Optimal Static Pricing for a Service Facility with Holding Costs

Transcription

1 Optimal Static Pricing for a Service Facility with Holding Costs Idriss Maoui 1, Hayriye Ayhan 2 and Robert D. Foley 2 1 ZS Associates Princeton, NJ 08540, U.S.A. 2 H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta, GA , U.S.A. idriss.maoui@zsassociates.com, hayhan@isye.gatech.edu, rfoley@isye.gatech.edu Phone: Fax: March 4, 2007 Abstract We study a service facility modelled as a single server queueing system with Poisson arrivals and limited or unlimited buffer size. In systems with unlimited buffer size, the service times have general distributions, whereas in finite buffered systems service times are exponentially distributed. Arriving customers enter if there is room in the facility and if they are willing to pay the posted price. The same price is charged to all customers at all times (static pricing). The service provider is charged a holding cost proportional to the time that the customers spend in the system. We demonstrate that there is a unique optimal price that maximizes the long-run average profit per unit time. We also investigate how optimal prices vary as system parameters change. Finally, we consider buffer size as an additional decision variable and show that there is an optimal buffer size level that maximizes profit. Keywords: Applied Probability, Pricing, Queueing, Revenue Management 1

2 1 Introduction Determining the optimal price to be charged for a service facility is a critical decision for a manager. There is a trade-off between prices and demand that greatly influences revenue. Moreover, there are penalties induced by the congestion of the system that affect the relationship between price and revenue. By appropriately pricing service, the service provider not only maximizes profit but also controls the congestion level in the facility. Although it has applications in other service industries, this paper was originally motivated by the pricing problem of outsourced computer services. These services offer processing power, server time or bandwidth resources and are provided to businesses that do not have sufficient in-house capabilities and hence, present an inexpensive and flexible way to handle spikes in computing needs for businesses with limited resources (see for example +computing+on+demand/ _ html). As businesses computing needs grow larger, these products give smaller companies access to supercomputing power that only very large corporations could afford (see The most prominent providers of such services include IBM, Hewlett-Packard, Cisco Systems, AT&T and Schlumberger. In the same fashion as utilities, the prices of these services should increase with congestion and usage. Our objective is to develop a better understanding of how congestion affects the optimal pricing decisions of the provider of such services. Without some sort of cost related to congestion, the optimal price may result in the arrival rate exceeding the service rate so that the number of customers and the waiting time in the system go to infinity. There are several natural choices to model congestion costs. In this paper, congestion penalties are captured by holding costs. We assume that the service provider is charged a fixed holding cost h per unit time that the customers spend in the system. In Section 3, with a numerical example we will illustrate that when h = 0, the number of customers and the waiting time in the system go to infinity. In some applications, the holding cost may be a cost incurred for storing jobs, particularly if the jobs require large amounts of space or specialized conditions such as being stored in temperature controlled conditions. If the service provider is charged a penalty proportional to the length of time from the start until the finish of a job, then this can be formulated as a holding cost using Little s formula. In other applications, the holding cost may be a surrogate for the loss of goodwill incurred when jobs spend long times in the system, resulting in customers who are unlikely to either return to the service system or be willing to pay as much for service. One advantage of using holding costs to capture congestion penalties is that the system is usually easier to analyze than when congestion penalties are modelled using balking or abandonment. Our objective is to maximize the long-run average profit per unit time of such a service system. We model the service facility as a single server queueing system with Poisson arrivals and finite or infinite buffer size. The revenue manager can only advertise one price at all times. Customers have independent identically distributed valuations of service and enter the system when their valuation is greater than the current advertised price. We will refer to the distribution of the service valuation as willingness-to-pay distribution, and we assume that the associated process is independent of arrival and service times and the fee is paid upon arrival. We demonstrate that there is a unique optimal price that maximizes the long-run average profit per unit time. We provide an expression for determining the optimal price. We also investigate how optimal prices vary as system parameters change. Finally, we conclude our work by considering buffer size as an additional decision variable. We demonstrate that there is an optimal buffer size that maximizes profit. Much of the literature in the area considers the pricing problem in queues in terms of flow and congestion control. Naor [8] and many papers extending his work such as Knudsen [5] and Yechiali 2

3 [11] focus on systems where customers decide to join according to the congestion of the system when they enter. Entering customers obtain a fixed reward and are charged a holding cost function of their time spent in the system. In order to maximize their self-interest, they decide to join or balk (join-balk rule). Stidham [9] develops an admission control model for single server systems, where the service provider controls the entry rate and also analyzes a multi-server system, where each server has its own buffer. In this setting, the service provider sets the routing probabilities of entering customers into the servers queues in order to minimize the average queue length. Larsen [6] and Hassin [2] consider the impact of releasing the exact system congestion status to potential customers as opposed to the expected queue length as they make their join-balk decision. Ittig [3] develops a model in which congestion is treated as a form of price. His objective is not optimal pricing but determining the optimal number of servers for the service facility. He introduces a general demand function relating average waiting time and demand rate as well as a cost of service capacity. He sets up a nonlinear constrained optimization problem where the queueing link between demand rate and average waiting time is a constraint. Ittig [4] is also interested in estimating the optimal number of servers through transaction data when the relationship between demand and congestion is not explicitly known. Lautenbacher and Stidham [7] and Subramanian et al. [10] present the connection between airline yield management and queueing admission control problems. Under the assumption that there are no cancellations, overbookings and discounts, Lautenbacher and Stidham [7] present a coherent framework linking dynamic and static seat allocation models through the underlying dynamic program that is common to both. Subramanian et al. [10] analyze a Markov decision process model for airline seat allocation on a single-leg flight with multiple fare classes, overbookings, cancellations and no shows. Ziya [12] and Ziya et al. [14] focus on optimal static pricing for systems without holding costs in M/G/1/ and M/M/1/N queueing systems. Instead of using a congestion-based join-balk rule, they link the customers arrival rate with the posted price through a random service valuation by each customer. They use a willingness-to-pay distribution to capture the proportion of customers willing to pay the posted price and shows the existence of a unique optimal price that maximizes the long-run average profit. Ziya [12] and Ziya et al. [14] also exhibit how the optimal price changes as system parameters vary. We extend the work of Ziya [12] and Ziya et al. [14] by introducing holding costs in our analysis and by considering buffer size as a decision variable. The inclusion of holding costs enables us to capture the customers sensitivity to waiting times in our optimal pricing decision. Disregarding holding costs when pricing service leads to setting lower prices that do not offset the loss incurred by making customers wait. This is particularly relevant in practice when the service in question is commoditized and customers can easily switch providers when they are dissatisfied with their waiting times. Holding costs make the analysis of the problem more difficult but yield different properties of optimal prices especially with regard to the buffer size. Rest of the paper is organized as follows. In Section 2, we introduce the model and the notation used in our analysis. Section 3 provides the expression of the optimal price in an M/G/1/ system and studies its properties. In Section 4, we focus on an M/M/1/N system. We investigate how the optimal price varies as the system parameters change and also show that there is an optimal buffer size that maximizes profit. Section 5 concludes the paper. Proofs of the results not given in the text are provided in the Appendix. 3

4 2 Model Description The service provider can only advertise one price at all times for all customers. Let y denote the mark-up charged for service. Note that the price to be charged is the sum of the mark-up and the variable cost of service. Without loss of generality, we assume that the variable cost of service is zero, so the mark-up is equal to the price. We model the service facility as a single server system, where N is the maximum number of customers allowed in the system at any time. Arriving customers enter if they are willing to pay the price charged by the service provider. Let N(t) be the number of arrivals in the time interval (0, t]. We assume that {N(t) : t 0} is a Poisson process with rate Λ. We call Λ the maximum arrival rate. For y 0, let F(y) be the proportion of customers willing to pay a price of y. We call F( ), the willingness-to-pay distribution. We assume that the cumulative distribution function F( ) is absolutely continuous with density f( ), support (α, β) where α 0, and β and finite f(y) 1 F(y) mean. Let r( ) denote the hazard rate function of F( ); that is, r(y) = for α < y < β and we define r(y) = 0 for y α and r(y) = for y β. In what follows, we assume that F has IGHR (Increasing Generalized Hazard Rate); that is, yr(y) is strictly increasing for all y in [α, β). As discussed in Ziya et al. [13, 14], IGHR assumption is equivalent to having a demand function with increasing price elasticity. Many common distribution functions (such as exponential and uniform distributions) have this property that simply states that the demand becomes more elastic as prices increase. Let N(y, t) be the number of customers who are willing to pay a price of y and arrive during (0, t]. If we let λ(y) denote the arrival rate of customers who are willing to pay a price of y, then λ(y) = Λ(1 F(y)) = lim t N(y,t) t. Service times are independent, identically distributed random variables with distribution G( ), mean 1 µ and squared coefficient of variation c2 s. The service process, the arrival process and the process associated with the amounts successive customers are willing to pay are assumed to be independent. When the price is y, the number of customers in the system forms a queueing process with Poisson arrival process {N(y, t) : t 0} and independent, identically distributed service times with c.d.f G( ). When an arriving customer is willing to pay the posted price, the customer enters the system if the system is not full; otherwise, the customer is lost. Moreover, we assume that each entering customer pays the posted price at the time of arrival and the service provider is charged $ h per unit time while the customer is in the system. To ensure that a positive long-run profit is attainable, we will assume that h µ < β. Let ρ(y) = Λ µ (1 F(y)) denote the traffic intensity when the price is y. Let ŷ be the maximum price under which we have a traffic intensity of 1; that is, ŷ = sup{y : ρ(y) = 1} when Λ µ 1. Note that when Λ µ < 1, ŷ =. When they exist, {π n(ρ(y), N)} and L(ρ(y), N) denote the stationary distribution and the expected number of customers in the system for traffic intensity ρ(y) and buffer size N. Let R(y, N) be the long-run average profit per unit time for a posted price y and buffer size N. When it exists and is unique, we let yn denote the optimal price to be charged to maximize R(y, N) and RN = R(y N, N) denote the optimal objective value. 4

5 3 Optimal Pricing for M/G/1/ Queues 3.1 Existence and Uniqueness of Optimal Prices In the following, we derive expressions for R(y, ) and y when no further assumptions are made on the service time distribution. Only customers who are willing to pay the posted price y enter the system and they pay y immediately. Since the service provider is charged an additional cost of h per unit time that the customers spend in the system, N(y,t) yn(y, t) h R(y, ) = lim t t k=1 D k where {D k : k = 1, 2,... } is the sequence of the total waiting times for successive customers. Note that from Little s Law we have N(y,t) N(y,t) k=1 D k k=1 D k N(y, t) lim = lim L(ρ(y), ). t t t N(y, t) t Therefore, we can write the long-run average reward per unit time as R(y, ) = yλ(y) hl(ρ(y), ). Clearly, if ρ(y) 1 and h > 0, then L(ρ(y), ) = and R(y, ) =. From the Pollaczek- Khinchin formula [1], if ρ(y) < 1, Therefore, if ρ(y) < 1, L(ρ(y), ) = ρ(y)(2 ρ(y)(1 c2 s)). 2(1 ρ(y)) R(y, ) = yλ(y) h ρ(y)(2 ρ(y)(1 c2 s)). 2(1 ρ(y)) Note that the long-run average reward function consists of two terms : the first describing the revenue through the arrival rate regardless of the service times, whereas the second accounts for the additional holding cost through the steady-state average number of customers in the system. Theorem 3.1 shows the existence and the uniqueness of an optimal price. Theorem 3.1 Under the IGHR assumption, there exists a unique optimal price given by : y = inf{y : r(y)(y h ϕ(ρ(y))) 1}, (1) µ, where 1+ρ(c 2 s 1)(1 ρ 2 ) if ρ < 1, h > 0, (1 ρ) 2 ϕ(ρ) = if ρ 1, h > 0, 0 if h = 0. The following lemma provides a structural result that is needed in the proof of Theorem 3.1. Lemma 3.1 Under the IGHR assumption, r(y)(y h µ ϕ(ρ(y))) is strictly increasing in y [inf{y : r(y)(y h µ ϕ(ρ(y))) 1}, β). 5

6 Proof Note that ϕ(ρ(y)) y = r(y)ρ(y)(c2 s +1) 0, which implies that ϕ(ρ(y)) is nonincreasing in (1 ρ(y)) 3 y [ˆα, β) where ˆα = max(α,ŷ). Pick y 1, y 2 [inf{y : r(y)(y h µ ϕ(ρ(y))) 1}, β) such that y 1 < y 2. If r(y 1 ) r(y 2 ), the result follows immediately since ϕ(ρ(y)) is nonincreasing in y. On the other hand, if r(y 1 ) > r(y 2 ), then we have 1 r(y 1 )(y 1 h µ ϕ(ρ(y 1))) < r(y 2 )y 2 r(y 1 ) h µ ϕ(ρ(y 1)) < r(y 2 )y 2 r(y 2 ) h µ ϕ(ρ(y 2)), where the first inequality follows from the IGHR assumption and the second inequality follows from the fact that ϕ(ρ(y)) is nonincreasing in y and r(y 1 ) > r(y 2 ). Hence, the proof is complete. We can use Theorem 3.1 to derive the following result for M/M/1/ queueing systems. Corollary 3.1 If the service times are exponentially distributed and h 0, then there exists a unique optimal price as defined in (1), and in this case 1 if ρ < 1, h > 0, (1 ρ) 2 ϕ(ρ) = if ρ 1, h > 0, 0 if h = 0. Before presenting our next result, we will show in a simple example how the optimal price and traffic intensity vary with the holding cost h and how having a positive holding cost h stabilizes the queueing system. Consider an M/M/1/ queueing system with service rate µ = 1, arrival rate Λ = 10, and willingness-to-pay distribution F(y) = 1 e y. First, if h is zero, the optimal price from Corollary 3.1 is y = 1, and the traffic intensity is ρ(1) Note that the server cannot keep up, the system is unstable and the queue length explodes. If h > 0, then the server must be able to keep up at the optimal price, which means that ρ(y ) < 1. In this example, ρ(y) < 1 if and only if the price y > ρ 1 (1) Figure 1 shows the amount the optimal price increases from ρ 1 (1) and how the traffic intensity changes as h > 0 increases. The graph is not continuous at zero, and the values of both functions at h = 0 are not shown on the graph. Note that as h decreases to zero, the optimal price y decreases to ρ 1 (1), and the traffic intensity increases to 1. At the other extreme (not shown in the figure) as h increases, the optimal price y increases (asymptotically to h + 1) and the traffic intensity converges to zero. For many willingness-to-pay distributions, it can be difficult to obtain a closed form expression for y. The next proposition whose proof is given in the Appendix provides crude bounds on the optimal price. The lower bound indicates that the optimal price should be greater than or equal to the smallest expected holding cost that the service provider is charged and the upper bound is obtained by replacing ϕ(ρ(y)) with ϕ( Λ µ ) in the expression of y. These bounds will be sufficient to compare the properties of systems with and without holding costs in the next section. Proposition 3.1 The unique optimal price y satisfies h µ y. Moreover, if Λ < µ, then y inf{y : y(r(y) h µ ϕ(λ )) 1}. µ 3.2 Properties of Optimal Prices in M/G/1/ Queues In this section, we compare the optimal price and the optimal reward in two M/G/1/ systems (indexed by 1 and 2). These two systems differ by marginal holding cost, maximum arrival rates, 6

7 y ρ 1 (1) ρ(y ) h Figure 1: The traffic intensity ρ(y ) and y ρ 1 (1) as a function of 0 < h < 1 service rates and squared coefficients of variation. Moreover, we also compare systems where the willingness-to-pay distributions are ordered in the stochastic ordering and hazard rate ordering. Recall that distribution F 1 is greater than or equal to distribution F 2 in the stochastic ordering (F 1 ST F 2 ) if and only if F 1 (y) F 2 (y), y 0. Furthermore, distribution F 1 is greater than or equal to distribution F 2 in the hazard rate ordering (F 1 HR F 2 ) if and only if r 1 (y) r 2 (y), y 0. Our objective is to compare the optimal prices y,1 and y,2 for these two systems. In systems without holding cost, Ziya [12] shows that if F 1 HR F 2, then y,1 y,2. We show in the next proposition that this result still holds when holding costs are incurred. However, stochastic ordering of the willingness-to-pay distributions does not necessarily guarantee ordered optimal prices (see Section 3.4 in Ziya [12] for a counterexample). In the remainder of this section, parameters relative to system i = 1, 2 are indicated by subscript i. Proposition 3.2 Consider two systems 1 and 2 such that Λ 1 Λ 2, F 1 HR F 2, h 1 h 2, c 2 s,1 c2 s,2, and µ 2. Then y,1 y,2. Proof From Theorem 3.1, we have y,i = inf{y : r i(y)(y h i µ i ϕ i (ρ i (y))) 1} for system i = 1, 2. Since hazard rate ordering implies stochastic ordering, F 1 ST F 2. In conjunction with conditions 1 and 5, this implies that ρ 1 ( ) ρ 2 ( ). Moreover, note that ϕ( ) is nondecreasing. Therefore, we have ϕ 1 (ρ 1 ( )) ϕ 2 (ρ 2 ( )). Suppose that y is such that r 1 (y)(y h 1 ϕ 1 (ρ 1 (y))) 1. Then, r 2 (y)(y h 2 µ 2 ϕ 2 (ρ 2 (y))) 1. Thus, y,2 y,1. As shown in the proof of Proposition 3.2, it is intuitive that systems with higher maximum arrival rate, smaller service rates, higher service variance and higher marginal holding cost yield 7

8 higher long-run average holding cost. Therefore, it is not surprising that the optimal price should be higher when higher holding costs are incurred. For systems with no holding cost, Ziya [12] shows that if Λ 1 Λ 2 µ 2, then y1 y 2. However, this result does not extend to facilities with holding costs. Consider two M/M/1/ systems, with = 1, µ 2 =.1, Λ 1 = 0.5, and Λ 2 = So, Λ 2 µ 2 = Λ 1 = 0.5. For both systems, assume that the willingness-to-pay distribution is exponential with rate 1. According to Proposition 3.1 the optimal solution for system i = 1, 2 satisfies: h y hµ i,i 1 + µ i (µ i Λ i ) 2. So, y,1 4h + 1 and 10h y,2. Consider h = 1. Therefore, y,1 < y,2, although Λ 1 Λ 2 µ 2. In the next proposition, whose proof is given in the Appendix, we analyze how the optimal reward varies as parameters change. Proposition 3.3 Consider two systems 1 and 2 such that Λ 1 Λ 2, F 1 ST F 2, h 1 h 2, c 2 s,1 c2 s,2, and µ 2. Then R,1 R,2. 4 Optimal Pricing for M/M/1/N Queues 4.1 Existence and Uniqueness of Optimal Prices In this section, we study optimal pricing for capacitated queues. We focus on M/M/1/N queueing systems for which we can easily quantify the long-run average queue length and the long-run average reward function. We prove the existence of a unique optimal price under the IGHR assumption and derive ordering properties as system parameters change. Only customers who are willing to pay the posted price y and find fewer than N customers in the system are allowed to enter. Therefore, Nin (y,t) yn in (y, t) h k=1 D k R(y, N) = lim t t where N in (y, t) denotes the number of customers allowed in the system up to time t. It follows from PASTA that π N (ρ(y), N) is the fraction of arrivals who find the system full. Thus, we have R(y, N) = yλ(y)(1 π N (ρ(y), N)) hl(ρ(y), N). Recall from Gross and Harris [1] that { 1 ρ if ρ 1, π 0 (ρ, N) 1 ρ N+1 1 N+1 if ρ = 1, π n (ρ, N) = ρ n π 0 (ρ, N) for n = 1,...,N, and L(ρ(y), N) = ρ(y)(1 (N + 1)ρ(y)N + Nρ(y) N+1 ) (1 ρ(y))(1 ρ(y) N+1. ) One can also express the long-run average reward per unit time as R(y, N) = yµ(1 π 0 (ρ(y), N)) hρ(y) (1 (N + 1)ρ(y)N + Nρ(y) N+1 ) (1 ρ(y))(1 ρ(y) N+1. ) We demonstrate the existence and the uniqueness of an optimal price in Theorem

9 Theorem 4.1 There exists a unique optimal price given by : where and ϕ N (ρ) = y N = inf{y : r(y)γ N (ρ(y))(y h µ ϕ N(ρ(y))) 1}, L(ρ,N) ρ π 0 (ρ,n) ρ γ N (ρ) { 1 (N+1) 2 ρ N (1+ρ 2 )+2N(N+2)ρ N+1 +ρ 2N+2 (1 ρ) 2 (1 (N+1)ρ N +Nρ N+1 ) if ρ 1, 1 6 N N if ρ = 1, { 1+Nρ N+1 (N+1)ρ N (1 ρ N+1 )(1 ρ N ) if ρ 1, 1 2 if ρ = 1. The following lemma is needed in the proof of Theorem 4.1. Lemma 4.1 The function ϕ N ( ) is nondecreasing on [0, ). Since in general it might be difficult to get a closed form expression for the optimal price, the next proposition provides some crude bounds on yn. The lower bound again indicates that the optimal prices is greater than or equal to the minimal holding cost and the upper follows immediately by replacing ϕ N (ρ(y)) with ϕ N ( Λ µ ) and γ N(ρ(y)) with γ N ( Λ µ ) in the expression of yn. These bounds will again be sufficient to compare the properties of systems with and without holding costs in the next section. Proposition 4.1 The unique optimal price y N satisfies h µ y N. Moreover, yn inf{y : r(y)γ N ( Λ µ )(y h µ ϕ N( Λ )) 1}. µ 4.2 Properties of Optimal Prices in M/M/1/N Queues In this section, as in the M/G/1/ case, we compare the optimal prices of two systems with different parameters. In the remainder of this section, parameters relative to system i = 1, 2 are indexed by i. First, we study how the optimal price yn changes as buffer size N increases. From Proposition 4.2 in Ziya [14], we know that when h = 0, the optimal price is increasing (decreasing) with respect to the buffer size when Λ µ > (<)ρc, where ρ c is called the critical traffic intensity (ρ c = (1 F(inf{y : yr(y) 2})) 1 ). However, when h > 0, this is not always the case. Let F(y) = 1 e βy, with β = 0.1 and Λ = 8, µ = 2 and h = 1. In this case, when buffer size is 5,6 and 7, the optimal prices y5, y 6 and y 7 are , , , respectively. Hence, the optimal price is not monotone in buffer size. However, the next result shows that optimal prices are ordered with respect to other system parameters. Proposition 4.2 Consider two systems 1 and 2 such that F 1 HR F 2, Λ 1 Λ 2, µ 2, and h 1 h 2. Then y N,1 y N,2. Proof Suppose conditions 1 through 4 hold. We have ρ 1 (y) ρ 2 (y), y in [α, β) and h 1 h 2 µ 2. Moreover, as shown in Lemma 4.1, ϕ N ( ) is nondecreasing and γ N ( ) is nonincreasing. Therefore, ϕ N (ρ 1 (y)) ϕ N (ρ 2 (y)) and γ N (ρ 2 (y)) γ N (ρ 1 (y)) for all y in [α, β). 9

10 Let y [α, β) be such that r 1 (y)γ N (ρ 1 (y))(y h 1 ϕ N (ρ 1 (y))) 1. Using the properties shown above, we have r 2 (y)γ N (ρ 2 (y))(y h 2 µ 2 ϕ N (ρ 2 (y))) r 1 (y)γ N (ρ 1 (y))(y h 1 ϕ N (ρ 1 (y))) 1. Hence, y N,2 y N,1. Similar to the infinite buffer size case, Proposition in Ziya [12] shows that in systems with no holding cost, Λ 1 Λ 2 µ 2 implies that yn,1 y N,2. This result cannot be extended to systems with holding costs. To see this, consider two M/M/1/2 systems, 1 and 2, where = 1, µ 2 = Λ.1, Λ 1 = 0.5, Λ 2 = So, 2 µ 2 = Λ 1 = 0.5. For both systems, assume that the willingness-to-pay distribution is exponential with rate 1. Therefore, from Proposition 4.1, we have yn, h and 10h y N,2. When h = 1, y,1 < y,2. We can claim that the arrival rate, service rate and hazard rate orderings that hold when h = 0 in the M/M/1/N case still hold when h > 0. However, as in the infinite buffer size case, the traffic intensity ordering without holding costs cannot be extended when h > 0. The following proposition shows that the optimal rewards are also ordered as the system parameters change. Proposition 4.3 Consider two systems 1 and 2 such that Λ 1 Λ 2, F 1 ST F 2, h 1 h 2, and µ 2. Then, R N,1 R N,2. Theorem 4.2 shows that the infinite buffer size model can be approximated by a finite buffer size model of large size provided that it is stable for all prices. We show that both the optimal reward and optimal price of a finite buffer size model converge to those of an infinite buffer size system as the system size grows to infinity. The proof of Theorem 4.2 requires the following lemma. Lemma 4.2 Under the stability condition Λ < µ, R(y, N) R(y, ) uniformly in y as N converges to infinity. Theorem 4.2 Under the stability condition Λ < µ, R N R and y N y as N. 4.3 Optimal Buffer Size We showed that the infinite buffer size model can be approximated by a finite (but large) buffer size model under the condition Λ < µ. A natural question that stems from this result is whether there is a buffer size level that maximizes the reward. Indeed, in our analysis so far, buffer size is a given parameter. Now, we relax this constraint by allowing the service provider to set the buffer size of the service facility in addition to the price. Note that the chosen buffer size could be finite or infinite. Ziya [14] shows that systems with larger capacities always perform better when there is no holding cost. In this case, the service provider should have an infinite buffer size system in order to maximize revenue. Thus, no customer is ever turned down due to buffer size limitations. However, when h > 0, there is a trade-off between large buffer size and high holding costs. In the following, we show the existence of a buffer size level N < that maximizes revenue when Λ < µ and h > 0. Proposition 4.4 If Λ < µ and h > 0, then there exists a buffer size level N <, such that R N = sup N R N. Consequently, there exists an optimal solution to sup y,n R(y, N). We need the following lemma in order to prove Proposition

11 Lemma 4.3 If Λ < µ and h > 0, then, B 0, there exists N IN such that for all y B, R(y, N) is nonincreasing for N N. Proof of Proposition 4.4 We showed in Theorem 4.2 that yn converges to y. Therefore, let y = sup N {yn } <. We use Lemma 4.3 to define N = 1 + max{n : y y, R(y, N + 1) > R(y, N)}. For y y and N N, we have R(y, N + 1) R(y, N). So, for N N, R N+1 = sup y y R(y, N + 1) sup R(y, N) = RN. y y Therefore, R N is nonincreasing for N N, which implies that R N = sup N R N exists. 5 Summary In this paper, we study the optimal pricing problem of a service facility modelled as a single-server queueing system with holding costs. Our objective is to maximize the service provider s long-run average profit per unit time under static pricing. We assume that the service provider is charged a fixed cost per unit of time that the customers spend in the system and that each customer randomly chooses to pay the advertised price according to a willingness-to-pay distribution. We show the existence of a unique optimal price in M/G/1/ and M/M/1/N queues. We provide expressions and bounds for the optimal prices we derive. Moreover, we analyze the sensitivity of the optimal prices and profits to the system parameters, ie., arrival rate, service rate and willingness-to-pay distribution. Then, we specifically focus on the effect of the system buffer size N on optimal prices and profits in M/M/1/N queues. Although there is no optimal price ordering with respect to buffer size, we show that there is a finite buffer level that maximizes profit in M/M/1/N queues when holding costs are incurred. In this paper, we captured congestion penalties via holding cots. In our current research, we are investigating two other methods of including congestion penalties in our models, namely, customer reneging and customer balking and studying the relationship among these three different methods. Appendix Proof of Theorem 3.1 Since the proof of h = 0 case is shown in Proposition of Ziya [12], assume h > 0 and let ˆα = max(α,ŷ). Note that for all y less than or equal to ŷ, the reward function is equal to. Therefore, an optimal price, if it exists, has to be greater than ŷ. Since F( ) is absolutely continuous, for all y in [ˆα, β), R(y, ) is continuous and a.e. differentiable on [ˆα, β). We can rewrite R(y, ) as R(y, ) = ρ(y)(µy h (2 ρ(y)(1 c2 s)) ). 2(1 ρ(y)) Note that µy h (2 ρ(y)(1 c2 s)) 2(1 ρ(y)) µβ h > 0 as y tends to β. 11

12 Therefore, there exists y in [ˆα, β) such that R(y, ) > 0. Moreover, for all y in [ˆα, β), R(y, ) < and R(y, ) 0 as y β. So, there exists an optimal price in [ˆα, β). Note that R(y, ) y > 0(< 0) if and only if r(y)(y h µ (ϕ(ρ(y)))) < 1(> 1). Since there exists y in [ˆα, β) such that R(y, ) > 0 and R(y, ) 0 as y β, there exists y in [ˆα, β) such that r(y)(y h µ ϕ(ρ(y))) > 1. It follows from Lemma 3.1 that R(y, ) is decreasing in the interval (inf{y : r(y)(y h µ ϕ(ρ(y))) 1}, β). In the same fashion, R(y, ) is increasing in the interval (ˆα,inf{y : r(y)(y h µ ϕ(ρ(y))) 1}). Therefore, y = inf{y : r(y)(y h µ ϕ(ρ(y))) 1}. Proof of Proposition 3.1 First, suppose that y < h µ. If h = 0, then there is clearly a contradiction. Assume now that h > 0. Since L(ρ(y), ) ρ(y), we have R(y, ) y λ(y ) h µ λ(y ) < 0, which is a contradiction since we proved in Theorem 3.1 that there exists y in [ˆα, β) such that R(y, ) > 0. Therefore, h µ y. Now suppose Λ < µ. From Theorem 3.1, y = inf{y : r(y)(y h µ ϕ(ρ(y))) 1}. Since ϕ(ρ( )) is nonincreasing, 1+Λ µ (c2 s 1)(1 Λ 2µ ) = ϕ( Λ (1 Λ µ )2 µ ) ϕ(ρ(y)) for y 0. Therefore, for all y in (α, β) such that r(y)(y h µ ϕ(λ µ )) 1, we have r(y)(y h µ ϕ(ρ(y))) 1. This completes the proof. Proof of Proposition 3.3 To prove this result, we split our proof into two parts. First, we show that the result holds when conditions 1 and 2 are changed to equalities. Second, we show that it holds when conditions 3,4 and 5 are changed to equalities. By composition, the result holds under all the conditions as well. Suppose that conditions 3,4 and 5 hold and Λ 1 = Λ 2 and F 1 ( ) = F 2 ( ). Recall that for all y 0, R i (y, ) = yλ i (y) h i L i (ρ i (y), ). Conditions 3,4 and 5 imply that h 1 L 1 (ρ 1 (y), ) h 2 L 2 (ρ 2 (y), ). Therefore, R 1 (y, ) R 2 (y, ) and R,1 R,2. Now suppose that 1 and 2 hold, whereas 3,4 and 5 are equalities. Since F 1 ( ) is absolutely continuous and λ 1 ( ) λ 2 ( ), there exists δ > 0 such that λ 1 (y,2 + δ) = λ 2(y,2 ). Therefore, system 1 with price y,2 + δ has the same arrival and service rates as system 2 with price y,2. Therefore, system 1 with price y,2 + δ performs better than system 2 with optimal price. Hence, R,1 R,2 and the proof is complete. Proof of Lemma 4.1 In what follows, all derivatives are with respect to ρ. For simplicity, we omit the arguments of the functions. Since ϕ N is differentiable, But we also have π 0 = π2 0 F ρ and ϕ N = L π 0 L π 0, π 0 2 L = π 0 F, where F(ρ) = L = π 0F + F π 0, N nρ n, L = π 0 F + 2F π 0 + Fπ 0 and L π 0 L π 0 = 2F π F π 0 π 0 F π 0 π 0. π 0 = F π 2 0 ρ + Fπ2 0 ρ 2 2 Fπ 0π 0 ρ = F π 2 0 ρ + Fπ2 0 ρ π 2 0. π 0 12

13 Since F π 0 π 0 = F 2 π 3 0 ρ + FF π 3 0 ρ 2 + 2F π 2 0, L π 0 L π 0 =F π 0 π 0 + F 2 π 3 0 ρ FF π0 3 ρ 2 = FF π0 3 ρ + F 2 π 3 0 ρ FF π 3 0 ρ 2, Therefore, = π3 0 ρ (FF + F ( F + F )) and ρ F + F N ρ = N N nρ n 1 n 2 ρ n 1 = n(n 1)ρ n 1. n=2 L π 0 L π 0 = π3 N 0 ρ ( N nρ n 1 n 2 (n 1)ρ n 1 = π3 N 0 ρ 3 ( N nρ n n 2 (n 1)ρ n = π3 2N 0 ρ 3 ( k=2 ρ k min(n,k) n=max(0,k N) N N n(n 1)ρ n 1 n 2 ρ n 1 ), N N n(n 1)ρ n n 2 ρ n ), n(k n)(k n 1)(k 2n)). Let a k = min(n,k) n=max(0,k N) n(k n)(k n 1)(k 2n). We will prove that a k is positive. Using the fact that max(0, k N) = k min(n, k), a k = min(k,n) k 2 ( n= k 2 min(n,k) k 2 n)(k 2 + n)(k 2 + n 1)2n = min(k,n) k 2 n= k 2 min(n,k) b n,k. Since b n,k = ( k 2 n)(k 2 + n)(k 2 + n 1)2n b n,k( k 2 n)(k 2 + n)(k 2 n 1)2n, we have a k 0. Thus, L π 0 L π 0 0 and ϕ N (ρ) 0. Proof of Theorem 4.1 First, we will prove that there exists an optimal solution. Note that for all y in [α, β), L(ρ(y), N) L(ρ(y), ) ρ(y) 1 ρ(y). When y is in the neighborhood of β, ρ(y) < 1 and L(ρ(y), ) <. So, for y in the neighborhood of β, Under the assumption that h µ < β, R(y, N) yλ(y)(1 π N (ρ(y), N)) h ρ(y) 1 ρ(y) = ρ(y) 1 ρ(y) (µy(1 π N(ρ(y), N))(1 ρ(y)) h). µy(1 π N (ρ(y), N))(1 ρ(y)) h µβ h > 0, as y β. Therefore, there exists y in [α, β) such that R(y, N) > 0. Note that R(, N) is continuous and also bounded on [α, β) since R(y, N) 0 as y β. Hence, there exists an optimal price in [α, β). Note that R(y,N) y > 0(< 0) if and only if r(y)γ N (ρ(y))(y h µ ϕ N(ρ(y))) < 1(> 1). Since there exists y in [α, β) such that R(y, N) > 0 and R(y, N) 0 as y β, there exists y in [α, β) such 13

14 that r(y)γ N (ρ(y))(y h µ ϕ N(ρ(y))) 1. It remains to prove that r(y)γ N (ρ(y))(y h µ ϕ N(ρ(y))) is increasing in [inf{y : r(y)γ N (ρ(y))(y h µ ϕ N(ρ(y))) 1}, β). According to Lemma A.1 in Ziya [14], γ N (ρ(y)) is nondecreasing for y > 0. Since ϕ N ( ) is also nondecreasing, using arguments similar to the ones in the proof of Lemma 3.1, one can show that under the IGHR assumption r(y)γ N (ρ(y))(y h µ ϕ N(ρ(y))) is increasing in the interval [inf{y : r(y)γ N (ρ(y))(y h µ ϕ N(ρ(y))) 1}, β). Hence, R(y, N) is decreasing in (inf{y : r(y)γ N (ρ(y))(y h µ ϕ N(ρ(y))) 1}, β). In the same fashion, R(y, N) is increasing in (α,inf{y : r(y)γ N (ρ(y))(y h µ ϕ N(ρ(y))) 1}). Therefore, y N = inf{y : r(y)γ N(ρ(y))(y h µ ϕ N(ρ(y))) 1}. Proof of Proposition 4.1 First, suppose that y N < h µ. Since L(ρ(y), N) ρ(y), R(y N, N) yn λ(y N ) h µ λ(y N ) < 0. We proved in Theorem 4.1 that there exists y in [α, β) such that R(y, N) > 0. Thus, yn can not be optimal. Therefore, h µ y N. Recall from Theorem 4.1 that y N = inf{y : r(y)γ N (ρ(y))(y h µ ϕ N(ρ(y))) 1}. Moreover, it follows from Lemma 4.1 that ϕ N (ρ(y)) is nonincreasing with respect to y. Thus, ϕ N (ρ(y)) ϕ N ( Λ µ ) In the same fashion, since γ N (ρ(y)) is nondecreasing with respect to y, γ N ( Λ µ ) γ N(ρ(y)). Let y be in [α, β) such that r(y)γ N ( Λ µ )(y h µ ϕ N( Λ µ )) 1. Using the previous orderings, we have r(y)γ N (ρ(y))(y h µ ϕ N(ρ(y))) r(y)γ N ( Λ µ )(y h µ ϕ N( Λ )) 1. µ Proof of Proposition 4.3 To prove this result, we split our proof into two parts as we did in the M/G/1/ case. First, we show that the result holds when conditions 1 and 2 are replaced by equalities. Second, we show that it holds when conditions 3 and 4 are equalities. By composition, the result holds under all conditions as well. Suppose that conditions 3 and 4 hold and Λ 1 = Λ 2 and F 1 ( ) = F 2 ( ). Recall that for all y 0, R i (y, N) = yλ i (y)(1 π N (ρ i (y), N)) h i L(ρ i (y), N). Conditions 3 and 4 imply that h 1 L(ρ 1 (y), N) h 2 L(ρ 2 (y), N) and that π N (ρ 1 (y), N) π N (ρ 2 (y), N). Therefore, R 1 (y, N) R 2 (y, N) and RN,1 R N,2. When conditions 1 and 2 hold but conditions 3 and 4 are equalities, the proof is similar to the proof of Proposition 3.3 and is omitted. Proof of Lemma 4.2 Let y 0. Consider R(y, ) = λ(y)y h 14 nπ n (ρ(y), ). n=0

15 Now consider a system with buffer size N with posted price y and reward corresponding to this price N R(y, N) = λ(y)y(1 π N (ρ(y), N)) hπ 0 (ρ(y), N) n λ(y)n µ n. We observe that π N (ρ(y), N) = ρ(y) N 1 ρ(y) 0 uniformly in y as N tends to infinity. In 1 ρ(y) N+1 1 ρ(y) the same fashion, π 0 (ρ(y), N) (1 ρ(y)) = π 1 ρ(y) N+1 0 (ρ(y), ) uniformly in y as N goes to. Finally, n=n+1 nλ(y)n µ n n=n+1 nλn µ 0 uniformly in y. Therefore, N n n=0 nλ(y)n µ n n=0 nλn µ uniformly in y, and hence, π n 0 (ρ(y), N) N n=0 nλ(y)n µ π n 0 (ρ(y), ) n=0 nλ(y)n µ = n n=0 nπ n(ρ(y), ) where the convergence is again uniform in y. The result now follows. Proof of Theorem 4.2 According to the previous lemma, R(y, N) R(y, ) uniformly in y. Therefore, RN converges to R as N goes to infinity. Let {yn(m) } be a converging (to y) subsequence of {y N }. Before we proceed, we need to show that such a subsequence exists and that y <. Since yn 0 for all N, it suffices to show that yn does not converge to infinity as N tends to infinity. Suppose that limy N =. Note that RN = R(y N, N) λ(y N )y N. Since λ(y N )y N 0 as N tends infinity (recall that F is assumed to have finite mean), RN 0. But this is a contradiction since we showed that R N R > 0. Therefore, {yn(m) } exists and has a finite limit. To simplify the notation in the remainder of the proof, we use N instead of N(m). Next, we show that R(y, ) RN 0 as N goes to infinity. Since R N R, this implies that R(y, ) = RN. Therefore, y = y since y is unique as shown in Theorem 3.1. We have n=0 N π 0 (ρ(yn), N) 1 π 0 (ρ(y), ) 1 ρ(yn) n ρ(y) n. Therefore, for an arbitrary integer M between 1 and N, Thus, M π 0 (ρ(yn), N) 1 π 0 (ρ(y), ) 1 (ρ(yn) n ρ(y) n ) + π 0 (ρ(y N), N) 1 π 0 (ρ(y), ) 1 N n=m+1 ρ(y N) n M ρ(yn) n ρ(y) n + 2 n=m+1 n=m+1 Λ n µ n. ρ(y) n. First, let N go to infinity and then let M go to infinity. Consequently, π 0 (ρ(y N ), N) π 0(ρ(y), ). We also have R(y, ) RN =µy(1 π 0 (ρ(y), )) µyn(1 π 0 (ρ(yn), N)) N hπ 0 (ρ(y), ) nρ(y) n + hπ 0 (ρ(yn), N) nρ(yn) n. 15

16 So, for an arbitrary integer M between 1 and N, Then, R(y, ) RN µ y(1 π 0 (ρ(y), )) yn(1 π 0 (ρ(yn), N)) M + h n π 0 (ρ(yn), N)ρ(yN) n π 0 (ρ(y), )ρ(y) n + hπ 0 (ρ(y), ) n=m+1 nρ(y) n + hπ 0 (ρ(y N), N) N n=m+1 R(y, ) RN µ y(1 π 0 (ρ(y), )) yn(1 π 0 (ρ(yn), N)) M + h n π 0 (ρ(yn), N)ρ(yN) n π 0 (ρ(y), )ρ(y) n + 2h nρ(y N) n. n=m+1 n( Λ µ )n. First, let N go to infinity and then M go to infinity. We have RN R(y, ) for any converging subsequence. Therefore, R(y, ) = R, so y = y is optimal for the infinite buffer size system. Since the limit is unique, any converging subsequence yn has the limit y. Hence, y = y = lim N yn. Proof of Lemma 4.3 Let B 0. Instead of N being restricted to integer values, let N attain real values. Note that R(y, N) is differentiable with respect to N. We will show that for all y B and N large enough, R(y,N) N 0. With some algebra, R(y, N) N ρ(y) N+1 = (1 ρ(y) N+1 ) 2( µy(1 ρ(y))ln(ρ(y)) + h(1 ρ(y)n+1 + ln(ρ(y) N+1 ))) ρ(y) N+1 (1 ρ(y) N+1 ) 2( µb ln(ρ(b)) + h(1 + (N + 1)ln(Λ µ ))) µb h 0, if N ln(ρ(b)) + 1 ln( Λ µ ). Therefore, there exists N IN such that for all y B and h > 0, R N (y) is nonincreasing in N for N N. References [1] Gross, D., and Harris, C. Fundamentals of Queueing Theory. John Wiley and Sons, New York, [2] Hassin, R. Consumer information in markets with random product quality : the case of queues and balking. Econometrica 54 (1986), [3] Ittig, P. T. Planning service capacity when demand is sensitive to delay. Decision Sciences 25 (1994), [4] Ittig, P. T. The real cost of making customers wait. International Journal of Service Industry Management 13 (2002),

17 [5] Knudsen, N. C. Individual and social optimization in a multiserver queue with a general cost-benefit structure. Econometrica 40 (1972), [6] Larsen, C. Investigating sensitivity and the impact of information on pricing decisions in an M/M/1/ queueing model. International Journal of Production Economics (1998), [7] Lautenbacher, C. J., and Stidham, S. The underlying markov decision process in the single-leg airline yield-management problem. Transportation Science 33 (1999), [8] Naor, P. The regulation of queue size by levying tolls. Econometrica 37 (1969), [9] Stidham, S. Optimal control of admission to a queueing system. IEEE Transactions on Automatic Control 30 (1985), [10] Subramanian, J., Stidham, S., and Lautenbacher, C. J. Airline yield management with overbooking, cancellations, and no-shows. Transportation Science 33 (1999), [11] Yechiali, U. On optimal balking rules and toll charges in the G/M/1/s queue. Operations Research 19 (1971), [12] Ziya, S. Optimal pricing for a service facility. PhD thesis, Georgia Institute of Technology, [13] Ziya, S., Ayhan, H., and Foley, R. D. Relationships among three assumptions in revenue management. Operations Research 52 (2004), [14] Ziya, S., Ayhan, H., and Foley, R. D. Optimal prices for finite capacity queueing systems. Operations Research Letters 34 (2006),