ° £ Q ¸ O (w)¡1 ¤ + dw ¶¶ = ~ N ¸ (¿(t))¡ ~ N ¸ (¿(s))+° Z t s £ Q ¸ O (w)¡1 ¤ + p ¸ dw · ¯ ¯ ¯ ~ N ¸ ¡ ¿ ¸ (t) ¢ ¯ ¯ ¯+ ¯ ¯ ¯ ~ N ¸ ¡ ¿ ¸ (s) ¢ ¯ ¯ ¯+°(t¡s) sup 0·t·T ~ Q ¸ (t): We conclude that P 0 B B B @ sup 0·s·t·T sup 0·u;v·s max n ~ U ¸ 1 (t;s;u;v); ~ U ¸ 2 (t;s;u;v) o + w O 1¡® 1 p ¸ 0 @ N ³ R t 0 ° £ Q ¸ O (w)¡1 ¤ + dw ´ ¡N ³ R s 0 ° £ Q ¸ O (w)¡1 ¤ + dw ´ 1 A > ² 2 1 C C C A (A.8) · P 0 @ sup 0·s·t·T max ³ ~ B ¸ I (t); ~ B ¸ O (t) ´ + ¯ ¯ ¯ ~ N ¸ ¡ ¿ ¸ (t) ¢ ¯ ¯ ¯+ ¯ ¯ ¯ ~ N ¸ ¡ ¿ ¸ (s) ¢ ¯ ¯ ¯ +°(t¡s)sup 0·t·T ~ Q ¸ (t)+ p ¸(t¡s) ³ µ p ¸ (w I +w O )¡w ´ > ² 2 1 A : From (A.1), ~ N ¸ ¡ ¿ ¸ (T) ¢ )0 as¸!1. Hence for large enough¸ P µ sup 0·t·T ~ N ¸ ¡ ¿ ¸ (t) ¢ > ² 8 ¶ < ´ 4 ; and so P 0 @ sup 0·s·t·T max ³ ~ B ¸ I (t); ~ B ¸ O (t) ´ + ¯ ¯ ¯ ~ N ¸ ¡ ¿ ¸ (t) ¢ ¯ ¯ ¯+ ¯ ¯ ¯ ~ N ¸ ¡ ¿ ¸ (s) ¢ ¯ ¯ ¯ +°(t¡s)sup 0·t·T ~ Q ¸ (t)+ p ¸(t¡s) ³ µ p ¸ (w I +w O )¡w ´ > ² 2 1 A · P 0 @ sup 0·s·t·T 0 @ max ³ ~ B ¸ I (t); ~ B ¸ O (t) ´ +°(t¡s)sup 0·t·T ~ Q ¸ (t) + p ¸(t¡s) ³ µ p ¸ (w I +w O )¡w ´ 1 A > ² 4 1 A + ´ 4 : Furthermore, it follows from Lemma 4.2 that there exists a finite positive constantM such that P µ sup 0·t·T ~ Q ¸ (t)>M ¶ < ´ 4 110 for all large enough¸. Therefore, P 0 @ sup 0·s·t·T 8 < : max ³ ~ B ¸ I (t); ~ B ¸ O (t) ´ +°(t¡s)sup 0·t·T ~ Q ¸ (t) + p ¸(t¡s) ³ µ p ¸ (w I +w O )¡w ´ 9 = ; > ² 4 1 A · P 0 @ sup 0·s·t·T 8 < : max ³ ~ B ¸ I (t); ~ B ¸ O (t) ´ +°(t¡s)M + p ¸(t¡s) ³ µ p ¸ (w I +w O )¡w ´ 9 = ; > ² 4 1 A + ´ 4 : We conclude that P 0 @ sup 0·s·t·T 8 < : max ³ ~ B ¸ I (t); ~ B ¸ O (t) ´ + ¯ ¯ ¯ ~ N ¸ ¡ ¿ ¸ (t) ¢ ¯ ¯ ¯+ ¯ ¯ ¯ ~ N ¸ ¡ ¿ ¸ (s) ¢ ¯ ¯ ¯ +°(t¡s)sup 0·t·T ~ Q ¸ (t)+ p ¸(t¡s) ³ µ p ¸ (w I +w O )¡w ´ 9 = ; > ² 2 1 A ·P 0 @ sup 0·s·t·T 8 < : max ³ ~ B ¸ I (t); ~ B ¸ O (t) ´ +°(t¡s)M + p ¸(t¡s) ³ µ p ¸ (w I +w O )¡w ´ 9 = ; > ² 4 1 A + ´ 2 : Hence, from (A.8), to complete the proof, it is sufficient to show that P 0 @ sup 0·s·t·T 0 @ max ³ ~ B ¸ I (t); ~ B ¸ O (t) ´ +°(t¡s)M + p ¸(t¡s) ³ µ p ¸ (w I +w O )¡w ´ 1 A > ² 4 1 A < ´ 2 : (A.9) This argument is similar to the paragraph following (10) in the proof of Theorem 3.2 in (Reiman, 1984), and we show the details in the next two paragraphs for the reader’s convenience. Details of the argument that (A.9) holds For large enough¸, sincew I >0 andw O >0, µ p ¸ (w I +w O )¡w <0: For anyº2(0;T), for large enough¸, sup 0·s·t·T max ³ ~ B ¸ I (t); ~ B ¸ O (t) ´ +°(t¡s)M + p ¸(t¡s) µ µ p ¸ (w I +w O )¡w ¶ · max ³ ~ B ¸ I (º); ~ B ¸ O (º) ´ +°ºM +max ³ ~ B ¸ I (T); ~ B ¸ O (T) ´ +°TM + p ¸º µ µ p ¸ (w I +w O )¡w ¶ : 111 Hence P 0 @ sup 0·s·t·T 8 < : max ³ ~ B ¸ I (t); ~ B ¸ O (t) ´ +°(t¡s)M + p ¸(t¡s) ³ µ p ¸ (w I +w O )¡w ´ 9 = ; > ² 4 1 A · P ³ max ³ ~ B ¸ I (º); ~ B ¸ O (º) ´ +°ºM > ² 8 ´ +P µ max ³ ~ B ¸ I (T); ~ B ¸ O (T) ´ +°TM > ² 8 + p ¸º · w¡ µ p ¸ (w I +w O ) ¸¶ : (A.10) The functional central limit theorem shows that ~ S ¸ O , ~ S ¸ I , and ~ A ¸ all weakly converge to mean 0 - Brownian motions. Hence ~ B ¸ I (t) ) 0 and ~ B ¸ O (t) ) 0 as t ! 1. This means that we can chooseº small enough so that P ³ max ³ ~ B ¸ I (º); ~ B ¸ O (º) ´ +°ºM > ² 8 ´ < ´ 4 : It is also true that ~ B ¸ I (T) and ~ B ¸ O (T) weakly converge to mean 0 normal random variables. Hence we can choose¸ large enough so that P µ max ³ ~ B ¸ I (t); ~ B ¸ O (t) ´ +°TM > ² 8 + p ¸º µ w¡ µ p ¸ (w I +w O ) ¶¶ < ´ 4 : Then, it follows from (A.10) that P 0 @ sup 0·s·t·T 8 < : max ³ ~ B ¸ I (t); ~ B ¸ O (t) ´ +°(t¡s)M + p ¸(t¡s) ³ µ p ¸ (w I +w O )¡w ´ 9 = ; > ² 4 1 A < ´ 2 ; which shows that (A.9) is valid, and so completes the proof. Proof of (ii) We first represent ~ Q ¸ using the one-sided linearly generalized regulator mapping given in Definition 4.1. Let ~ X ¸ (t) ´ ~ A ¸ (t)¡ ~ S ¸ I ¡ T ¸ I (t) ¢ ¡ ~ S ¸ O ¡ T ¸ O (t) ¢ ¡ ~ N ¸ ¡ ¿ ¸ (t) ¢ +µt ~ ² ¸ (t) ´ ° Z t 0 Ã (1¡®)w I ®w O +(1¡®)w I ~ Q ¸ (s)¡ · ~ Q ¸ O (s)¡ 1 p ¸ ¸ + ! ds: Then, for allt¸0, ~ Q ¸ (t)= ~ X ¸ (t)+~ ² ¸ (t)¡° Z t 0 (1¡®)w I ®w O +(1¡®)w I ~ Q ¸ (s)+ ~ I ¸ (t)¸0: 112 Since also ~ I ¸ is non-decreasing, ~ I ¸ (0)=0, and Z 1 0 ~ Q ¸ (t)d ~ I ¸ (t)= Z 1 0 Q ¸ (t)1fQ ¸ (t)=0gdt=0; it follows that ³ ~ Q ¸ ; ~ I ¸ ´ ´(Á · ;Ã · ) ³ ~ X ¸ +~ ² ¸ ´ : (A.11) Suppose we can show ~ X ¸ ) ~ X; (A.12) as ¸!1, where ~ X is a Brownian motion with drift µ and variance ¾ 2 = ¾ 2 A +¾ 2 S as in (2.11). Suppose we can also show that ~ ² ¸ )0; (A.13) as¸!1. Proposition 4.1 part (iv) in Reed and Ward (2008) establishes that the mapping (Á · ;Ã · ) is continuous. Therefore, by the continuous mapping theorem (Á · ;Ã · ) ³ ~ X ¸ +~ ² ¸ ´ )(Á · ;Ã · ) ³ ~ X ´ ; as¸!1. The representation( ~ Q; ~ I) in terms of the one-sided linearly generalized regu- lator mapping shows( ~ Q; ~ I)=(Á · ;Ã · )( ~ X), and so ³ ~ Q ¸ ; ~ I ¸ ´ )( ~ Q; ~ I) as¸!1. We now establish (A.12). The sequence f(T ¸ O ;T ¸ I )g is tight in D because jT ¸ I (t)¡ T ¸ I (s)j·jt¡sj andjT ¸ O (t)¡T ¸ O (s)j·jt¡sj. Consider any subsequencef¸ k g on which ³ T ¸ k O ;T ¸ k I ´ )(T O ;T I ) as¸ k !1. By Lemma 4.1, the limit process satisfies T O +T I =e: LetB 1 ,B 2 , andB 3 be independent, standard Brownian motions. On the subsequencef¸ k g, by the functional central limit theorem and the continuous mapping theorem ~ A ¸ k ¡ ~ S ¸ k I ±T ¸ k I ¡ ~ S ¸ k O ±T ¸ k O +µe)¾ A B 1 ¡¾ S B 2 ±T I ¡¾ S B 3 ±T O +µe; as¸ k !1. By (A.1) ~ N ¸ k ±¿ ¸ k )0; 113 as¸ k !1. Therefore, ~ X ¸ k )¾ A B 1 ¡¾ S B 2 ±T I ¡¾ S B 3 ±T O +µe D = ~ X; as ¸ k !1, where the symbol D = denotes equality in distribution. Since the subsequence f¸ k g was arbitrary, we conclude ~ X ¸ ) ~ X; as¸!1. Finally, to establish (A.13) and complete the proof, let¸ k be a subsequence along which ~ Q ¸ k ) ~ Q as ¸ k ! 1. Such a subsequence exists by Lemma 4.2. On this subsequence, by part (i) and the fact that ~ Q ¸ k = ~ Q ¸ k I + ~ Q ¸ k O , for anyT > 0, sup 0·t·T 1 p ¸ k ¯ ¯ ¯ ¯ w I ® ³ Q ¸ k (t)¡Q ¸ k O (t) ´ ¡ w O 1¡® Q ¸ k O (t) ¯ ¯ ¯ ¯ !0; in probability as¸ k !1 or equivalently, sup 0·t·T 1 p ¸ k ¯ ¯ ¯ ¯ w I (1¡®) w I (1¡®)+w O ® Q ¸ k (t)¡Q ¸ k O (t) ¯ ¯ ¯ ¯ !0; in probability as¸ k !1: It now follows that ~ ² ¸ k ) 0 as ¸ k ! 1. Since the subsequence ¸ k was arbitrary, we conclude that~ ² ¸ )0 as¸!1. ¤ Proof of Corollary 1 The fact that Q ¸ (t) p ¸ = Q ¸ I (t) p ¸ + Q ¸ O (t) p ¸ combined with Theorem 2.1 part (i) gives that for any T > 0, sup 0·t·T 1 p ¸ ¯ ¯ ¯ ¯ w I ® ¡ Q ¸ (t)¡Q ¸ O (t) ¢ ¡ w O 1¡® Q ¸ O (t) ¯ ¯ ¯ ¯ !0; in probability as¸!1 or equivalently, sup 0·t·T 1 p ¸ ¯ ¯ ¯ ¯ w I (1¡®) w I (1¡®)+w O ® Q ¸ (t)¡Q ¸ O (t) ¯ ¯ ¯ ¯ !0; in probability as¸!1: But by Theorem 2.1 part (ii), we have that 1 p ¸ Q ¸ ) ~ Q, as¸!1 and so Slutsky’s theorem implies that Q ¸ O p ¸ ) w I (1¡®) w I (1¡®)+w O ® ~ Q; as¸!1: (A.14) 114 Following a similar argument, we can conclude that Q ¸ I p ¸ ) w O ® w I (1¡®)+w O ® ~ Q; as¸!1: For the last convergence result, first observe that for anyt¸0 1 p ¸ N µZ t 0 °[Q ¸ O (s)¡1] + ds ¶ = ~ N ¸ ¡ ¹ ¿ ¸ (t) ¢ +° Z t 0 [Q ¸ O (s)¡1] + p ¸ ds: By (A.1), ~ N ¸ ±¿ ¸ )0 as¸!1. It now follows from (A.14) above that 1 p ¸ N µZ ¢ 0 °[Q ¸ O (s)¡1] + ds ¶ )° (1¡®)w I ®w O +(1¡®)w I Z ¢ 0 ~ Q(s)ds as¸!1: ¤ Proof of Theorem 2 Recalling thatW I (t)=Q I (t)=[¹(¸)®], it then follows from Corollary 2.1 that ~ W ¸ I (t)) w O (1¡®)w I +®w O ~ Q: Hence it also follows from Lemma 4.3 that sup 0·t·T ¯ ¯ ¯ ~ W ¸ I (t)¡ ~ W ¸ I (t) ¯ ¯ ¯!0 in probability as¸!1; A very similar argument shows that for anyT >0 sup 0·t·T ¯ ¯ ¯ ~ W ¸ O (t)¡ ~ W ¸ O (t) ¯ ¯ ¯!0 in probability as¸!1; except that sinceW O (t) = O(t)=[¹(¸)(1¡®)], recalling that O(t) is upper-bounded in (2.10), we must additionally establish that 1 p ¸ Ã N µZ t 0 °[Q O (s)¡1] + ds ¶ ¡N Ã Z [t¡sup 0·s·T W O (s)] + 0 °[Q O (s)¡1] + ds !! )0: (A.15) Note that 1 p ¸ Ã N µZ t 0 °[Q O (s)¡1] + ds ¶ ¡N Ã Z [t¡sup 0·s·T W O (s)] + 0 °[Q O (s)¡1] + ds !! = 115 ~ N ¸ ¡ ¹ ¿ ¸ (t) ¢ ¡ ~ N ¸ Ã ¹ ¿ ¸ Ã · t¡ sup 0·s·t W ¸ O (s) ¸ + !! +° Z t [ t¡sup 0·s·t W ¸ O (s) ] + [Q ¸ O (s)¡1] + p ¸ ds: The first two terms in the above weakly converge to the zero process by the weak convergence in (A.1). Next, since the definition of the workload process P ¸ O implies W ¸ O (t) · P ¸ O (t) 1¡® for all t ¸ 0, it follows from Lemma 4.1 that W ¸ O ! 0 a.s., u.o.c., as ¸!1. Hence the third term also weakly converges to the zero process by the continuous mapping theorem and the weak convergence ofQ ¸ = p ¸ in Corollary 2.1. ¤ Proofs of Lemmas 4.1 - 4.3 The proofs of Lemmas 4.1-4.3 require use of the well-known conventional one-sided regu- lator mapping. This mapping is defined exactly as in Definition 4.1 for·=0. Proof of Lemma 4.1 We first represent the process Q ¸ using the conventional one-sided regulator mapping. Define X ¸ (t)´A ¸ (t)¡S ¸ I ¡ T ¸ I (t) ¢ ¡S ¸ O ¡ T ¸ O (t) ¢ ¡N ¸ ¡ ¿ ¸ (t) ¢ + µ p ¸ t: Then, for allt¸0, ¹ Q ¸ (t)= ¹ X ¸ (t)¡¹ ¿ ¸ (t)+ µ 1¡ µ p ¸ ¶ I ¸ (t): Since I ¸ is non-decreasing, I ¸ (0) = 0 and R 1 0 Q ¸ (t)d ³³ 1¡ µ p ¸ ´ I ¸ (t) ´ = 0, it follows that µ Q ¸ ; µ 1¡ µ p ¸ ¶ I ¸ ¶ =(Á;Ã) ³ X ¸ ¡¿ ¸ ´ : Since¿ ¸ is a non-decreasing process, Lemma 5.1 in Kruk et al. (2007) establishes Á ³ X ¸ ¡¿ ¸ ´ ·Á ³ X ¸ ´ : The functional strong law of large numbers establishes X ¸ !0 a.s., u.o.c.; as¸!1, and so, by the continuous mapping theorem, Á ³ X ¸ ´ !0 a.s., u.o.c.. 116 SinceQ ¸ is a non-negative process bounded above byÁ ³ X ¸ ´ , we conclude Q ¸ !0 a.s., u.o.c.; as¸!1, and so also ¿ ¸ !0 a.s., u.o.c.; as ¸ ! 1. Since (Á;Ã)(0) = (0;0) and Ã is a continuous function in D, we can also conclude that I ¸ = p ¸ p ¸¡µ Ã ³ X ¸ ¡¿ ¸ ´ !0 a.s., u.o.c., as¸!1. The condition (2.8) then implies T ¸ I +T ¸ O !e a.s., u.o.c.; as¸!1. It remains to show P ¸ O !0 andP ¸ I !0 a.s., u.o.c.; as¸!1. Define the total time required to process all the customers in the offline queue, ignoring any partial processing that may have already occurred on the customer in service U ¸ O (t)´ S ¸ O ( T ¸ O (t) ) +Q ¸ O (t) X j=S ¸ O (T ¸ O (t))+1 v O j ¹(¸) : Then, P ¸ O (t)·U ¸ O (t) for allt¸0: Define V ¸ O (t)´ 1 ¸ b¸tc X i=1 ¡ v O i ¡1 ¢ ; and observe that U ¸ O (t)= p ¸ p ¸¡µ ½· V ¸ O µ 1 ¸ S ¸ O ¡ T ¸ O (t) ¢ +Q ¸ O (t) ¶ ¡V ¸ O µ 1 ¸ S ¸ O ¡ T ¸ O (t) ¢ ¶¸ +Q ¸ O (t) ¾ : Since0·Q ¸ O (t)·Q ¸ (t) for allt¸0 and we have already establishedQ ¸ !0 a.s., u.o.c. as¸!1, it follows that Q ¸ O !0 a.s., u.o.c.; 117 as¸!1. Therefore, because alsoV ¸ O !0 a.s., u.o.c. as¸!1, it follows thatU ¸ O !0 a.s., u.o.c. as¸!1. We conclude P ¸ O !0 a.s., u.o.c., as¸!1. The argument thatP ¸ I !0 a.s., u.o.c. is identical and so is omitted. ¤ Proof of Lemma 4.2 The representation (A.11), and the continuity of the mapping(Á · ;Ã · ) established in Propo- sition 4.1 part (iv) in Reed and Ward imply that it is enough to show that the familiesf ~ X ¸ g andf~ ² ¸ g defined in the proof of part (ii) of Theorem 2.1 are tight in D. The tightness of the family f ~ X ¸ g is immediate from the weak convergence established in (A.12), which requires only Lemma 4.1. Hence we need only show that the familyf~ ² ¸ g is tight in D. (Note that this does not follow from the weak convergence in (A.13) because that argu- ment relies on the fact that the sequencef ~ Q ¸ g is tight.) For this, we must verify conditions (16.17) and (16.18) in Billingsley. Suppose we can show that for any T > 0 and ² > 0 arbitrarily small, there existsB and¸ 0 large enough such that P µ sup 0·t·T ~ Q ¸ (t)>B ¶ <² (A.16) for all¸¸¸ 0 . (B16.17) We must show that for´ >0 arbitrarily small, there exists ana and a¸ 0 large enough such that P µ sup 0·t·T ¯ ¯ ~ ² ¸ (t) ¯ ¯ ¸a ¶ <´; ¸¸¸ 0 : Since ~ ² ¸ (t)·°T (1¡®)w I ®w O +(1¡®)w I sup 0·t·T ~ Q ¸ (t); this follows from (A.16). (B16.18) It is sufficient to show that for ° > 0 and ´ > 0 arbitrarily small, there exists a ± small enough and a¸ 0 large enough such that P Ã sup 0·t·T¡± sup v;s2[t;t+±] ¯ ¯ ~ ² ¸ (s)¡~ ² ¸ (v) ¯ ¯ ¸° ! <´; ¸¸¸ 0 : Since ¯ ¯ ~ ² ¸ (s)¡~ ² ¸ (v) ¯ ¯ ·° Z s v ¯ ¯ ¯ ¯ (1¡®)w I ®w O +(1¡®)w I ~ Q ¸ (³) ¯ ¯ ¯ ¯ d³ +° Z s v ~ Q ¸ O (³)d³ 118 and ~ Q ¸ O (t)· ~ Q ¸ (t) for allt¸0, sup 0·t·T¡± sup v;s2[t;t+±] ¯ ¯ ~ ² ¸ (s)¡~ ² ¸ (v) ¯ ¯ ·° µ (1¡®)w I ®w O +(1¡®)w I +1 ¶ ± sup 0·t·T ~ Q ¸ (t): Hence the condition (B16.18) also follows from (A.16). Finally, it remains to establish (A.16). Fix T > 0 and ² > 0. We first represent the process ~ Q ¸ using the conventional one-sided regulator mapping. Define ~ Â ¸ ´ ~ A ¸ (t)¡ ~ S ¸ I ¡ T ¸ I (t) ¢ ¡ ~ S ¸ O ¡ T ¸ O (t) ¢ +µt ~ A ¸ (t) ´ 1 p ¸ N µZ t 0 °[Q ¸ O (s)¡1] + ds ¶ : Then, ~ Q ¸ (t)= ~ Â ¸ (t)¡ ~ A ¸ (t)+ ~ I ¸ (t)¸0 for allt¸0: Since ~ I ¸ is non-decreasing, ~ I ¸ (0)=0, it follows that ³ ~ Q ¸ ; ~ I ¸ ´ =(Á;Ã) ³ ~ Â ¸ ¡ ~ A ¸ ´ : (A.17) Since ~ A ¸ is a non-decreasing process, Lemma 5.1 in Kruk et al. (2007) establishes that Á ³ ~ Â ¸ ¡ ~ A ¸ ´ (t)·Á ¡ ~ Â ¸ ¢ (t) for allt¸0: (A.18) The functional central limit theorem and the continuous mapping theorem establish Á ¡ ~ Â ¸ ¢ )Á ³ ~ X ´ ; as ¸!1, where ~ X is a Brownian motion with drift µ and variance ¾ 2 = ¾ 2 A +¾ 2 S as in (2.11). Since weak convergence implies the random variable sup 0·t·T Á ¡ ~ Â ¸ ¢ (t) is tight, there existsB and¸ 0 large enough so that for all¸¸¸ 0 P µ sup 0·t·T Á ¡ ~ Â ¸ ¢ (t)>B ¶ <²: Therefore, it follows from the representation (A.17) and the upper bound (A.18) that for all ¸¸¸ 0 P µ sup 0·t·T ~ Q ¸ (t)>B ¶ <²: ¤ 119 Proof of Lemma 4.3 An argument very similar to Theorem 5.3 in Reiman (1984) shows that ~ P ¸ I ) ®w O (1¡®)w I +®w O ~ Q as¸!1: Suppose we can also show that ~ P ¸ O ) (1¡®)w I (1¡®)w I +®w O ~ Q as¸!1: (A.19) Next note that it follows from Corollary 2.1 that for anyt> 0 P ¡ Q ¸ I (t)>0 ¢ !1 andP ¡ Q ¸ O (t)>0 ¢ !1; as ¸ ! 1. Since d dt T ¸ I (t) = ® and d dt T ¸ O (t) = (1¡ ®) if and only if Q ¸ I (t) > 0 and Q ¸ O (t)>0, we conclude P µ d dt T ¸ I (t)=® ¶ !1 andP µ d dt T ¸ O (t)=1¡® ¶ !1; as ¸ ! 1. The convergences in (A.2) and (A.3) now follow by the definition of the workload process and the converging together Lemma. It remains to show (A.19). Since the offline service queue receives at least (1¡ ®) proportion of the server’s effort when the queue is non-empty,(1¡®) ¡1 P ¸ O (t) upper bounds the amount of time required to finish serving all customers in the offline queue that will eventually receive service. Therefore, at timet>0, the number of customers in the offline queue that will eventually abandon is less than or equal to A ¸ (t)´N Ã Z t+(1¡®) ¡1 P ¸ O (t) 0 °[Q ¸ O (s)¡1] + ds ! ¡N µZ t 0 °[Q ¸ O (s)¡1] + ds ¶ : Then,Q ¸ O (t)¡A ¸ (t) is a lower bound on the number of customers in the offline queue that will eventually receive service, and so L ¸ O (t)´ S ¸ O (T ¸ O (t))+Q ¸ O (t)¡A ¸ (t) X j=S ¸ O ( T ¸ O (t) ) +2 v O j ¹(¸) ·P ¸ O (t): 120 Also, Q ¸ O (t) is an upper bound on the number of customers in the offline queue that will eventually receive service, and so U ¸ O (t)´ S ¸ O (T ¸ O (t))+Q ¸ O (t) X j=S ¸ O (T ¸ O (t))+1 v O j ¹(¸) ¸P ¸ O (t): Note that to get the upper bound of the workload process we include in the summation the customer in service, whereas to get the lower bound, we do not. We conclude 0· p ¸P ¸ O (t)¡ p ¸L ¸ O (t)· p ¸U ¸ O (t)¡ p ¸L ¸ O (t): (A.20) Define ~ V ¸ O (t)´ 1 p ¸ b¸tc X i=1 ¡ v O i ¡1 ¢ for allt¸0: Observe that p ¸U ¸ O (t)¡ p ¸L ¸ O (t) (A.21) = p ¸ ¹(¸) v O S ¸ O (T ¸ O (t))+1 + p ¸ ¹(¸) A ¸ (t) + ¸ ¹(¸) µ ~ V ¸ O µ S ¸ O (T ¸ O (t)) ¸ + Q ¸ O (t) ¸ ¶ ¡ ~ V ¸ O µ S ¸ O (T ¸ O (t)) ¸ + Q ¸ O (t) ¸ ¡ A ¸ (t) ¸ ¶¶ and p ¸L ¸ O (t) (A.22) = ¸ ¹(¸) ~ Q ¸ O (t)¡ p ¸ ¹(¸) ¡ p ¸ ¹(¸) A ¸ (t) + ¸ ¹(¸) µ ~ V ¸ O µ S ¸ O (T ¸ O (t)) ¸ + Q ¸ O (t) ¸ ¡ A ¸ (t) ¸ ¶ ¡ ~ V ¸ O µ S ¸ O (T ¸ O (t)) ¸ + 1 ¸ ¶¶ : We will first show that p ¸U ¸ O ¡ p ¸L ¸ O )0 as¸!1, and then show p ¸L ¸ O ) (1¡®)w I (1¡®)w I +®w O ~ Q (A.23) as¸!1. The inequality (A.20) and the converging together lemma then establish (A.19). To show p ¸U ¸ O ¡ p ¸L ¸ O ) 0 as ¸ ! 1, first note that it follows from Lemma 3 in Iglehart and Whitt (1970) that for anyt>0 p ¸ ¹(¸) v O S ¸ O (T ¸ O (t))+1 = ¸ ¸¡ p ¸µ 1 p ¸ v O S ¸ O (T ¸ O (t))+1 !0 121 in probability, as¸!1. Next, since 1 p ¸ A ¸ (t)= ~ N ¸ µ ¿ ¸ µ t+ P ¸ O (t) 1¡® ¶¶ ¡ ~ N ¸ ¡ ¿ ¸ (t) ¢ +° Z t+(1¡®) ¡1 P ¸ O (t) t [Q ¸ O (s)¡1] + p ¸ ds; and Lemma 4.1 establishes¿ ¸ ! 0 andP ¸ O ! 0 a.s., u.o.c., it follows from the functional central limit theorem, continuous mapping theorem, and the weak convergence of Q ¸ O p ¸ in Corollary 2.1 that¸ ¡1=2 A ¸ )0 and so p ¸ ¹(¸) A ¸ = ¸ ¸¡ p ¸µ 1 p ¸ A ¸ )0 (A.24) as¸!1. Now, the sequencefT ¸ O g is tight inD becausejT ¸ O (t)¡T ¸ O (s)j·jt¡sj. On any subsequencef¸ k g on which T ¸ k O )T O as¸ k !1, the functional strong law of large numbers and random time change theorem establish S ¸ k O ±T ¸ k O ¸ k )T O as ¸ k ! 1. Furthermore, on this same subsequence, by the convergences in (A.24) and Lemma 4.1,¸ ¡1 k A ¸ k ) 0 and¸ ¡1 k Q ¸ k O ! 0 a.s., u.o.c. as¸ k !1. Therefore, because by Donsker’s theorem ~ V ¸ O weakly converges to a continuous limit process, ~ V ¸ k O 0 @ S ¸ k O ³ T ¸ k O (¢) ´ ¸ k + Q ¸ k O (¢) ¸ k 1 A ¡ ~ V ¸ k O 0 @ S ¸ k O ³ T ¸ k O (¢) ´ ¸ k + Q ¸ k O (¢) ¸ k ¡ A ¸ k (¢) ¸ k 1 A )0 as¸ k !1. Since the subsequencef¸ k g was arbitrary, it follows that ~ V ¸ O Ã S ¸ O ¡ T ¸ O (¢) ¢ ¸ + Q ¸ O (¢) ¸ ! ¡ ~ V ¸ O Ã S ¸ O ¡ T ¸ O (¢) ¢ ¸ + Q ¸ O (¢) ¸ ¡ A ¸ (¢) ¸ ! )0 as¸!1. We conclude from (A.21) that as¸!1 p ¸U ¸ O ¡ p ¸L ¸ O )0: We now establish the weak convergence in (A.23). An argument similar to that in the above paragraph shows ~ V ¸ O Ã S ¸ O ¡ T ¸ O (¢) ¢ ¸ + Q ¸ O (¢) ¸ ¡ A ¸ (¢) ¸ ! ¡ ~ V ¸ O Ã S ¸ O ¡ T ¸ O (¢) ¢ ¸ + 1 ¸ ! )0 122 as¸!1. Hence, the representation of p ¸L ¸ O in (A.22), Corollary 2.1, the convergence in (A.24), and the continuous mapping theorem establish (A.23). ¤ 123 B Managing Service Systems with an Offline Waiting Option & Customer Abandonment: Companion Note The Amusement Park Ride Setting An amusement park ride departs at deterministically spaced intervals, and can carry only a certain number of customers. Hence it is desirable to extend our analysis to include situations in which customers are served in batches at set time intervals. The parameter regime we have considered, in which the arrival and service rates are large, is applicable to most popular amusement park rides, because generally hundreds or thousands of customers arrive to the ride, and board the ride, each hour. Recall that we are considering a system in which the arrival rate¸ becomes large, and the service rate is ¸¡ p ¸µ for some µ 2 <. We superscript any process or quantity associated with the system having arrival rate¸ and service rate¹(¸)=¸¡ p ¸µ by¸. The required modification to the model is the service process definition. In a slight abuse of notation, we use S ¸ I and S ¸ O to denote the cumulative number of customers that have boarded the ride from the inline and offline queues, even though the service processes are no longer renewal. The reader is to understand that in this Companion Note,S ¸ I andS ¸ O refer to the processes defined below. Let l ¸ ´ µ 1 ¸ ¶ 2=3 ; and assume service occurs only at discrete time pointsl ¸ ;2l ¸ ;3l ¸ ;:::, which represent ride departure times. At each discrete time il ¸ , the number of customers that can enter into service (board the ride) isn ¸ ´ ¸ 1=3 ¡¸ ¡1=6 µ. Then, the service rate is¹(¸) = n ¸ =l ¸ = ¸¡ p ¸µ customers per hour. The service process is defined recursively as follows. At time 0, no customers have boarded the ride, so that S ¸ I (0)´S ¸ O (0)´0: Suppose that at discrete time il ¸ , there are Q I customers in the inline queue and Q O cus- tomers in the offline queue. Then, the number of customers served from each queue that board the ride is B ¸ I (il ¸ )´ ( b®n ¸ c+min ³ £ d(1¡®)n ¸ e¡Q O ¤ + ;Q I ¡b®n ¸ c ´ ; Q I ¸b®n ¸ c Q I ; Q I <b®n ¸ c and B ¸ O (il ¸ )´ 8 < : d(1¡®)n ¸ e+min ½ £ b®n ¸ c¡Q I ¤ + ; Q O ¡d(1¡®)n ¸ e ¾ ; Q O ¸d(1¡®)n ¸ e Q O ; Q O <d(1¡®)n ¸ e : 124 Hence S ¸ I (il ¸ ) ´ S ¸ I ((i¡1)l ¸ )+B ¸ I (il ¸ ) S ¸ O (il ¸ ) ´ S ¸ O ((i¡1)l ¸ )+B ¸ O (il ¸ ): No customers board the ride in between the discrete time pointsl ¸ ;2l ¸ ;3l ¸ ;:::, and so for anyt> 0, S ¸ I (t)=S ¸ I µ¹ t l ¸ º l ¸ ¶ andS ¸ O (t)=S ¸ O µ¹ t l ¸ º l ¸ ¶ : The evolution equations for the queue-length process are very similar to (2.3) and (2.4) in Section 2.2 Q ¸ I (t) ´ A ¸ (t) X i=1 1fw I W ¸ (t ¡ i )·w O W ¸ (t ¡ i )g¡S ¸ I (t) (B.1) Q ¸ O (t) ´ A ¸ (t) X i=1 1fw I W ¸ (t ¡ i )>w O W ¸ (t ¡ i )g¡S ¸ O (t)¡N µZ t 0 °Q ¸ O (s)ds ¶ :(B.2) The difference is that now, because the service process counts the cumulative number of customers that have boarded the ride (and not the number of customers that have departed after riding), the processesQ ¸ I andQ ¸ O track only the customers waiting to ride (and do not include the customers riding or in service). Hence the wait time estimates W ¸ I (t)´ Q ¸ I (t) ¹(¸)® andW ¸ O (t)´ O ¸ (t) ¹(¸)(1¡®) do not include any customers currently on the ride. This is reasonable because at timet the time remaining until the next ride departs is (dt=l ¸ e)l ¸ ¡t), which becomes negligible as ¸ increases. Finally, note that the bound onO ¸ in (2.10) is now Q ¸ O (t)·O ¸ (t)·Q ¸ O (t)+N µZ t 0 °Q ¸ O (s)ds ¶ ¡N Ã Z [ t¡sup 0·s·t W ¸ O (s) ] + 0 °Q ¸ O (s)ds ! ; (B.3) where W ¸ O (t) represents the actual time a customer arriving to the offline queue at time t must wait to board the ride. We expect that the discrete review system behaves similarly to the continuous time system. The following proposition shows that Theorems 2.1 and 2.2, and hence also Corol- lary 2.1, remain valid for the discrete review system. Proposition 4.2. Theorems 2.1 and 2.2 remain valid for the model defined through (B.1)- (B.2). The process ~ Q appearing in Theorems 2.1 and 2.2 again solves the stochastic equa- tion (2.11) but the infinitesimal variance of the Brownian motion ~ X is¾ 2 A . 125 We end the body of this Companion Note by showing how to apply Proposition 4.2 to one popular roller coaster ride at Six Flags Magic Mountain, Tatsu. Suppose the arrival rate¸ has been estimated. To use the approximation, we must determine the parameterµ. Tatsu has capacity for approximately 1600 people to ride every hour, and so ¸¡ p ¸µ =1600; orµ = ¸¡1600 p ¸ : The approximation is not very sensitive to l ¸ , because Proposition 4.2 remains valid for any review period of size ¸ ¡f for 1=2 < f < 1. The key is that the time between ride departures is roughly on the order of seconds if the number of people that can ride every hour is around one or two thousand (which is true for most roller coasters). Proof of Proposition 4.2 For the proof of Proposition 4.2 we will need the following lemma whose proof we defer at the end of this Companion Note. This final Lemma states that Lemmas 4.1-4.3 remain valid in the modified model in the Amusement Park Ride Setting, in which customers are served in batches at set time intervals. In this setting, as is true for the processes Q ¸ I and Q ¸ O , the workload processesP ¸ I andP ¸ O , and the actual waiting time processesW ¸ I andW ¸ O , refer to the customers waiting to board the ride (and do not include the customers currently riding). Furthermore, ¿ ¸ (t)´ 1 ¸ Z t 0 °Q ¸ O (s)ds: Lemma 4.4. Lemmas 4.1-4.3 also hold when the system evolution equations are specified through (B.1)-(B.2). Note that in this setting the processesT ¸ I ,T ¸ O , andI ¸ no longer appear in the system evolution equations, and so Lemma 4.1 is modified to state that as¸!1, ¡ ¹ Q ¸ ;P ¸ I ;P ¸ O ;¹ ¿ ¸ ¢ !(0;0;0;0); a:s:; u:o:c:: (B.4) We must show the following. (i) For anyT >0,sup 0·t·T ¯ ¯ ¯ w I ® ~ Q ¸ I (t)¡ w O 1¡® ~ Q ¸ O (t) ¯ ¯ ¯!0, in probability, as¸!1. (ii) As¸!1; ³ ~ Q ¸ ; ~ I ¸ ´ )( ~ Q; ~ I). (iii) As¸!1, sup 0·t·T p ¸jW I (t)¡W I (t)j!0 and sup 0·t·T p ¸jW O (t)¡W O (t)j!0; in probability. 126 Proof of (i) Modify the definitions of ~ U ¸ 1 and ~ U ¸ 2 in the proof of Theorem 2.1(i) so that ~ U ¸ 1 (t;s) = ¡ w O 1¡® ³ ~ A ¸ (t)¡ ~ A ¸ (s) ´ + ½ µ p ¸ (w I ¡w O )¡ ® 1¡® w O ¡w I + w O 1¡® 1 ¸l ¸ ¾ p ¸(t¡s) ~ U ¸ 2 (t;s) = ¡ w I ® ³ ~ A ¸ (t)¡ ~ A ¸ (s) ´ + ½ µ p ¸ (w O ¡w I )¡ 1¡® ® w I ¡w O + w I ® 1 ¸l ¸ ¾ p ¸(t¡s): With» ¸ and» ¤ ¸ defined exactly as in the proof of Theorem 2.1(i), observe that when w I ® ~ Q ¸ I (» ¤ ¸ )>(·) w O 1¡® ~ Q ¸ O (» ¤ ¸ ); because the inline (offline) queue does not become empty during[» ¤ ¸ ;» ¸ ], the offline (inline) queue may become empty, and service occurs in discrete time intervals S ¸ I (t)¡S ¸ I (» ¤ ¸ ¡) ¸(·) ¹ t¡» ¤ ¸ l ¸ º b®n ¸ c S ¸ O (t)¡S ¸ O (» ¤ ¸ ¡) ·(¸) » t¡» ¤ ¸ l ¸ ¼ d(1¡®)n ¸ e: Then, substitution of the above bounds into the equivalent of (A.6) in the proof of The- orem 2.1(i) in this setting (specifically, replace S ¸ I (T ¸ I (t))¡S ¸ I (T ¸ I (» ¤ ¸ ¡)) with S ¸ I (t)¡ S ¸ I (» ¤ ¸ ¡) and S ¸ O (T ¸ O (t))¡ S ¸ O (T ¸ O (» ¤ ¸ ¡)) with S ¸ O (t)¡ S ¸ O (» ¤ ¸ ¡)) shows that for large enough¸ ¯ ¯ ¯ ¯ w I ® ~ Q ¸ I (t)¡ w O 1¡® ~ Q ¸ O (t) ¯ ¯ ¯ ¯ · ² 2 +1+max n ~ U ¸ 1 (t;» ¤ ¸ ¡); ~ U ¸ 2 (t;» ¤ ¸ ¡) o + w O 1¡® 1 p ¸ N µZ t 0 °Q ¸ O (s)ds ¶ : Since(¸l ¸ ) ¡1 = ¸ ¡1=3 ! 0 as¸!1, the remainder of the proof proceeds exactly as the proof of Theorem 2.1 (i), noting that by Lemma 4.4 1 ¸ Z t 0 °Q ¸ O (s)ds!0 a.s., u.o.c.; as¸!1. 127 Proof of (ii) We first obtain a useful equivalent representation for the processQ ¸ O (t) = Q ¸ I (t)+Q ¸ O (t). Define ² ¸ (t) ´ ° Z t 0 µ (1¡®)w I ®w O +(1¡®)w I Q ¸ (s)¡Q ¸ O (s) ¶ ds; so that Q ¸ (t) = A ¸ (t)¡S ¸ I (t)¡S ¸ O (t)¡ Z t 0 ° (1¡®)w I ®w O +(1¡®)w I Q ¸ (s)ds +² ¸ (t)¡N µZ t 0 °Q ¸ O (s)ds ¶ + Z t 0 °Q ¸ O (s)ds: Next we define recursively the process that tracks the cumulative number of empty ride seats I ¸ (0) ´ 0 I ¸ (il ¸ ) ´ I ¸ ¡ (i¡1)l ¸ ¢ + £ n ¸ ¡Q ¸ (il ¸ ¡) ¤ + : The processI ¸ does not increase in between the discrete time pointsl ¸ ;2l ¸ ;3l ¸ ;:::, and so for anyt> 0, I ¸ (t)=I ¸ µ¹ t l ¸ º l ¸ ¶ : Then, S ¸ I (t)+S ¸ O (t) = b t=l ¸ c X i=1 ³ B ¸ I (il ¸ )+B ¸ O (il ¸ ) ´ = bt=l ¸ c X i=1 µ n ¸ 1fQ ¸ (il ¸ )¸n ¸ g+Q ¸ (il ¸ )1fQ ¸ (il ¸ )<n ¸ g ¶ ; and so S ¸ I (t)+S ¸ O (t)+I ¸ (t)= ¹ t l ¸ º n ¸ : It follows that Q ¸ (t)=X ¸ (t)+² ¸ (t)¡ Z t 0 ° (1¡®)w I ®w O +(1¡®)w I Q ¸ (s)ds+I ¸ (t) (B.5) 128 where X ¸ (t)´A ¸ (t)¡ ¹ t l ¸ º n ¸ ¡N µZ t 0 °Q ¸ O (s)ds ¶ + Z t 0 °Q ¸ O (s)ds: Furthermore, because the ride will depart with empty seats only when no customers are waiting, 1 X i=1 Q ¸ (il ¸ ) ¡ I ¸ (il ¸ )¡I ¸ ((i¡1)l ¸ ) ¢ =0: (B.6) By Lemma 4.4, there exists a convergent subsequence µ Q ¸ i p ¸ i ; I ¸ i p ¸ i ¶ ) ³ ~ Q; ~ I ´ as¸ i !1: We show that the limit( ~ Q; ~ I) satisfies ³ ~ Q; ~ I ´ =(Á · ;Ã · ) ³ ~ X ´ ; (B.7) where ~ X is a Brownian motion with driftµ and variance¾ 2 A . We first show (C1) in Defini- tion 4.1 is satisfied. On the subsequence¸ i , because X ¸ i (t) p ¸ i = ~ A ¸ i (t)¡ ~ N ¸ i ¡ ¿ ¸ i (t) ¢ + p ¸ i t µ 1¡ ¹ t l ¸ i º l ¸ i t ¶ +µt ¹ t l ¸ i º l ¸ i t ; the functional central limit theorem shows X ¸ i p ¸ i ) ~ X: Also on the subsequence¸ i , as in the proof of part (ii) of Theorem 2.1 (which requires the tighness of ~ Q ¸ established in Lemma 4.4), ² ¸ i p ¸ i )0; as¸ i !1. We conclude from (B.5) that the limit( ~ Q; ~ I) satisfies ~ Q(t)= ~ X(t)¡ Z t 0 ° (1¡®)w I ®w O +(1¡®)w I ~ Q(s)ds+ ~ I(t): (B.8) For (C2), note that on the subsequence ¸ i , the definition of the Reimann-Stieltjes integral implies that when we take limits as¸ i !1 on both sides of the equality in (B.6), Z 1 0 ~ Q(t)d ~ I(t)=0: 129 Since furthermore I ¸ i (0) = 0 and I ¸ i is non-decreasing, it follows that ~ I(0) = 0 and ~ I is non-decreasing. We conclude that (B.7) is valid. From the representation (A.4), this is equivalent to the stochastic equation for ~ Q in (2.11). Since the subsequence ¸ i was arbitrary, this part of the proof is complete. Proof of (iii) The argument is very similar to the proof of Theorem 2.2 in the Technical Appendix ((Kostami & Ward, 2008b)), and so is omitted. The exception is that the upper-bound onO ¸ in (B.3) replaces the upper-bound onO ¸ in (2.10). ¤ Proof of Lemma 4.4 We divide the proof of Lemma 4.4 into three parts, with each part re-proving Lem- mas 4.1, 4.2, and 4.3 for the modified model in the Amusement Park Ride Setting, in which customers are served in batches at set time intervals. Proof of (B.4) (Lemma 4.1 equivalent) We require defining the following two comparison systems. Comparison system 1 is the model in the Amusement Park Ride Setting without abandonments. In particular, the inline and offline queue-length processes,Q ¸ B;I andQ ¸ B;O , satisfy equations (B.1) and (B.2) with° =0. Under the same arrival sequence, on a sample path basis, Q ¸ (t)·Q ¸ B;I (t)+Q ¸ B;O (t); for allt¸0: Comparison system 2 is a conventional single-server queue with no abandonments and deterministic, non-batched service. In particular, the queue-length process evolution equa- tion is Q ¸ C (t)´A ¸ (t)¡¹(¸) Z t 0 1fQ ¸ C (s)>0gds: Under the same arrival sequence, on a sample path basis Q ¸ B;I (il ¸ )+Q ¸ B;O (il ¸ )=Q ¸ C (il ¸ ) for everyi=0;1;2;:::: We conclude that on a sample path basis Q ¸ (t)·Q ¸ C µ¹ t l ¸ º l ¸ ¶ +A ¸ (t)¡A ¸ µ¹ t l ¸ º l ¸ ¶ : (B.9) 130 It is well known thatQ ¸ C =¸!0 a.s., u.o.c., as¸!1. Furthermore, A ¸ (t)¡A ¸ ¡¥ t l ¸ ¦ l ¸ ¢ ¸ =A ¸ (t)¡A ¸ µ¹ t l ¸ º l ¸ ¶ + µ t¡ ¹ t l ¸ º l ¸ ¶ ; and, as¸!1,A ¸ !0 a.s., u.o.c. and ¡ t¡bt=l ¸ cl ¸ ¢ !0. Hence Q ¸ !0 a.s., u.o.c., as¸!1. It then follows that ¿ ¸ !0 a.s., u.o.c., as¸!1. Proof that the sequencef ~ Q ¸ ; ~ I ¸ g is tight inD (Lemma 2 equivalent) It is sufficient to verify that the sequencef ~ Q ¸ g is tight inD. Tightness of the sequence f ~ Q ¸ ; ~ I ¸ g then follows from equation (B.5), because the functional central limit theorem shows the sequence fX ¸ = p ¸g is tight, and tightness of the sequence f² ¸ = p ¸g can be established very similarly to Lemma 4.2. Let T > 0. We verify conditions (16.17) and (16.18) in Theorem 16.8 in Billingsley. (B16.17) We must show that for´ >0 arbitrarily small, there exists ana and a¸ 0 large enough such that P µ sup 0·t·T ¯ ¯ ¯ ~ Q ¸ (t) ¯ ¯ ¯¸a ¶ <´; ¸¸¸ 0 : (B.10) It is well-known that forQ ¸ C defined as in the first part of this proof,Q ¸ C = p ¸ weakly converges to a reflected Brownian motion with driftµ and variance¾ 2 A . Furthermore, A ¸ (t)¡A ¸ ¡¥ t l ¸ ¦ l ¸ ¢ p ¸ = ~ A ¸ (t)¡ ~ A ¸ µ¹ t l ¸ º l ¸ ¶ + p ¸ µ t¡ ¹ t l ¸ º l ¸ ¶ : The functional central limit theorem implies that ~ A ¸ weakly converges to a Brow- nian motion, and the definition of l ¸ implies p ¸ ¡ t¡bt=l ¸ cl ¸ ¢ ! 0 as ¸ ! 1. Therefore, the condition (B.10) follows from the bound in (B.9). (B16.18) It is sufficient to show that for ° > 0 and ´ > 0 arbitrarily small, there exists a ± small enough and a¸ 0 large enough such that P Ã sup 0·t·T¡± sup v;s2[t;t+±] ¯ ¯ ¯ ~ Q ¸ (s)¡ ~ Q ¸ (v) ¯ ¯ ¯¸° ! <´; ¸¸¸ 0 : (B.11) 131 Without loss of generality, assumes<v. We require an upper and a lower bound on the processQ ¸ (v)¡Q ¸ (s). For the upper bound, define a comparison system Q ¸ C (t)´A ¸ (s+t)¡A ¸ (s)¡¹(¸) Z t s 1fQ ¸ C (³)>0gd³: By similar reasoning as in the previous paragraph, Q ¸ (v)¡Q ¸ (s)·Q ¸ C µ¹ v¡s l ¸ º l ¸ ¶ +A ¸ (v)¡A ¸ ³j v l ¸ k l ¸ ´ : (B.12) For the lower bound, since Q ¸ (v)¡Q ¸ (s) = A ¸ (v)¡A ¸ ³j v l ¸ k l ¸ ´ +A ¸ ³j v l ¸ k l ¸ ´ ¡A ¸ (s) +A ¸ ³l s l ¸ m l ¸ ´ ¡A ¸ ³l s l ¸ m l ¸ ´ ¡S ¸ I (v)+S ¸ I (s)¡S ¸ O (v)+S ¸ O (s) ¡N µZ v 0 °Q ¸ O (³)d³ ¶ +N µZ s 0 °Q ¸ O (³)d³ ¶ ; and at mostn ¸ customers are served everyl ¸ time units, Q ¸ (v)¡Q ¸ (s) ¸ bv=l ¸ cl ¸ ¡1 X i=ds=l ¸ el ¸ ¡ A ¸ ((i+1)l ¸ )¡A ¸ (il ¸ )¡n ¸ ¢ ¡2n ¸ (B.13) ¡N µZ v 0 °Q ¸ O (³)d³ ¶ +N µZ s 0 °Q ¸ O (³)d³ ¶ : Noting that ~ Q ¸ O (t)· ~ Q ¸ (t) for allt¸0, 1 p ¸ ¡ A ¸ ((i+1)l ¸ )¡A ¸ (il ¸ )¡n ¸ ¢ = ~ A ¸ ((i+1)l ¸ )¡ ~ A ¸ (il ¸ )¡¸ ¡2=3 µ; and 1 p ¸ µ N µZ v 0 °Q ¸ O (³)d³ ¶ ¡N µZ s 0 °Q ¸ O (³)d³ ¶¶ = ~ N ¸ ¡ ¿ ¸ (v) ¢ ¡ ~ N ¸ ¡ ¿ ¸ (s) ¢ ¡ Z v s ° ~ Q ¸ O (³)d³; it follows from (B.12) and (B.13) that ¯ ¯ ¯ ~ Q ¸ (v)¡ ~ Q ¸ (s) ¯ ¯ ¯·max(M ¸ U ;M ¸ L ); (B.14) 132 where M ¸ U ´ ~ Q ¸ C µ¹ v¡s l ¸ º l ¸ ¶ + ¯ ¯ ¯ ~ A ¸ (v)¡ ~ A ¸ ³j v l ¸ k l ¸ ´¯ ¯ ¯+ p ¸ ³ v¡ j v l ¸ k l ¸ ´ M ¸ L ´ ¯ ¯ ¯ ~ A ¸ ³j v l ¸ k l ¸ ´ ¡ ~ A ¸ ³l s l ¸ m l ¸ ´¯ ¯ ¯+ ¯ ¯ ¯ ~ N ¸ ¡ ¿ ¸ (v) ¢ ¯ ¯ ¯+ ¯ ¯ ¯ ~ N ¸ ¡ ¿ ¸ (s) ¢ ¯ ¯ ¯ +°(v¡s) sup 0·t·T ¯ ¯ ¯ ~ Q ¸ (t) ¯ ¯ ¯+¸ ¡2=3 µ ³j v l ¸ k l ¸ ¡ l s l ¸ m l ¸ ´ +2 n ¸ p ¸ : The condition (B.11) follows because every term on the right-hand side of (B.14) becomes arbitrarily small with high probability as ± converges to 0, forjv¡sj < ± and large enough¸. In particular, ¥ (v¡s)=l ¸ ¦ l ¸ ! v¡s as¸!1, and so, since ~ Q ¸ C weakly converges to a continuous limit process (a reflected Brownian motion) with initial position 0, it follows that ~ Q ¸ C ¡ b(v¡s)=l ¸ cl ¸ ¢ can be made arbitrarily small with high probability as ± becomes small. Furthermore, ~ A ¸ conveges to a continuous lmiit process and so the terms ¯ ¯ ¯ ~ A ¸ (v)¡ ~ A ¸ ³j v l ¸ k l ¸ ´¯ ¯ ¯ and ¯ ¯ ¯ ~ A ¸ ³j v l ¸ k l ¸ ´ ¡ ~ A ¸ ³l s l ¸ m l ¸ ´¯ ¯ ¯ become arbitrarily small with high probability as ± becomes small. The constant terms all converge to 0, and, because we have shown the convergence in (B.4) in Lemma 4.4, which implies (A.1) remains valid in this setting, ~ N ¸ ±¿ ¸ weakly con- verges to 0. Finally, because we have already shown condition (B16.17) is satisfied, the term °(v¡s)sup 0·t·T j ~ Q ¸ (t)j becomes arbitrarily small with high probability as± becomes small. Proof of (A.2) and (A.3) (Lemma 4.3 equivalent) As in the proof of Lemma 4.3, it is sufficient to show that as¸!1 ~ P ¸ I ) ®w O (1¡®)w I +®w O ~ Q and ~ P ¸ O ) (1¡®)w I (1¡®)w I +®w O ~ Q: For the inline queue, note that the number of batches required to serve all customers in the inline queue exceedsbQ ¸ I (t)=n ¸ c and is less thandQ ¸ I (t)=n ¸ e. Since each batch requires l ¸ time units to process l ¸ ¹ Q ¸ I (t) n ¸ º ·P ¸ I (t)·l ¸ » Q ¸ I (t) n ¸ ¼ ; and so 0· p ¸P ¸ I (t)¡ p ¸l ¸ ¹ Q ¸ I (t) n ¸ º · p ¸l ¸ : 133 Since p ¸l ¸ ! 0 as¸!1 and by parts (i) and (ii) of this Proposition the weak conver- gence in Corollary 2.1 remains valid, p ¸l ¸ Q ¸ I n ¸ = ¸l ¸ n ¸ ~ Q ¸ I ) ®w O (1¡®)w I +®w O ~ Q: We conclude ~ P ¸ I ) ®w O (1¡®)w I +®w O ~ Q as¸!1. Since whenever the number of customers in the offline queue exceeds (1¡®)n ¸ at a discrete review time point, at least(1¡®)n ¸ customers are served, µ Q ¸ O (t) (1¡®)n ¸ +1 ¶ l ¸ exceeds the amount of time required for all customers in the offline queue that do not abandon to be served. Hence the number of customers in the offline queue that eventually do abandon must be less than or equal to A ¸ (t) ´ N 0 @ Z t+ µ Q ¸ O (t) (1¡®)n ¸ +1 ¶ l ¸ 0 °Q ¸ O (s)ds 1 A ¡N µZ t 0 °Q ¸ O (s)ds ¶ : Therefore, l ¸ ¹ Q ¸ O (t)¡A ¸ (t) n ¸ º ·P ¸ O (t)·l ¸ » Q ¸ O (t) n ¸ ¼ : It follows from the observation that µ Q ¸ O (t) (1¡®)n ¸ +1 ¶ l ¸ = Q ¸ O (t) (1¡®)¹(¸) +l ¸ !0 as¸!1 that 1 p ¸ A ¸ )0 as ¸ ! 1 by identical argument as that in the proof of Lemma 4.3. As in the preceding paragraph, we conclude ~ P ¸ O ) (1¡®)w I (1¡®)w I +®w O ~ Q as¸!1. ¤ 134 C Speed Quality Tradeoffs in a Dynamic Model: Techni- cal Appendix In this Technical Appendix we provide the proofs for Theorem 3.1, Theorem 3.2, Theo- rem 3.3, Theorem 3.4, Theorem 3.5, Theorem 3.6 and Theorem 3.7 stated in Chapter 3. Proofs of Section 3.3 Proof of Theorem 3.1 We start first with showing the optimal price and service speed decisions and at the end we prove concavity. Recall that the expected profit is R(¹;p)=p(¤¡®p)¡° ¤¡®p ¹¡¤+®p ¡±(¤¡®p)(¹¡ ^ ¹): The optimal pricing and service speed decisions will satisfy the first and second order conditions. Solving the first order conditions @R(¹;p) @p =0 and @R(¹;p) @¹ =0 gives ¤¡2®p ¤ + °¹ ¤ ® (¹ ¤ ¡¤+®p ¤ ) 2 +®±(¹ ¤ ¡ ^ ¹)=0 (¹ ¤ ¡¤+®p ¤ ) 2 = ° ± and so (3.1) and (3.2) follow. Moreover since¸ ¤ >0, we always need to have that ¸ ¤ = ¤+®±^ ¹¡2® p °± 2(1+®±) >0 To verify that these are maxima, we check also that the second order conditions are true at the optima. We have that @ 2 R(¹;p) @p 2 ¯ ¯ ¯ ¯ ¹=¹ ¤ ;p=p ¤ = ¡2® µ 1+ ®°¹ (¹¡¤+®p) 3 ¶¯ ¯ ¯ ¯ ¹=¹ ¤ ;p=p ¤ = ¡2® Ã 1+ ± p ±®¹ ¤ p ° ! <0 @ 2 R(¹;p) @¹ 2 ¯ ¯ ¯ ¯ ¹=¹ ¤ ;p=p ¤ = ¡ 2°(¤¡®p) (¹¡¤+®p) 3 ¯ ¯ ¯ ¯ ¹=¹ ¤ ;p=p ¤ =¡2± Ã ¹ ¤ s ± ° ¡1 ! <0 @ 2 R(¹;p) @¹@p ¯ ¯ ¯ ¯ ¹=¹ ¤ ;p=p ¤ = ®°(¡¹¡¤+®p) (¹¡¤+®p) 3 +®± ¯ ¯ ¯ ¯ ¹=¹ ¤ ;p=p ¤ =¡2®± Ã ¹ ¤ s ± ° ¡1 ! 135 and we should verify that @ 2 R(¹;p) @p 2 @ 2 R(¹;p) @¹ 2 > µ @ 2 R(¹;p) @¹@p ¶ 2 ; or equivalently 1+®± > 0: Substituting (3.1) and (3.2) in the expected profit gives (3.3). Now since(¹ ¤ ;p ¤ ) is the only extremum and satisfies @ 2 R(¹;p) @p 2 @ 2 R(¹;p) @¹ 2 > ³ @ 2 R(¹;p) @¹@p ´ 2 , and the function is continuous and differentiable, it will also be concave. ¤ Sensitivity Analysis In this subsection we provide the derivatives of the optimal pricing and service speed deci- sions along with the optimal revenue with respect to the several parameters. To determine the sign of each expression, we have assumed that¤> ^ ¹. with respect to± @¹ ¤ @± = ®^ ¹¡ p ° ± 3 ¡3® p ° ± ¡®¤ 2(1+®±) 2 <0 @p ¤ @± = ¤¡ ^ ¹¡® p °±+ p ° ± 2(1+®±) 2 >0 @¸ ¤ @± = ¡® ¤¡ ^ ¹¡® p °±+ p ° ± 2(1+®±) 2 <0 @R ¤ @± = ¤+®±^ ¹¡2® p °± 4(1+®±) 2 µ ¡¤+®±^ ¹+2^ ¹¡2 r ° ± ¶ <0 if ^ ¹<¹ ¤ >0 if ^ ¹>¹ ¤ For the last inequalities observe than ^ ¹<¹ ¤ means ¡¤+®±^ ¹+2^ ¹¡2 r ° ± <0: 136 with respect to° @¹ ¤ @° = 1 2 p °±(1+®±) >0 @p ¤ @° = s ± ° 1 2(1+®±) >0 @¸ ¤ @° = ¡® s ± ° 1 2(1+®±) <0 @R ¤ @° = ¡ s ± ° ¤¡2® p °±+®±^ ¹ 2(1+®±) <0 with respect to ^ ¹ @¹ ¤ @^ ¹ = ®± 2(1+®±) >0 @p ¤ @^ ¹ = ¡ ± 2(1+®±) <0 @¸ ¤ @^ ¹ = ®± 2(1+®±) >0 @R ¤ @^ ¹ = ± ¡ ¤+ ^ ¹®±¡2® p °± ¢ 2(1+®±) >0 with respect to¤ @¹ ¤ @¤ = 1 2(1+®±) >0 @p ¤ @¤ = 1+2®± 2®(1+®±) >0 @¸ ¤ @¤ = 1 2®(1+®±) >0 @R ¤ @¤ = ¤+ ^ ¹®±¡2® p °± 2®(1+®±) >0 137 with respect to® @¹ ¤ @® = ± ¡ ¡¤+ ^ ¹¡2 p ° ± ¢ 2(1+®±) 2 <0 @p ¤ @® = ¡ ¤+2®¤±+2® 2 ¤± 2 ¡® 2 ± 2 ^ ¹+2® 2 ± p °± 2® 2 (1+®±) 2 <0 @¸ ¤ @® = ¡ ±¤¡±^ ¹+2 p °± 2(1+®±) 2 <0 @R ¤ @® = ¤+®±^ ¹¡2® p °± 4® 2 (1+®±) 2 n ®±^ ¹¡2® p °±¡¤¡2®±¤ o <0 ¤ Proofs of Section 3.4 Proof of Theorem 3.2 We want to prove that if¹ i < ^ ¹ andN is large, then¹ i will increase with time approaching ^ ¹. A similar argument is true in the case when ¹ i > ^ ¹. The demand potential ¤ j can be also written as a function of the decision variables and the parameters due to the following reasoning ¤ j ¡®p = ¤ j¡1 ¡®p¡±¸ j¡1 (¹ j¡1 ¡ ^ ¹) = ¸ j¡1 ¡±¸ j¡1 (¹ j¡1 ¡ ^ ¹) = [1¡±(¹ j¡1 ¡ ^ ¹]¸ j¡1 = ¸ 1 j¡1 Y k=1 [1¡±(¹ k ¡ ^ ¹)] and so, ¤ j =(¤ 1 ¡®p) j¡1 Y k=1 [1¡±(¹ k ¡ ^ ¹)]+®p; 1·j·N +1 138 The derivative with respect to¹ i will then be @¤ j @¹ i = ¡±¸ 1 j¡1 Y k=1;k6=i [1¡±(¹ k ¡ ^ ¹)] = ¡ ±¸ 1 1¡±(¹ i ¡ ^ ¹) j¡1 Y k=1 [1¡±(¹ k ¡ ^ ¹)] = ¡ ±¸ j 1¡±(¹ i ¡ ^ ¹) fori+1·j·N, @¤ N+1 @¹ i =¡ ±¸ N [1¡±(¹ N ¡ ^ ¹)] 1¡±(¹ i ¡ ^ ¹) and 0, otherwise . Now the optimal speed ¹ i at each period i will satisfy the first order condition @R @¹ i =0; which can be written as N X j=i+1 µ p¡ °¹ j (¹ j ¡¸ j ) 2 ¶ @¤ j @¹ i +µ @¤ N+1 @¹ i + °¸ i (¹ i ¡¸ i ) 2 =0 or equivalently N X j=i+1 ½µ p¡ °¹ j (¹ j ¡¸ j ) 2 ¶ ¡±¸ j 1¡±(¹ i ¡ ^ ¹) ¾ +µ ¡±¸ N [1¡±(¹ N ¡ ^ ¹)] 1¡±(¹ i ¡ ^ ¹) + °¸ i (¹ i ¡¸ i ) 2 =0 that leads to ¡± ( N X j=i+1 µ p¡ °¹ j (¹ j ¡¸ j ) 2 ¶ ¸ j +µ¸ N [1¡±(¹ N ¡ ^ ¹)] ) + °¸ i+1 (¹ i ¡¸ i ) 2 =0 (C.1) where the last equality holds wheni<N. Fori=N, we will get that ¡±µ¸ N [1¡±(¹ N ¡ ^ ¹)]+ °¸ N [1¡±(¹ N ¡ ^ ¹)] (¹ N ¡¸ N ) 2 =0: (C.2) 139 Writing (C.1) fori andi¡1 (i<N), we have ¡± N X j=i+1 µ p¡ °¹ j (¹ j ¡¸ j ) 2 ¶ ¸ j ¡±µ¸ N [1¡±(¹ N ¡ ^ ¹)]+ °¸ i+1 (¹ i ¡¸ i ) 2 = 0 ¡± N X j=i µ p¡ °¹ j (¹ j ¡¸ j ) 2 ¶ ¸ j ¡±µ¸ N [1¡±(¹ N ¡ ^ ¹)]+ °¸ i (¹ i¡1 ¡¸ i¡1 ) 2 = 0 and combining them leads to ¡ ±p ° + ±(2¹ i ¡ ^ ¹)¡1 (¹ i ¡¸ i ) 2 + 1 (¹ i¡1 ¡¸ i¡1 ) 2 =0 (C.3) fori<N. If we further combine (C.2) with (C.1) wheni=N¡1, we get that (C.3) holds fori·N. Notice that (C.3) can be written as 1 (¹ i ¡¸ i ) 2 ¡ 1 (¹ i¡1 ¡¸ i¡1 ) 2 =¡ ±p ° + ±(2¹ i ¡ ^ ¹) (¹ i ¡¸ i ) 2 : (C.4) If we assume that ¹ i is constant or decreasing over time given that ¤ i > ¤ i¡1 and so ¸ i >¸ i¡1 , then we should have ¡ ±p ° + ±(2¹ i ¡ ^ ¹) (¹ i ¡¸ i ) 2 >0; or equivalently p< °(2¹ i ¡ ^ ¹) (¹ i ¡¸ i ) 2 < °¹ i (¹ i ¡¸ i ) 2 for alli<N; Now suppose N is large enough that the term µ¸ N [1¡ ±(¹ N ¡ ^ ¹)] is small compared to the summation¡± P N j=i+1 ³ p¡ °¹ j (¹ j ¡¸ j ) 2 ´ ¸ j in (C.1). Then (C.1) cannot be satisfied because all the remaining terms in (C.1) will be positive and we have a contradiction. Hence¹ i >¹ i¡1 and ¡ ±p ° + ±(2¹ i ¡ ^ ¹) (¹ i ¡¸ i ) 2 <0; (C.5) (C.4) and (C.5) imply that ¹ i ¡ ¹ i¡1 > ¸ i ¡ ¸ i¡1 . Further, since ^ ¹ > ¹ i > ¹ i¡1 by assumption, j¹ i ¡ ^ ¹j<j¹ i¡1 ¡ ^ ¹j 140 Furthermore, recall that the demand potential is defined as ¤ i+1 = ¤ i +±¸ i (^ ¹¡¹ i ) and so the demand rate when price is constant will satisfy¸ i+1 +®p=¸ i +®p+±¸ i (^ ¹¡¹ i ). Therefore, we have that ¸ i+1 ¸ i = 1+±(^ ¹¡¹ i ) ¸ i ¸ i¡1 = 1+±(^ ¹¡¹ i¡1 ) and hence, ¸ i+1 ¸ i < ¸ i ¸ i¡1 : That is the rate of increase of ¸ i (and the demand potential) is decreasing over time. To complete the proof it remains to show that this happens until ¹ i = ^ ¹. Once ¹ i = ^ ¹, no further change will happen to ¤ i , ¹ i , and ¸ i . Suppose that the service speed converges to a value _ ¹ < ^ ¹. But then the market potential will increase and so the demand rate will increase. If¸ i increases, then¹ i increases and we have a contradiction because¹ i does not converge to a value. A similar argument holds and in the case that _ ¹> ^ ¹. The optimal price should satisfy the first order condition. The derivative of the demand potential with respect to the price is @¤ j @p = ¡® j¡1 Y k=1 [1¡±(¹ k ¡ ^ ¹)]+® = ¡® ¸ j ¸ 1 +®; whenj¸2. The first order condition will be N X i=1 ½ ¸ i ¡® ¸ i ¸ 1 µ p¡ °¹ i (¹ i ¡¸ i ) 2 ¶¾ +®µ µ 1¡ ¸ N [1¡±(¹ N ¡ ^ ¹)] ¸ 1 ¶ =0: If we further use (C.1), then it becomes N X i=1 ¸ i ¡® ½ p¡ °[1¡±(2¹ 1 ¡ ^ ¹)] ±(¹ 1 ¡¸ 1 ) 2 ¾ +µ® =0: (C.6) Therefore, the optimal price will satisfy ¤ 1 ® ( 1+ N X i=2 i¡1 Y k=1 [1¡±(¹ k ¡ ^ ¹)] ) +p N X i=2 i¡1 Y k=1 [1¡±(¹ k ¡ ^ ¹)]¡ °[1¡±(2¹ 1 ¡ ^ ¹)] ±(¹ 1 ¡¤ 1 +®p) 2 =0; 141 that is an equivalent expression of (C.6). ¤ Proofs of Section 3.5 Proof of Theorem 3.3 The demand potential can also be written as ¤ i = ¤ i¡1 ¡±(¹¡ ^ ¹)¸ i¡1 = ¤ 1 ¡±(¹¡ ^ ¹) i¡1 X j=1 ¸ j for2·i·N +1, and its derivative with respect to¸ j will be @¤ i @¸ j = 8 < : ¡±(¹¡ ^ ¹) 0 forj <i otherwise : Eliminating the prices p i by using the fact that p i = ¤ i ¡¸ i ® the expected profit can be also written as R= N X i=1 · ¤ i ¡¸ i ® ¸ i ¡° ¸ i ¹¡¸ i ¸ +µ¤ N+1 The optimal demand rate¸ i will satisfy @R @¸ i =0; or equivalently ¤ i ¡2¸ i ® ¡ °¹ (¹¡¸ i ) 2 ¡±µ(¹¡ ^ ¹)¡ ±(¹¡ ^ ¹) ® N X j=i+1 ¸ j =0: (C.7) We first consider the case when¹ > ^ ¹. A similar argument holds for¹ < ^ ¹. (C.7) for i and(i¡1) is, ¤ i ¡2¸ i ® ¡ °¹ (¹¡¸ i ) 2 ¡±µ(¹¡ ^ ¹)¡ ±(¹¡ ^ ¹) ® N X j=i+1 ¸ j = 0 ¤ i¡1 ¡2¸ i¡1 ® ¡ °¹ (¹¡¸ i¡1 ) 2 ¡±µ(¹¡ ^ ¹)¡ ±(¹¡ ^ ¹)¸ i ® ¡ ±(¹¡ ^ ¹) ® N X j=i ¸ j = 0 142 and combining both, we have ¤ i¡1 ¡±¸ i¡1 (¹¡ ^ ¹) ® ¡ 2¸ i ® ¡ °¹ (¹¡¸ i ) 2 = ¤ i¡1 ¡2¸ i¡1 ® ¡ °¹ (¹¡¸ i¡1 ) 2 ¡ ±(¹¡ ^ ¹)¸ i ® : By rearranging the terms, we end up with ±¸ i¡1 (¹¡ ^ ¹) ® + 2¸ i ® + °¹ (¹¡¸ i ) 2 = ±(¹¡ ^ ¹)¸ i ® + 2¸ i¡1 ® + °¹ (¹¡¸ i¡1 ) 2 : In the above equation, both the right and the left hand side are positive with the same form. Therefore, we can conclude that ¸ i = ¸ i¡1 . The same is true for every i. Therefore, ¸ N =¸ N¡1 =:::=¸ i+1 =¸ and the result follows. ¤ Proof of Theorem 3.4 Writing (C.7) fori and using Theorem 3.3, we have that ¤ i ¡2¸ ¤ ® ¡ °¹ (¹¡¸ ¤ ) 2 ¡±µ(¹¡ ^ ¹)¡ ±(¹¡ ^ ¹) ® (N¡i)¸ ¤ =0: Now we can use the fact that¸ ¤ =¤ i ¡®p i and so p i ¡ ¸ ¤ ® ¡ °¹ (¹¡¸ ¤ ) 2 ¡±µ(¹¡ ^ ¹)¡ ±(¹¡ ^ ¹) ® (N¡i)¸ ¤ =0: Therefore, the optimal pricing policy will be p ¤ i = ¹ ¤ ° (¹ ¤ ¡¸ ¤ ) 2 +±µ(¹¡ ^ ¹)+ ¸ ¤ ® + (N¡i)¸ ¤ ±(¹ ¤ ¡ ^ ¹) ® : (C.8) As we proved in Theorem 3.3, demand rate remains constant and thus ¤ 1 ¡ ®p ¤ 1 = ¸ ¤ . Therefore, we have that¸ ¤ will satisfy ¤ 1 ¡2¸ ¤ ¡ ®¹ ¤ ° (¹ ¤ ¡¸ ¤ ) 2 ¡®±µ(¹¡ ^ ¹)¡(N¡1)¸ ¤ ±(¹ ¤ ¡ ^ ¹)=0: (C.9) optimal speed: The optimal speed of the system will satisfy the first order condition, @R @¹ =0 that is 143 N X i=1 ½µ p i ¡ °¹ (¹¡¸ i ) 2 ¶ @¤ i @¹ + °¸ i (¹¡¸ i ) 2 ¾ +µ @¤ N+1 @¹ =0: Taking into consideration that¸ i =¸ and that fori>0 @¤ i @¹ = ¡±(i¡1)[1¡±(¹¡ ^ ¹)] i¡2 ¤ 1 +±® i¡1 X j=1 [1¡±(¹¡ ^ ¹)] i¡j¡2 [1¡(i¡j)±(¹¡ ^ ¹)]p j ; we end up with N X i=1 ± · p i ¡ °¹ (¹¡¸) 2 ¸½ ¡(i¡1)[1¡±(¹¡ ^ ¹)] i¡2 ¤ 1 +® P i¡1 j=1 [1¡±(¹¡ ^ ¹)] i¡j¡2 [1¡(i¡j)±(¹¡ ^ ¹)]p j ¾ +±µ ( ¡N[1¡±(¹¡ ^ ¹)] N¡1 ¤ 1 + ® P N j=1 [1¡±(¹¡ ^ ¹)] N¡j¡1 [1¡(N +1¡j)±(¹¡ ^ ¹)]p j ) + N°¸ (¹¡¸) 2 =0: (C.10) If we use (C.8) in the above, we get, N X i=1 [¸ ¤ +(®µ+(N¡i)¸ ¤ )±(¹ ¤ ¡ ^ ¹)]A i¡1 (¸ ¤ ;¹ ¤ ) + ®µA N (¸ ¤ ;¹ ¤ )+ ®°N¸ ¤ ±(¹ ¤ ¡¸ ¤ ) 2 =0: (C.11) Suppose that¹ ¤ = ^ ¹, then (C.8) becomes p ¤ i = ¹ ¤ ° (¹ ¤ ¡¸ ¤ ) 2 + ¸ ¤ ® and i¡1 X j=1 p ¤ j =(i¡1) µ ¹ ¤ ° (¹ ¤ ¡¸ ¤ ) 2 + ¸ ¤ ® ¶ and (C.9) becomes ¤ 1 ¡2¸ ¤ ¡ ®¹ ¤ ° (¹ ¤ ¡¸ ¤ ) 2 =0: (C.12) Using this, (C.10) becomes N X i=1 ± µ ¸ ¤ ® ¶ " ¡(i¡1)¤ 1 +® i¡1 X j=1 p ¤ j # +±µ " ¡N¤ 1 +® N X j=1 p ¤ j # + °¸N (¹ ¤ ¡¸ ¤ ) 2 =0: 144 We now substitute P i¡1 j=1 p ¤ j and so, N X i=1 ± µ ¸ ¤ ® ¶· ¡¤ 1 +® ¹ ¤ ° (¹ ¤ ¡¸ ¤ ) 2 +¸ ¤ ¸ (i¡1) + ±µ · ¡N¤ 1 + ®N¹ ¤ ° (¹ ¤ ¡¸ ¤ ) 2 +N¸ ¤ ¸ + °¸ ¤ N (¹ ¤ ¡¸ ¤ ) 2 =0 and by rearranging the terms, we have ± ¸ ¤ ® · ¡¤ 1 +® ¹ ¤ ° (¹ ¤ ¡¸ ¤ ) 2 +¸ ¤ ¸ N X i=1 (i¡1) + ±µN · ¡¤ 1 + ®¹ ¤ ° (¹ ¤ ¡¸ ¤ ) 2 +¸ ¤ ¸ + °¸ ¤ N (¹ ¤ ¡¸ ¤ ) 2 =0 Using (C.12) leads to ± ¸ ¤ ® (¡¸ ¤ ) N(N¡1) 2 +±µN(¡¸ ¤ )+ °¸ ¤ N (¹ ¤ ¡¸ ¤ ) 2 =0: that is equivalently written as ± µ ¸ ¤ ® N¡1 2 +µ ¶ = ° (¹ ¤ ¡¸ ¤ ) 2 : But (C.12) implies that ° (¹ ¤ ¡¸ ¤ ) 2 = ¤ 1 ¡2¸ ¤ ®¹ ¤ ; and so, ¹ ¤ = ^ ¹= 2(¤ 1 ¡2¸ ¤ ) ±(¸ ¤ (N¡1)+2®µ) : 145 Let ^ ^ ¹ be defined as the value of ^ ¹ that satisfies the above and so, if ^ ^ ¸ is the correspond- ing demand rate when service rate is ^ ^ ¹, then ^ ^ ¸ and ^ ^ ¹ will solve the following system of equations ¤ 1 ¡2 ^ ^ ¸¡ ® ^ ^ ¹° ( ^ ^ ¹¡ ^ ^ ¸) 2 =0 ^ ^ ¹= 2(¤ 1 ¡2 ^ ^ ¸) ± ³ ^ ^ ¸(N¡1)+2®µ ´: Now let us discuss the case when ^ ¹> ^ ^ ¹ and observe the following two equations ¤ 1 ¡2 ^ ^ ¸ ® ¡ ° ^ ^ ¹ ( ^ ^ ¹¡ ^ ^ ¸) 2 = 0 (C.13) ¤ 1 ¡2¸ ¤ ® ¡ °¹ ¤ (¹ ¤ ¡¸ ¤ ) 2 ¡±µ(¹ ¤ ¡ ^ ¹)¡(N¡1)¸ ¤ ±(¹ ¤ ¡ ^ ¹) ® = 0: (C.14) where the second one is the optimality condition satisfied when ^ ¹ > ^ ^ ¹. Now suppose that ^ ¹ > ^ ^ ¹ but ¹ ¤ < ^ ¹ is not true in the optimality equation, i.e. (C.14) is not satisfied when ¹ ¤ < ^ ¹. Consider the following solution to the scenario ^ ¹ > ^ ^ ¹. Let¹ ¤ > ^ ^ ¹ be the optimal service speed and let¸ ¤ be such that ^ ^ ¹ ( ^ ^ ¹¡ ^ ^ ¸) 2 = ¹ ¤ (¹ ¤ ¡¸ ¤ ) 2 ; or equivalently, ¸ ¤ = ^ ^ ¸ s ¹ ¤ ^ ^ ¹ +¹ ¤ ¡ q ¹ ¤^ ^ ¹: Since¹ ¤ > ^ ^ ¹, the above implies that¸ ¤ > ^ ^ ¸. Subtracting (C.13) from (C.14) gives 2 ^ ^ ¸ ® ¡ 2¸ ¤ ® + ° ^ ^ ¹ ( ^ ^ ¹¡ ^ ^ ¸) 2 ¡ °¹ ¤ (¹ ¤ ¡¸ ¤ ) 2 ¡±µ(¹ ¤ ¡ ^ ¹)¡(N¡1)¸ ¤ ±(¹ ¤ ¡ ^ ¹) ® =0:(C.15) 146 But 2 ^ ^ ¸ ® ¡ 2¸ ¤ ® < 0; because¸ ¤ > ^ ^ ¸ ° ^ ^ ¹ ( ^ ^ ¹¡ ^ ^ ¸) 2 ¡ °¹ ¤ (¹ ¤ ¡¸ ¤ ) 2 = 0; due to the definition of¸ ¤ : (C.15) can be satisfied only if¹ ¤ · ^ ¹ and so we have a contradiction. A similar argument holds for the opposite case and the result follows. ¤ Proofs of Section 3.6 Proof of Theorem 3.5 The objective function is concave since we have already shown that the single period profit function is concave. The multi-period profit function is the sum of concave functions and therefore, it will also be a concave function. For the demand potential, we have that ¤ i =¤ 1 ¡± i¡1 X j=1 (¹ j ¡ ^ ¹)¸ j and its derivative with respect to¹ i andp i are, @¤ j @¹ i = 8 < : ¡±¸ i Q j¡1 k=i+1 [1¡±(¹ k ¡ ^ ¹)] 0 ; j¸i+1 , else @¤ j @p i = 8 < : ®±(¹ i ¡ ^ ¹) Q j¡1 k=i+1 [1¡±(¹ k ¡ ^ ¹)] 0 ; i<j , else : We now show that¹ i andp i are increasing over time if¹ i < ^ ¹: A similar argument is true in the case when¹ i > ^ ¹. Consider a 2-period problem with N as the last period and (N ¡1) as the first period with ¤ N and ¤ N¡1 as the market potential at the beginning of periodsN and(N¡1) andR denote the expected profit of this problem. We introduce some additional notation for this proof. Let R 1 (¤); ¹ 1 (¤) and p 1 (¤) denote respectively the profit function, optimal speed and price in the single-period problem with market potential ¤: That is, (¹ 1 (¤);p 1 (¤))= argmax ¹;p (R 1 (¤)) 147 Note thatR 1 (¤)=p(¤¡®p)¡° ¤¡®p ¹¡¤+®p ¡±(¤¡®p)(¹¡ ^ ¹) from the single-period model. Since¤ N+1 =¤ N ¡±(¤ N ¡®p N )(¹ N ¡^ ¹) andµ >0, optimizing periodN profit in the 2-period problem is equivalent to optimizingR 1 (¤ = ¤ N ) except for the additional termµ¤ N which is increasing in¤ N : Now, suppose¹ ¤ N¡1 andp ¤ N¡1 are the optimal speed and price in period (N¡1) in the two-period problem. Consider a solution to the 2-period problem with ¹ N¡1 = ¹ 1 (¤ = ¤ N¡1 ) and p N¡1 = p 1 (¤ = ¤ N¡1 ). Any value of ¹ N¡1 > ¹ 1 (¤ = ¤ N¡1 ) will decrease period (N¡1) profit and will also decrease¤ N and¤ N+1 since @¤ j @¹ i <0: From the single- period analysis, decreasing in ¤ N will decrease period N profit. Hence, ¹ N¡1 > ¹ 1 (¤ = ¤ N¡1 ) will never be an optimal solution and so ¹ ¤ N¡1 · ¹ 1 (¤ = ¤ N¡1 ): The same logic holds can be used to show thatp ¤ N¡1 (¤ N¡1 )· p 1 (¤ = ¤ N¡1 ) except that @¤ j @p i < 0 is true only if¹ N¡1 < ^ ¹: Now, if¹ N¡1 < ^ ¹, we also know that¤ N > ¤ N¡1 and from the single- period analysis,¹ 1 (¤= ¤ N )>¹ 1 (¤ = ¤ N¡1 ): Hence,¹ ¤ N (¤ N ) =¹ 1 (¤ = ¤ N )¸¹ ¤ N¡1 : The same logic can be used to show thatp ¤ N ¸p ¤ N¡1 : We can use induction and the fact that @¤ j @p i <0 when¹ i < ^ ¹ and @¤ j @¹ i <0 to show that¹ ¤ i+1 ¸¹ ¤ i andp ¤ i+1 ¸p ¤ i when¹ i < ^ ¹. Next, we show that service speed increases faster that the demand rate and that the service speed will eventually reach ^ ¹. The optimal service speed will satisfy the first order condition @R @¹ i =0 that is N X j=i+1 µ p j ¡ °¹ j (¹ j ¡¸ j ) 2 ¶ @¤ j @¹ i +µ @¤ N+1 @¹ i + °¸ i (¹ i ¡¸ i ) 2 =0 or equivalently ¡± ( P N j=i+1 ³ p j ¡ °¹ j (¹ j ¡¸ j ) 2 ´ Q j¡1 k=i+1 [1¡±(¹ k ¡ ^ ¹)] +µ Q N k=i+1 [1¡±(¹ k ¡ ^ ¹)] ) + ° (¹ i ¡¸ i ) 2 =0: (C.16) If we write (C.16) fori¡1, ¡± ( P N j=i ³ p j ¡ °¹ j (¹ j ¡¸ j ) 2 ´ Q j¡1 k=i [1¡±(¹ k ¡ ^ ¹)] +µ Q N k=i [1¡±(¹ k ¡ ^ ¹)] ) + ° (¹ i¡1 ¡¸ i¡1 ) 2 =0; and combine the last two equations, we get ¡± µ p i ¡ °¹ i (¹ i ¡¸ i ) 2 ¶ ¡± N X j=i+1 µ p j ¡ °¹ j (¹ j ¡¸ j ) ¶ [¡±(¹ i ¡ ^ ¹)] j¡1 Y k=i+1 [1¡±(¹ k ¡ ^ ¹)] +± 2 µ(¹ i ¡ ^ ¹) N Y k=i+1 [1¡±(¹ i ¡ ^ ¹)]+ ° (¹ i¡1 ¡¸ i¡1 ) 2 ¡ ° (¹ i ¡¸ i ) 2 =0: (C.17) 148 Let us now derive the optimal price for periodi. The derivative of the demand potential over the price is The optimal pricep i will satisfy the first order condition @R @p i =0 that is ¤ i ¡2®p i + ®°¹ i (¹ i ¡¸ i ) 2 + N X j=i+1 µ p j ¡ °¹ j (¹ j ¡¸ j ) 2 ¶ @¤ j @p i +µ @¤ N+1 @p i =0 and using the derivative of the demand potential this gives ¤ i ¡2®p i + ®°¹ i (¹ i ¡¸ i ) 2 + ®±(¹ i ¡ ^ ¹) " P N j=i+1 h p j ¡ °¹ j (¹ j ¡¸ j ) 2 i Q j¡1 k=i+1 [1¡±(¹ k ¡ ^ ¹)] +µ Q N k=i+1 [1¡±(¹ k ¡ ^ ¹)] # =0: (C.18) Furthermore, (C.17) can be written as p i ¡ °¹ i (¹ i ¡¸ i ) ¡ ° ±(¹ i¡1 ¡¸ i¡1 ) 2 + ° ±(¹ i ¡¸ i ) 2 + ±(^ ¹¡¹ i ) ( P N j=i+1 h p j ¡ °¹ j (¹ j ¡¸ j ) 2 i Q j¡1 k=i+1 [1¡±(¹ k ¡ ^ ¹)] +µ Q N k=i+1 [1¡±(¹ k ¡ ^ ¹)] ) =0: We use the above in (C.18), ¤ i ¡2®p i + ®°¹ i (¹ i ¡¸ i ) 2 +®p i ¡ ®°¹ i (¹ i ¡¸ i ) 2 ¡ ®° ±(¹ i¡1 ¡¸ i¡1 ) 2 + ®° ±(¹ i ¡¸ i ) 2 =0 and so 0<¸ i = ®° ± ½ 1 (¹ i¡1 ¡¸ i¡1 ) 2 ¡ 1 (¹ i ¡¸ i ) 2 ¾ : (C.19) The last equation implies that ¹ i ¡¸ i > ¹ i¡1 ¡¸ i¡1 , or that the service speed increases faster than the demand rate. Finally, it is left to show that the service speed will eventually converge to ^ ¹. Till now, we have assumed that for everyi,¹ i < ^ ¹. Suppose that¹ i¡1 ¡ ^ ¹ < 0, then (C.18) implies that ¤ i ¡2®p i + ®°¹ i (¹ i ¡¸ i ) 2 >¤ i¡1 ¡2®p i¡1 + ®°¹ i¡1 (¹ i¡1 ¡¸ i¡1 ) 2 : (C.20) Combining (C.16) and (C.18) gives that ¤ i ¡2®p i + ®°¹ i (¹ i ¡¸ i ) 2 = ®°(^ ¹¡¹ i ) (¹ i ¡¸ i ) 2 for everyi. If ^ ¹¡¹ i <0 but ^ ¹¡¹ i¡1 >0, using the above, (C.20) implies that0>0 that leads to contradiction. Therefore, if¹ 1 < ^ ¹, then¹ i < ^ ¹ for alli. 149 Further, since ^ ¹ > ¹ i > ¹ i¡1 ,j¹ i ¡ ^ ¹j <j¹ i¡1 ¡ ^ ¹j. To complete the proof it remains to show that this happens until ¹ i = ^ ¹. Once ¹ i = ^ ¹, no further change will happen to ¹ i and thus to¸ i and¤ i . Suppose that the service speed converges to a value _ ¹ < ^ ¹. But then the market potential will increase and so the demand rate will increase. If¸ i increases, then¹ i increases and we have a contradiction because¹ i does not converge to a value. A similar argument holds and in the case that _ ¹> ^ ¹. ¤ Proofs of Section 3.7 Proof of Theorem 3.6 The expected profit under the G/G/1 framework will be R=p¸¡° ¸ ¹¡¸ ¡° ¸ 2 (C¡1) ¹(¹¡¸) ¡±¸(¹¡ ^ ¹): and the optimal pricing and service speed decisions will satisfy the first order conditions that give @R @¹ =0 , ° (¹¡¸) 2 + °¸(C¡1)(2¹¡¸) ¹ 2 (¹¡¸) 2 ¡± =0; (C.21) @R @p =0 , ¤¡2®p+ ®°¹ (¹¡¸) 2 +°¸®(C¡1) 2¹¡¸ ¹(¹¡¸) 2 +®±(¹¡ ^ ¹)=0:(C.22) Using (C.21) in (C.22), we get that p= ¤+®±(2¹¡ ^ ¹) 2® (C.23) and plugging it back to (C.22) we have ¡±(2¹¡ ^ ¹)+ °¹ (¹¡¸) 2 +°¸(C¡1) 2¹¡¸ ¹(¹¡¸) 2 +±(¹¡ ^ ¹)=0 that after some algebra leads to ¹¡¸=¹ s °C °(C¡1)+±¹ 2 150 and using the fact that¸=¤¡®p, leads to p= 1 ® ( ¤¡¹+¹ s °C °(C¡1)+±¹ 2 ) : Hence, the optimal speed will be the solution of ®±(2¹¡ ^ ¹)=¤¡2¹+2¹ s °C °(C¡1)+±¹ 2 ¤ Proof of Theorem 3.7 Constant Price: We want to show that if¹ i < ^ ¹, then¹ i increases over time until it reaches^ ¹ whenN >>0. A similar argument is true in the case when¹ i > ^ ¹. The demand potential is ¤ j =¤ j¡1 ¡±(¹ j¡1 ¡ ^ ¹)¸ j¡1 ; j¸2 (C.24) and its derivative with respect to¹ i is @¤ j @¹ i = 8 > > > > > < > > > > > : ¡ ±¸ j 1¡±(¹ i ¡^ ¹) ¡ ±¸ N [1¡±(¹ N ¡^ ¹)] 1¡±(¹ i ¡^ ¹) 0 ; i+1·j·N i=N +1 , else : The optimal service speed will satisfy the first order condition @R @¹ i =0 that leads to N X j=i+1 · p¡ °(1¡C) ¹ j ¡ ¹ j °C (¹ j ¡¸ j ) 2 ¸ @¤ j @¹ i +µ @¤ N+1 @¹ i + °¸ i (1¡C) ¹ 2 i + °¸ i C (¹ i ¡¸ i ) 2 =0: Substituting the derivative of the demand potential gives ¡± ( P N j=i+1 h p¡ °(1¡C) ¹ j ¡ ¹ j °C (¹ j ¡¸ j ) 2 i ¸ j +µ¸ N [1¡±(¹ N ¡ ^ ¹)] ) + °¸ i+1 (1¡C) ¹ 2 i + °¸ i+1 C (¹ i ¡¸ i ) 2 =0;(C.25) where the last equality holds fori<N. Fori=N, we have that ¡±µ¸ N [1¡±(¹ N ¡ ^ ¹)]+ °¸ N [1¡±(¹ N ¡ ^ ¹)](1¡C) ¹ 2 N + °¸ N [1¡±(¹ N ¡ ^ ¹)]C (¹ N ¡¸ N ) 2 =0: 151 Combining (C.25) fori andi+1 gives °¸ i+1 (1¡C) ¹ 2 i + °¸ i+1 C (¹ i ¡¸ i ) 2 = ¡ ± · p¡ °(1¡C) ¹ i ¡ ¹ i °C (¹ i ¡¸ i ) 2 ¸ ¸ i + °¸ i (1¡C) ¹ 2 i¡1 + °¸ i C (¹ i¡1 ¡¸ i¡1 ) 2 (C.26) and this is true fori·N. We will focus on the case when¹ i < ^ ¹. First observe that¸ i > ¸ i¡1 since¤ i > ¤ i¡1 due to the fact that¹ i < ^ ¹. We distinguish two cases 1¡C>0. Suppose that¹ i over time is decreasing then °¸ i+1 (1¡C) ¹ 2 i increases withi and °¸ i+1 C (¹ i ¡¸ i ) 2 increases withi: Using this in (C.26) implies that 0 · °¸ i+1 (1¡C) ¹ 2 i + °¸ i+1 C (¹ i ¡¸ i ) 2 ¡ °¸ i (1¡C) ¹ 2 i ¡ °¸ i C (¹ i¡1 ¡¸ i¡1 ) 2 = ¡± · p¡ °(1¡C) ¹ i ¡ ¹ i °C (¹ i ¡¸ i ) 2 ¸ ¸ i for all i. Now suppose that N is large enough so that the term µ¸ N [1¡ ±(¹ N ¡ ^ ¹)] is small compared to¡± P N j=i+1 h p¡ °(1¡C) ¹ j ¡ ¹ j °C (¹ j ¡¸ j ) 2 i ¸ j in (C.25). Then (C.25) cannot hold since all the terms of the left hand side are positive. Hence, the service speed will increase over time when1¡C > 0. 1¡C<0. In this case, notice that if we assume that¹ i is decreasing then ¸ i+1 (1¡C) ¹ 2 i + ¸ i+1 C (¹ i ¡¸ i ) 2 =¸ i+1 ½ 1 ¹ 2 i +C¸ i 1 ¹ i (¹ i ¡¸ i ) 2 +C¸ i 1 ¹ 2 i (¹ i ¡¸ i ) ¾ is increasing withi. Then, (C.25) implies that p¡ °(1¡C) ¹ j ¡ ¹ j °C (¹ j ¡¸ j ) 2 <0 and so we will also have that °¸ i+1 (1¡C) ¹ 2 i + °¸ i+1 C (¹ i ¡¸ i ) 2 <0 152 or equivalently µ ¹ i ¸ i ¡1 ¶ 2 ¡C +2C ¹ i ¸ i <0: Defineu = ¸ i =¹ i , the utilization of the system at periodi, thenu· 1 and so we will have that µ 1 u ¡1 ¶ 2 +C 2¡u u <0; that is a contradiction. Therefore,¹ i increases over time and moreover, ¸ i+1 (1¡C) ¹ 2 i + ¸ i+1 C (¹ i ¡¸ i ) 2 should decrease over time. Since 1¡C < 0 and ¹ i increases over time, this implies that ¹ i ¡¸ i >¹ i¡1 ¡¸ i¡1 . Thus,¹ i >¹ i¡1 under both cases. Further, since ^ ¹>¹ i >¹ i¡1 , j¹ i ¡ ^ ¹j<j¹ i¡1 ¡ ^ ¹j: Furthermore, recall that the demand potential is defined like¤ i+1 = ¤ i +±¸ i (^ ¹¡¹ i ) and so the demand rate when price is constant will satisfy¸ i+1 +®p=¸ i +®p+±¸ i (^ ¹¡¹ i ). Therefore, we have that ¸ i+1 ¸ i = 1+±(^ ¹¡¹ i ) ¸ i ¸ i¡1 = 1+±(^ ¹¡¹ i¡1 ) and hence, ¸ i+1 ¸ i < ¸ i ¸ i¡1 : That is the rate of increase of ¸ i (and the demand potential) is decreasing over time. To complete the proof it remains to show that this happens until ¹ i = ^ ¹. Once ¹ i = ^ ¹, no further change will happen to ¤ i , ¹ i , and ¸ i . Suppose that the service speed converges to a value _ ¹ < ^ ¹. But then the market potential will increase and so the demand rate will increase. If¸ i increases, then¹ i increases and we have a contradiction because¹ i does not converge to a value. A similar argument holds and in the case that _ ¹> ^ ¹. 153 The optimal price should satisfy the first order condition. The derivative of the demand potential with respect to the price is @¤ j @p = ¡® j¡1 Y k=1 [1¡±(¹ k ¡ ^ ¹)]+® = ¡® ¸ j ¸ 1 +®; whenj¸2. The first order condition will be N X i=1 ½ ¸ i ¡® ¸ i ¸ 1 µ p¡ °¹ i (¹ i ¡¸ i ) 2 ¡ °¸ i (C¡1)(2¹ i ¡¸ i ) ¹ i (¹ i ¡¸ i ) 2 ¶¾ + ®µ µ 1¡ ¸ N [1¡±(¹ N ¡ ^ ¹)] ¸ 1 ¶ =0: Constant Service Speed: (a) By eliminating the prices in the expected profit , we can have an equivalent form R= N X i=1 ¤ i ¡¸ i ® ¸ i ¡° ¸ i ¹¡¸ i ¡° ¸ 2 i (C¡1) ¹(¹¡¸ i ) +µ¤ N+1 : The demand potential is ¤ i =¤ i¡1 ¡±(¹¡ ^ ¹)¸ i¡1 =¤ 1 ¡±(¹¡ ^ ¹) i¡1 X j=1 ¸ j (C.27) for2·i·N +1 and its derivative with respect to¸ j @¤ i @¸ j = 8 < : ¡±(¹¡ ^ ¹) 0 ; j <i , else : The demand rate will satisfy the first order condition @R @¸ i =0, that is ¤ i ¡2¸ i ® ¡ °¹ (¹¡¸ i ) 2 ¡±µ(¹¡ ^ ¹)¡ ±(¹¡ ^ ¹) ® N X j=i+1 ¸ j +° C¡1 ¹ ¡° (C¡1)¹ (¹¡¸ i ) 2 =0 or equivalently, ¤ i ¡2¸ i ® ¡ °¹C (¹¡¸ i ) 2 +° C¡1 ¹ ¡±µ(¹¡ ^ ¹)¡ ±(¹¡ ^ ¹) ® N X j=i+1 ¸ j =0: (C.28) 154 We will assume that¹ > ^ ¹. A similar argument holds for¹ < ^ ¹. Writing (C.28) fori and i¡1, we get ¤ i ¡2¸ i ® ¡ °¹C (¹¡¸ i ) 2 +° C¡1 ¹ ¡±µ(¹¡ ^ ¹)¡ ±(¹¡ ^ ¹) ® N X j=i+1 ¸ j = 0 ¤ i¡1 ¡2¸ i¡1 ® ¡ °¹ (¹¡¸ i¡1 ) 2 +° C¡1 ¹ ¡±µ(¹¡ ^ ¹)¡ ±(¹¡ ^ ¹) ® N X j=i ¸ j = 0: Combining them together leads to ¤ i ¡2¸ i ® ¡ °¹C (¹¡¸ i ) 2 = ¤ i¡1 ¡2¸ i¡1 ® ¡ °¹ (¹¡¸ i¡1 ) 2 ¡ ±(¹¡ ^ ¹) ® ¸ i : We can now use the fact that¤ i = ¤ i¡1 ¡±(¹¡ ^ ¹)¸ i¡1 and by rearranging the terms, we end up with ±¸ i¡1 (¹¡ ^ ¹) ® + 2¸ i ® + °¹C (¹¡¸ i ) 2 = ±(¹¡ ^ ¹)¸ i ® + 2¸ i¡1 ® + °¹C (¹¡¸ i¡1 ) 2 : In the above equation, both the right and the left hand side are positive with the same form. Therefore, we can conclude that and so ¸ i+1 = ¸ i and this is true for every i. Hence, ¸ N =¸ N¡1 =:::=¸ 0 =¸ ¤ and the result follows. (b) Now given that the demand rate remains constant over time, (C.28) will give that p i = ¸ ¤ ® + ¹ ¤ °C (¹ ¤ ¡¸ ¤ ) 2 ¡ °(C¡1) ¹ ¤ ¡±µ(¹ ¤ ¡ ^ ¹)+(N¡i)¸ ¤ ±(¹ ¤ ¡ ^ ¹) ® (C.29) and notice that ¹ ¤ °C (¹ ¤ ¡¸ ¤ ) 2 ¡ °(C¡1) ¹ ¤ =°¹ ¤ ½ 1 ¹ ¤2 +C¸ ¤ 1 ¹ ¤ (¹ ¤ ¡¸ ¤ ) 2 +C¸ ¤ 1 ¹ ¤2 (¹ ¤ ¡¸ ¤ ) ¾ >0; where¹ ¤ is the optimal service speed. Furthermore,¸ ¤ = ¤ 1 ¡®p 1 and so¸ ¤ will be the solution of ¤ 1 ¡2¸ ¤ ¡ ®¹ ¤ °C (¹ ¤ ¡¸ ¤ ) 2 + ®°(C¡1) ¹ ¤ ¡®±µ(¹ ¤ ¡ ^ ¹)¡(N¡1)¸ ¤ ±(¹¡ ^ ¹)=0: The optimal speed of the system will satisfy the first order condition, @R @¹ =0 155 that is N X i=1 ½µ p i ¡ °¹ (¹¡¸) 2 ¡°(C¡1) ¸(2¹¡¸) ¹(¹¡¸) ¶ @¤ i @¹ ¾ +µ @¤ N+1 @¹ + N°¸ (¹¡¸ i ) 2 + N°(C¡1)¸ 2 (2¹¡¸) ¹ 2 (¹¡¸) 2 =0: Taking into consideration that¸ i =¸ and that fori>0 @¤ i @¹ = ¡±(i¡1)[1¡±(¹¡ ^ ¹)] i¡2 ¤ 1 +±® i¡1 X j=1 [1¡±(¹¡ ^ ¹)] i¡j¡2 [1¡(i¡j)±(¹¡ ^ ¹)]p j ; we end up with N X i=1 ± " p i ¡ °¹ (¹¡¸) 2 ¡ °(C¡1)¸(2¹¡¸) ¹(¹¡¸) # ½ ¡(i¡1)[1¡±(¹¡ ^ ¹)] i¡2 ¤ 1 + ® P i¡1 j=1 [1¡±(¹¡ ^ ¹)] i¡j¡2 [1¡(i¡j)±(¹¡ ^ ¹)]p j ¾ +±µ ( ¡N[1¡±(¹¡ ^ ¹)] N¡1 ¤ 1 +® P N j=1 [1¡±(¹¡ ^ ¹)] N¡j¡1 [1¡(N +1¡j)±(¹¡ ^ ¹)]p j ) + N°¸ (¹¡¸) 2 + N°(C¡1)¸ 2 (2¹¡¸) ¹ 2 (¹¡¸) 2 =0 : (C.30) We can derive (3.5) from the above using (C.29). Suppose that¹ ¤ = ^ ¹, then (C.29) becomes p ¤ i = ¸ ¤ ® + ¹ ¤ °C (¹ ¤ ¡¸ ¤ ) 2 ¡ °(C¡1) ¹ and so, i¡1 X j=1 p ¤ j =(i¡1) · ¸ ¤ ® + ¹ ¤ °C (¹ ¤ ¡¸ ¤ ) 2 ¡ °(C¡1) ¹ ¸ and (3.4) becomes ¤ 1 ¡2¸ ¤ ¡ ®¹ ¤ °C (¹ ¤ ¡¸ ¤ ) 2 + ®°(C¡1) ¹ ¤ =0: (C.31) 156 Using the expression forp ¤ i and the above in (C.30), we have that N X i=1 ± ¸ ¤ ® · ¡¤ 1 +¸ ¤ + ®¹ ¤ °C (¹ ¤ ¡¸ ¤ ) 2 ¡ ®°(C¡1) ¹ ¤ ¸ (i¡1) + ±µ · ¡N¤ 1 + ®N¹ ¤ °C (¹ ¤ ¡¸ ¤ ) 2 ¡ ®°N(C¡1) ¹ ¤ ¸ + N°¸ ¤ (¹ ¤ ¡¸ ¤ ) 2 · 1+ ¸ ¤ (C¡1)(2¹ ¤ ¡¸ ¤ ) ¹ ¤2 ¸ =0: But we know that P N i=1 (i¡1)= N(N¡1) 2 and using (C.31) we get ±¸ ¤ (N¡1) 2 +®±µ = ®° (¹ ¤ ¡¸ ¤ ) 2 · 1+ ¸ ¤ (C¡1)(2¹ ¤ ¡¸ ¤ ) ¹ ¤2 ¸ : Now, from (C.31) with some algebra, we get ®° (¹ ¤ ¡¸ ¤ ) 2 · 1+ ¸ ¤ (C¡1)(2¹ ¤ ¡¸ ¤ ) ¹ ¤2 ¸ = ¤ 1 ¡2¸ ¤ ¹ ¤ and so, ¹ ¤ = ^ ¹= 2(¤ 1 ¡2¸ ¤ ) ±[¸ ¤ (N¡1)+2®µ] : We define ^ ^ ¹ this specific ^ ¹ that satisfies the above and so, if ^ ^ ¸ is the corresponding demand rate when service rate is ^ ^ ¹, then ^ ^ ¸ and ^ ^ ¹ will solve the following system ^ ^ ¹= 2(¤ 1 ¡2 ^ ^ ¸) ± h ^ ^ ¸(N¡1)+2®µ i ¤ 1 ¡2 ^ ^ ¸¡ ® ^ ^ ¹°C ( ^ ^ ¹¡ ^ ^ ¸) 2 + ®°(C¡1) ^ ^ ¹ =0: Now let us discuss the case when ^ ¹> ^ ^ ¹ and observe the following two equations ¤ 1 ¡2 ^ ^ ¸ ® ¡ ° ^ ^ ¹C ( ^ ^ ¹¡ ^ ^ ¸) 2 + °(C¡1) ^ ^ ¹ = 0 (C.32) ¤ 1 ¡2¸ ¤ ® ¡ °¹ ¤ C (¹ ¤ ¡¸ ¤ ) 2 + °(C¡1) ¹ ¤ ¡±µ(¹ ¤ ¡ ^ ¹)¡(N¡1)¸ ¤ ±(¹ ¤ ¡ ^ ¹) ® = 0;(C.33) 157 where the second one is the optimality condition satisfied when ^ ¹ > ^ ^ ¹. Now suppose that ^ ¹ > ^ ^ ¹ but ¹ ¤ < ^ ¹ is not true in the optimality equation, i.e. (C.33) is not satisfied when ¹ ¤ < ^ ¹. Consider the following solution to the scenario ^ ¹ > ^ ^ ¹. Let¹ ¤ > ^ ^ ¹ be the optimal service speed and let¸ ¤ be such that ^ ^ ¹C ( ^ ^ ¹¡ ^ ^ ¸) 2 ¡ C¡1 ^ ^ ¹ = ¹ ¤ C (¹ ¤ ¡¸ ¤ ) 2 ¡ C¡1 ¹ ¤ ; or equivalently written 1 ^ ^ ¹ ¡ C( ^ ^ ¹¡ ^ ^ ¸) 2 ^ ^ ¹ ¡ C ^ ^ ¹ ( ^ ^ ¹¡ ^ ^ ¸) 2 = 1 ¹ ¤ ¡ C(¹ ¤ ¡¸ ¤ ) 2 ¹ ¤ ¡ C¹ ¤ (¹ ¤ ¡¸ ¤ ) 2 : Suppose¸ ¤ · ^ ^ ¸, then C(¹ ¤ ¡¸ ¤ ) 2 ¹ ¤ ¡ C( ^ ^ ¹¡ ^ ^ ¸) 2 ^ ^ ¹ = 1 ¹ ¤ ¡ 1 ^ ^ ¹ + C ^ ^ ¹ ( ^ ^ ¹¡ ^ ^ ¸) 2 ¡ C¹ ¤ (¹ ¤ ¡¸ ¤ ) 2 <0 and so, C(¹ ¤ ¡¸ ¤ ) 2 ¹ ¤ < C( ^ ^ ¹¡ ^ ^ ¸) 2 ^ ^ ¹ or, ¸ ¤ ¸ ^ ^ ¸ s ¹ ¤ ^ ^ ¹ +¹ ¤ ¡ q ¹ ¤^ ^ ¹¸ ^ ^ ¸, since¹ ¤ ¸ ^ ^ ¹; and this leads to a contradiction. Therefore, we should have that ¸ ¤ ¸ ^ ^ ¸. Subtracting (C.32) from (C.33) gives 2 ^ ^ ¸ ® ¡ 2¸ ¤ ® + ° ^ ^ ¹C ( ^ ^ ¹¡ ^ ^ ¸) 2 ¡ °¹ ¤ C (¹ ¤ ¡¸ ¤ ) 2 ¡ °(C¡1) ^ ^ ¹ + °(C¡1) ¹ ¤ ¡ ±(¹ ¤ ¡ ^ ¹) · µ+ (N¡1)¸ ¤ ® ¸ =0: (C.34) But 2 ^ ^ ¸ ® ¡ 2¸ ¤ ® · 0; because¸ ¤ ¸ ^ ^ ¸ ° ^ ^ ¹C ( ^ ^ ¹¡ ^ ^ ¸) 2 ¡ °¹ ¤ C (¹ ¤ ¡¸ ¤ ) 2 ¡ °(C¡1) ^ ^ ¹ + °(C¡1) ¹ ¤ = 0; due to the definition of¸ ¤ : 158 The only way that (C.34) can be satisfied is if ¹ ¤ · ^ ¹ and so we have a contradiction. A similar argument holds for the opposite case and the result follows. ¤ 159 D Analysis and Comparison of Inventory Systems with Dynamic vs Static Policies: Technical Appendix In this Technical Appendix we provide the proofs for Theorem 4.1, Theorem 4.2, Proposi- tion 4.1, Theorem 4.3, Theorem 4.4 stated in Chapter 4. Proof of Theorem 4.1 First recall that the arrival rate is¸ 0 =¤ v¡p ? 0 ¡¯L E (p ? 0 ) v whereL E (p ? 0 ) satisfies L E (p ? 0 )= µ ¸ 0 ¹ ¶ B+1 1 ¹¡¸ 0 : Now the optimal static pricep ? 0 will be the solution of the following optimization prob- lem max p 0 v 0 s.t. 0·¸ 0 ·¹: The Lagrangian function is L= ¸ 0 p 0 ¡c¹ ± +l(¹¡¸ 0 ) and the Kuhn-Tucker conditions will be @L @p 0 ·0 and p 0 @L @p 0 =0; @L @l ¸0 and l @L @l =0: By (4.9), we have that @L E (p 0 ) @p 0 =¡ ¤ v ³ ¸ 0 ¹ ´ B 1 ¹ [(B+1)¹¡B¸ 0 ] (¹¡¸ 0 ) 2 +¯ ¤ v ³ ¸ 0 ¹ ´ B 1 ¹ [(B+1)¹¡B¸ 0 ] : We know thatp 0 >0 and so @L @p 0 =0 There are two different scenarios for the value ofl: 1. l6=0 and @L @l =0 Then @L @l =0 implies that¹=¸ 0 and so, @L E (p 0 ) @p 0 =¡ 1 ¯ 160 We further know that @L @p 0 =0 that can be also written as ³ ¡ ¤ v ¡ ¯¤ v @L E (p 0 ) @p 0 ´ p 0 +¸ 0 ± +l µ ¤ v + ¯¤ v @L E (p 0 ) @p 0 ¶ =0; that implies¸ 0 =0 (contradiction). 2. l =0 and @L @l >0 The above implies that¹>¸ 0 and so @L @p 0 =0 will lead to µ ¤ v + ¯¤ v @L E (p 0 ) @p 0 ¶ p 0 =¸ 0 : Substituting @L E (p 0 ) @p 0 gives that ¸ 0 p 0 = (¹¡¸ 0 ) 2 v(¹¡¸ 0 ) 2 ¤ + ³ ¸ 0 ¹ ´ B+1 ¯ ¹ [(B+1)¹¡B¸ 0 ] and after some algebra, we can derive (v¡2p ? 0 )(¹¡¸ 0 )+¯L E (p ? 0 )B¹=0 or equivalently, p ? 0 = ¡ 1 2 ¹ ¤ v¡ 1 2 ¯L E (p ? 0 )+ 3 4 v + s · 3 4 v¡ 1 2 ¹ ¤ v¡ 1 2 ¯L E (p ? 0 ) ¸ 2 + 1 2 v¯L E (p ? 0 )(¤¡B¹) We can also get an expression for the arrival rate in terms ofL E (p 0 ), ¸ 0 = 1 4 ¤¡ 1 2 ¯ ¤ v L E (p ? 0 )+ 1 2 ¹ + s · 3 4 v¡ 1 2 ¹ ¤ v¡ 1 2 ¯L E (p ? 0 ) ¸ 2 + 1 2 v¯L E (p ? 0 )(¤¡B¹) ¤ 161 Proof of Theorem 4.2 We first establish that (4.12) is valid, and then argue that the closed form expressions in (4.13) and (4.14) are valid. The argument that (4.12) holds. Ifp2P policy under whichv d (x) is maximized, then v ? d (x)= p ?¤ v h v¡p ? ¡¯ x + ¹ i + ¡c¹¡h([x] ¡ ) ±+¤+¹ + ¤+¹ ±+¤+¹ X y2fx¡1;x;x+1g q d (yjx;p ? )v ? d (y); (D.1) and v ? d (¡B)= p ?¤ v [v¡p ? ] + ¡c¹¡h(B) ±+¤ + ¤ ±+¤ X y2¡B;¡B+1 q d (yj¡B;p ? )v ? d (y): There are three cases 1. 0<p ? <v¡¯ x + ¹ , 2. p ? ·0, 3. p ? ¸v¡¯ x + ¹ . In case 1, since the optimal value functionv d has a maximum atp, regardingv d as a function ofx andp and taking the partial derivative with respect top, it is true that @v d (x;p) @p ¯ ¯ ¯ ¯ ¯ p=p ? =0: Taking the partial derivative of (D.1) with respect top shows that 0= v¡¯ x + ¹ ¡2p ? ±+¤+¹ ¡ 1 ±+¤+¹ v d (x+1;p)+ 1 ±+¤+¹ v d (x;p) whenx2f¡B+1;¡B+2;:::g and 0= v¡2p ? ±+¤ ¡ 1 ±+¤ v d (¡B+1;p)+ 1 ±+¤ v d (¡B;p): 162 We can therefore conclude that p ? =¡ v d (x+1;p)¡v d (x;p) 2 + v¡¯ x + ¹ 2 : (D.2) Hence in case 1, substituting in the expression forp ? in (D.2) v ? d (x;p) = max 0<p<v¡¯ x + ¹ 8 < : p ¤ v h v¡p¡¯ x + ¹ i + ¡c¹¡h([x] ¡ ) ±+¤+¹ + ¤+¹ ±+¤+¹ P y q d (yjx;p)v ? d (y;p) 9 = ; = 1 ±+¤+¹ 8 < : ¤v ³ 1¡ ¯ v x + ¹ ´ 2 4 ¡c¹¡h([x] ¡ )+ ¤ ³ 1¡ ¯ v x + ¹ ´ 2 v ? d (x+1) + ¤ ³ 1+ ¯ v x + ¹ ´ 2 v ? d (x;p)+ ¤[v ? d (x+1;p)¡v ? d (x;p)] 2 4v +¹v ? d (x¡1;p) 9 = ; : The optimality equation (4.10) is equivalently written as v d (x;p) = max 8 > > > > > < > > > > > : max 0<p<v¡¯ x + ¹ v d (x;p); max p¸v¡¯ x + ¹ v d (x;p); max p·0 v d (x;p) 9 > > > > > = > > > > > ; = 1 ±+¤+¹ max 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : ¤v ³ 1¡ ¯ v x + ¹ ´ 2 4 ¡c¹¡h([x] ¡ ) + ¤ ³ 1¡ ¯ v x + ¹ ´ 2 v d (x+1;p)+ ¤ ³ 1+ ¯ v x + ¹ ´ 2 v d (x;p) + ¤[v d (x+1;p)¡v d (x;p)] 2 4v +¹v d (x¡1;p); ¡c¹¡h([x] ¡ )+¤v d (x;p)+¹v d (x¡1;p); ¡c¹¡h([x] ¡ )+¤ ³ 1¡ ¯ v x + ¹ ´ + v d (x+1;p) +¤ ¯ v x + ¹ v d (x;p)+¹v d (x¡1;p) 9 > > > > > > > > > > > > > > = > > > > > > > > > > > > > > ; : It follows that v d (x;p) = v ? d (x;p) for x 2 f¡B +1;¡B +2;:::g, where v ? d is given in (D.2) because 163 8 < : 1 ±+¤+¹ 2 4 ¤v ³ 1¡ ¯ v x + ¹ ´ 2 4 ¡c¹¡h([x] ¡ )+ ¤ ³ 1¡ ¯ v x + ¹ ´ 2 v d (x+1;p) + ¤ ³ 1+ ¯ v x + ¹ ´ 2 v d (x;p)+ ¤[v d (x+1;p)¡v d (x;p)] 2 4v +¹v d (x¡1;p) 3 5 9 = ; ¡ ½ ¡c¹¡h([x] ¡ ) ±+¤+¹ + ¤ ±+¤+¹ v d (x;p)+ ¹ ±+¤+¹ v d (x¡1;p) ¾ = ¤ 4v(±+¤+¹) · v¡¯ x + ¹ +v d (x+1;p)¡v d (x;p) ¸ 2 ¸0; and 8 < : 1 ±+¤+¹ 2 4 ¤v ³ 1¡ ¯ v x + ¹ ´ 2 4 ¡c¹¡h([x] ¡ )+ ¤ ³ 1¡ ¯ v x + ¹ ´ 2 v d (x+1;p) + ¤ ³ 1+ ¯ v x + ¹ ´ 2 v d (x;p)+ ¤[v d (x+1;p)¡v d (x;p)] 2 4v +¹v d (x¡1;p) 3 5 9 = ; ¡ 8 > < > : ¡c¹¡h([x] ¡ ) ±+¤+¹ + ¤ ³ 1¡ ¯ v x + ¹ ´ + ±+¤+¹ v d (x+1;p) + ¤ ¯ v x + ¹ ±+¤+¹ v d (x;p)+ ¹ ±+¤+¹ v d (x¡1;p) 9 > = > ; = ¤ 4v(±+¤+¹) · v d (x+1;p)¡v d (x;p)¡ µ v¡¯ x + ¹ ¶¸ 2 ¸0 Following a similar argument we can show thatv d (¡B;p)=v ? d (¡B;p), where v d (¡B;p) = max 8 > > > > < > > > > : max 0<p<v v d (¡B;p); max p¸v v d (¡B;p); max p·0 v d (¡B;p) 9 > > > > = > > > > ; : Hence, the optimal value function is maximized for0<p(x)<v¡¯ x + ¹ . Using (D.2), the optimal pricing policy (4.12) follows. The argument that the closed-form expressions (4.13) and (4.14) hold. We hypothesize thatv ? d has the form v ? (x;p)=rx 2 +sx+t; (D.3) 164 and then show thatv ? d satisfies the optimality equations in (4.10) for the stated values ofr, s, andt. Substituting (4.12) into (4.10), we have that v ? d (x;p)= 1 ±+¤+¹ 8 < : v¤ ³ 1¡ ¯ v x + ¹ ´ 2 4 ¡c¹+ ¤ ³ 1¡ ¯ v x + ¹ ´ 2 v ? d (x+1;p) + ¤ ³ 1+ ¯ v x + ¹ ´ 2 v ? d (x;p)+ ¤[v ? d (x+1;p)¡v ? d (x;p)] 2 4v +¹v ? d (x¡1;p) 9 = ; and using (D.3), we find v ? d (x;p)= 1 ±+¤+¹ 8 > > < > > : v¤ ³ 1¡ ¯ v x + ¹ ´ 2 4 ¡c¹+ ¤ ³ 1¡ ¯ v x + ¹ ´ 2 [v ? d (x;p)+2rx+r+s] + ¤ ³ 1+ ¯ v x + ¹ ´ 2 v ? d (x;p)+ ¤[2rx+r+s] 2 4v +¹[v ? d (x;p)¡2rx+r¡s] 9 > > = > > ; ; or equivalently, ±(rx 2 +sx+t) = ¤v 4 ¡ ¤¯ 2¹ x+ ¤¯ 2 4v¹ 2 x 2 ¡c¹+¤rx¡ ¤¯r v¹ x 2 + ¤ 2 (r+s) ¡ ¤¯ 2v¹ (r+s)x+ ¤ v r 2 x 2 + ¤ 4v (r+s) 2 + ¤ v rx(r+s)¡2r¹x +r¹¡s¹: We want the above equality to hold for allx¸0 and sor;s andt should satisfy ±r = ¤¯ 2 4v¹ 2 ¡ ¤¯r ¹v + ¤ v r 2 ±s = ¡ ¤¯ 2¹ +¤r¡ ¤¯ 2v¹ (r+s)+ ¤ v r(r+s)¡2r¹ ±t = ¤v 4 ¡c¹+ ¤ 2 (r+s)+ ¤ 4v (r+s) 2 +r¹¡s¹ which implies r = ¤¯ ¹ +v(±¡ p ¢) 2¤ s = ¤¯ ¹ (±¡2¹)+ ³ ¤v+ ¤¯ 2¹ +±v¡2¹v ´ (±¡ p ¢) ¤(±+ p ¢) t = ¤v 4 ¡c¹+ ¤ 2 (r+s)+ ¤ 4v (r+s) 2 +r¹¡s¹ ± for¢=±(±+ 2¤¯ v¹ ). 165 Combining (D.3) and (4.12), we have that the optimal pricing policy forx¸0 will be p ? (x)=¡ 2rx+r+s 2 + v¡¯ x ¹ 2 : ¤ Proof of Proposition 4.1 It is sufficient to show that for any²>0, there existsB ² such that ¤v 4 ¡c¹ ± ¡²·v ? d (¡B ² )· ¤v 4 ¡c¹ ± +² and v 2 ¡²·p ? (¡B ² )· v 2 +²: It follows from the optimality equation (4.11) thatv ? d (¡B) satisfies v ? d (¡B) = sup p2P 8 < : p¤ v¡p v ¡c¹ ±+¤ + ¤ ±+¤ 0 @ ¤ v¡p v ¤ v ? d (¡B+1) + ³ 1¡ ¤ v¡p v ¤ ´ v ? d (¡B) 1 A 9 = ; = sup p2P ( p¤ v¡p v ±+¤ + ¤ ±+¤ µ v¡p v ¶ [v ? d (¡B+1)¡v ? d (¡B)] ) ¡ c¹ ±+¤ + ¤ ±+¤ v ? d (¡B): Let f(p):= p¤ v¡p v ±+¤ + ¤ ±+¤ µ v¡p v ¶ [v ? d (¡B+1)¡v ? d (¡B)]: Since f 0 (p)= ¤ ±+¤ 1 v (v¡2p)¡ ¤ ±+¤ 1 v [v ? d (¡B+1)¡v ? d (¡B)]=0 implies that p= v+v ? d (¡B)¡v ? d (¡B+1) 2 ; (D.4) we can substitutep into the expression v ? d (¡B)=sup p2P ( p¤ v¡p v ¡c¹ ±+¤ + ¤ ±+¤ Ã ¤ v¡p v ¤ v ? d (¡B+1)+ Ã 1¡ ¤ v¡p v ¤ ! v ? d (¡B) !) 166 to find that v ? d (¡B) = ¤ 4v ¡c¹ ±+¤ + ¤ 4v £ 2(v ? d (¡B+1)¡v ? d (¡B))+(v ? d (¡B+1)¡v ? d (¡B)) 2 ¤ ±+¤ + ¤ ±+¤ v ? d (¡B): Multiplying both sides by±+¤ shows that ±v ? d (¡B)= ¤ 4v ¡c¹+ ¤ 4v £ 2(v ? d (¡B+1)¡v ? d (¡B))+(v ? d (¡B+1)¡v ? d (¡B)) 2 ¤ ; so that v ? d (¡B)= ¤ 4v ¡c¹ ± + ¤ 4v± £ 2(v ? d (¡B+1)¡v ? d (¡B))+(v ? d (¡B+1)¡v ? d (¡B)) 2 ¤ : (D.5) Next, note that under any static pricing policy, for anyB > 0, v ? d (¡B) · E ·Z 1 0 e ¡±t vdA(t) ¸ = vE ·Z 1 0 µZ 1 t ±e ¡±s ds ¶ dA(t) ¸ = vE ·Z 1 0 Z s 0 ±e ¡±s dA(t)ds ¸ = vE ·Z 1 0 ±e ¡±s A(s)ds ¸ · v Z 1 0 ±e ¡±s ¤sds = ¤v ± is finite. Hence lim B!1 v(¡B) is finite. Since lim B!1 v(¡B+1)= lim B!1 v(¡B); it follows that for any², there existsB ² such that ¯ ¯ ¯ ¯ ¤ 4v± £ 2(v ? d (¡B ² +1)¡v ? d (¡B ² ))+(v ? d (¡B ² +1)¡v ? d (¡B ² )) 2 ¤ ¯ ¯ ¯ ¯ <²: 167 It then follows from (D.5) that ¤v 4 ¡c¹ ± ¡²·v ? d (¡B ² )· ¤v 4 ¡c¹ ± +²; and from (D.4) that v 2 ¡²·p ? (¡B ² )· v 2 +²: ¤ Proof of Theorem 4.3 Define r 0 (x;w 0 )= p 0 ¸ 0 ¡c¹w 0 (x) ±+¤+¹ : To show the optimality of a monotone policy, we have to show that 1. r 0 (x;w 0 ) is nonincreasing inx for everyw, 2. r 0 (x;w 0 ) is superadditive, 3. ^ q 0 (kjx;w 0 )= P 1 j=k q 0 (jjx;w 0 ) is nondecreasing inx, and 4. P 1 j=0 q 0 (jjx;w 0 )v ? 0 (j) is superadditive. r 0 (x;w 0 ) will satisfies r 0 (x;w 0 )= p 0 ¸ 0 ¡c¹w 0 (x) ±+¤+¹ : We have to show each of the four properties mentioned. We start with 1. r 0 (x+1;1)¡r 0 (x;1) = 0; r 0 (x+1;0)¡r 0 (x;0) = 0 and so,r 0 (x;w 0 ) is nonincreasing inx for everyw 0 . Let us move to 2. From Lemma 4.7.6 in Puterman (2005), we need r 0 (x+1;1)¡r 0 (x+1;0)¡[r 0 (x;1)¡r 0 (x;0)]¸0: But r 0 (x+1;1)¡r 0 (x+1;0)¡[r 0 (x;1)¡r 0 (x;0)]= p 0 ¸ 0 ¡c¹ ±+¤+¹ ¡ p 0 ¸ 0 ±+¤+¹ ¡ · p 0 ¸ 0 ¡c¹ ±+¤+¹ ¡ p 0 ¸ 0 ±+¤+¹ ¸ =0: 168 Therefore, r 0 (x;w 0 ) is superadditive. Now we have to show that ^ q 0 (kjx;w 0 ) = P 1 j=k q 0 (jjx;w 0 ) is nondecreasing in x. We distinguish five different cases to compute ^ q 0 (kjx;w 0 ). Forx<k¡1; Forx=k¡1; Forx=k; Forx=k+1; Forx>k+1; ^ q 0 (kjx;1)=0 ^ q 0 (kjk¡1;1)= ¸ 0 ¤+¹ ^ q 0 (kjk;1)= ¤ ¤+¹ ^ q 0 (kjk+1;1)=1 ^ q 0 (kjx;1)=1 and ^ q 0 (kjx;0)=0: and ^ q 0 (kjk¡1;0)= ¸ 0 ¤+¹ : and ^ q 0 (kjk;0)=1: and ^ q 0 (kjk+1;0)=1: and ^ q 0 (kjx;0)=1: Apparently, ^ q 0 (kjx;w 0 ) = P 1 j=k q 0 (jjx;w 0 ) is nondecreasing in x. Finally, it is left to show that P 1 j=0 q 0 (jjx;w 0 )v ? 0 (j) is superadditive, or equivalently, it is left to show that 1 X j=0 q 0 (jjx+1;1)v ? 0 (j)¡ 1 X j=0 q 0 (jjx+1;0)v ? 0 (j)¡ 1 X j=0 q 0 (jjx;1)v ? 0 (j)+ 1 X j=0 q 0 (jjx;0)v ? 0 (j)¸0: We have that 1 X j=0 q 0 (jjx+1;1)v ? 0 (j)¡ 1 X j=0 q 0 (jjx+1;0)v ? 0 (j)¡ 1 X j=0 q 0 (jjx;1)v ? 0 (j)+ 1 X j=0 q 0 (jjx;0)v ? 0 (j)= = ¸ 0 ¤+¹ v ? 0 (x+2)+ ¤¡¸ 0 ¤+¹ v ? 0 (x+1)+ ¹ ¤+¹ v ? 0 (x) ¡ ¸ 0 ¤+¹ v ? 0 (x+2)¡ ¤+¹¡¸ 0 ¤+¹ v ? 0 (x+1)¡ ¸ 0 ¤+¹ v ? 0 (x+1) ¡ ¤¡¸ 0 ¤+¹ v ? 0 (x)¡ ¹ ¤+¹ v ? 0 (x¡1)+ ¸ 0 ¤+¹ v ? 0 (x+1) + ¤+¹¡¸ 0 ¤+¹ v ? 0 (x) = ¡ ¹ ¤+¹ v ? 0 (x+1)+ ¹ ¤+¹ v ? 0 (x)+ ¹ ¤+¹ v ? 0 (x)¡ ¹ ¤+¹ v ? 0 (x¡1): 169 We will need to show that¡v ? 0 (x+1)+2v ? 0 (x)¡v ? 0 (x¡1) ¸ 0, that is the definition ofv ? 0 (x) being concave. We will show it using the value iteration algorithm and induction. We have that v 0 0 (x)= p 0 ¤(1¡ 1 v p 0 ) ±+¤+¹ and so @ 2 v 0 0 @x 2 =0: Given @ 2 v n 0 (x) @x 2 <0, we want to show that @ 2 v n+1 0 (x) @x 2 <0. We have that if we produce atx, @ 2 v n+1 0 (x) @x 2 = ¤ ¡ 1¡ 1 v p 0 ¢ ±+¤+¹ @ 2 v n 0 (x+1) @x 2 + ¤ v p 0 ±+¤+¹ @ 2 v n 0 (x) @x 2 + ¹ ±+¤+¹ @ 2 v n 0 (x¡1) @x 2 <0; and @ 2 v n+1 0 (x) @x 2 = ¤ ¡ 1¡ 1 v p 0 ¢ ±+¤+¹ @ 2 v n 0 (x+1) @x 2 + ¹+ ¤ v p 0 ±+¤+¹ @ 2 v n 0 (x) @x 2 <0; if we do not produce atx. Therefore, P 1 j=0 q 0 (jjx;w ) )v ? 0 (j) is superadditive. In the proof, we have not study the statex=¡B since it should be clear that it is optimal not to produce at this state considering that a component produced at this state will be discarded. ¤ Proof of Theorem 4.4 The optimality equation (4.17) can be rewritten as v ? (x;p)=max p fF 1 (p);F 2 (p)g where F 1 (p) = p(x)¸(x;p)¡c¹¡h([x] ¡ ) ±+¤+¹ + ¸(x;p) ±+¤+¹ v ? d (x+1;p) + ¤ v p(x)+ ¤¯ v x + ¹ ±+¤+¹ v ? d (x;p)+ ¹ ±+¤+¹ v ? d (x¡1;p) [whenw d (x)=1]; F 2 (p) = p(x)¸(x;p)¡h([x] ¡ ) ±+¤+¹ + ¸(x;p) ±+¤+¹ v ? d (x+1;p) + ¹+ ¤ v p(x)+ ¤¯ v x + ¹ ±+¤+¹ v ? d (x;p) [whenw d (x)=0]: 170 Thereforew d (x)=1 when ¡ c¹ ±+¤+¹ + ¹ ±+¤+¹ v ? d (x¡1;p)> ¹ ±+¤+¹ v ? d (x;p); or equivalently, when v ? d (x;p)¡v ? d (x¡1;p)<¡c: (D.6) In order to determine the optimal pricing policy we need to compute max p F 1 (p) and max p F 2 (p). We have already shown that max p F 1 (p)= 1 ±+¤+¹ 8 < : ¤v ³ 1¡ ¯ v x + ¹ ´ 2 4 ¡c¹¡h([x] ¡ )+ ¤ ³ 1¡ ¯ v x + ¹ ´ 2 v ? d (x+1;p) + ¤ ³ 1+ ¯ v x + ¹ ´ 2 v ? d (x;p)+ ¤[v ? d (x+1;p)¡v ? d (x;p)] 2 4v +¹v ? d (x¡1;p) 9 = ; : Formax p F 2 (p), the optimal pricing policy should satisfy the first order condition, that is, 0= ¤ ³ 1¡ ¯ v x + ¹ ¡ 2 v p(x) ´ ±+¤+¹ ¡ ¤ v(±+¤+¹) v ? d (x+1;p)+ ¤ v(±+¤+¹) v ? d (x;p); and so the optimal pricing policy under no production will again be p ? (x)=¡ v ? d (x+1;p)¡v ? d (x;p) 2 + v¡¯ x + ¹ 2 : Therefore,max p F 2 (p) will be max p F 2 (p)= 1 ±+¤+¹ 8 > < > : ¤v ³ 1¡ ¯ v x + ¹ ´ 2 4 ¡h([x] ¡ )+ ¤ ³ 1¡ ¯ v x + ¹ ´ 2 v ? d (x+1;p) + · ¤ ³ 1+ ¯ v x + ¹ ´ 2 +¹ ¸ v ? d (x;p)+ ¤[v ? d (x+1;p)¡v ? d (x;p)] 2 4v 9 > = > ; : When x > 0, we know that the firm does not have any option but to produce in order to be consistent with the leadtime quotation. The customer is expecting to receive the product after waiting for an average of x + ¹ and this can only happen if we always produce when x > 0. We will show that for x < 0 there is an increasing monotone policy on the production decision. In other words, there existsS d <0 such that w ? d (x)= 8 < : 1 0 forx¸S d forx<S d : 171 To show that the optimal production decision is a monotone policy, we first need to show thatv ? d is concave whenx < 0. We will use the value iteration algorithm and induc- tion. We start withv 0 d (x;p)=0 and so we have that v 1 d (x;p)= 1 ±+¤+¹ 8 > < > : ¤v ³ 1¡ ¯ v x + ¹ ´ 2 4 9 > = > ; : We need to focus on the case whenx<0 and so we have that @v 1 d (x;p) @x =0 and @ 2 v 1 d (x;p) @x 2 =0: Assume that @v 1 d (x;p) @x <0 and @ 2 v 1 d (x;p) @x 2 <0. We have that v n+1 d (x;p)=max © F n+1 1 (x); F n+1 2 (x) ª ; where F n+1 1 (x) = 1 ±+¤+¹ 8 > > < > > : ¤v ³ 1¡ ¯ v x + ¹ ´ 2 4 ¡c¹+ ¤ ³ 1¡ ¯ v x + ¹ ´ 2 v n d (x+1;p) + ¤ ³ 1+ ¯ v x + ¹ ´ 2 v n d (x;p)+ ¤[v n d (x+1;p)¡v n d (x;p)] 2 4v +¹v n d (x¡1;p) 9 > > = > > ; ; F n+1 2 (x) = 1 ±+¤+¹ 8 > < > : ¤v ³ 1¡ ¯ v x + ¹ ´ 2 4 + ¤ ³ 1¡ ¯ v x + ¹ ´ 2 v n d (x+1;p) + · ¤ ³ 1+ ¯ v x + ¹ ´ 2 +¹ ¸ v n d (x;p)+ ¤ [ v n d (x+1;)¡v n d (x;p) ] 2 4v 9 > = > ; 172 and so, @F n+1 1 (x) @x = 1 ±+¤+¹ 8 < : ¤ 2 @v n d (x+1;p) @x + ¤ 2 @v n d (x;p) @x +¹ @v n d (x¡1;p) @x + ¤[v n d (x+1;p)¡v n d (x;p)] 2v ³ @v n d (x+1;p) @x ¡ @v n d (x;p) @x ´ 9 = ; <0; @F n+1 2 (x) @x = 1 ±+¤+¹ 8 < : ¤ 2 @v n d (x+1;p) @x + £ ¤ 2 +¹ ¤ @v n d (x;p) @x + ¤[v n d (x+1;p)¡v n d (x;p)] 2v ³ @v n d (x+1;p) @x ¡ @v n d (x;p) @x ´ 9 = ; <0; @ 2 F n+1 1 (x) @x 2 = 1 ±+¤+¹ 8 > > > < > > > : ¤ 2 @ 2 v n d (x+1;p) @x 2 + ¤ 2 @ 2 v n d (x;p) @x 2 +¹ @v n d (x¡1;p) @x + ¤ 2v ³ @v n d (x+1;p) @x ¡ @v n d (x;p) @x ´ 2 + ¤[v n d (x+1;p)¡v n d (x;p)] 2v ³ @ 2 v n d (x+1;p) @x 2 ¡ @ 2 v n d (x;p) @x 2 ´ 9 > > > = > > > ; <0; @ 2 F n+1 2 (x) @x 2 = 1 ±+¤+¹ 8 > > > < > > > : ¤ 2 @ 2 v n d (x+1;p) @x 2 + £ ¤ 2 +¹ ¤ @ 2 v n d (x;p) @x 2 + ¤ 2v ³ @v n d (x+1;p) @x ¡ @v n d (x;p) @x ´ 2 + ¤ [ v n d (x+1;p)¡v n d (x;p) ] 2v ³ @ 2 v n d (x+1;p) @x 2 ¡ @ 2 v n d (x;p) @x 2 ´ 9 > > > = > > > ; <0; that implies thatv n+1 d is concave forx<0 and sov ? is also concave. Define r d (x;w d )= p(x)¸(x;p)¡c¹w d (x) ±+¤+¹ : To show the optimality of a monotone policy, we have to show that 1. r d (x;w d ) is nonincreasing inx for everyw d , 2. r d (x;w d ) is superadditive, 3. ^ q d (kjx;w d )= P 1 j=k q d (jjx;w d ) is nondecreasing inx, and 4. P 1 j=0 q d (jjx;w d )v ? d (j;p) is superadditive. r(x;w d ) satisfies r d (x;w d )= 1 ±+¤+¹ 8 > < > : ¤v ³ 1¡ ¯ v x + ¹ ´ 2 4 ¡ ¤ 4v [v ? d (x+1;p)¡v ? d (x;p)] 2 ¡c¹w d (x) 9 > = > ; : 173 We have to show each of the four properties mentioned whenx<0. We start with 1. r d (x+1;1)¡r d (x;1) = ¡ ¤ ¡ [v ? d (x+2;p)¡v ? d (x+1;p)] 2 ¡[v ? d (x+1;p)¡v ? d (x;p)] 2 ¢ 4v(±+¤+¹) = ¡ ¤[v ? d (x+2;p)¡v ? d (x;p)][v ? d (x+2;p)¡2v ? d (x+1;p)+v ? d (x;p))] 4v(±+¤+¹) <0; r d (x+1;0)¡r d (x;0) = ¡ ¤[v ? d (x+2;p)¡v ? d (x;p)][v ? d (x+2;p)¡2v ? d (x+1;p)+v ? d (x;p))] 4v(±+¤+¹) <0; sincev ? d (x+2;p)¡v ? d (x;p) < 0 andv ? d (x+2;p)¡2v ? d (x+1;p)+v ? d (x;p) < 0 and so r d (x;w d ) is nonincreasing inx < 0 for everyw d . Let us move to 2. From Lemma 4.7.6 in Puterman (2005), we need r d (x+1;1)¡r d (x+1;0)¡[r d (x;1)¡r d (x;0)]¸0; but we have r d (x+1;1)¡r d (x+1;0)¡[r d (x;1)¡r d (x;0)]=0: Therefore, r d (x;w d ) is superadditive. Now we have to show that ^ q d (kjx;w d ) = P 1 j=k q d (jjx;w d ) is nondecreasing in x. We distinguish five different cases to compute ^ q d (kjx;w d ). Forx<k¡1; Forx=k¡1; Forx=k; Forx=k+1; Forx>k+1; ^ q d (kjx;1)=0 ^ q d (kjk¡1;1)= ¸(k¡1;p) ¤+¹ ^ q d (kjk;1)= ¤ ¤+¹ ^ q d (kjk+1;1)=1 ^ q d (kjx;1)=1 and ^ q d (kjx;0)=0: and ^ q d (kjk¡1;0)= ¸(k¡1;p) ¤+¹ : and ^ q d (kjk;0)=1: and ^ q d (kjk+1;0)=1: and ^ q d (kjx;0)=1: Therefore, ^ q d (kjx;w d )= P 1 j=k q d (jjx;w d ) is nondecreasing inx. Finally, it is left to show that P 1 j=0 q d (jjx;w d )v ? d (j;p) is superadditive, or equivalently, it is left to show that 1 X j=0 q d (jjx+1;1)v ? d (j;p)¡ 1 X j=0 q d (jjx+1;0)v ? d (j;p) ¡ 1 X j=0 q d (jjx;1)v ? d (j;p)+ 1 X j=0 q d (jjx;0)v ? d (j;p)¸0: 174 We have that 1 X j=0 q d (jjx+1;1)v ? d (j;p)¡ 1 X j=0 q d (jjx+1;0)v ? d (j;p) ¡ 1 X j=0 q d (jjx;1)v ? d (j;p)+ 1 X j=0 q d (jjx;0)v ? d (j;p)= = ¸(x+1;p) ¤+¹ v ? d (x+2;p)+ ¤¡¸(x+1;p) ¤+¹ v ? d (x+1;p) + ¹ ¤+¹ v ? d (x;p)¡ ¸(x+1;p) ¤+¹ v ? d (x+2;p) ¡ ¤+¹¡¸(x+1;p) ¤+¹ v ? d (x+1;p)¡ ¸(x;p) ¤+¹ v ? d (x+1;p) ¡ ¤¡¸(x;p) ¤+¹ v ? d (x;p)¡ ¹ ¤+¹ v ? d (x¡1;p) + ¸(x;p) ¤+¹ v ? d (x+1;p)+ ¤+¹¡¸(x;p) ¤+¹ v ? d (x;p) = ¡ ¹ ¤+¹ v ? d (x+1;p)+ ¹ ¤+¹ v ? d (x;p)+ ¹ ¤+¹ v ? d (x;p)¡ ¹ ¤+¹ v ? d (x¡1;p): We will need to show that¡v ? d (x+1;p)+2v ? d (x;p)¡v ? d (x¡1;p)¸0, that is the definition ofv ? d (x;p) being concave and the result follows. Once more, at statex=¡B, it should be straightforward that it is optimal not to produce. ¤ 175

Abstract (if available)

Abstract This thesis focuses on the analysis of stochastic models that frequently arise in the management of service and manufacturing operations. Determining the waiting time, the routing choice or proposing a dynamic pricing policy in stochastic systems are of great importance in practice but can be challenging. Waiting is an integral part of the service experience and customers joining a line to acquire a product can rarely estimate their waiting time so any relevant information is appreciated by them. But sometimes disclosing delay information to the customers may not benefit the provider. In other settings, a manager has to trade off speed and quality. Speeding up a service or manufacturing operation may reduce congestion or increase production but at the expense of quality and customer satisfaction. In my dissertation, I focus on modeling these phenomena and obtaining significant managerial insights on how firms can improve the service experience and maximize their revenue. My main findings should be of interest to managers

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Asset Metadata

Creator Kostami, Vasiliki (author)

Core Title Essays on dynamic control, queueing and pricing

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Publication Date 08/04/2010

Defense Date 04/26/2010

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

Tag dynamic pricing,FASTPASS,inventory control,leadtime quotation,OAI-PMH Harvest,quality vs speed,stochastic modeling

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ward, Amy R. (committee chair), Rajagopalan, Sampath (committee member), Ross, Sheldon M. (committee member)

Creator Email kostami@usc.edu,vkostami@london.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3265

Unique identifier UC1175643

Identifier etd-Kostami-3750 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-370385 (legacy record id),usctheses-m3265 (legacy record id)

Legacy Identifier etd-Kostami-3750.pdf

Dmrecord 370385

Document Type Dissertation

Rights Kostami, Vasiliki

Type texts

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

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu