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Essays on revenue management
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Essays on revenue management
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ESSAYS ON REVENUE MANAGEMENT by Chunyang Tong A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful¯llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (BUSINESS ADMINISTRATION) May 2007 Copyright 2007 Chunyang Tong Acknowledgments Iwouldliketotakethisopportunitytoexpressmydeepestgratitudetomyadvisor, Professor Sriram Dasu and to thank him for the guidance, patience, and support, which have so helped to make this thesis possible. I am also grateful to Professors YehudaBassok,S.Rajagopalan(Raj),GreysSosicandGuofuTan,whogenerously gave of their time to advise and support me throughout my Ph.D. studies. My thanks also go to all current and former sta® and to my fellow Ph.D. students in theIOMDepartmentfortheirsupportandfriendshipandforthesharedmemories I will long cherish. Finally, this thesis would not have been possible without the ¯nancial support of the Marshall School of Business, University of Southern California. ii Table of Contents Acknowledgments ii List of Tables v List of Figures vi Abstract vii 1 Introduction 1 1.1 General Revenue Management Issues and My Position . . . . . . . 1 1.2 Customers' Strategic Behavior . . . . . . . . . . . . . . . . . . . . . 4 1.3 Renewable Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Dynamic Pricing under Customers' Strategic Behavior 19 2.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Equilibrium Buying Behavior . . . . . . . . . . . . . . . . . . . . . 22 2.2.1 Posted Pricing Scheme . . . . . . . . . . . . . . . . . . . . . 22 2.2.2 Contingent Pricing . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Some Properties of the Pricing Schemes. . . . . . . . . . . . . . . . 27 2.3.1 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . 28 2.3.2 Optimal Stocking Levels . . . . . . . . . . . . . . . . . . . . 30 2.3.3 Value of Inventory Information . . . . . . . . . . . . . . . . 30 2.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 Revenue Management When Assets Are Renewable 41 3.1 Markov Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.1 A Dual Formulation . . . . . . . . . . . . . . . . . . . . . . 43 3.1.2 Solving the optimality equation . . . . . . . . . . . . . . . . 48 3.1.3 Value of dynamic control . . . . . . . . . . . . . . . . . . . . 49 3.2 Large Size Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2.1 Static control on Erlang B system . . . . . . . . . . . . . . . 53 3.2.2 A di®usion approximation model . . . . . . . . . . . . . . . 55 iii 3.2.3 Stochastic Control over the Di®usion Process. . . . . . . . . 60 3.3 Extension I: Heterogenous service requirement . . . . . . . . . . . . 63 3.3.1 An Admission Control Sub-problem . . . . . . . . . . . . . . 65 3.4 Extension II: Di®erentiated Service . . . . . . . . . . . . . . . . . . 66 4 Conclusion 72 References 74 Appendix 78 iv List of Tables 2.1 Comparison of Optimal Prices, Buyers Distribution are U(0,1) . . . 24 2.2 N=20,Valuation U(0,1) . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1 Comparison of Blocking Probability with Intensity Less Than 1 . . 54 3.2 Comparison of Blocking Probability with Intensity Greater Than 1 55 v List of Figures 2.1 Drop in expected revenues when seller uses two posted prices with- out considering strategic behavior . . . . . . . . . . . . . . . . . . . 34 2.2 % loss in optimality: 2 posted prices and uniform distribution . . . 35 2.3 % loss in optimality, 3 posted prices and uniform distribution. . . . 35 2.4 % loss in optimality, 2 posted prices, Beta(2,8):left skewed . . . . . 36 2.5 % loss in optimality, 3 posted prices, Beta(2,8):left skewed . . . . . 36 2.6 % loss in optimality, 2 posted prices Beta(8,2):right skewed . . . . . 37 2.7 Comparison of Posted and Contingent Pricing Schemes, Fixed N . . 38 2.8 Comparison of Posted and Contingent Pricing Scheme, Uncertain N 39 2.9 Value of Revealing Initial Sales Quantity, N=20, Valuations U(0,1). 39 2.10 Value of Concealing Sales Information. Valuations: U(0,1) . . . . . 40 3.1 Optimal arrival rate and value di®erence as function of state s . . . 49 3.2 Optimal dynamic control policy and static policy for moderate sys- tem size N=6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Average Pro¯t increase for moderate system size N=6 . . . . . . . 51 3.4 TheOptimalRevenueRatederivedfromSMDPandDi®usionModel When Increasing System Size, N=6 . . . . . . . . . . . . . . . . . . 63 vi Abstract For this thesis, I combined two of my research works on revenue management and dynamic pricing problems and built upon them with current literature and con- tribute to these themes in two ways. The ¯rst way extends the existing, pricing decision model by relaxing commonly used assumptions about demand|and that is customers tend to be strategic, rather than myopic, in the sense that they time their purchases to maximize their payo®s. The second contribution is to take a di®erent view on analyzing pricing issues that relate to assets, which are renew- able, rather than perishable. Thus, this second contribution concerns a di®erent modeling for supply. Chapter 2 studies dynamic pricing policies for a monopolist selling perishable productsovera¯nitetimehorizontobuyerswhoarestrategic. Buyersarestrategic in the sense that they anticipate the ¯rm's pricing policies. I am interested in situations where auctions are not feasible and it is costly to change prices. I begin by showing that unless strategic buyers expect shortages dynamic pricing will not increase revenues. I investigate two pricing schemes that I call posted and contingent pricing. In the posted pricing scheme at the beginning of the horizon the¯rmannouncesasetofprices. Inthecontingentpricingschemepriceevolution depends on demand realization. My focus is on the posted pricing scheme because ofitseaseofimplementation. Inequilibrium,buyerswillemployathresholdpolicy vii in both pricing regimes, i.e. they will buy only if their private valuations are above a particular threshold. I show that a multi-unit auction with a reservation price provides an upper bound for the expected revenues for both pricing schemes. Numerical examples suggest that a posted pricing scheme with two or three price changes is enough to achieve revenues that are close to the upper bound. I ¯nd thatneitherpostedpricingschemenorcontingentpricingschemeisdominant. The di®erence in expected revenues of these two schemes is small. I also investigate whether it is optimal for the seller to conceal inventory and sales information from buyers. Chapter3studiesanarrivalcontrolproblemforasystemwithrenewableassets. Customers belong to di®erent classes and the arrival rate of each class depends on the price being charged for that class. Each customer uses the resource for a random duration. The objective is to maximize the long run average pro¯t by dynamically changing prices, subject to a constraint on the quality of service, stated in terms of probability that demand is turned away because all units are being used. I show that this problem can be converted into a problem in which there is a penalty cost for blockage. The alternative formulation is more tractable and equivalent to the original model. I derive several structural properties of the optimal policy. I also provide a heuristic solution procedure to the original dynamic problem when the number of assets is large. Speci¯cally,I formulate a di®usion model to approximate the original system. Numerical results show that this heuristic procedure performs very well. I also study the optimal policy when the only options available are either to admit customers from a particular class or deny admissions. My conclusion and some interesting future extensions are incorporated in the last chapter. All proofs are found in the appendices. viii Chapter 1 Introduction 1.1 General Revenue Management Issues and My Position With liberalization of the economy and deregulation of many industry, today's ¯rms operate in a highly competitive and dynamic business environment. With this trend, comes the challenge of managing demand in more e®ective ways than simply meeting a relatively static demand, which is the feature of regulated indus- tries. Revenue management(RM), as a quantitative tool of enhancing revenue by dynamic control of pricing and/or capacity availability, is attracting the attention of both practitioners and academicians. To practitioners, RM is the art of sell- ing the right product to the right customer at the right time for the right price. And the scope of its application spans the domains of the airline and hotel indus- tries and such more recent areas as retail, manufacturing, telecommunications, entertainment, ¯nancial services, and health care. Among academicians, revenue management is the topic that has commanded increasing attention over the past decade. Researchersinoperationsmanagement,operationsresearch,statisticsand marketing science, think of revenue management as the scienti¯c way of dynam- ically managing prices and/or capacity based on demand-forecasting tools and market segmentation techniques. Starting from \Littlewood' rule" for the two- fareseatallocationproblem, tomorerecenttheories ondynamicpricing, customer 1 choice and auctions and such, revenue management for many ¯rms is becoming a more widely used solution. Inthesuccessfulapplicationofrevenuemanagement, demand uncertaintyplays animportantrole. Forexample, thedemandforelectricitycorrelateswithweather conditions, and weather conditions cannot be forecasted with certainty. In addi- tion to demand °uctuations, short-term capacity is ¯xed, and capacity can only be changed in the long term. Also what is commonly observed, although not absolute, is that capacity is perishable|a railway seat is lost if it is not sold by departure time, and the cargo shipper loses revenue if the containers are empty at loading time. In the implementation of revenue management,Market segmenta- tion is another critical factor.In the airline industry, ¯rms charge time-conscious, business executives di®erently, as compared to price-sensitive, leisure customer. Depending upon business type, revenue management can be classi¯ed roughly in four control categories: dynamic pricing,capacity allocation,market segment identi¯cation and overbooking. This thesis focuses on pricing problems and also discussescapacityallocationasspecialcasesofdynamicpricing.Thethesisextends currentliteratureintwoways. Oneisthedirectstudyofdemandsiderelatedprob- lem,i.e., what if customers are strategic in timing their purchase when prices are dynamically changing. The second extension is the direct study of the supply side related problem,i.e., what if the product or service is renewable, rather than perishable{as is usually assumed in the literature. While the practice of revenue management is to enhance revenue by changing price dynamically, it might be risky ignoring customers' choice decisions when for- mulating the pricing model. Broadly speaking, consideration of customers' choice involves a two dimensional problem, one of which is cross product choice deci- sion. In the presence of di®erentiated products or services, customers will choose 2 accordingtotheirpreference. Theotherdimensionalproblemconcerns time-series choice decision,i.e., customers decide what is best timing for the purchase of a product. In the hotel industry,for example, one would not assume that customers would be resigned to the fact that hotel charges di®erent rates for the same room and would not react accordingly. My ¯rst research work, as discussed in Chapter 2, was concerned with this second dimension of customer choice and attempted to answer the following questions: How will a varying pricing patterns change the demand curve? To what degree does ignorance of customer's timing decision negatively impact upon a ¯rms' revenue? What is the structural property of an optimalpricingdecisioninthepresenceofcustomer'stimingchoice? Whetherrev- elation of information about initial capacity and/or sales a®ects a ¯rm's revenue. To this end, I formulated a dynamic principle-multi-agent model to characterize customers' equilibrium behavior and to further study pricing decisions based on these equilibrium behaviors. My second research work, as discussed in Chapter 3, was a study of pricing decisions on renewal assets. For any one night, a hotel room may be a perishable asset, but from the general manager's point of view, the room may be, in fact, a renewable asset. While discouraging or rejecting customers in order to leave a room empty may seem to be a negative management decision in terms of active use of individual hotel rooms, holding a room empty may actually be a calculated decision on the part of the manager in order to have an empty room available for customers who may be willing to pay higher rates on the following day. While dynamicpricingpolicymayboosta¯rm'sshort-termpro¯tability,apracticewhich discourages or rejects customer may lead to strong customer dissatisfaction result- ing in their refusal to patronize the hotel in the future. In a highly competitive market, where quality of service is the key to success, such short term revenue 3 maximizing practices may not be in the only best interest of business. In evalu- ating the e±ciency of a RM system, the tradeo® between generating short-term pro¯t and enhancing long-term customer loyalty needs careful study. My study on renewable assets o®ers solutions for pricing decisions, when quality of service and revenue are both factors under consideration. In the following sections, I introduce these two problems in greater detail and extend the discussion into subsequent chapters. Literature review for each topic is also included. 1.2 Customers' Strategic Behavior Inchapter2,westudyamonopolistsellinga¯xedquantityofaperishableproduct overa¯nitetimehorizon. Examplesofsuchproductsincludeairlinetickets,fashion goods, and tee times on golf courses. Two important feature common to these products are (i) that replenishment may not be possible and that even if it is, costs may be too high, and (ii) that any unsold product at the end of the horizon is either worthless or of considerably lower value. One strategy for increasing revenues is to segment the market and to o®er the product at a di®erent price to each segment. In the airline industry, the market can be segmented on the basis of purchase time { willingness to pay for a °ight ticket increases as you get closer to departure time. When the seller cannot e®ectivelysegmentthemarket,analternativeapproachforincreasingrevenueisto dynamically change prices. Of course, if segmentation is possible, dynamic pricing can also be used in conjunction with segmentation strategies. Dynamic pricing is inter-temporal price discrimination, with a focus on demand uncertainty. 4 Work that has preceded ours in the sub-¯eld of dynamic pricing 1 , implicitly assumes that customers do not anticipate prices and behave myopically (Bitran & Caldentey 2003). We argue that many customers are aware of pricing paths and that they time their purchases. Evidence of customers' strategic behavior abound; customerswaitingforafter-Christmassales,anticipatingpricemark-downsoffash- ion goods and electronic products (McWilliams, 2004), and tracking prices of air- linetickets,arejustafewexamples. Lazear(1986)suggeststhatconsumerscanbe divided into (i) shoppers who are exploring prices, and (ii) buyers who are ready to purchase. Wetrytoshedlightonthein°uenceofcustomers'strategicbehavioronseller's equilibrium pricing policies. Auction theory, and more speci¯cally literature on mechanism design is directly concerned with this question (Krishnan ). There are, however, many retail settings in which mechanisms such as auctions are not practical. Changing prices in retail outlets is a complex process that consumes resources(Levyetal., 1997)andbrickandmortarretailerarenotsetuptohandle bids from buyers. Therefore, we are interested in mechanisms that involve a small numberofpricechanges. Weconsideranumberofpricingstrategies. Thesimplest policy would be to announce and commit to a set of prices at the beginning of the selling horizon 2 . At the other extreme, the ¯rm can change prices after observing sales. In addition, it can either reveal its sales and inventory levels to all buyers or conceal this information. In the following chapter, we study two di®erent pricing policies we call posted and contingent. In the posted pricing policy, the ¯rm 1 Dynamic pricing is a sub-¯eld for general pricing theory. In pricing theory there is a long tradition of incorporating strategic consumer behavior. In our literature review we elaborate on this point. 2 Filene's basement store in Boston is famous for using this approach (www.¯lenesbasement.com) 5 commits to a price path at the beginning of the season. In the contingent pricing policy, the ¯rm determines its price based on realized sales. In both schemes, the ¯rm commits to the number of price changes 3 .We assume that all buyers are presentatthebeginningofthesellinghorizonandignoreuncertaintyinthearrival process. In short we assume that all buyers are strategic and not just a subset as suggestedbyLazear (1986). In our model all the buyers are constantlymonitoring prices. This is the polar extreme of the assumption traditionally made in the revenue management literature that none of the buyers are strategic. We believe thatexaminingthisextremeisvaluabletoisolatetheimpactofstrategicbehavior. We assume that a single product is being sold and consumers di®er only in the value they place on the product. The buyers and sellers have information about the number of units for sale, the size of the market, the number of price changes, and the distribution of valuations. Our analysis readily incorporates uncertainty in the size of the market. We assume that the duration of the selling horizon is short and ignore discounting. Because all buyers are present at the beginning, the actual duration of the selling period is inconsequential and all that matters is the number of price changes. Presence of strategic buyers raises a number of interesting questions. What are the consequences on revenues if a ¯rm ignores strategic behavior? What is the impact of strategic behavior on pricing policies? What is the loss in optimality if the ¯rm commits to posted prices? How many price changes are needed? How do factors such as the size of the market, level of supply, uncertainty in the size of themarket, andthedistributionofthevaluationsin°uencetheperformanceofthe di®erent pricing policies? How many units should the ¯rm stock? Should the ¯rm 3 Pricingstrategiesinwhichnumberofpricechangesisrandommayincreasea¯rm'sexpected revenues. We ignore that possibility and view the number of price changes as an industry norm. 6 reveal the number of units it has for sale? Should the ¯rm conceal information about the number of units sold at di®erent prices? The characteristics of optimal dynamic pricing policies and the ability of the ¯rm to extract consumer surplus are signi¯cantly altered when consumers antici- patepricingpolicies. Foronething, itiscostlyforthe¯rmtoignoresuchbehavior whendevelopingpricingpolicies. Alsothepricingschemesthat accountfor strate- gic buyers have a narrower range. The di®erence between the lowest price and the highest price is reduced. If consumers are not strategic, it is well known (Gallego & van Ryzin 1994) that dynamic pricing will increase a ¯rm's revenues if there is uncertainty in the customer arrival process. On the other hand, if buyers are strategic then even if there is uncertainty in demand, dynamic pricing need not increase the ¯rm's revenues. For dynamic pricing to be useful it is essential that consumers anticipate a shortage. Static pricing is optimal regardless of whether or not demand is uncertain, as long as buyers are assured of supply. Further, when buyers are strategic and shortages are perceived, dynamic pricing is better than static pricing even if demand is deterministic. A strategic buyer, except in the terminal period, will buy the product only if his or her valuation is above a threshold. This threshold, except at the lowest price, is strictly greater than the prevailing price. When the valuation is close to the prevailing price, a buyer may ¯nd it worth while to wait for a lower price, even thoughthereisariskofastock-out. Modelsthatignorestrategicbehaviorassume that everyone whose valuation is above the given price will buy the product. The threshold depends on the size of the market, and the level of scarcity. We use the ratio of the number of units for sale(K) to the number of customers (N) as a measure of the level of scarcity. 7 Inthelimitasthenumberofpricechangesapproachesin¯nity,postedandcon- tingent pricing schemes maximize expected revenues. They are revenue-equivalent to a multi-unit auctions with reservation prices. This linkage enables us to derive an upper bound for expected revenues. When the number of prices changes is restricted the loss in optimality depends on the level of scarcity ( K N ), the distrib- ution of valuations, and the size of the market. If K N ¸ 1 or K N is close to zero, a single price is optimal or nearly optimal, respectively. Also as N !1, regardless of the ratio K N , no more than one price change is needed to maximize expected revenues. When prices can be changed limitlessly, expected revenues are not e®ected by whether or not buyers are aware of the number of units remaining unsold. However, when the number of price changes is ¯nite information about the sales and inventories in°uence expected revenues. Based on our numerical examples we conjecture that the ¯rm should reveal the number of units that are available for sale at the beginning of the season, but subsequently conceal the inventory levels. Contingent pricing clearly dominates posted pricing when consumers are not strategic. This is no longer true when buyers anticipate prices. Neither scheme is dominant. The di®erence in expected revenues, however, is quite small. Overall ournumericalexamplessuggestthatsimplepoliciesofpre-announcingafewprices in advance achieves near optimal revenues. This is particularly true when there are more than 30 buyers or the number of units for sale is in excess of 50% of the market size. In the sequel, we review related literature. In the ¯rst section of chapter 2, we introduce our model and show that customers have to perceive a shortage for dynamic pricing to be useful. In section 2 of chapter 2, we analyze buyers' equilibrium strategies and show that in all of the pricing schemes there is a unique 8 equilibrium. In section 3 of chapter 2, we derive properties of the pricing policies and show that in the limit as the number of price changes approaches in¯nity both schemes are optimal and are equivalent to a descending price auction with a reservation price. We also derive su±cient conditions under which it is optimal for the ¯rm to conceal sales information. Section 4 of chapter 2 contains numerical examples that assess the value of multiple price changes, compares the two pricing schemes, and explores the bene¯ts of concealing inventories. All proofs are in the appendix. Economists were among the earliest researchers to study pricing when cus- tomers behave strategically. This work was concerned with consumer durable, and ignored capacity and inventory constraints. The well-known Coase theorem ( Coase, 1972) considers a monopolist selling a consumer durable. In equilibrium, a declining price path is subgame perfect, and rational customers anticipating the pricing path would not buy the product until the last period (Stokey, 1981). The price steadily declines to marginal cost, thereby eliminating all monopoly power. If customers tradeo® the value of consuming right away against the bene¯ts of di®ering consumption for a lower price then customers with high valuations would buyintheearlyperiodswhilelowvaluationcustomerswouldbuyinalaterperiod. Researchersinmarketinghavealsostudiedoptimalpricingstrategieswhenbuy- ers anticipate prices. The price-skimming phenomenon has been investigated by BesankoandWinston(1990). Intheirmodel,customers'valuationsareassumedto beuniformlydistributed. TheauthorscharacterizeasubgameperfectNashequilib- riuminvolvingarationalsellerandrationalconsumersandestablishtheoptimality ofadecliningpricepath. Throughnumericalexamples,theyalsodemonstratethat a seller who ignores consumer strategic behavior will observe a substantial decline 9 in pro¯ts. Unlike our work, papers in marketing are not concerned with perish- able assets (Besanko & Winston, 1990; Moorthy, 1988). They assume that the manufacturer can produce additional product in each period. The dynamic pricing and revenue management literature is concerned with pricing perishable assets that cannot be replenished. This literature is vast, and a recent book by van Ryzin and Taluri provides an excellent overview of this literature. Readers are also referred to reviews by McGil and van Ryzin (1999), BitranandCaldentey(2003),andElmaghrabyandKeskinocak(2003). Earlywork in dynamic pricing did not consider strategic behavior. In recent years several papers have been developed on dynamic pricing that take into account consumer behavior. Several papers consider how consumers select among di®erent products (Talluri and van Ryzin (2004), Zhang and Cooper (2005), Netessine, Savin and Xiao (2006),and Maglaras and Meissner (2006)). These papers are concerned with selection among alternative products, but do not worry about consumers timing their purchases. Arnold and Lipmann (2001) study a posted pricing problem. The ¯rm announces a price and spends money to stimulate demand. A random number of customers arrive, and all buyers whose valuations exceed the prevailing price buy the product. They show that a declining price path is optimal. The key di®erence between our work and theirs is that in their model consumers do not anticipate price changes. We argue that the number of price changes are ¯nite because it is expensive for the ¯rm to change prices. Arnold and Lipmann explic- itly consider this cost and we do not. Nevertheless, in our model the seller can determine the expected revenues for di®erent number of price changes and then select the optimum number. 10 Xu(2006) considers strategic buyers who time their purchases. In his model buyersarriveataconstantdeterministicrate. Buyersfallintofourgroupsdepend- ing on their valuation (high or low) and patience level (high or low). The number of buyers in each of the four groups is known for certain. He shows that whether or not prices increase or decrease depends on how the total population is distrib- uted among these four groups. A declining price path is optimal if there are many impatient high value customers or there are many patient low value customers. Theconverseistrueintheothercases. Inourmodelpricingdecisionsbytheseller and the purchase decisions by the buyer are driven by the uncertainty in demand. Xu (2006) model is a deterministic model. Due to these di®erence, the insights obtained by him are also very di®erent from those in our work. Liu and van Ryzin(2005) also study a deterministic demand model that incor- porates customers' strategic behavior. They are concerned with rationing policies. Buyers are risk averse and decide when to buy a single unit of the product. The prices are exogenous to their model, and are known to everyone. Thus they also study a posted pricing scheme. They explore whether or not a ¯rm should ration the amount of stock available for sale at di®erent prices. We derive optimal prices under uncertainty and do not address whether or not a ¯rm should ration. We, however, show that revenues are maximized under both pricing schemes we study here, if we permit the number of price changes are allowed to grow to in¯nity. Dynamicpricingwhenconsumersarestrategicisakintoauctionswitharestric- tiononthenumberofpricechanges. Wethereforedrawfromtheauctionliterature (seeMilgrom,2004). HarrisandRaviv(1981)werethe¯rsttoidentifymechanisms that maximize seller's earnings. They showed that when the number of buyers exceeds the number of items available for sale a Vickery auction with an appro- priate reserve price is optimal. Employing the revenue equivalence principle we 11 show quite easily that if prices can be changed an in¯nite number of times then in the limit both of the dynamic pricing schemes we consider here maximize seller's expected revenues. 1.3 Renewable Assets We study a dynamic arrival control policy for a system with renewable assets in Chapter 3. Examples of such assets include telecommunication routers, rental cars,andhotelrooms. Thesystemfacesdemandfromdi®erentclassesofcustomers who di®er in their willingness to pay. Each arriving customer occupies one unit of the asset for a random duration. The arrival rate of each class depends on the prevailing price for that class, but the length of usage is not class dependent. The fee received by the system is based on the price at the time of admission and the duration of occupation. Quality of service is measured by the long term probabilitythatanarrivingcustomerisrejectedbecauseallunitsareoccupied. We are interested in maximizing the long term expected revenues subject to an upper bound on the probability of rejecting demand. The system dynamically changes pricesforeachclassbasedonthenumberofoccupiedunit. Wemodelthisproblem asamulti-serverqueuingsystemwithnowaitingspace. Inshortifweassumethat the inter-arrival times and usage duration are exponentially distributed, we are dealing with an Erlang B loss M c =M=N=0 model, where M c denotes a controlled arrival process. Henceforth we will refer to the assets as servers and the duration of usage as service time. Theproblemispartiallymotivatedbyapplicationsintelecommunication,where each customer class represents a di®erent type of data { voice, video, and text. Despite these di®erences they all use the same set of processors. Our analysis 12 also applies to rental cars and hotels. Here each class of customer is assumed to arrive through a di®erent distribution channel such as wholesalers and web basedresellers,orhavedi®erenta±liationssuchasloyaltyprograms,AAA,AARP, etc. Since we are using price as a control variable, blocking probabilities become important. We contend that when a price is announced there is an expectation in the market that the system has capacity to meet demand at that price. Thus turning arriving customers away is likely to result in a loss of goodwill for the system. In telecommunications, unavailability of switching services will result in a loss in data. For this reason in our model, we explicitly incorporate constraint on blocking probability. We address three basic questions. First, what is the structure of the opti- mal dynamic pricing policy that balances revenue generation and quality of ser- vice? Second, how does the presence of multiple customer classes impact the problem structure? In our model the system demand is regulated by changing prices. Because prices are continuous variables, our control rule is continuous and dynamic. An alternate policy is to maintain static prices but accept or reject demandfrom di®erentclasses basedon the state of the system. Wecall this policy a static binary policy. Third question of interest to us is, what are the condi- tions under which a dynamic control policy is preferable to a static control policy? Clearly the static policy is easier to implement. We begin by modeling the optimization problem as a Markov decision problem (MDP). We show the optimal policy is monotone in the number of busy servers. Thevaluefunctionisconcave,andtheaggregatearrivalrateandthecorresponding \aggregateprice"arenon-increasingandnon-decreasingfunctionsofthenumberof occupied servers, respectively. Second,we show that the dimension of the problem canbereduced. Undersomemildassumptionaboutthearrivalprocesses, weshow 13 that the multiple customer classes can be aggregated into a single class. Third, we use numerical experiments to determine whether or not it is valuable to consider dynamic control in place of static control. Quality of service drives this choice. The value of dynamic control is signi¯cant when we have a stringent constraint on quality of service. Conversely, dynamic control policy only marginally improves the system's pro¯tability when constraint on the quality of service is loose. This observation holds even when the number of servers is small; i.e. two or three. Next we develop a di®usion model to approximate the system dynamics when the number of servers (N) is large. Although the Markov decision problem can be analyzed to derive several structural properties, it poses computational problems when the number of servers is large. Also if we relax the assumption that the inter-arrival times are exponential, the Markov model is intractable. To address theselimitations, weexploreapproximationsthatarebasedondi®usionprocesses. Under this approximation scheme we can use stochastic control theory to directly derive the optimal long-run revenue rate. If we know the optimal revenue rate, derivingtheoptimalcontrolpolicybecomesaneasytask. Anotherbene¯tofusing di®usion models is that the original Markov model can be extended to incorporate general independent identically distributed arrival processes. Insection1ofchapter3, weformulatetheMarkovdecisionproblem. Asstated earlier,tofacilitateanalysisweassumethattheinter-arrivaltimesandservicetimes are exponentially distributed. In the following section we consider a sequence of models that are indexed by the number of servers N. The arrival rate is scaled such that as N grows, in the limit we achieve heavy tra±c conditions, while main- tainingaconstraintonblockingprobability. Weshowthatunderthisheavytra±c regimethenumberofbusyserversisapproximatedbyare°ectedBrownianprocess with state-dependent drift. As a result, the original Markov decision problem is 14 replacedbyadi®usioncontrolproblem. Weusecontinuous-timestochasticcontrol theory to characterize the optimality condition as a partial di®erential equation (PDE) with boundary conditions. Through numerical examples we show that the di®usion approximation provides near optimal solutions. In Section 3 of chapter 3 we discuss the implications of permitting service time requirements to depend on the customer class.In section 4 therein, we also discuss dynamic control problems on di®erentiated service types. Our work contributes to the revenue management literature. Early develop- ments in revenue management were motivated by applications in industries in which capacity is perishable. Commonly cited examples include airplane seats, cruise ships, and fashion goods. Revenue management literature also contains analysis of renewable assets such as hotel rooms and rental cars, but in these papers the product is sold once. Demand arrives over a ¯nite time horizon but the assets are sold only once. In these situations, the main challenge is to dynamically match demand and supply in real-time to maximize the terminal expected prof- its.(Gallego and van Ryzin 1994, 1997; McGill and van Ryzin 1999, Weatherford and Bodily 1992). For a complete review of the revenue management literature, readers are referred to the recent book by Talluri and van Ryzin(2004). Unlike research in this body of work, we consider an in¯nite horizon problem where the assets can be sold repeatedly. We derive stationary policies that maximize average revenues. Wealsocontributestotheliteratureoncontrolofqueueingsystems. Ourmodel is essentially an Erlang B loss model, with a control on arrivals and constraints on blocking probabilities. Bulk of the literature on queuing control studies optimal strategies for controlling servers. Stidham and Weber (1989) provide a compre- hensive survey of resource allocation problems. Readers are also refered to Gans 15 and van Ryzan (1997), Stidham (2001) and George and Harrison(2001) for more recent development. In all these papers, just as we do in the next section, the control problem is formulated as a semi-Markov decision process. However, in our workweareconcernedwitharrivalcontrolandthecongestionconcernisexpressed as an upper bound on blocking probability instead of waiting costs. Another key di®erence is that our state space is a ¯nite set because we do not permit waiting. There is a growing body of research on di®usion approximations for queueing models (Whitt, 2002). One set of papers study systems in which the number of servers is ¯xed,in most cases a single server, and the probability of the customers waiting is almost 100% . These types of models are not suitable for our work because we want to avoid stock-outs. We do not want customers to ¯nd all servers busy with probability of one. Papers by Harrison(1985,1988,1996,1997) provide a verythoroughcoverageofsingleserverheavytra±cqueueingsystems. Thesecond setofheavytra±cmodelsareduetoHal¯nandWhitt(1981). Theydevelopalim- iting regime by considering a sequence of systems in which the arrival rate and the number of servers both increase so that the system is always in one of three heavy tra±c conditions. In these systems the number of servers grows to in¯nity but the blocking probability remains constant. This limiting regime is better suited for the system we analyze. The Hal¯n-Whitt regime is for system in which arrival are not regulated. Mandelbaum and Pats (1995) develop di®usion approximations for systems with arrival control but they only consider a single server system. Deriva- tion of di®usion limits for the multiple server systems with dynamic continues arrival control is an open problem. We adopt a heuristic approach to justify the usage of the Hal¯n-Whit regime, and place more emphasis on optimal control over this process. We validate our di®usion approximation model via numerical exper- iments. Work by Harrison and Zeevi(2004) is closely related to our work. They 16 studiedadynamicschedulingprobleminthetypeofregimewestudy. Theirmajor focus is on scheduling policies. In their paper, the queuing model is an Erlang A delay model, unlike the Erlang loss model in our work. Arrivals in their paper are controlled through a binary admission policy (o® or on) while we use dynamic continuous control. 1.4 Methodology Chapter 2 utilizes game theory. It is based on game theoretic work under incom- plete information{an auction-alike analysis of pricing. In my model, where multi agents compete against each other, additional assumptions about inter-agent con- sistency are imposed: e.g., the agents hold common beliefs about market set- tings: capacity to sell and number of potential competing customers, and allo- cating scheme adopted by the principle. The collective behavior of agents is assumed to lead to a state of equilibrium in which each agent's position is opti- mal (expected-utility-maximizing) given the positions of the other agents, and the existence of such equilibria is proved via standard tools adopted in game the- ory literature.Mathematically, my ¯rst topic on dynamic pricing with customers strategic behavior is coined as a constrained dynamic programming with multiple controllers(stochasticgames). Speci¯cally, foranygivenpricingschemeproposed by the seller, customers, being sel¯sh and non-cooperative, choose the price they will purchase. Under this game, it is shown that a unique Bayesian Nash equilib- rium exists. Furthermore, we show that if the market is \thick" in the sense that both the number of units to sell and potential customers are large, by the Law of Large Number, a two-price schemeis asymptotically optimal. We also numerically 17 illustrate the e®ect of information revelation about capacity as well as number of the potential customers. Instead of a model with a static, ¯nite number of potential customers adopted in chapter 2, we model demand as a stochastic arrival process in chapter 3 when studying revenue management problems on renewable assets.We use queueing the- orytomodelsystemdynamics. Ourfocusisontheoptimalcontroloverthesystem dynamics in order to obtain the maximum long term average revenue subject to a constraint on quality of service. Theory of stochastic control is used to obtain the optimal control policy.Furthermore, depending on the size and the nature of the system, we adopt both discrete-state and an approximating continuous-state modelstherein. Fordiscretestatesystem,methodologyonstochasticdynamicpro- gramming with constraints is used. For continuous-state model, we resort to It^ o's lemma to derive a di®erential equation to characterize the optimality conditions. We use numerical experiments to show its e®ectiveness. 18 Chapter 2 Dynamic Pricing under Customers' Strategic Behavior 2.1 Model Formulation Consider a monopolist selling K perishable objects to N risk neutral consumers over a ¯nite time horizon T. Each customer wants to buy at most one unit of the product. In order to highlight the e®ect of consumer's surplus maximizing behavior, we assume the following: (1) All customers arrive at the start of the selling season. (2) Consumption takes place immediately upon purchase and there is no discounting. (3) Since the initial capacity is assumed to be exogenous, the cost of the product is zero. (4) Consumer's values are independent identically distributedandareprivate. Thesevaluesremainconstantthroughoutthehorizon. The seller does not know each consumer's valuation, but she and all the buyers know the distribution of the valuations. Without loss of generality, we normalize the support of the valuations to the range [0,1] and denote the valuation distrib- ution by G(v), for v 2 [0;1]. The seller and all the buyers at the start of selling horizon are aware of the number of units for sale (K) and the size of the mar- ket (N), number of price changes (T), and distribution of private valuations. For ease of exposition we develop our analysis assuming that the number of buyers is known. Our analysis extends to situations in which N is a random variable, provided everyone has the same prior distribution. Since all buyers are present 19 at the beginning and there is no discounting, the length of the selling horizon is irrelevant. With a bit of abuse of notation we will use the term periods to denote price changes. We begin by identify conditions under which dynamic pricing is preferred to staticpricing. We¯rststudytheroleofcapacity. Ifsupplyexceedmarketsize(K¸ N) in a posted pricing scheme, it is trivial to show that the optimal price should be a single price. In the contingent pricing scheme things appear a little more tricky because buyers see an uncertain price path and the \lowest" price may not be obvious. However, adopting the concept of rational expectation equilibrium ¯rst proposed by Stockey(1979), we show below that a single price is still optimal. In a two period model the seller's pricing decision is a vector (P 1 ;P 2 ), where P i is the price in period i, i=1;2. Note that period 2 follows period 1. In contingent pricing scheme, the seller's pricing scheme should be denoted as (P 1 ;f(P 2 )), where f(:) is the probability density function of period 2 price, and would depend on P 1 . Inrationalexpectationequilibrium,buyersandthesellersharethesameknowledge about the distribution of period 2 price. Lemma 1. If K ¸ N, for any given price P 1 , if a consumer with valuation v b ¯nds it optimal to buy in period 1, then all consumers with valuation v ¸ v b will also buy in period 1; if a consumer with valuation v b decides to postpone purchase to period 2, all consumers with valuation v < v b will also postpone purchases to period 2 even if their valuation exceeds the current price. The following proposition establishes the fact that no matter what pricing scheme is adopted, a single price is optimal when supply exceeds demand. Proposition 1. Let ¼ r be the optimal revenue for the seller under the pricing scheme (P 1 ;f(P 2 )) , and ¼ s be the optimal revenue for the seller in the single price scheme. If K¸N then E(¼ r )·E(¼ s ). 20 We see that dynamic pricing is not useful when there is no shortage. The consumers' ability to time the purchase eliminates the possible bene¯ts of inter- temporal price discriminating by the seller. Therefore, in the remainder of this chapter 2, we restrict ourselves to the case when supply is limited (K < N). Since the number of price changes is ¯nite, it is conceivable that at some price the number of o®ers received by the seller exceeds the number of units for sale. Since the buyer can not rank the buyers on their valuations we assume that a proportional rationing mechanism is employed to allocate the objects. We next study the role of demand uncertainty. Uncertainty in demand can be duetouncertaintyinvaluationsanduncertaintyinthenumberofbuyers. Herewe assume the number of buyers is known. We investigate whether dynamic pricing is better than static pricing when demand is deterministic. The following example provides us some insights. Example 1. Suppose a seller has 2 units of product to sell, and there are 10 buyers. The valuation vector is: (100,40,35,30,28,26,25,23,21,20). It is trivial to show that the optimal single price for the seller is 100, with realized sale of 1. The total revenue would be 100. The clearing price of 40 yields a revenue of 40£2 = 80. Alternatively, the seller can also pre-announce prices (82,20). The consumer with valuation 100 can get a surplus of 100-82=18 for sure in period 1, or get an expected surplus of (100¡ 20)£ 2¥ 10 = 16 in period 2, provided a proportional rationing mechanism is used. Obviously, the ¯rst unit will be sold at price 82 to the customer with valuation 100. As a result, the total revenue of the seller will be 82+20, which is larger than 100. We conclude that for this particular case, a dynamic price-path (82,20) is superior to the optimal single price policy. This example at least shows that a single price policy may not be optimal. 21 Since a single pricing policy is not optimal in this deterministic demand case, what is the structure of optimal pricing policy? How many price changes are needed? We answer this question in the following lemma, which is interesting in its own right. Lemma 2. If demand is deterministic and demand exceeds supply, the optimal pricing scheme has no more than one price change. Based on these ¯ndings, in the remainder of this chapter, we assume that K <N and demand is uncertain. 2.2 Equilibrium Buying Behavior We ¯nd that in both posted and contingent pricing schemes, whether or not the number of buyers is certain or uncertain the equilibrium strategy is a threshold policy. At any price P all buyers with valuations that exceed a threshold will bid for the product. This threshold is almost always greater than the current price. 2.2.1 Posted Pricing Scheme In the posted pricing scheme, the seller announces and commits to a price path (P 1 ;P 2 ;::P T ). Clearly P 1 ¸ P 2 ¸ ::P T¡1 ¸ P T . Let us begin with a two period problem. We show next that the equilibrium strategy is for all buyers with valua- tion above a threshold (y) to purchase in the ¯rst period. Let: i;j : Number of bidders in period 1 and period 2, respectively; Pr 1 (i) : Probability that i¡out¡of¡(N¡1) consumers bid in period 1; 22 Pr 2 (jji) : Probability that j¡out¡of¡(N¡1¡i) consumers bid in period 2 given that i consumers bid in period 1; ¼ 1 (y) : Probability of the N th buyer getting the product in period 1; and ¼ 2 (y) : Probability of the N th getting the product in period 2. Proposition 2. For any set of prices (P 1 ;P 2 ), the following characterizes the unique Bayesian Nash Equilibrium: Let y ¤ be the smallest solution for equation(2.1) in the range [P 1 ;1]. All buyers withvaluationsv2[y ¤ ;1] will bid in period1and buyers with valuation v2[P 2 ;y ¤ ) bid in period 2. If equation(2.1) has no solution within the range [P 1 ;1] , no one will bid for the product in period 1 and those with valuations in the range [P 2 ;1] will bid in period 2. ¼ 1 (y)(y¡P 1 )=¼ 2 (y)(y¡P 2 ); (2.1) where: ¼ 1 (y) = k¡1 X i=0 Pr 1 (i)+ N¡1 X i=k k i+1 Pr 1 (i) ¼ 2 (y) = E I;J [min( k¡i j+1 ;1)]= k¡1 X i=0 Pr 1 (i) j=N¡1¡i X j=0 Pr 2 (jji)min( K¡i j+1 ;1) 23 Pr 1 (i) = µ N¡1 i ¶ (1¡G(y)) i G(y) (N¡1¡i) for i=0;1;:::::N¡1 Pr 2 (jji) = µ N¡1¡i j ¶ (1¡ P 2 G(y) ) j ( P 2 G(y) ) (N¡1¡i¡j) for i=0;::K¡1; j =0;::(N¡1¡i) Pr 2 (jji) = 0; for i=K;:::N¡1; j =0;1::(N¡1¡i) (2.2) The fact that only buyers whose valuations are strictly greater than P 1 are willing to purchase at the higher price is one of the consequences of strategic behavior. This clearly decreases the demand at the higher price and the expected revenues. The di®erence between the threshold level y and P 1 depends on the di®erencebetweenP 1 andP 2 . Fora givenP 2 as weincreaseP 1 , y keepsincreasing. If the gap between P 1 and P 2 increases beyond a critical point, no one will o®er to buyatthehigherprice. Thus,asecondmajorconsequenceofstrategicbehaviorisa compressionintheprices. Atoptimality,thegapbetweenP 1 andP 2 willbesmaller relative to what it would be if buyers were not strategic. Table 2.1 illustrates this point. In this example the valuations of all the buyers are uniformly distributed between 0 and 1. N K Pricing for Myopic Consumers Pricing for Strategic Consumers 10 2 0.85,0.64 0.76,0.64 20 4 0.86,0.68 0.78,0.68 30 6 0.86,0.70 0.80,0.71 40 8 0.86,0.71 0.78,0.68 50 10 0.87,0.72 0.77,0.65 Table 2.1: Comparison of Optimal Prices, Buyers Distribution are U(0,1) Aconsumerwithvaluationv willexpectasurplusof(v¡P 1 )¼ 1 (y)ifhedecides to purchase in period 1 and a surplus of (v¡P 2 )¼ 2 (y) if he decides to wait till period 2. In both periods the probability of getting a unit of product depends on 24 theinitialcapacitylevelK, totalpopulationN, aswellasthedecisionsofallother consumers. Buyers with valuation su±ciently above P 1 prefer to buy in period 1 and reduce the likelihood of not getting the product. The threshold policy derived above extends to a general T period problem. Proposition3. For a T-period pricing scheme (P 1 ;:::;P T ), in each period t, only buyers with valuation greater than or equal to a threshold y ¤ t (k t ) will bid for the product. The threshold depends on the number of units (k t ) available for sale. For 1·t<T, y ¤ t (k t )¸P t and for t=T, y ¤ T (k T )=P T . Proofs of these propositions hinge on the fact that regardless of the strategy employed by other buyers, for any given buyer the expected probability of getting the product does not increase as the prices drop. This remains true even if N is a random variable. Hence a threshold policy remains an equilibrium strategy when N is a random variable. Proposition4. Let N be a random variable with probability density function f(:). This is distribution is common knowledge among all the buyers and the seller. For aT-periodpricingscheme(P 1 ;:::;P T ), ineachperiodt, onlybuyerswithvaluation greater than or equal to a threshold y ¤ t (k t ) will bid for the product. The threshold depends on the number of units (k t ) available for sale. For 1·t<T, y ¤ t (k t )¸P t and for t=T, y ¤ T (k T )=P T . 2.2.2 Contingent Pricing In the contingent pricing scheme the seller determines the price based on the inventory on hand and does not commit to a price path. The buyers anticipate pricesbasedonrealizedsales. Despitethedi®erencebetweencontingentandposted pricing schemes, the structure of the equilibrium remains the same. For brevity 25 we derive the equilibrium for a two price setting. The analysis can be extended to multiple prices and uncertain N in a manner similar to that for the posted pricing scheme. Let P 1 be the price in period 1, and: P 2i : the optimal price to be charged in period 2 when i¡out¡of¡(N¡1) other buyers bid in period 1;i.e. P 2i = fp:max p p[ K¡i X j=0 µ N¡i j ¶ (1¡F 1 (p)) j F 1 (p) (N¡i¡j) j+ N¡i X j=K¡i+1 µ N¡i j ¶ )(1¡F 1 (p)) j F 1 (p) (N¡i¡j) (K¡i)]g (2.3) ¯ i : theprobabilitythattheN th buyerwillgettheproductifi¡out¡of¡N¡1 other buyers bid in period 1 and and he decides to bid in period 2 , i.e. ¯ i = E j [min( k¡i j+1 ;1)]= N¡1¡i X j=0 Pr 2 (jji)min( K¡i j+1 ;1) for i=0;:::;K¡1 ¯ i = 0 for any i=K;:::N¡1 Proposition5. For any prices P 1 the following characterizes the unique Bayesian Nash Equilibrium. Let y ¤ be the smallest solution for equation (2.4) in the range [P 1 ;1]. All buyers with valuations v2 [y ¤ ;1] will bid in period 1. If equation(2.4) has no solution within the range [P 1 ;1] , no one will bid for the product in period 1. In period 2 all those whose valuations exceed P 2 but are less than y ¤ will bid. P 2 is contingent on sales at price P 1 and solves (2.3). ¼ 1 (y ¤ )(y¡P 1 )= k¡1 X i=0 Pr 1 (i)¯ i (y¡P ¤ 2i ) (2.4) 26 2.3 Some Properties of the Pricing Schemes One of our goals is to understand types of pricing policies that ¯rms can employ when consumers anticipate prices. In particular we want to evaluate the e®ec- tiveness of a posted pricing scheme that involves one or two price changes. More generally we would like to shed light on the following questions: 1. What is the loss in revenues if a ¯rm ignores strategic behavior? 2. Since scarcity drives dynamic pricing, what is the relationship between scarcity and value of dynamic pricing? 3. What is the optimum stocking level when buyers are strategic? 4. How many price changes are needed? 5. Howe®ectiveisthepostedpricingschemecomparedtothecontingentpricing scheme? 6. Howdoestheuncertaintyinthesizeofthemarketin°uencetheperformance of di®erent pricing schemes? 7. What is the value of withholding inventory information from the buyers? Determiningoptimalpostedpricesentailssolvingnon-linearoptimizationprob- lems that involve polynomial equations of high order. Unfortunately, this makes it di±culttogainanalyticinsightsintomanyofthesequestions. Asaresultwehave toresorttonumericalexperiments, andwedosointhenextsection. Nevertheless, there are a handful of analytic insights that we are able to elicit. We begin with asymptotic properties of these policies. We explore the structure of the policy when the number of price changes T approaches 1, or the number of players in the market N approaches 1 while holding the ratio K=N constant. Asymptotic 27 analysis, besides being of independent interest, provide us with an upper bound on expected revenues. This in turn allows us to evaluate di®erent pricing schemes. We ¯nd that the expected revenues are concave increasing in the initial stocking levels and provide a bound on the maximum stock level for large markets (N). This section also contains su±cient conditions under which the ¯rm is better o® concealing inventory levels. 2.3.1 Asymptotic Analysis AsthenumberofpricechangesT approaches1thepostedandcontingentpricing schemes resemble auctions. Accordingly we draw on ¯ndings in auction theory to establish limiting properties. Based on work by Milgrom and Weber(2000) it is fairly straight-forward to show a ¯rm maximizes its revenues if it uses a ¯rst price simultaneous auction with a reserve price 1 . The posted and contingent pricing schemes also have the objective of maximizing ¯rm's revenues and we are able to show that both these schemes are revenue equivalent to the optimal mechanism. Further, the lowest price charged in the limit in both schemes will be strictly greater than zero and will correspond to the reservation price in the ¯rst price auction. These results are formally presented below and for brevity we focus only on the posted pricing scheme. Let J(v i )=[v i ¡ 1¡G i (v i ) g i (v i ) ], V : vector of valuations of all N buyers V ¡i : vector of valuations of N¡1 buyers other than i, V j ¡i : j th largest value in the vector V ¡i , and 1 This is one of several allocation mechanisms that maximize revenues. 28 y i (V)=Max(J ¡1 (0);V K ¡i ) Proposition 6. If distribution G(.) is such that J(.) is non-decreasing, then in a posted pricing scheme, as the number of price changes T !1: 1. the lowest posted price P T ! v ¤ , where v ¤ =fv :J(v)=0;for v2(0;1) g, 2. the buyers with K highest valuations will get the product, provided their val- uations are greater than v ¤ , and 3. the seller expected revenues are maximized Let Q(v i ;V ¡i ) denote the probability that the i th buyer gets the product when his valuations is v i and the valuation of the others is V ¡i . Set Q(v i ;V ¡i ) = 1 if v i >y i (v i ;V ¡i ) and 0 otherwise. In the limit the seller expected revenues are given by: X i2N E V [J(v i )Q i (v i ;V ¡i )] (2.5) Proposition 6 provides us with an upper bound on the expected revenues. We will use this upper bound for numerically evaluating pricing schemes. Wehaveseeninsection2.1thatwhendemandisdeterministicweneedatmost two prices. In the limit as N becomes large we would intuitively expect, due to law of large numbers, the e®ect of uncertainty to progressively diminish. Thus it is useful to understand the impact of the size of the market on the pricing policies. We show in the following proposition, that in the limit as N approaches 1 and K=N is held constant, we need at most two prices. This is the same result we had when there was no uncertainty in valuations. Proposition 7. For any ¯xed ratio K=N, as N approaches1 the optimal pricing scheme involves at most two prices. 29 In the next section we use numerical experiments to understand how rapidly a two price scheme converges to the optimal value as the market size grows. 2.3.2 Optimal Stocking Levels We have seen in section 2.3.1 that as the ratio K N approaches 0 or 1, a single price is adequate. In our numerical experiments we ¯nd that the performance of the pricing schemes depends on this ratio. In this context the optimal stocking ratios becomerelevant. Aretheoptimalstockinglevelssuchthatonlyafewpricechanges result in near optimal revenues? We can show that optimal revenues are concave increasing in the number of units K available for sale. We also ¯nd that as the number of buyers in the market N grows large the ¯rm ¯nds it optimal to stock no more than the fraction of the market that can a®ord the optimal reservation price. Proposition 8. The optimal expected revenues are concave increasing in K for ¯xed N. Proposition 9. As N approaches 1, the optimal stocking level approaches N ¤ (1¡G(v ¤ )), where v ¤ =fv :[v i ¡ 1¡G i (v i ) g i (v i ) ]=0;for v2(0;1) g 2.3.3 Value of Inventory Information Shouldthesellerrevealthenumberofunitsshehasforsale? Shouldthesellerreveal the number of units for sale and then conceal the actual sales information? If the seller chooses not to reveal the number of units available for sale then she forgoes the ability to optimally price based on available stock. Thus hiding initial sales quantitymaynotbene¯ttheseller. Ontheotherhandinapostedpricingscheme, concealing number of units sold at di®erent price impacts the seller and buyer in 30 di®erent ways. Under this scheme the seller is setting prices based on anticipated sales. By concealing sales information the seller is forcing the buyers to work with the same limited information set that the buyer used to determine prices. We conjecture that it always bene¯cial for the seller to conceal sales information. Howeverweareunabletoestablishthisanalytically. Wecanonlyprovidesu±cient conditions under which hiding sales information bene¯ts a seller. When buyers know the initial quantity for sale but do not know the number of units sold then it also means that once the selling period begins buyers do not know the number of other buyers in the market. We concede that it is awkward to assumethatbuyerscanobservethenumberinthemarketatthebeginningbutnot subsequently. It may be more appropriate to assume that the number of buyers is also uncertain to start with. We leave this analysis for future work. The question of whether or not a ¯rm should conceal sales data is equivalent to studying the impact on pro¯ts if prices are ¯xed but the number of units for sales is a random variable. For simplicity, and without loss of generality, let us suppose that inventory levels are believed to be either high or low. Probability of inventories being high will increase the propensity to wait for lower prices and the possibility of the inventories being low will cause buyers to bid at higher prices. Since we are considering situations in which initial sales quantities (K), market size (N) and distribution of the valuations(G(:)) are known, beliefs about number of unsold units should be unbiased and well calibrated. Below we provide su±cient conditions under which concealing inventories increases expected revenues. Let ¦(K;N;P 1 ;P 2 ;y) be expected pro¯ts for the ¯rm when the threshold is y and the number units for sale is known to be K. 31 Proposition 10. Let us suppose that the buyers believe that with probability ® the inventory level is K 1 and with probability 1¡® the inventory level is K 2 . For these beliefs and posted prices P 1 ;P 2 let the threshold value be given by y(®). If y(®) is convex and ¦(K;N;P 1 ;P 2 ;y) is concave in y, then a ¯rm's expected pro¯ts are higher if it does not reveal inventory levels. Properties of two factors y(:) and ¦(:) are central to proposition 10. Inter- estingly, y(:) is related to buyers' behavior and ¦(:) depends on the valuation distribution G(:). As ® increases the likelihood of the inventory being high increases. Conse- quentlyas®increases,foragivensetofpricesthethresholdlevely(®)willincrease. If y(®) is convex then it means that the threshold levels will decrease rapidly as ® drops below one, on the other hand as ® increases from zero the threshold level rises very slowly. Convexity of y(:) implies that for most values of ® the threshold level is "low". This in turn means that buyers bid more aggressively, or are averse to stock-outs. 2.4 Numerical Experiments We begin this section with examples that measure the cost of ignoring strategic behavior. Next, we investigate the loss in optimality due to limiting the number of price changes. We know from proposition 6 that as the number of price changes increases the performance of both the posted and contingent pricing scheme will improve, and in the limit attain optimality. This proposition also gives us the upper bound that will be used to determine the loss in optimality from limited price changes. 32 Thereareseveralfactorsthatin°uencetheperformanceofthepricingschemes. Two of them are market size and scarcity levels (K=N). We therefore vary these twoparameters. Thethirdfactorthathasabearingonrevenuesisthedistribution of customer valuations (G(:)). We consider three di®erent valuation distributions: Uniform,Beta(8,2),andBeta(2,8). Beta(8,2)isskewedtotherightandunderthis distribution most of the buyers have high valuations. B(2,8) is skewed to the left. The fourth factor is uncertainty in the market size. Intuitively we would expect the performance of a limited price change scheme to deteriorate if we introduce uncertainty in the number of buyers. Finally we present examples that explores whether or not the ¯rm should con- ceal inventory levels. We have seen in the previous section that there are two di®erent approaches to concealing inventory information. It can reveal the num- berofunitsforsaleatthebeginningofthesellinghorizonandsubsequentlyconceal sales information. Alternate approach is to hide the number of units for sale at the beginning but reveal sales information. For uniformly distributed valuations we study these two situations. Figure 2.1 shows the consequences of ignoring strategic behavior. A ¯rm that ignores strategic behavior will set prices assuming that a buyer will purchase if his or her valuation exceeds the current price. We refer to prices set under this assumption as naive prices. Figure 2.1 shows the percentage drop in expected revenues if a seller employs naive prices instead of the optimal posted prices that consider strategic behavior. Here valuations are assumed to be distributed uni- formly between 0 and 1. For this distribution, the percentage loss in revenues varies between 5.5% and 13%. As the market size N increases the percentage loss decreases. For higher K=N ratios the loss in expected revenues also appears to be lower. 33 10 15 20 25 30 35 40 45 50 5 6 7 8 9 10 11 12 13 Number of Buyers % Decrease Comparison of Revenue from Optimal 2 Prices and Naive 2 Prices for Differen K/N Ratios K/N=10% K/N=20% K/N=50% Figure 2.1: Drop in expected revenues when seller uses two posted prices without considering strategic behavior InFigure2.2{2.6wegraphthe%lossinoptimalitywhena¯rmusestwoorthree posted prices. We compare the expected revenue from the posted price scheme withthemaximumpossiblerevenues(upperbound). Foruniformdistributionand 3 prices (2.3 the loss is no more than 3%. When K=N ratio is close to 50% the percentagelossisgenerallyunder1%for3prices. Figures2.4through2.6showthe e®ect of the valuation distribution (G(:)). It is interesting to compare ¯gures 2.5 and 2.6. For Beta(8,2), that is right skewed we ¯nd that we are close to the upper boundwithjusttwoprices. Ontheotherhand, whenvaluationsareBeta(2,8)and bulk of the buyers have lower valuations, the performance of a two price scheme is considerably worse. In this case it appears that you need more price changes. On the positive side, for all three distributions for N greater than 30, and stocking levels of 50%, it appears that 2 posted prices are adequate. The stocking decision will of course depend on the product cost. Table 2.2 below contains the optimal stocking levels for di®erent costs, when N = 20 and the valuations are U(0,1). 34 10 15 20 25 30 35 40 45 50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Number of Buyers % Decrease 2 Posted Prices, Uniform Distribution K/N=10% K/N=20% K/N=50% Figure 2.2: % loss in optimality: 2 posted prices and uniform distribution 10 15 20 25 30 35 40 45 50 0 0.5 1 1.5 2 2.5 Number of Buyers % Decrease 3 Posted Prices, Uniform Distribution K/N=10% K/N=20% K/N=50% Figure 2.3: % loss in optimality, 3 posted prices and uniform distribution Cost Optimal K/N 0.1 50% 0.2 40% 0.3 35% 0.4 30% 0.5 25% Table 2.2: N=20,Valuation U(0,1) 35 10 15 20 25 30 35 40 45 50 0 1 2 3 4 5 6 7 8 Number of Buyers % Decrease 2 Prices Beta(2,8) K/N=10% K/N=20% K/N=50% Figure 2.4: % loss in optimality, 2 posted prices, Beta(2,8):left skewed 10 15 20 25 30 35 40 45 50 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Number of Buyers % Decrease 3 Prices Beta(2,8) K/N=10% K/N=20% K/N=50% Figure 2.5: % loss in optimality, 3 posted prices, Beta(2,8):left skewed We know from proposition 9 that for large N the stocking level depends on the reservation price level v ¤ . When N is large even if the cost of the product is zero, the ¯rm will not stock much more than N ¤(1¡G(v ¤ )), and the corresponding K=N ratio is 1¡G(v ¤ ). For U(0;1) this limit is 50%, for Beta(8,2) (right skewed) the ratio is 13.5% and for Beta(2,8) (left skewed) the ratio is 92%. This suggests 36 10 15 20 25 30 35 40 45 50 0 0.5 1 1.5 2 2.5 3 Number of Buyers % Decrease 2 Prices Beta(8,2) K/N=10% K/N=20% K/N=50% Figure 2.6: % loss in optimality, 2 posted prices Beta(8,2):right skewed that when the distribution is left skewed the stocking levels are going to be higher, decreasing the loss in optimality from limited number of price changes. Letusnowcomparepostedpricingschemetocontingentpricingscheme. Ifbuy- ers are not strategic then clearly the contingent pricing scheme dominates posted pricing scheme. This need not be true when buyers are strategic. A seller who adopts a posted pricing forgoes some °exibility. This commitment however, also eliminates options for the buyer. Thus on balance it is di±cult to predict the impact on expected revenues. Our computational experiments suggest that nei- therproceduredominates. Figure2.7comparesthe%di®erenceinexpectedpro¯ts betweenpostedandcontingentpricingschemes 2 forN =20. Bothschemesemploy 2 prices. The di®erence between the two is small. The maximum di®erence we found is 1.6%. This di®erence will decrease further if either N or the number of pricechangesincreases. Thissuggeststhatpostedpricingschememaybeadequate if the only uncertainty is due to buyer valuations. 2 The ¯gure plots (posted¡contingent) posted against K 37 0 5 10 15 20 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 K (posted−contingent)/posted% Contingent vs Posted Pricing: 2 Prices U(0,1) Beta(2,8):Left Skewed Beta(8,2):Right Skewed Figure 2.7: Comparison of Posted and Contingent Pricing Schemes, Fixed N Fortheselleristheregreatervaluetothe optionof delayingpricingdecisionsif there is uncertainty in the size of the market? Figure 2.8 sheds some light on this question. Here weassume that N is either small(7) or large (20) with probabilities ® and 1¡®, respectively. The number of units for sale is ¯xed at 5. Both schemes are within one or two percent of each other and the relative performance seems to dependon expected valueofthe scarcityratio (K=N). For higher levels ofscarcity or lower values of K=N contingent pricing appears to be marginally better than postedpricingscheme. Thislittleexampledoesnot,however,presentacompelling reasontoabandonpostedpricingschemeinfavorofthecontingentpricingscheme. The last two examples explore the role of inventory information. We begin with the question of whether or not a ¯rm should conceal the number of units available for sale at the start of the season. We assume that the valuations are U(0,1) and there are 20 buyers. They all believe that the stock level is either 5 or 15 with probabilities ® and 1¡®, respectively. These beliefs are unbiased and well calibrated. The ¯rm however knows the actual stock level. In this situation 38 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Alpha (posted−contingent)/posted% Contingent vs Posted Pricing: 2 Prices U(0,1) Beta(2,8):Left Skewed Beta(8,2):Right Skewed Figure 2.8: Comparison of Posted and Contingent Pricing Scheme, Uncertain N 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Alpha (Revealed−Hidden)/Revealed% Value of Revealing Initial Sales Quantity Figure 2.9: Value of Revealing Initial Sales Quantity, N=20, Valuations U(0,1) the ¯rm can either price based on the actual stock level or act as if it too does not know the true stock level but holds the same beliefs as the buyers. Once the prices are announced by the ¯rm buyers can discern the approach adopted by the ¯rm. Figure 2.9 shows the change in expected revenues if the ¯rm bases its prices on actual stock levels instead of treating stock levels as random variables. In this example the ¯rm is marginally better o® if it reveals the stock levels. 39 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 Alpha (Hidden−Revealed)/Hidden % Value of Concealing Sales Information Figure 2.10: Value of Concealing Sales Information. Valuations: U(0,1) In the last example we consider the question of whether the ¯rm should hide salesinformation. Sincethisisanintermediarystageinthesellingprocess, andwe are assuming posted prices, the ¯rm can not change price based on the inventory level. We therefore ¯x prices and compute expected pro¯ts for di®erent estimates of the number of units available for sale. Here N =12 and K =2 with probability ® and N = 10 and K = 4 with probability 1¡®. The valuations are assumed to be U(0,1). In this case (¯gure 2.10) we ¯nd that it is bene¯cial for the ¯rm to hide sales information. Overall the value of inventory information does not seem signi¯cant. 40 Chapter 3 Revenue Management When Assets Are Renewable 3.1 Markov Model We begin by modeling our problem as a birth and death continuous time Markov chains (CTMC) with state space S = (S(t) : t ¸ 0 on Z + = f0;:::;Ng), the numberofoccupiedservers. ThereareN identicalserversworkinginparallel. The system faces demand from M customer classes that di®er in their arrival(demand) rate functions. The service times do not depend on the customer class, and are assumedtobeexponentiallydistributedwithmean1=¹. Notethattheassumption of identical service requirement across all customer classes allows us to have a one- dimensional state variable. This assumption is made only for sake of tractability and is otherwise innocuous. The arrival processes are all Poisson processes with rates¸ i (i=0;:::;M)thatareafunctionsofthecorrespondingpricesP i . Forease ofexposition,weassumethatthearrivalprocessareindependent. Thisassumption is not crucial and we show how it can be relaxed. Prices quoted at any point in time depend on the number of occupied servers and are used to control the arrival streams. We assume that the inverse demand functions exist and treat arrival rates ¸ i as the control variables. In our system when all servers are occupied arriving customers are turned away. To account for the loss in goodwill that results from customer who are denied service we impose 41 an upper bound ± on the blocking probability. Later in this section we show how this constraint can be transformed into an endogenously derived penalty cost that is charged whenever the system is fully occupied. This transformation consid- erably simpli¯es analysis, both in the Markov formulation and in the di®usion approximation based models. We make the following additional assumptions: (A.1) For each i = 1;:::;M, ¸ i (p i ) is strictly decreasing in p i . The prices are all non-negative and there exists a ¯nite price ¹ p such that the arrival rate ¸ i (¹ p)=0. (A.2) ThereexistsacontinuousinversedemandfunctionP(¸):A!Pthatmaps the vector of demand rates ¸ = (¸ 1 ;:::;¸ m ) into a corresponding vector of prices P(¸)=(P 1 (¸);:::;P N (¸)). Moreover, !(¸ i ):= M P i=1 ¸ i P i (¸ i ) is concave in ¸ i for any i=1::::;M. The¯rmcontrolsthethevectorofinstantaneousarrivalratesbypostingprices, so as to maximize the long-run average pro¯ts. When s = N, the ¯rm will post a price ¹ P to block all customer classes. We restrict ourselves to work-conserving, stationary policies U. The policies are such that when the system state is in s, the total arrival rate is ¸(s)¸0 , for 0·s<N. Since the state space is ¯nite, under any policy u 2 U there exists an ergodic distribution ®(u) = (® 0 (u);:::;® N (u)) (Cooper, 1981). Here ® i (u) is the ergodic probability of the state i=0;:::;N. The long-run average pro¯t rate associated with the admission policy u is as follows. Z u = M X i=1 N¡1 X n=0 ® n (u)P i (¸ in )¸ in =¹ subject to ® N (u)·± 42 Let Z ¤ =supfZ u :u2Ug An admissible policy u is optimal if Z u =Z ¤ . 3.1.1 A Dual Formulation Note that the above formulation is a chance constrained stochastic program. Although it is natural, it is not amenable to exact analysis. Using theory of constrained MDPs (Altman 1998), we translate this formulation into one with a penaltycost. Inthetransformedproblem,apenaltyrateofF isimposedwhenever all servers are occupied. We show next that there is a one-to-one correspondence between these two formulations. For each ± there exists a unique F, such that the solutions of both problems are identical. To derive this result we ¯rst analyze the model with penalty costs and then delve into its one-to-one correspondence with the original model. The penalty cost based model is given below. Z ¤ =max u M X i=1 N¡1 X n=0 ® n (u)P i (¸ in )¸ in =¹¡® N (u)F Optimality Equation To derive the optimal policy, we starts with the Bellman equation for this semi- Markov decision process. In this problem it is straight-forward to show that a new control is needed only when a new customer arrives or an existing customer completes his or her service, but not at any other time. By employing the uni- formizationtechnique(seepage254-261ofBertsekas2001),weobtainthefollowing set of equations. The normalization is such that M P i=1 ¸ i +¹N =1. 43 v 0 = max ¸ [ M X i=1 P i (¸ i ) ¸ i ¹ ¡° ¤ + M X i=1 ¸ i v 1 ]+(1¡ M X i=1 ¸ i )v 0 (3.1) v s = max ¸ [ M X i=1 P i (¸ i ) ¸ i ¹ ¡° ¤ + M X i=1 ¸ i v s+1 ]+¹sv s¡1 +(1¡ M X i=1 ¸ i ¡¹s)v s for all s such that 1·s<N, (3.2) v N = (0¡F ¡° ¤ )+0+¹Nv N¡1 +(1¡¹N)v N when s=N (3.3) Thevectorv =(v 0 ;:::;v N )istherelativecostfunctionand° ¤ isthemaximum long-run average pro¯t (Bertsekas, 2001). Let the aggregate revenue rate be as follows: !(¸):= M X i=1 P i (¸ i ) ¸ i ¹ (3.4) State Space Reduction Aggregaterevenuerateissuchthatexpectedrevenuesfromanadmittedconsumer are independent of future system dynamics. As is conventionally done for MDPs with long run reward criteria, we de¯ne relative pro¯t di®erence as follows: h(s)=v(s)¡v(s+1); for s=0;1;:::;N¡1 (3.5) Then, the (3.1{3.3) can be re-written as follows: ° ¤ = max ¸ [!(¸)¡ M X i=1 ¸ i h(0)] (3.6) ° ¤ = max ¸ [!(¸)¡ M X i=1 ¸ i h(s)]+¹sh(s¡1) for s such that 0 <s<N (3.7) ° ¤ = ¹Nh(N¡1)¡F when s=N (3.8) 44 Due to (3.6{3.8), choosing the optimal arrival rates reduces to the following: ¸(s) ¤ =argmax ¸ [!(¸)¡ M X i=1 ¸ i h(s)] (3.9) Let q(¸) := M P i=1 ¸ i . Then given a vector of arrival rates ¸ = (¸ 1 ;:::;¸ M ), the quantity q(¸) represents the total arrival rate. The maximum revenue achievable from a given total arrival rate q(¸) is: ¼(q)=max ¸ f!(¸):q(¸)=q;¸2Z M g for q2R + (3.10) Note that the derived arrival rates ¸ for any given q in (3.10) represents an e±cient frontier. To determine the optimal solution we can restrict ourselves to this e±cient frontier 1 . Again, ¸=0 when the state s=N. Hence forth, we denote q and ¼(q) as the aggregate arrival rate and corre- sponding revenue rate, respectively. Once an optimal aggregate rate is obtained, the optimal price vector and the optimal arrival rate vector for all classes are readily obtained be solving the optimization problem (3.10). Properties of Optimal Policy For later use, we ¯rst show the following lemma 3. Lemma 3. ¼(q) is concave in q for any q2R + This is a direct consequence of assumption (A.2). Proof is skipped. 1 This approach parallels the one adopted by Eberly & Van Mieghem(1997) to analyze mean- variance analysis of portfolios. Van Ryzin (2003) also uses this techinque. 45 Thus,theaboveBellmanequation(3.1)to(3.3)canbereducedtothefollowing problem with one dimensional control and one dimensional state variable. ° ¤ = max q f¼(q)¡qh 0 g (3.11) ¹sh s¡1 = ° ¤ ¡max q¸0 f¼(q)¡qh s g for s such that 0<s<N (3.12) h N¡1 = 1 ¹N (° ¤ +F) when s=N (3.13) We summarize the results above in the following propositions and lemma. Proposition 11. The Bellman equation (3.1{3.3) can be reduced to (3.11{3.13). For later use, we now show some results associated with the concavity of ¼(q) if !(¸) is concave in ¸. Lemma 4. De¯ne G(x) = max q¸0 f¼(q)¡qxg,for q ¸ 0 , and x ¸ 0. We have that G(x) is non-increasing and convex in x. Proof can be found in appendix. BecausewehaveconvexityofG(x), theoptimizerin(3.11)and(3.12)isunique and has the following speci¯cs q ¤ (s)=max[0;¼ 0 (h s ) ¡1 ] (3.14) Proposition12. The relative value di®erence function h(s) in nonnegative and is non-decreasinginthestatevariables, i,e, 0·h(s)·h(s+1), fors=0;1;:::;N¡ 1. Equivalently, the value function v(s) is non-increasing and concave in s. See appendix for proof. We conclude this section by the following Theorem 1: 46 Theorem 1. The optimal total arrival rates q(s) is an non-increasing function of state s. That is, the more the occupied servers, the lower the optimal aggregate arrival rates would be. The proof is a direct consequence of proposition 12 and (3.14). Duality Relationship To show the duality relationship between original model and the transformed model,we start with the following proposition.. Proposition13. The optimal average pro¯t °(F) as a function of the penalty cost F is non-increasing. Moreover,the optimal arrival rate q s (F) is non-increasing function of F. The following proposition 14 shows an one-to-one correspondence of a penalty cost F and the blocking probability ®(u);u=(q 0 ;q 1 ;:::;q N¡1 ). Proposition 14. The blocking probability under the optimal arrival rate (q 0 ;q 1 ;:::;q N¡1 ), as a function of penalty F is non-increasing. Duetothemonotonicrelationshipbetweenthepenaltycostandblockingprob- ability, we can think of our original model in this way: for any given constraint on blockingprobability±,wecan¯ndapenaltycostF(±)thatisuniquelydetermined. For a more stringent requirement on blocking probability, an greater dual penalty cost F is expected. The advantage of adopting penalty cost rather than blocking probabilityhas beenrealized in theabovesection. Wewillalso see its bene¯tlater when we approach large size problem using di®usion control approximation. 47 3.1.2 Solving the optimality equation Having derived the equations (3.11{3.13), we can compute the optimal long run average pro¯t ° ¤ , and the corresponding relative di®erence value function h s ;(s= 0;:::;N¡1) in the following way: for any given positive °, we start from (3.13). By plugging (3.13) into right hand side of (3.12), we obtain h N¡1 . And by doing it recursively, up to (3.11), we would ¯nally obtain an expression, say ° =g(°;F). In most cases, obtaining a close-form solution to ° = g(°;F) is complicated, if possible, foragenericdemandfunction. Computationally, wecanhaveasearching for the optimal ° within some range. Technically, we can restrict ourselves by setting bounds on °. The most natural bound is easily identi¯ed as ° 2 (0;¹ °), where ¹ ° =sup ¸ !(¸) de¯ned in (3.4). As a ¯nal step, we need to use (3.10) to give an implementable control policy. We can numerically verify the monotonicity property about optimal arrival rates as function of state variables as established in theorem 1. Here we give a toy example in the following. Example 2. Figure 3.1 illustrates the optimal solutions for dynamic arrival con- trol under a range of demand function characterized by the number of servers N. Speci¯cally, we assume the demand function is p=10¡ 5q N , where p and q denotes an non-negative price and arrival rate, respectively. We allow N =2;:::;12. Note that for these demand curves, the optimal arrival rates without congestion concerns are simply q ¤ = N. We arbitrarily choose F = 3. By employing these demand curves, we emphasize the situations of constrained capacity. For each given N, we observe that the optimal arrival rate and the value di®erence function h(:) de¯ned above is decreasing and increasing function of the state variables, respectively. 48 0 2 4 6 8 10 0 2 4 6 8 10 12 number of servers optimal arrival rates and the associated value difference numerical examples for N=2,...10 value difference optimal arrival rate Figure 3.1: Optimal arrival rate and value di®erence as function of state s Note that the computation process is trivial in principle because we have a boundary condition (3.13) that gives (3.11{3.13) an unique solution. As com- parison, the problem analyzed by George & Harrison (2001) does not have such boundary condition. The approximationapproachinitiated in their workprecisely aimed to \creating" such an auxiliary condition. 3.1.3 Value of dynamic control For any given passive demand pattern (see chapter 2), an optimal dynamic pricing policy always dominates any static pricing policy because a static policy is simply a subset of the set of dynamic policies. Since a dynamic control policy often involves some implementation cost, an natural question arises: whether dynamic pricing policy signi¯cantly improves the system pro¯tability to justify its use. In the sequel, we try to answer this question by some numerical experiments. We represent the value of a dynamic policy as the di®erence between the maximal pro¯t achievable with state-dependent arrival rates against the maximum pro¯t achievable with a static(single) arrival rate. 49 Example3. We¯xthenumberofserversN =6, anddemandfunctionisgivenby p=18¡q where p and q denotes non-negative prices and arrival rate, respectively. We assume the penalty F = 10. In ¯gure 3.3, we will also allow the value of F change to illustrate a variety of trade-o®s between revenue stream and penalty. Aswesee,therevenue-maximizingarrivalrateandpriceareq ¤ =9andP ¤ =9, respectively, which means we have an under-capacitated problem. The following ¯gure 3.2 shows the optimal dynamic trajectory of arrival rates derived from (3.11{ 3.13) as well as the static arrival rate derived by optimizing (3.16) for the cases that the penalty F = 10 and F = 100, repsectively. Note that when penalty is large F =100, it is optimal to shut o® arrivals even when there are still 2 servers available. 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 Dynamic arrival rate and static arrival rate for penalty F=10 state: number of occupied servers, N=6 arrival rate 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 Dynamic arrival rate and static arrival rate for penalty F=100 state: number of occupied servers, N=6 arrival rate Figure 3.2: Optimal dynamic control policy and static policy for moderate system size N=6 Furthermore, we show the value of dynamic control, represented by the average pro¯t percentage increase between dynamic policy and static policy, as the function of penalty cost F. As shown in ¯gure 3.3, the heavier the penalty is, the greater pro¯t increase that dynamic control materializes. For instance, when the penalty F = 800, and therevenue-maximizingpriceP ¤ =9, dynamiccontrolpolicycanalmostdoublethe average pro¯t as compared to a static control policy. In other words, when system 50 0 200 400 600 800 1000 0 20 40 60 80 100 120 Average profit increase(%) as function of penalty cost Penalty cost F Average Profit increase % Figure 3.3: Average Pro¯t increase for moderate system size N=6 managers put a very high standard of quality of service or simply face a serious consequence of rejection on arrivals( as ¯le transfer in telecommunication system requires), a dynamic control policy is su±ciently justi¯ed. The above examples show that a dynamic control policy as a comparison to a static control policy has a signi¯cant value when the penalty cost F is su±ciently large. Ifqualityofserviceisnotofagreatconcern,thusF isnotlarge,thevalueof dynamiccontrolissmall. Weobservethateventhesystemsizeissmall,sayN =2, thebene¯tofdynamiccontrolisstillverysmallforsomeregulardemandfunctions. Intuitively, when the system state is in a state of \very available"(occupied server numberissmall),thesystemencouragesafastarrivalrate,thusgeneratesagreater revenuerate. Astheresult,thesystemstateischangingtoalargerstateinafaster pace. This makes the sojourn time in this state relatively short. In the long run, thistradeo®e®ectissmoothedout. Dynamiccontrolpolicyworksbetterwhenthe 51 penalty cost is high. This is because dynamic policy can e®ectively avoid events of blocking while keeping revenue rate high in most time. Consequentially, we will focus on the problems featured with large penalty cost, or equivalently with stringent requirement on quality of service in the sequel. The above procedure works well when the system size N is small or in a mod- erate range. When the problem size (number of servers N) is large, computational e®ort becomes quite demanding on the machine precision. For large size N, arbi- trarilychoosingsome° insome¯niterangeisindeednotane±cientalgorithm. In addition, the steady state performance is somewhat di®erent when the system size is large due to economies of scale. To this end, we will utilize an approximation method to directly obtain an near-optimal solution about ° and use it to derive the optimal control policy. Before proceeding on, we need to point out an important issue. It is obvious that the value of dynamic control versus a simple static control is much rele- vant to the fact that whether the system is under-capacitated or over-capacitated. This is particularly true for large size system. Speci¯cally, let us de¯ne revenue- maximizingarrivalrateas¸ ¤ :=argmax ¸¸0 f!(¸)g. Wehaveanunder-capacitated systemwhen¸ ¤ >N andanover-capacitatedsystemwhen¸ ¤ ·N. Whenthesys- tem is over-capacitated and also of large size, i.e. ;¸ ¤ ·N and N !1, the value of dynamic control diminishes| a static control at the revenue-maximizing arrival rate ¸ ¤ gives a point-wise optimal revenue stream and asymptotic zero blocking probability (as will be shown in the following section ). Therefore, problems to be analyzed in the following section is restricted to the under-capacitated setting. It is worthwhile to emphasize the fact that for small or moderate size system, congestion is not negligible at point-wise revenue-maximizing arrival rate even for over-capacitated problem. 52 3.2 Large Size Problem 3.2.1 Static control on Erlang B system Beforedelvingintothedynamiccontrolmodel,westudyastaticmodel. Ourstudy on this static model serves two purposes: we will use static model as a benchmark tojustifyouruseofheavytra±cregimetobespeci¯edlater; Andastudyonstatic model when the size of the system is large is by its own right interesting due to the economies of scale. For any given arrival rate and number of servers, there always exists some non-negative probability of blocking. For a M/M/N/0 queueing model, the steady state blocking probability B(¸;N) as function of arrival rate ¸ and server number N (without loss of generality, we always assume service rate ¹ = 1 from now on )is: B(¸;N)= ¸ N =N! j=N P j=o ¸ j =j! (3.15) An explicit formulation on static control problem is given by: max ¸¸0 [1¡B(¸;N)]!(¸)¡B(¸;N)F (3.16) subject to (3:15) where !(¸):=¸P(¸) While the formula on blocking probability B(¸;N) is explicit under given assumptions, it is not insightful and also not computationally friendly when the N is large. There is a nice recursion procedure for an e±cient computation on B(¸;N) (See Cooper(1981)). Furthermore, analytical work can be much hurdled if B is one component of some optimization problem. To this end, we ¯rst show 53 some asymptotic property about B when the system scale is large, i.e, when both arrival rate and server numbers is large and meanwhile tra±c intensity ½=¸=N is constant. Forpurposeofexposition,wecategorizethecasessuchthattheintensity ½ is and is not around the critical number 1 when the number of servers are large. The following lemma 5 shows the \°uid-like" behavior about blocking probability when the system size is large. Lemma 5. lim N!1 B(¸;N)= lim N!1 B(½)= 8 > > > < > > > : 0 if 0<½<1; 1 p N '(¯) Á(¯) !0 if ½=1¡ ¯ p N ;¡1<¯ <1; 1¡ 1 ½ if ½>1: where '(:), and Á(:) are pdf and cdf of standard normal distribution, respectively. See appendix for proof. Lemma5showsthatwhenthetra±cintensity½islessthanone,thereisalmost no block when the system size is large; and when the intensity is larger than one, thesystembehavesasifallN serversnevergetidled,thustheblockingprobability converges to 1¡ 1 ½ . Thefollowingtableillustratesthiseconomiesofscalephenomenonbyexamples based on the case of ½=0:6 and ½=5=3, respectively. Arrival Rate ¸ Number of Servers N Blocking Probability 3 5 11% 30 50 0.2% 300 500 0% Table 3.1: Comparison of Blocking Probability with Intensity Less Than 1 Wenowinvestigatetheoptimalstaticcontrolpolicyforlargesizedsystem. For simplicity of illustration, let's consider the a linear demand model. We assume a 54 Arrival Rate ¸ Number of Servers N Blocking Probability 5 3 53% 50 30 42% 500 300 40% Table 3.2: Comparison of Blocking Probability with Intensity Greater Than 1 demand function q = (¤¡p)s where p ¸ 0 and q ¸ 0. The number of servers N = ·s, where s = 1;:::; denotes a real positive number acting as index of a sequence of system, and ·2 (0;¤=2). Note that the de¯ned range of · indicates the under-capacitated setting. We show in the following lemma 6 that when the indexing parameter s approaches in¯nity, the optimal static control policy will have a tra±c intensity being 1 asymptotically. Lemma 6. When the system size index s approaches in¯nity, we have an optimal static control policy such that lim s!1 ½ s =1, where ½ s denotes the tra±c intensity when the index is s. 3.2.2 A di®usion approximation model So far, the stationary distribution of the above Markov model is derived \explic- itly",simplybecausethesystemstate(numberofoccupiedserversinourparticular formulation) is a birth-and-death process. However, if we relax the assumption on the Markovian property about arrival process, a direct task on the analysis and computation becomes quite involved , if possible. In the following, we introduce a di®usion model that aims to approximate the original process with great accuracy, and meanwhile provide a concise and clear representation of system dynamics and its control. The basic idea about approximation adopted in this work is to translate the discrete process of the number of customers described in the preceding section to 55 a continuous counterpart. Under a particular re-scaling, this translated process turns out to be a di®usion process with state-dependent in¯nitesimal drift rate. Our ¯nal goal is to adopt stochastic control theory to derive asymptotic optimal dynamic policy. We will use numerical examples to verify its e®ectiveness in the end. The regimes under which a di®usion approximation is justi¯ed usually involve heavy tra±c conditions under which the overall arrival rate approaches the overall service rate. In the classical heavy tra±c regime, a model with a ¯xed number N of servers (normally N = 1)gives rise to a re°ected Brownian process that has linear rate, if the arrival and service process is stationary. This regime is used for heavy tra±c analysis when customers are expected to wait almost surely. In systems where the number of servers is large in commensurate with the arrival rate, it is reasonable to consider an alternative heavy tra±c regime: Hal¯n-Whitt regime that is more suitable for our purpose. 1 We shall directly use the di®usion approximation model on loss system due to chapter 10 of Whitt (2002)(at the beginning of appendix B, we also show the part that is of direct relevance to our purpose). Speci¯cally, we normalize our state variable as X(t) N = S(t) N ¡N p N for all t such that 0·t<1 Here, the capital S(:) denotes a random number of occupied servers, with s rep- resents its realizations. The superscript N denotes the scaled process associated with index N, which is also the total number of servers. 1 It is useful to think about this parameter regime in an example of parking lot. In a well- designed parking lot, the average parking request should be approximately equal to its average accommodation capacity, which could be de¯ned as number of parking slot times the average sojourn time for every car. Even the parking slot is highly utilized, the probability that an arrival parking request observes full occupancy should be a non-degenerate number.Given that a customer has to wait, the waiting time should be small comparing to a typical parking time. 56 Similarly, we denote the normalized arrival rate µ x as µ x = q(s) p N whereq(s)denotestheoriginalcustomerarrivalratewhensystemstateiss. Alter- natively, µ x denotes normalized customer arrival rate when the normalized system state is x. Since the normalization is conducted in a linear form, it is without loss of generality that a transformed revenue rate R(µ x ) is de¯ned as the function of normalized state variable x. R(µ x ):=f¼(q)= p Nsuch thatµ x = q p N ;x= s(t)¡N p N g (3.17) The controlled stochastic process X has the following form: X(t)=X(0)+B(t)+ Z t 0 b(x s )ds+L(t)¡U(t)for any s2[0;N] (3.18) Here X(t) is normalized system state with initial state being X(0). B(t) is a driftless Brownian process with variance ¾ 2 = 2¹, and b(x) =¡¹(x+µ t ), and L, U are \pushing processes" associated with the lower boundary X(:) =¡ p N and upper boundary X(:) = 0, respectively. Note that L and U represents singular control when the actual system state \hits" 0 and N, respectively. In our model, there is a penalty F imposed on each time the upper boundary is hit. Hence, in 57 consistency with section 3.2, we have the following accumulated pro¯t over time [0,t] »(t)= Z t 0 R(µ x )ds¡FU(t) t¸0 here R(:) is a non-decreasing concave function (3.19) The ¯rm's objective is to min° = lim t!1 E[»(t)] t (3.20) In addition to the equation(3.18) which is re-written as follows, we have the following system equations to completely de¯ne the di®usion process: dX(t)=¡(¹X(t)+µ x )dt+¾dB(t)+dL¡dU here ¾ 2 =2¹ (3.21) X(t)2[¡ p N;0]; t¸0 (3.22) L(t);U(t) are nondecreasing and continous withL(0)=U(0)=0 (3.23) Z t 0 1 f X(s)>¡ p NgdL(s)= Z t 0 1 f X(s)<0gdU(s)=0; t¸0 (3.24) TheprocessfX t gareadaptedwithrespecttothe¯ltrationF(X t ). Thecontrols areclosedloopcontrols. Thecontrolvaluechosenatanytimetonlydependsonthe state of the system at that time. Hence, we hereafter have a Markov control. Due to the nature of \ergodic control" problem, we hereafter eliminate the subscript of time t on state variable µ, and explicitly express µ x as function of the current state x. As observed from above, we have \pushing" at each barriers of L and U. This corresponds to the singular control discussed in Harrison 1985. However, the \re°ecting" occurred when the pre-normalized state variable reaches 0 or N is 58 not a controlling process, thus not discretionary, it is more suitable to associate our model with the classical theory of drift control for di®usion process. Given any feasible control µ x , we have an associated set of processes (X;L;U characterized by (3.21{3.24)) and the corresponding expected long run average pro¯t de¯ned in (3.20). This stochastic control problem subject to boundary re°ecting conditions are also called \Skorohod problem". Thereasonwhywekeepthenotationforlowerbarrierat¡ p N sofar,ispurely to clearly demonstrate the dynamic range of the normalized state variable. Since in the limiting case, N is very large, we will replace this notation simply by¡1. Thus, the lower barrier ( when the number of empty servers is N) \disappears" in the heavy tra±c limit. Ourdi®usionapproximationisinthespiritofthecanonicalHal¯n-Whittregime with the addition of state-dependent arrival rate. A rigorous proof of the sequence of system processes converging to the limit process is beyond the scope of our work. While we assume ergodicity of this state-dependent queueing system, we will have to study the transient behavior of the system due to the need to obtain stochasticoptimalcontrolpolicyonareal-timeform. Readersarereferredto(Ser- fozo 1999) that has a comprehensive study on the steady-state analysis of various state-dependent Markovian queueing network. However, the transient analysis of state-dependent Markovian queueing system is far less common. Mandelbaum & Pats (1995) studied on time-varying and state-dependent single M » =M » =1 queue- ing. Usingstrongapproximationframworkandfunctionallawoflargenumberand functional central limit theorem, some ordinary and partial di®erential equations describing the transient system behavior are derived for its °uid approximation 59 and di®usion re¯nement. We will rely on their work to approximate our state- dependent di®usion model. Also see Kurtz(1978) for a detailed study on limit process of time as well as state dependent stochastic process. 3.2.3 Stochastic Control over the Di®usion Process In the following, we follow mechanically the derivation of optimality condition shown in Chapter 5 of Harrison (1985), with only variant being non-constant drif. With a common use of It^ o formula, the following HJB equation characterizes opti- mal policy analytically. That is, there exists a continuous, twice di®erentiable, value function v(x);x2[¡1;0], such that ° = max µ t f¡v(x)+R(µ x )g for all x2[¡1;0] ; (3.25) v 0 (0) = ¡F; (3.26) v 0 (¡1) = 0 (3.27) Here ¡v is the generator of the process v(:). In particular, we have ¡v(x)= 1 2 ¾ 2 v 00 (x)¡(¹x+µ x )v 0 (x) (3.28) Thus, our problem reduces to solving a system of PDE. Numerical solution of the above PDE has the advantage that it applies to state and/or time dependent models as well as the transient behavior of stationary models. By de¯ning a new 60 function f(x) = v 0 (x) for all x 2 [¡ p N;0], we rewrite the above second order di®erential equation as the following ¯rst order, ordinary di®erential equations: ° = max µ x [ 1 2 ¾ 2 f 0 (x)¡(¹x+µ x )f(x)+R(µ x )] (3.29) f(0) = ¡F (3.30) f(¡1) = 0 (3.31) The maximization operations in equation (3.29) gives the following: f(x)=R 0 (µ x ) (3.32) The following proposition follows: Proposition 15. If the value function v(x) is assumed to be concavely decreasing, theoptimalpolicyisintheformthattheoptimalarrivalrateq ¤ (x)isnon-increasing function of the state x. See appendix for the proof. Now, let's illustrate the procedure to solve the long-run average pro¯t rate ° ¤ . Let's ¯rst de¯ne a new function £(f(x))=max µ x [R(µ x )¡(¹x+µ x )f(x)] (3.33) Based on the assumption on the revenue function R(:), we know that £(x) is a well-de¯ned function in sense that there exists an unique µ(x) that achieves the optimality of (3.33). 61 Our target now is to obtain the optimal long-run average pro¯t ° ¤ based on (3.29{3.36): ° = [ 1 2 ¾ 2 f 0 (x)+£(f(x))] (3.34) f(0) = ¡F (3.35) f(¡1) = 0 (3.36) By looking at the above ordinary di®erential equations, we know that we have two boundary conditions to su±ciently solve ° and f(:). We can use the result of ° to the computation of exact analysis investigated in subsection 3.2. In the sequel, we will give numerical study to verify the accuracy of di®usion model. Example 4. We will use the same demand curve as that of numerical example 2. However, the system is scaled up by a parameter · that indicate the size of the system. Speci¯cally, the demand will be p = (18¡ q · )q, and the number of servers N = 6·.Thus, the point-wise optimal revenue rate is q ¤ = 9· and still we have undercapacitated system that is of our interest. We let · = 1;1:5;2;2:5;3 and compare the optimal revenue rate derived from our original SMDP model and that of the di®usion model adopted. The following ¯gure 3.4 illustrates both per- formances. We see that when the system size is small (N =6), the approximation performs poorly with error rate being 21.2%. When the system size N = 18, the performance of di®usion approximation could be near 1.32%. 62 1 1.5 2 2.5 3 40 60 80 100 120 140 160 180 s: scale parameters. Number of servers, N=6s optimal revenue rate numerical comparision between diffusion approximation and original model original model diffusion model Figure 3.4: The Optimal Revenue Rate derived from SMDP and Di®usion Model When Increasing System Size, N=6 3.3 Extension I: Heterogenous service require- ment The above formulation is general enough to contain multi-class customers with homogenous service requirement. However, it is often arguably true that cus- tomers having di®erent price-related characteristics also have di®erent form of service requirement. In the example of car rental business, customer class that is less sensitive to the price might need a shorter rental duration. It is not surprising to see that business travelers is located within this category. In this section, we increase a bit our ambition to a problem that has heterogenous service rate,i.e. ¹ 1 6= :::;¹ M , where ¹ i is a ¯nite positive number. Since the service rates for di®erentcustomerclassesarenotidentical, thesystemstatecannotbesu±ciently 63 characterized by the total number of occupied servers without recording the occu- pied servers from each customer class. From now on in this section, we restrict ourself to the case of such multi-dimensional state s = (s 1 ;:::;s M ), where s i is the number of servers occupied by class i customers, i = 1;:::;M. As will be shown, this state indicator leads to a signi¯cant complexity of equality conditions in (3.1){(3.3). As a result, computations of optimal control policy become more involved. Forthemulti-dimensionalcase, eventhetotalnumberofoccupiedserver is1, thereisMdi®erentstatesaccordingly, notmentioningthecasewhenthetotal number of occupied server is greater than 1. In this section, we aim to derive somestructuralpropertyofheterogenousservicecase, andleaveitsapproximation computation unanswered. Recall the Bellman equation derived in (3.1{3.3),we now incorporate the sub- script to indicate the class-dependency of state variables. Therefore, we can write thefollowingasBellmanequationsforourheterogeneousservicerequirementcase. v 0 = max ¸ [ M X i=1 P i (¸) ¸ i ¹ i ¡° ¤ ]+ M X i=1 ¸ i v ² i +(1¡ M X i=1 ¸ i )v 0 (3.37) v s = max ¸ [ M X i=1 P i (¸) ¸ i ¹ i ¡° ¤ ]+ M X i=1 ¸ i v s+² i + M X i=1 ¹ i s i v s¡² i +(1¡ M X i=1 ¸ i ¡ M X i=1 ¹ i s i )v s for all s such that 0<jsj<N, (3.38) v N = (0¡F ¡° ¤ )+0+ M X i=1 ¹ i s i v N¡² i +(1¡ M X i=1 ¹ i s i )v N when s=N (3.39) Here we make some explanation on notations used above: The vector v = (v 0 ;:::;v N )is,asusual,calledarelativecostfunction. v s istheminimumexpected 64 cost incurred from state s to the ¯rst hitting into some arbitrary reference point, says 0 . ² i denotesavectorwhichhasallitscomponentaszeroexceptthe i¡th.For notational convenience,we also de¯ne aggregated revenue rate as: !(¸):= M X i=1 P i (¸) ¸ i ¹ i (3.40) As convention, we de¯ne relative pro¯t di®erence: h i (s)=v(s)¡v(s+² i );for i=1;:::;M (3.41) Then, the (3.37{3.38) can be modi¯ed as follows: ° ¤ = max ¸ [!(¸)¡ M X i=1 ¸ i h i (0)] (3.42) ° ¤ = max ¸ [!(¸)¡ M X i=1 ¸ i h i (s)]+ M X i=1 ¹ i s i h i (s¡² i ) for s such that 0<jsj<N (3.43) ° ¤ = M X i=1 ¹ i s i h i (N¡1)¡F for all s such thatjsj=N (3.44) Similarto(3.9),theoptimizationreducestothefollowingasfarasitisfeasible: ¸(s) ¤ =argmax ¸ [!(¸)¡ M X i=1 ¸ i h i (s)] (3.45) 3.3.1 An Admission Control Sub-problem In many business setting, particularly in highly competitive market, ¯rms do not havetheluxurytodynamicallychangeitspriceandhencesticktosomepre-de¯ned prices. However, the ¯rm can open or close each customer class as a control over 65 itsarrivalprocess. Thissortofdynamicallocationruleisoftencalledanadmission controlrule. Undersomeoptimalpolicy, the¯rmmaynotalwaysadmitcustomers even when it is feasible to do so. This is termed as an option e®ect. Assume that there exists price (P 1 ;:::;P M ) associated with each customer class,respectively. The¯rm'scontrolu2UisintheformofU=[0; ¹ ¸ 1 ]£:::[0; ¹ ¸ M ] forjsj<N, and of courseU=f0g ifjsj=N. Then the optimality equation (3.2) can be modi¯ed as ° ¤ =max ¸ [ M X i=1 ¸ i P i ¹ i ¡ M X i=1 ¸ i h i (s)]+ M X i=1 ¹ i s i h i (s¡² i ) (3.46) Proposition 16. For the admission control problem, in state s,the optimal policy is to open the class j such that P j ¹ j ¸h j (s), and close the rest of the class otherwise. Sometimes, it is not implausible to further assume that for any i;j = 1;_;M, if we have ¹ i ¸ ¹ j , we also have P i ¸ P j , i.e., the shorter the service duration, the higher charge is expected. Moreover, if we assume P i ¹ i ¸ P j ¹ j , by the proposition 11 combinedwiththeresultfromproposition12, wehavethat, instates, theoptimal policytoacceptclass1throughK ¤ (s):=argmax k : P k ¹ k ¸h k (s),andrejectclassed K ¤ (s) through M. Note that a threshold policy is applied under this special case, as it is always indicated for admission control policy. 3.4 Extension II: Di®erentiated Service While our work so far concerns a problem with multi market channels for the same service/product, we now study a problem with multi market channels with di®erentiated service type. This extension is not trivial as illustrated below. As shown in previous work, for the problem with multi-class demand process with single service type, the technique of state dimension reduction technique 66 applies. This is due to the idea of e±cient frontier borrowed from the Capital Asset Pricing Model(CAPM) in ¯nance literature. Speci¯cally, the multi-class demand process with single service type problem is translated into a problem as "single demand channel, single service type" problem, where the single demand channel is reformulated as an aggregate demand process and single service is the onecorrespondingtoanaggregaterevenueratefunction. Thedemandratecontrol, or equivalently the pricing control, at the channel level, are then obtained from the control decision derived from aggregate level in a straightforward manner. While the above policy works well in some technology engineering problems, it is not implementable in revenue management problem for industries like hotel, rentals, etc. because by law it is not legal to list di®erent prices to di®erent customer class at the same time. Firms, subject to this law constraint, however, can di®erentiate their product or service and charge di®erent prices to di®erent class of customers at the same time. By listing a vector of prices, customers from di®erent class choose the preferred product or service. Our focus in this extension is on how the ¯rm can e®ectively control its prices to manipulate the multi-class demand process in a way that maximize its long term revenue rate. Due to the varying \prospect" of each type of product or service asset, the state variable would be a vector instead of a single scalar adopted in the previ- ous work. By studying the interaction between di®erent product/service on their pricingdecision,wearedealingwithamulti-dimensionaldynamiccontrolproblem. Two type of service N1 servers for type 1, and N2 servers for type 2. For all demand channels, it boiled down to a two demand curves for each service types ¸ 1 (P1;P2), and ¸ 2 (P1;P2), where P1;P2 denotes the listed prices for type 1 service and type 2 service, respectively. Similar to the previous one, we assume on 67 a prior base the existence of penalty cost rates F1 andF2 for each blocking events on service type 1 and type 2, respectively. The state variable is two dimensional (S 1 ;S 2 ) 2 [(0;N1);(0;N2)]. Bellman Equation is as follows: v(0;0) = max (¸ 1 ;¸ 2 ) [P1(¸ 1 ;¸ 2 )¸ 1 +P2(¸ 1 ;¸ 2 )¸ 2 ¡°+¸ 1 v(1;0)+¸ 2 v(0;1)] +(1¡¸ 1 ¡¸ 2 )v(0;0) (3.47) v(0;1) = max (¸ 1 ;¸ 2 ) [P1(¸ 1 ;¸ 2 )¸ 1 +P2(¸ 1 ;¸ 2 )¸ 2 ¡°+¸ 1 v(1;1)+¸ 2 v(0;2)] +v(0;0)+(1¡¸ 1 ¡¸ 2 ¡1)v(0;1) (3.48) v(1;0) = max (¸ 1 ;¸ 2 ) [P1(¸ 1 ;¸ 2 )¸ 1 +P2(¸ 1 ;¸ 2 )¸ 2 ¡°+¸ 1 v(2;0)]+¸ 2 v(1;1) +v(0;0)+(1¡¸ 1 ¡¸ 2 ¡1)v(1;0) (3.49) v(S 1 ;S 2 ) = max (¸ 1 ;¸ 2 ) [P1(¸ 1 ;¸ 2 )¸ 1 +P2(¸ 1 ;¸ 2 )¸ 2 ¡°+¸ 1 v(S 1 +1;S 2 )] +¸ 2 v((S 1 ;S 2 +1))+S 1 v(S 1 ¡1;S 2 )+S 2 v(S 1 ;S 2 ¡1) +(1¡¸ 1 ¡¸ 2 ¡S 1 ¡S 2 )v(S 1 ;S 2 ) (3.50) v(N1;S 2 ) = max (¸ 1 ;¸ 2 ) [¡F1+P2¸ 2 ¡°+N1v(N1;S 2 )+¸ 2 v(N1;S 2 +1)] +S 2 v(N1;S 2 ¡1)+(1¡¸ 2 ¡N1¡S 2 )v(N1;S 2 ) (3.51) v(S 1 ;N2) = max (¸ 1 ;¸ 2 ) [¡F2+P1¸ 1 ¡°+N2v(S 1 ;N2¡1)+¸ 1 v(S 1 +1;N2)] +S 1 v(S 1 ¡1;N2)+(1¡¸ 1 ¡S 1 ¡N2)v(S 1 ;N2) (3.52) v(N1;N2) = (¡F1¡F2¡°)+0+N1v(N1¡1;N2)+v(N1;N2¡1) +(1¡N1¡N2)v(N1;N2) (3.53) Here, o<S 1 <N1, 0<S 2 <N2, ¸ i ¸0, for i=1;2. For easy illustration, we ¯rst de¯ne the following expression: We de¯ne K 1 (S 1 ;S 2 )=v(S 1 ;S 2 )¡v(S 1 +1;S 2 ); 68 similarly K 2 (S 1 ;S 2 )=v(S 1 ;S 2 )¡v(S 1 ;S 2 +1) for o < S 1 < N1, 0 < S 2 < N2, ¸ i ¸ 0.Note K i denotes the marginal value for each unit of N i service, i=1;2. Furthermore, we de¯ne ¼(K 1 ;K 2 )= max (¸ 1 ;¸ 2 ) [P1(¸ 1 ;¸ 2 )¸ 1 +P2(¸ 1 ;¸ 2 )¸ 2 ¡K 1 ¸ 1 ¡K 2 ¸ 2 ]; again ¸ i ¸0, for i=1;2: Thus, the (3.47{3.53) is simpli¯ed as: ° = ¼(K 1 (0;0);K 2 (0;)) (3.54) ° = ¼(K 1 (0;1);K 2 (0;1)+K 2 (0;0) (3.55) ° = ¼(K 1 (1;0);K 2 (1;0)+K 1 (0;0) (3.56) ° = ¼(K 1 (S 1 ;S 2 );K 2 (S 1 ;S 2 ))+S 1 K 1 (S 1 ¡1;S 2 )+S 2 K 2 (S 1 ;S 2 ¡1)(3.57) ° = max (¸ 1 =0;¸ 2 ) [¡F1+P2¸ 2 +N1K 1 (N1¡1;S 2 )¡¸ 2 K 2 (N1;S 2 ) +S 2 K 2 (N1;S 2 ¡1)] (3.58) ° = max (¸ 1 ;¸ 2 =0) [¡F2+P1¸ 1 +N2K 2 (S 1 ;N2¡1)¡¸ 1 K 1 (S 1 ;N2) +S 1 K 1 (S 1 ¡1;N2)] (3.59) ° = (¡F1¡F2¡°)+0+N1K 1 (N1¡1;N2) +N2K 2 (N1;N2¡1) (3.60) Note that if we don't di®erentiate service type 1 and service type 2, and there is a common demand process for the service, we will have a one-dimensional state 69 descriptor. For multi-class demand process, the state dimension reduction tech- nique applies. In other words, our previous work is just a special case of current model. In order to obtain some interesting properties of the value function and struc- tural optimal control, we need to start with some assumption on the function ¼(K 1 ;K 2 ). Assumption 2: The demand curves P1(¸) and P2(¸), here ¸=(¸ 1 ;¸ 2 ) are the ones such that ¼(K 1 ;K 2 ) satis¯es the following: if ¼(K 0 1 ;K 0 2 ) ¸ ¼(K 1 ;K 2 ); then K 0 1 ·K 1 ;K 0 2 ·K 2 : Note this assumption 2 looks like the typical formulation of production eco- nomics: it helps to take ¸ i as input variables for i = 1;2. K 1 and K 2 then would betakenasperunitcostforinput1and2respectively. Theassumptionessentially says that a ¯rm's maximum pro¯t is increased means that both of the cost rates for the two input factors are decreased. Proposition 17. Under assumption 2, the marginal value for both service types K i (S) are nonnegative, and are non-decreasing in state variable S as well, i = 1;2.Speci¯cally, we have 0·K i (S)·K i (S +² i ), where ² 1 =(1;0),² 2 =(0;1). For general demand curves, there might not be monotonicity properties on optimalcontrol. However,ifwerestricttolineardemandcurves,underassumption 1 , we can also characterize a monotonic property. Assume demand curves are as follows: P1=a 1 ¡b 1 ¸ 1 +c 1 ¸ 2 ; P2=a 2 ¡b 2 ¸ 2 +c 1 ¸ 1 70 where, a i ,b i ,c i are all nonnegative real numbers, i=1;2: As de¯ned before, ¼(K 1 ;K 2 )=max ¸ 1 ;¸ 2 [a 1 ¸ 1 ¡b 1 ¸ 2 1 +c 1 ¸ 1 ¸ 2 +a 2 ¸ 2 ¡b 2 ¸ 2 2 +c 2 ¸ 1 ¸ 2 ¡K 1 ¸ 1 ¡K 2 ¸ 2 ] The optimization problem has the following solution: ¸ ¤ 1 =[ d 2 d+2b 2 d 1 d 2 ¡4b 1 b 2 ] + (3.61) ¸ ¤ 2 =[ d 2 d+2b 1 d 2 d 2 ¡4b 1 b 2 ] + (3.62) Where, d = c 1 +c 2 , d 1 = K 1 (S)¡a 1 , d 2 = K 2 (S)¡a 2 . If we can further give conditions( assumption on a,b,c) such that assumption 2 applies, we can conclude that both of the arrival rate to service 1 and 2, ¸ 1 and ¸ 2 are non-increasing functions of state variable. That is, each time an additional server, regardless type 1 and type 2, is occupied, under the optimal control policy, the arrival rate ¸ 1 and ¸ 2 decreases. 71 Chapter 4 Conclusion The objective of Chapter 2 was to understand optimal pricing policies when con- sumers are strategic and it is di±cult to frequently change prices. One of my main ¯ndingsisthatwhenbuyersarestrategicdynamicpricingisvaluableonlyifbuyers consider stock-outs a possibility. Strategic buyers purchase a product only if their valuations exceed a threshold. This threshold is higher than the prevailing price and depends on the perceived scarcity level. I also ¯nd that, given any level of supply, as the size of the market increases the need for price changes decreases. My computations suggest that a simple policy of announcing a small number of prices at the beginning of the sales season is close to optimal. My analysis is based on several restrictive assumptions. The most signi¯cant assumption is that buyers have the ability and ¯nd it worthwhile to compute their equilibrium strategies. It will be interesting to conduct experiments to see how buyers make purchase decisions in similar situations. Another assumption I make is that the number of buyers, or the distribution of the number of buyers in the market is common knowledge. It is quite likely that for many short life cycle products the biggest challenge may lie in determining the size of the market. There are a number of directions in which this work needs to be extended. These include incorporating di®erent types of customers some of whom are strate- gic and other are not, and allowing multiple substitutable products. InChapter3,Ihavestudieddynamicarrivalcontrolonasystemwithrenewable assets. Based on a Markov decision process model, I derived structural properties 72 ofoptimalpoliciestomaximizethelongrunaveragerevenuesubjecttoaconstraint on quality of service. For large size problem, I proposed a di®usion approximation modelandusedstochasticcontroltheorytoobtainanear-optimallongrunaverage revenue rate and the corresponding near-optimal control policy. In the model, I incorporatemulti-classcustomerarrivalprocessandalsodiscussadmissioncontrol as a special case of dynamic pricing policy. There are some works that are left unanswered. First, I showed the one-to- one corresponding relationship between constraint on blocking probability and an uniquely derived penalty cost that facilitates our model analysis. However,I have not provided an on-hand procedure to compute this equivalent penalty cost for any given constraint on blocking probability. Second, I have not given a rigorous treatmentontheproofofthedi®usionprocessasasymptoticapproximationtothe original model. Instead, I just show its justi¯cation on a heuristic level and use numerical testings to show its validity. These are all interesting future works. Thereareseveralinterestingdirectionstopursuearoundtheproblemstudiesin my dissertation. I have so far focused on the demand side management and taken thesupplysideasgiven. Itwouldbeinterestingtocombinethistwosideddecisions into one model to study the interaction between the ¯rm's long term decision on service capacity and its short term control policy. Besides, it is reasonable to assume that the incoming customer does not reveal its class category upon arrival. Howwillthisinformationasymmetryleadtodistortiononsystem'srevenue? These might be interesting topics in future work. 73 References [1] Afche,Philipp., B. Ata. 2005 Revenue Management in Queueing Systems with Unknown Demand Characteristics, working paper. [2] Altman, Eitan. 1998 Constrained markov decision processes, Chapman & Hall/CRC [3] Armony, A.,C. Maglaras 2004. Contact centers with a call-back option and a real-time delay information, Operations Research. 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Due to the continuity and monotonicity of the function Z(v), if there is an indi®erence point v ¤ 2 (P 1 ;1) such that Z(v ¤ ) = 0, then v ¤ must be the unique. Monotonicity of Z(:) implies that only consumers with valuation v ¸ v ¤ will bid in period 1. Proof of Proposition 1. We ¯rst consider the case when there exists a v ¤ such that Z(v ¤ ) = 0. Expected sales in period 1 is NPr(v ¸ v ¤ ) = N(1¡ G(v ¤ )) and expected sales in period 2 is NPr(P 2 < v < v ¤ ) = N[G(v ¤ )¡G(P 2 )]. Hence, the expected revenue of the contingent pricing scheme is: E(¼ R ) = N[1¡G(v ¤ )]P 1 + Z v 0 N[G(v ¤ )¡G(P 2 )P 2 f(P 2 )dP 2 subject to : (v ¤ ¡P 1 )¡ Z v ¤ 0 (v ¤ ¡P 2 )f(P 2 )dp 2 =0 79 Implying: P 1 = v ¤ ¡ Z v ¤ 0 (v ¤ ¡P 2 )f(P 2 )dP 2 E(¼ R ) = N[1¡G(v ¤ )]v ¤ [1¡F(v ¤ )]+[1¡G(v ¤ )] Z v ¤ 0 P 2 f(P 2 )dP 2 + Z v ¤ 0 [G(v ¤ )¡G(P 1 )]P 1 f(P 1 )dP 1 = N[1¡G(v ¤ )]v ¤ [1¡F(v ¤ )]+ Z v ¤ 0 [1¡G(P 1 )]P 1 f(P 1 )dP 1 · E(¼ s )[1¡F(v ¤ )]+E(¼ s )F(v ¤ ) = E(¼ s ) Here, E(¼ s ) is de¯ned as E(¼ s )=max p2(0;1) N[1¡G(p)]p. If the indi®erence point v ¤ does not exist, then because Z(:) is monotone increasing and Z(1) < 0 every buyer would be better o® if he postpones his bid to period 2. Therefore, the optimal pricing strategy for the seller is just a single price that maximizes her expected revenues E(¼ s ). Proof of Lemma 2. When demand is deterministic, if capacity is limited, there always exists a clearing price P c that equates supply and demand. Clearly the set of optimal prices can not all be lower than P c nor can they all be above P c . The range of the optimal prices must include the clearing price P c . Let's assume for the sake of a contradiction, that the optimal pricing scheme consists of three prices P 1 > P 2 > P 3 . First assume that P 1 > P c > P 2 > P 3 . In this case the seller can not be worse of if she employs a pricing scheme (P 1 ;P 2 ) instead of (P 1 ;P 2 ;P 3 ). At price P 2 all inventories can be cleared for sure. On the other hand if P 1 >P 2 >P c >P 3 , buyers know for sure that at price P 2 there will not be a stock-out. Hence no sales will be occur at price P 1 . The seller would be indi®erent between (P 2 ;P 3 ) and (P 1 ;P 2 ;P 3 ). 80 By the same logic, we can show that the optimal pricing schemes can never have more than three prices. Proof of Proposition 2. Averygeneralpolicyisforabuyerwithvaluationv tobid with probability p(v) in period 1. Clearly we must have the following: a1) p(v)=0 if v·P 1 , and a2) all buyers with valuation greater than P 2 bid either in period 1 or period 2. LetB be the set of values for which p(v)>0. We must have: a3) for v distributed as G(:), Prob(v2B)>0. AssumeN¡1buyersfollowthebiddingstrategygivenabove. Let¼ 1 and¼ 2 bethe probabilities that the N th buyer gets the product in periods 1 and 2, respectively. Let v ¤ 2[P 1 ;1] be a solution to the equation: (v¡P 1 )¼ 1 =(v¡P 2 )¼ 2 (1) Let Z(v) = (v¡ P 1 )¼ 1 ¡ (v¡ P 2 )¼ 2 . Z(:) represents the incremental value of bidding in period 1 instead of period 2. Since v ¤ solves equation(1), Z(v) = (v ¡ v ¤ )(¼ 1 ¡ ¼ 2 ). Due to (a1), (a2) and (a3) ¼ 1 > ¼ 2 . Thus Z() is strictly monotone increasing. Hence if the valuation of the N th buyer v ¸ v ¤ , Z(v) > 0 and it is optimal for the N th buyer to bid in period 1. Strict monotonicity of Z(:) also implies that there is at most one solution for equation(1). If equation(1) does not have a solution the N th buyer will not bid in period 1. What we have just shown is that regardless of the strategies followed by the other N¡1 buyers the optimal strategy for the N th buyer is to bid if his valuation exceeds y ¤ . This in 81 turn establishes that the unique equilibrium strategy is for all buyers to bid in the ¯rst period if and only if their valuations exceed some threshold. Solutionstoequation(2.1)representcandidatethresholdvalues. Ifthereisonly one solution y ¤ to this equation in the range [P 1 ;1] then everyone with valuation above y ¤ will bid in period 1. What if there are multiple solutions in the range [P 1 ;1]? In that case we let y ¤ be the smallest solution. Even if N¡1 buyers elect to bid if their valuations exceed y ¤ it is still attractive for the N th bidder to bid if his value exceed y ¤ . Let y 0 be another solution to equation(2.1). By de¯nition y 0 > y ¤ . If some of the buyers choose to bid only if their values are higher than y 0 then these buyers decrease competition in period 1, making it more attractive for those bidding in period 1. Thus everyone with valuations above y ¤ will bid in period 1. What if there is no solution to equation(2.1)? Since Z(P 1 ) < 0 and Z(y) is continuous for any y 2 (P 1 ;1), we know that Z(y) < 0 for any y 2 (P 1 ;1) (otherwise Z(y) = 0 would have solution). In particular, we have Z(1) < 0. Thus evenifall the other buyersdelay theirpurchase to period2 the N th buyer will also not bid in period 1. Proof of Proposition 3. Proposition 2 establishes the equilibrium for a 2 period problem. We use induction on the number of time periods to extend that result to a T period pricing problem. Assume for a T ¡1 period posted pricing scheme with prices (P 1 ;:::;P T¡1 ), whereP i >P j foranyi;j2f1;:::;T¡1gandi<j,buyers'behaviorischaracter- ized by a set of thresholds: y ¤ 1 (k 1 )¸ :::y ¤ j (k j )¸¢¢¢¸ y ¤ T¡1 (k T¡1 ) = P T¡1 . Only buyers with valuations within the range [y ¤ j (k j );y ¤ j¡1 (k j¡1 )] bid for the products at price P j . The threshold value (y ¤ j ) depends on the number of units available for 82 sale (k j ). Let us now consider a T period problem, and for ease of notation we will denote this additional period as period 0. The T periods pricing scheme is now given by: (P 0 ;P 1 ;:::;P T¡1 ). Let¼ i (y)denotetheprobabilitythatthefocalbuyergetstheproductgiventhat allbuyersfollowthepolicy[y ¤ 1 (k 1 );:::;y ¤ i (k i );:::;y ¤ T¡1 (k T¡1 )]andfollowabidding strategy described in Proposition 2 in period 0. As in the proof of Proposition 2, we have ¼ i (y)¸¼ j (y) for all i;j2f0;1;:::;T ¡1g and i<j. Next, consider the following equation: ¼ 0 (y)(y¡P 0 )= max j=1;:::;T¡1 f¼ j (y)(y¡P j )g (2) If equation (2) has a solution y =y ¤ 0 , then due the reasons employed in propo- sition 2, the focal buyer will buy in period 0 if his valuation y¸y ¤ 0 , otherwise this buyer will delay his bid to a future period. In case equation (2) has no solution in the range of (P 0 ;1], no buyer will bid for the product in period 0 as explained also in proof of proposition 2. Onceagainsince¼ i (y)¸¼ j (y)ifequation(2)hasmultiplesolution,thesmallest solution in the range [P 0 ;1] is chosen as the threshold. Thus, we see that in the ¯rst period of a T periods pricing scheme (P 0 ;:::;P T¡1 ), strategic buyers' equilibrium is also a threshold policy and propo- sition 2 extends fully to the T period case. Proof of Proposition 4. This proof is similar to that of proposition 3, and we omit the details. If N is uncertain then we merely have to condition probabilities of a successful bids on N. Proof of Proposition 5. Note that the right-hand side (RHS)of (2.4) is a piece- wise linear function of y for any given y ¤ . Also, we see that the slope of LHS is 83 greater than slope of RHS for each cut-o® value y ¤ , since ¼ 1 (y ¤ ) = K¡1 P i=0 Pr 1 (i)+ N¡1 P i=k Pr 1 (i) K i+1 > K¡1 P i=0 Pr 1 (i)¯ i . Let y ¤ be the smallest solution in [P 1 ;1] for the equation (2.4), for any buyer with valuation y >y ¤ , the di®erence of expected surplus between buying in period 1 and buying in period 2, given that all the other buyers follow the policy of threshold value y ¤ ,is : D(y) = ¼ 1 (y ¤ )(y¡ P 1 )¡ K¡1 P i=0 Pr 1 (i)¯ i (y¡ p ¤ 2r ) = (y¡ y ¤ )[¼ 1 (y ¤ )¡ P K¡1 i=0 Pr 1 (i)¯ i ] > 0. It is straightforward to see thatD(y) < 0 for any y <y ¤ . This concludes our proof. Proof of Proposition 6. Weneedtoderivetheoptimalmechanismandshowthata T period posted pricing scheme with P T converging to v ¤ yields the same revenues as the optimal mechanism. Given any mechanism, let m i (v) denote expected pay- mentbyabuyerwithvaluationv. Themechanismwillalsodetermineprobabilities Q i (v i ;V ¡i ). De¯ne q(v i )= R V ¡i Q(v i ;X ¡i )g(X ¡i )dX ¡i . Due to the revelation prin- ciple (Milgrom (2002,1998) and Krishnan(2002, page 63)), expected revenues for any mechanism under equilibrium are given by: E¦ Q = X i2N m i (0)+ X i2N Z v J(v i )Q i (v i ;V ¡i )g(v i ;V ¡i )dV (3) Design of optimal mechanism than becomes one of ¯nding Q i (v i ;V ¡i ) so as to maximizeE¦ Q , subject to (a) incentive compatibiliy, (b) individual rationality, and(c)capacityconstraints. Theseconstraintsinturncanbeformulatedasfollows (Krishnan(2002), page 63): 84 q(v)¡q(v 0 )¸0forallv¸v 0 (4) m i (0)·0 (5) N X i=1 Q i (v) · K (6) Any allocation scheme that sells the product to buyers with K highest valuations, provided their valuations are above the threshold v ¤ , will result in allocation prob- abilities Q(v i ;V ¡i ) = 1 if v i > y i (v i ;V ¡i ) and 0 otherwise. This allocation scheme ensures that m i (0) = 0, and satis¯es the capacity constraint (10). It also maxi- mizestheobjectivevaluebyallocatingunitweightstotheKlargest J(v i )forevery outcome V, provided J(v i )¸0 (See also Krishnan 2002, page 63). Only thing left to show is that constraint set (A.6) is satis¯ed. Observe that if Q(v i ;V ¡i ) = 1 for v i = v 0 , then for all v i > v 0 , Q(v i ;V ¡i ) = 1. This ensures that constraint set (A.6) is satis¯ed and establishes the optimality of the proposed allocation scheme. To complete the proof we have to show that in a posted price scheme in the limit as T ! 1 and P T ! v¤ buyers with the K highest valuation get the product, provided that their valuations are above v¤. Since in the limit P T equals v¤, only those with valuations above v¤ will get the product. All that is left to show is that those with the highest valuations get the product. Suppose there are K 0 objects available for sale and the current price is P. In the next "period" the price is going to be P¡¢P. A buyer with valuation y, will buy now if: ¼(y¡P) > (¼¡¢¼)(y¡P +¢P), where ¼ is the probability that the buyer will get the product at price P. Clearly if the inequality holds for some y 0 , then it holds for all y¸y 0 . Hence buyers with higher valuation will bid earlier in the process. 85 Proof of Proposition 7. Proof: Let µ > 0 be a large positive number that is used for scaling the market by having µN potential customers and µK units to sell. For any given pricing scheme (P µ 1 ;:::;P µ T ), where 1 ¸ P µ i > P µ i+1 for i = 1;:::;T¡1, we knowthere exist a series of threshold values y µ 1 ;::y µ T¡1 that charac- terize customer's purchasing behavior. The number of customers N(P µ i ) who bid at price P µ i is a random number with mean Nµ[F(y µ i¡1 )¡F(y µ i )]. By law of large numbers as µ!1, N(P µ i ) µ !N[F(P (i¡1)r )¡F(P i r)] almost surely. Thus,wehavedeterministicdemandinthelimit. Usinglemma2,weconcludethat at most two price are needed. Further, if two prices are needed then the clearing price is located between these two prices. Proof of proposition 8. The proof will be based on sample path arguments. For any given realization of value v i ;i=1;:::;N, the optimal revenue is R(K)= N X i=1 J i (v i )Q i (v i ;v ¡i ) Let us incense the capacity K by one unit. If the (K +2) th largest value in the realization v i ;i = 1;:::;N is no greater than the reservation price v ¤ de¯ned above, we have that R(K) = R(K + i);i = 1;:::;N ¡ K. Thus the expected revenue is constantfor all capacity over than K. If the (K + 2) th largest value v K+2 in the realization v i ;i = 1;:::;N is greater than the reservation price v ¤ de¯ned above, we have that ¢R(K) = J(v K+2 ).Furthermore, by the nature of order statistics and assumption about 86 increasingvirtualvaluefunction J(:), therevenueimprovement, ifpositive, willbe decreasing. Thus¯xedN expectedmarginalrevenueisanon-increasing,non-negativefunc- tion of K. Proof of Proposition 9. As shown in proposition 8, if we increase the number of units available for sale from K to K + 1, then revenues will increase provided the valuation of the buyer with the K +1 th highest valuation exceeds v ¤ . By the law of large number,the number of customers whose valuation is no less than v ¤ is asymptotically N[1¡G(v ¤ )] for sure as N !1. In other words, the optimal number of units sold asymptotically approaches N[1¡G(v ¤ )] as N !1. Proof of Proposition 10. Let us assume that the ¯rm has inventory level K 1 and K 2 , where N > K 1 > K 2 , with probabilities ® and (1¡®), respectively. If the ¯rm truthfully reveals its inventory levels then the expected revenues are ®¦(K 1 ;N;P 1 ;P 2 ;y 1 )+(1¡®)¤¦(K 2 ;N¡(K 1 ¡K 2 );P 1 ;P 2 ;y 2 ) (7) On the other hand if the ¯rm does not reveal its inventory levels, the expected pro¯ts are: ®¦(K 1 ;N;P 1 ;P 2 ;y ® )+(1¡®)¦(K 2 ;N¡(K 1 ¡K 2 );P 1 ;P 2 ;y ® ) (8) It will be optimal for the ¯rm to hide inventories if ®¦(K 1 ;N;P 1 ;P 2 ;y ® )+(1¡®)¦(K 2 ;N¡(K 1 ¡K 2 );P 1 ;P 2 ;y ® ) ¸®¦(K 1 ;N;P 1 ;P 2 ;y 1 )+(1¡®)¦(K 2 ;N¡(K 1 ¡K 2 );P 1 ;P 2 ;y 2 ) (9) 87 Right hand side of (14) is linear in ®, also for ® = 0 and® = 1, the left hand side and right hand side are equal. Therefor inequality (14) holds if we can show the left hand side is a concave function of ®. Let H(®) = ®¦(K 1 ;N;P 1 ;P 2 ;y ® )+(1¡®)¦(K 2 ;N ¡(K 1 ¡K 2 );P 1 ;P 2 ;y ® ). For ease of notation let ¦ i = ¦(K i ;N¡(K 1 ¡K i );P 1 ;P 2 ;y ® ). We need to show d 2 H(®) d® 2 ·0. Using the chain rule, we get: d 2 H(®) d® 2 =2[ d¦ 1 dy ¡ d¦ 2 dy ] dy d® +[® d 2 ¦ 1 d 2 y +(1¡®) d 2 ¦ 2 d 2 y ][ dy d® ] 2 +[® d¦ 1 dy +(1¡®) d¦ 2 dy ] d 2 y d 2 ® (10) As ® increases, the probability of higher inventory levels increase; this in turn implies that threshold levels will decreases. Therefore dy d® · 0. Also as inventory levels increase the loss for the ¯rm from increasing threshold levels is greater,(you aremorelikelytoloseasale),hence[ d¦ 1 dy ¡ d¦ 2 dy ]¸0. Consequentlythe¯rsttermon the right hand side is negative. Since the pro¯t function is assumed to be concave in y, the second term is negative, and the third term is negative because pro¯ts decrease with increase in threshold levels and we have assume that y ® is convex. Since all three terms are negative, the proof is complete. Page 361 of Whitt's book Theorem (FCLT for the G/M/m loss models) Consider a sequence of G/M/m loss models indexed by the number of servers, m, where the individual service rate is ¯xed at ¹.Suppose that the arrival process is fA(¸t) : t¸ 0g,where A is a general stationary rate-1 process satisfying A ¸ ) ¾ A B as ¸ ! 1 and B standard Brownian motion. Suppose that m ! 1 and 88 ° ´ ¸=¹ ! 1 with (m¡°)=° ! ¯ holding and ° ¡ 1 2 (Q m (0)¡m) ) y, where Q m (t):t¸0 is the queue-length process in model m. Then Q m )Q in (D;J 1 ) where Q m (t)´° ¡1=2 (Q m (t)¡m)t¸0 andQisare°ectedOrnstein-Uhlenbeckprocesswithin¯nitesimalmeanm(x)= ¡¹(x+¯)forx·0,in¯nitesimalvariance¾ 2 (x)=¹(1+¾ 2 A ),initialpositionQ=y and instantaneous re°ecting barrier above at 0. Proof of Lemma 4. Due to the concavity about ¼(q), we have that given any non- negative x, the optimal q would ne q(x) = max[0;¼ 0 (x) ¡1 ]. For the domain of x that makes q(x)=0, we have G(x)=0. From now on we restrict to the domain of x that has q(x) = ¼ 0 (x) ¡1 ¸ 0.In this domain, we have the following: G(x)=¼(q(x))¡q(x)x here ¼ 0 (q(x))=x¸0 Since ¼(q) is a concave function,we have that as x increases, q(x) will be non- increasing, i.e. q 0 (x)·0. Now, we have that G 0 (x)=¼ (q(x))q 0 (x)¡q 0 (x)x¡q(x)=¡q(x)·0. Furthermore, we have G 00 (x)=¡q 0 (x)¸0. We conclude that G(x) is a non-increasing, convex function inR + . Proof of Proposition 12. By the notion that h 0 =v(0)¡v(1). It is evidently true that h 0 ¸0. ( the worst scenario for the ¯rm is to have the service \free" when all 89 servers are empty and system switch to state of 1). By the direct use of (3.12), we have the following: ¹h 0 =G(h 0 )¡G(h 1 ))¸0)h 1 ¸h 0 ¸0 (11) The above result comes from Lemma 2 that says G(x) is non-increasing function. The rest of the proof proceeds by induction: By (3.12), we have that ¹s(h s¡1 ¡h s )=G(h s+1 )¡G(h s )+¹h s for s=1;:::;N¡1 (12) Suppose it is true that h s ¸ h s¡1 ¸ 0, it follows that we must also have h s+1 ¸ h s ¸ 0 because G(x) is non-increasing function by lemma 2. Since it is true that h 1 ¸h 0 ¸0, the proof concludes. Proof of Proposition 13. The proof is made by contradiction. Suppose for a given penaltycostF,theoptimalpro¯trateis°(F). AndforsomeF 0 >F,wehavethat the corresponding optimal pro¯t °(F 0 )>°(F). By the use of (3.11), we have that h 0 0 · h 0 due to lemma 2.This leads to the fact that G(h 0 1 )¸ G(h 1 ), thus h 0 1 · h 1 , since only a larger G(h 1 ) can make the (3.12) balanced for s = 1. Proceeding on this iteration until we have h 0 N¡1 · h N¡1 , we encounter contradiction, since the right-hand-side of (3.13)has °(F 0 )+F 0 >°(F)+F as assumed before. Since the optimal pro¯t rate °(F) is non-increasing in F, we have that as F increases, so does h 0 and h 1 ,...until h N¡1 by the same iteration represented by (3.11-3.13). Due to the regularity assumption about the revenue function ¼(q), we have that the optimal arrival rate q(F) is non-increasing function of F.This concludes the proof. 90 Proof of Proposition 14. For a given arrival rate (q o ;:::;q N¡1 ), by a standard birth-and-death analysis of this continuous time Markovian Chain, we have that the blocking probability B(:) is B(:)=1=(1+ N¡1 X j=0 g(j)); whereby g(j) = N!¹ N¡j =j! Q N¡1 k=j q k . As we know from proposition 3, as the penalty cost F increases, q = (q o ;:::;q N¡1 ) is non-increasing, and thus g(j) is non-decreasing for j =0;:::;N¡1. As a conclusion,we have that B(F) is non-increasing. Proof of Lemma 5. Bymultiplyinge ¡¸ ontoboththenumeratoranddenumerator on equation 3.15, we have that blocking probability B(¸;N)= P(X =N) P(X ·N) X isPoisson(¸) Let ¸ = ½N. We have E(X) = ½N, Var(x) = ½N. Furthermore, we have P(X ¸n)· E(X) N =½ by Markov's Inequality. When 0<½<1, we have the following 0·B· P(X =N) 1¡½ = 1 1¡½ ½ N N N e ½N N! By Sterling's formula lim n!1 N!= p 2¼Ne ¡N N N , we have that 0· lim N!1 B· 1 1¡½ (½=e ½¡1 ) N p 2¼N Since 0 < ½ < 1, we have ½=e ½¡1 2 (0;1). Thus we have lim N!1 B = 0, when 0<½<1. 91 When ½ > 1, the proof becomes more involved even the intuition is obvious. Fortunately, we can use the lower/upper bound on the blocking probability due to Harel(1988) to prove its convergence. As in Harel(1988), we have L(N;½) · B(N;½)·U(N;½), whereby L(N;½)= N(½¡1) 3 +2½(½¡1)+(½¡1) N½(1¡½) 2 +2½ 2 ; where ½>1 and U(N;½)= N(1¡½) 2 +2½+(½¡1) p 4N½+N 2 (1¡½) 2 N½(½¡1)+2½+½ p 4N½+N 2 (1¡½) 2 ;where ½>1 As N !1, we have that lim N!1 L(N;½)= (½¡1) 3 ½(½¡1) 2 = ½¡1 ½ ;where ½>1 and also lim N!1 U(N;½)= lim N!1 ½¡1 ½ N(½¡1)+ 2½ ½¡1 + p 4N½+N 2 (½¡1) 2 N(½¡1)+2+ p 4N½+N 2 (½¡1) 2 = ½¡1 ½ ;½>1 since we have that lim N!1 N(½¡1)+ p 4N½+N 2 (½¡1) 2 !1. Thus we have lim N!1 B(N;½)= ½¡1 ½ ;½>1. Forthecaseofheavytra±csuchthat ½=1¡ ¯ p N , asN !1, seeWhitt(2002) or Jagerman(1974). Proof of Lemma 6. We prove it by the method of contradiction. Suppose instead, we have a intensity ½ s 2(0;1) or ½ s >1 when s!1. Case 1: when we assume ½ s 2(0;1). 92 Based on lemma 3, we know in this case, we have that blocking probability lim s!1 B s =0. Thus the average pro¯t rate lim s!1 ° s =1¤(¤¡ q s )q¡0¤F: It can be easily found that we can always increase the average pro¯t rate by increasing the arrival rate q s as far as case 1 is valid. Thus, the optimal arrival rate could not have an intensity ½ s 2(0;1); Case 2: when we assume ½ s > 1. In this parameter regime, we will have lim s!1 B s = ½ s ¡1 ½s =1¡ ·s q ¤ . The average pro¯t rate follows as lim s!1 ° s = ·s q ¤(¤¡ q s )q¡(1¡ ·s q )F =·s¤¡F +·(Fs=q¡q): Also it can be easily found that we can always increase the average pro¯t rate by decreasing the arrival rate as far as case 2 is valid. Thus any arrival rate that has ½ s >1 would not be optimal. Proof of Proposition 15. Ifv(x)isconcavelydecreasing,thenf(x)byitsde¯nition above is increasing function of state x2 [¡ p N;0]. If x increases,by (35) we have thatR 0 (µ x )increases. Sincebyassumptionatthebeginningofthechapter3(A1){ (A5), we have that R(q) is concavely increasing. By checking the relationship shown in (20), we know R(µ) is convexly decreasing function. The conclusion follows by the facts that , when x increases, R 0 (µ) increases,then µ increases, and ¯nally q decreases. 93 Proof of Proposition 17. It is evidently true that K i (0;0)¸ 0 for i = 1;2(maybe need more work to convince it). By (3.54) and (3.55), we have ¼(K 1 (0;0);K 2 (0;0))¡¼(K 1 (1;0);K 2 (1;0))=K 1 (0;0) ¼(K 1 (0;0);K 2 (0;0))¡¼(K 1 (0;1);K 2 (0;1))=K 2 (0;0) By assumption 1, we have that 0· K 1 (0;0)· K 1 (1;0), 0· K 2 (0;0)· K 2 (1;0), 0·K 1 (0;0)·K 1 (0;1), 0·K 2 (0;0)·K 2 (0;1): We proceed on by induction. Suppose it is true that 0 · K 1 (S 1 ¡ 1;S 2 ) · K 1 (S 1 ;S 2 ), and 0·K 2 (S 1 ;S 2 ¡1)·K 2 (S 1 +1;S 2 ¡1). By (3.57), we have S 1 [K 1 (S 1 ¡1;S 2 )¡K 1 (S 1 ;S 2 )]+S 2 [K 2 (S 1 ;S 2 ¡1)¡K 2 (S 1 +1;S 2 ¡1)] =¼(K 1 (S 1 +1;S 2 );K 2 (S 1 +1;S 2 )¡¼(K 1 (S 1 ;S 2 );K 2 (S 1 ;S 2 )) +K 1 (S 1 ;S 2 ) (13) By the assumption made combined with assumption 1 on demand curves, we have that 0·K 1 (S 1 ;S 2 )·K 1 (S 1 +1;S 2 ), and 0·K 2 (S 1 ;S 2 )·K 2 (S 1 +1;S 2 ). This concludes the proof. 94
Abstract (if available)
Abstract
For this thesis, I combined two of my research works on revenue management and dynamic pricing problems and built upon them with current literature and contribute to these themes in two ways. The first way extends the existing, pricing decision model by relaxing commonly used assumptions about demand -- and that is customers tend to be strategic, rather than myopic, in the sense that they time their purchases to maximize their payoffs. The second contribution is to take a different view on analyzing pricing issues that relate to assets, which are renewable, rather than perishable. Thus, this second contribution concerns a different modeling for supply.
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Creator
Tong, Chunyang
(author)
Core Title
Essays on revenue management
School
Marshall School of Business
Degree
Doctor of Philosophy
Degree Program
Business Administration
Degree Conferral Date
2007-05
Publication Date
01/18/2009
Defense Date
11/27/2006
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
dynamic pricing,OAI-PMH Harvest,revenue management
Language
English
Advisor
Dasu, Sriram (
committee chair
), Bassok, Yehuda (
committee member
), Rajagopalan, Sampath (
committee member
), Sosic, Greys (
committee member
), Tan, Guofu (
committee member
)
Creator Email
chunyant@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m227
Unique identifier
UC1500649
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etd-Tong-20070118 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-158902 (legacy record id),usctheses-m227 (legacy record id)
Legacy Identifier
etd-Tong-20070118.pdf
Dmrecord
158902
Document Type
Dissertation
Rights
Tong, Chunyang
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
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Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
dynamic pricing
revenue management