Optimal Pricing for a Service Facility Serhan Ziya, Hayriye Ayhan, Robert D. Foley School of Industrial and Systems Engineering Georgia Institute of Technology 765 Ferst Drive, Atlanta, GA, 30332-0205 E-mail:
[email protected],
[email protected],
[email protected] November 22, 2002
Working Paper Abstract This paper investigates optimal pricing policies for a service facility modeled as a queueing system. Arriving customers are accepted if they are willing to pay the price charged by the service provider and if there is room in the waiting area. Capacity of the waiting area can be either finite or infinite. We determine expressions for the optimal prices that maximize the service provider’s long-run average profit and we prove some structural results on the optimal policies exploring their relationships with the customers’ willingness to pay and system parameters such as service speed and waiting room capacity. We show that the optimal price is not necessarily higher for systems where customers are willing to pay more and the relationship between the optimal price and waiting room capacity depends on customer demand and other system parameters. Under certain assumptions, we give necessary and sufficient conditions for the optimal price to be an increasing or decreasing function of the waiting room capacity.
1
1
Introduction
The goal of this paper is to determine the best price to charge customers in a service facility and to investigate how the best price changes as assumptions about the system change. By “best,” we mean the price that maximizes long run average profit per unit time. Our “system” is assumed to be a single server queue, and each customer has their own cut-off point for how much they are willing to pay for service. If the price charged is more than the amount the customer is willing to pay, the customer will not purchase service. If the advertised price is less than or equal to a customer’s cut-off point and there is room in the waiting area when the customer arrives, the customer joins the queue. In this paper, we restrict attention to the simplest pricing strategy in which there is one fixed, static price for all customers. Clearly, there is a trade-off. A higher price yields more revenue per customer, but fewer customers purchase service. Surprisingly, even if all customers are willing to pay more, it may be optimal to charge less; see the example in Section 7. The remainder of the paper is organized as follows. In Section 2, we review the relevant literature on pricing in queues. Section 3 describes the model in more detail and introduces notation. In Section 4, we derive expressions for the expected profit per unit time. In Section 5, we discuss the existence of the optimal prices and compare the optimal prices for infinite capacity systems with the optimal prices for finite capacity systems. In Section 6, we give expressions for optimal prices under certain assumptions. Section 7 discusses the relationships between the optimal prices and customers’ willingness-to-pay. In Section 8, we investigate how the optimal price changes as the customer demand rate or service speed changes. Section 9 contains our results on the relationships between the optimal prices and waiting room capacity. Section 10 contains the concluding remarks. Proofs of most of our results are given in the Appendix.
2
Literature Review
To our knowledge, the earliest analytical work on pricing in queueing systems is by Naor (1969). Naor’s motivation comes from the congestion control problem in queueing systems. Naor considers an M/M/1 queueing model and assumes that each customer served obtains a constant reward R while incur2
ring cost for each time unit spent waiting in the queue. An arriving customer decides to join the queue or not depending on the congestion in the system. In this formulation, a pricing policy is in effect equivalent to a threshold type admission control policy, according to which customers are not accepted if the number of customers waiting is above a certain value (threshold). As a result of this, even though there is not an external constraint on the number of waiting customers, the system behaves like a finite capacity queue with the capacity equal to the threshold. Note that in our case, queueing capacity is an externally set system parameter, which can possible be infinite. Naor considers two different objectives, self-optimization and overall optimization and shows under certain assumptions that the strategy maximizing each individual customer’s expected benefit (self-optimization) does not necessarily optimize the overall system. It is then suggested the use of tolls as a tool to help achieve the overall optimum and the optimal range for the toll is given. Naor also gives an expression for the optimal toll that would maximize the revenue obtained by the agency. Yechiali (1971), Knudsen (1972), Edelson and Hildebrand (1975) and Lippman and Stidham (1977) generalize or extend Naor’s model and results. Mendelson (1985) treats capacity (the term capacity is used to refer to service capability) as a long term decision variable and gives a methodology to find optimal pricing and capacity decisions with the objectives of the maximization of the overall organization benefit, cost recovery and maximization of service provider’s profit. Both delay costs (assumed to be linear) and capacity costs are included and there is no external restriction on the number of customers accepted to the system. The author studies the effects of queueing costs to the pricing and capacity policies. Mendelson and Whang (1990) consider an M/M/1 system with multiple customer classes. Each class has a different delay cost, expected service time and demand function. They generalize Mendelson’s (1985) model and derive an optimal incentive-compatible priority pricing mechanism. Dewan and Mendelson (1990) extends Mendelson’s model by assuming a general nonlinear delay cost function. Stidham (1992) considers a slightly different version of Dewan and Mendelson’s model by putting an upper bound on the arrival rate and by deriving an expression for the expected value received per unit time using the concept of random valuation for each customer. To be more precise, Stidham assumes that the arrival rate of customers with a valuation of more than x is given by λ = Λ(1−F (x)) where Λ is the maximal arrival rate and F is the distribution function for the value of service to a job. 3
This is also how we define the arrival rate as a function of the price in this paper. Stidham shows that the optimal solution may not be characterized by the first-order differential conditions and first-order conditions may also have more than one solution. Ha (1998) considers a variant of Mendelson’s model; however, in his model, service rate decision is made by the customers rather than the facility manager and the cost of capacity is not fixed; it depends on the service volume. Optimal policies for net-value maximization and profit-maximization are studied and also incentive-compatible pricing schemes (which induce the customers to choose the optimal service rates) are derived. Larsen (1998) studies pricing decisions in an M/M/1/∞ queueing model. The main focus is on the effect of the queue length information on pricing decisions. For that purpose two different models are developed. While one of the models assumes that arriving customers only know the steady state information about the queue length process, the other model assumes that an arriving customer knows exactly how many people there are already waiting. Larsen assumes that customer reward is uniformly distributed, derives expressions for the optimal prices (for both of the models and both for profit and welfare optimization) and compares the performance of the two models in terms of their optimal profit contributions. There is also some work related to dynamic pricing policies in queueing systems. Low (1974a, 1974b) considers the dynamic pricing policies for an M/M/s queue with both finite and infinite waiting room capacity and his objective is to maximize the long-run average expected reward per unit time. He assumes that the arrival rate is a decreasing function of the price and there is a holding cost for waiting customers. Low formulates the system as a semi-Markov decision process and gives some structural results on the optimal policies. He shows that under certain assumptions the optimal price to be charged is a non-decreasing function of the number of customers in the system. Lippman (1975) extends Low’s monotonicity result to the finite and infinite horizon discounted problem with the extra assumption that the holding cost is a convex function of the number of customers in the system. Johansen (1994) studies optimal pricing in an M/G/1 jobshop. Both profit maximization and welfare maximization (social optimization) are considered. The problem is formulated as a semi-Markov decision process (with and without discounting) and the price charged depends on the work backlog (the sum of the remaining processing times of the jobs already in the system) and processing time of the arriving job. Johansen focuses on value rates and 4
opportunity costs (expected future loss caused by accepting a customer) and gives some results on the structure of these functions. Paschalidis and Tsitsiklis (2000) study pricing policies within a network service provider context. Interarrival and service times are exponential and customers use a certain amount of a renewable resource (e.g. bandwidth) as long as they are in the system. Customers of the network belong to one of the service classes. Each class is characterized by its demand function, resource requirement and network usage duration. The arrival rate for a customer class is a function of the price charged to that class. The authors allow the price to be dependent on the system congestion level but they also consider static pricing policies under some limiting regimes. They conclude that in some cases static pricing policies are asymptotically optimal and provide computational results showing that optimal static prices can perform almost as well as optimal dynamic policies. Although all of the previous work mentioned above have similarities with our work, our interests are different. Since we are also interested in static pricing policies, our work seems to be closer to Mendelson (1985) and its extensions. However, differing from these, we are not interested in the effects of queueing costs on the optimal policies and service capacity is not a decision variable in our formulation. Our objective is to identify optimal pricing policies and investigate their relationships with customers’ willingness-topay and system parameters. Assuming that queueing costs are negligible, we derive optimal pricing policies and prove some structural results. Differing from the earlier work, we also consider systems with a finite capacity and analyze the effect of capacity changes on the optimal prices.
3
Model Description
We consider a single server service system. The queue may have either a finite or infinite capacity waiting area, and we let m ≤ ∞ denote the maximum number of customers in the system at any time. Customers who arrive when the system is full are lost. The arrival process is allowed to be quite general. We simply let N (t) be the random number of customer arrivals during (0, t] where 0 ≤ N (t) < ∞, and assume that N (t)/t converges to a strictly positive finite number Λ, almost surely. We call Λ, the arrival rate. (If the reader would like a particular example to keep in mind, a Poisson process with rate Λ is a good 5
choice.) Since the objective is to maximize long run average profit, we need some assumptions on the cost to serve customers. We assume that the total cost to serve a group of customers is simply a linear function of the number of customers served, and we let c denote the variable cost per customer. The amounts that customers are willing to pay for are assumed to be independent, identically distributed non-negative random variables. Let y denote the mark-up over the variable cost c per customer; hence, the price would be the sum of the mark-up and the variable cost c. Let F (y) denote the proportion of customers willing to pay a mark-up of at most y. The cumulative distribution function F will be referred to as the willingness-to-pay distribution. Thus, the probability that an arriving customer is willing to join the system is F ([y, ∞)) where y is the mark-up. The distribution F is assumed to be absolutely continuous with density f and support (α, β) with 0 ≤ α < β ≤ ∞ and F (y) < 1 for all y < β. Unless otherwise stated, the mean of F is assumed to be finite. Henceforth, we only consider mark-ups y that satisfy α ≤ y < β. (1) Let N (y, t) be the number of customers who are willing to pay (a mark-up of) at least y and arrive during (0, t]. If we let λ(y) denote the arrival rate of customers willing to pay y or more, then λ(y) = ΛF ([y, ∞)).
(2)
Note that λ(0) = Λ, and we are assuming that all arrivals are willing to pay at least the variable cost c (or, equivalently, that we ignore all customers who would not pay a price of at least c). Service times are assumed to be i.i.d. random variables with c.d.f. G and mean µ−1 . Thus, the service rate is µ, which is assumed to be 0 < µ < ∞. The service process, arrival process, and amounts that the customers are willing to pay are assumed to be mutually independent. To avoid complications in the construction of the queue length process, we assume that the probability of an arrival and service completion occurring simultaneously is zero. For each mark-up y ≥ 0, the number of customers in the system forms a G/GI/1/m queue with arrival rate λ(y), service distribution G with mean µ−1 , capacity m ≤ ∞, and traffic intensity ρ(y) = λ(y)/µ. Furthermore,when m < ∞ and the mark-up is y, we assume that the longrun fraction of customers blocked is a constant BN (λ(y), m). More precisely, 6
let N B (y, t) be the number of blocked customers during (0, t] among the customers who are willing to pay at least the mark-up y. We assume that B (y, t)/N (y, t) = BN (λ(y), m)a.s. lim Nm
t→∞
(3)
Note that the blocking probability only depends upon the mark-up y through the arrival rate λ(y). Under our assumptions, setting a mark-up y corresponds to randomly deleting each potential customer from the arrival process N with probability 1 − λ(y)/Λ. When the arrival process is stationary and ergodic, a sufficient condition for the blocking probabilities to exist is that at most one customer departs at any time; see Franken et. al (1981).
4
Objective Functions
We consider two different alternatives for collecting payments: • All customers pay upon joining the system (payment at arrivals), • All customers pay upon service completion (payment at departures). Fix the arrival process N , willingness-to-pay distribution F , service time distribution G, and let Ra (y, m) be the long-run average revenue per unit time with mark-up y and system capacity m ≤ ∞ assuming customers pay upon acceptance into the system. Similarly, define Rd (y, m) except that customers pay upon completing service. For m < ∞, it turns out that Rd (y, m) = Ra (y, m) so we will simply write R(y, m). The objective is to determine the mark-up y that maximizes Rd (y, m) for each m and also the mark-up y that maximizes Ra (y, m) for each m. We now derive expressions for Ra (y, ∞), Rd (y, ∞), and R(y, m).
4.1
m = ∞ and payment at arrival–Ra (y, ∞)
For the infinite capacity system and under the assumption that customers pay at the time of arrivals, the objective function is easy to define. Since every customer who is willing to pay at least y is accepted to the system and each customer pays y, we have: yN (y, t) = yλ(y). (4) t→∞ t Obviously, in this setting, assumptions on the service times are irrelevant. Ra (y, ∞) = lim
7
4.2
m = ∞ and payment at departure–Rd (y, ∞)
Under the assumption that customers pay at departures and m = ∞, the objective function can be expressed in two parts depending on the value of ρ(y). Since the customers pay as they leave the system, we are interested in the departure rate from the system. Let N d (y, t) denote the total number of customers who departed (and therefore paid for their service) from the system until time t. Then, it can be shown that (see Sigman 1995) N d (y, t) µ if ρ(y) ≥ 1 lim = . λ(y) if ρ(y) < 1 t→∞ t Thus, the objective function can be defined as yµ if ρ(y) ≥ 1 Rd (y, ∞) = . yλ(y) if ρ(y) < 1
(5)
The functions Ra (y, ∞) and Rd (y, ∞) agree for ρ(y) < 1 as one would expect. This means that if Λ < µ, then Ra (y, ∞) = Rd (y, ∞) for all y and therefore optimizing one system is equivalent to optimizing the other.
4.3
m < ∞ and payment at either arrivals or departures– R(y, m)
If the capacity m is finite, the long run average reward per unit time is the same regardless of whether payments occur at arrivals or departures since the difference in the amount of revenue under the two schemes at any time is smaller than my. This means that while deriving the objective function, we can assume either payment alternative. We derive two different expressions for R(y, m) by considering the two different payment schemes, and we will use both expressions. First, by considering payments at arrivals, B
m (y,t)) R(y, m) = limt→∞ y(N (y,t)−N t N (y,t)−N B (y,t) = y limt→∞ N (y,t) . t N (y,t)
Thus, using (3), we have R(y, m) = yλ(y)(1 − BN (λ(y), m)).
(6)
We now derive a second expression for R(y, m) by assuming that customers pay as they leave the system. Let S(y, t) be the total time the server 8
is busy until time t and let Sn denote the service time for the nth customer. d We use Nm (y, t) to denote the total number of customers departed (completed service) until time t given that the capacity is m and mark-up is y. d B d Note that limt→∞ Nm (y, t)/t exists since N (y, t) − Nm (y, t) − m ≤ Nm (y, t) ≤ B N (y, t) − Nm (y, t). Thus, we have PNmd (y,t) n=1
d (y,t) PNm
Sn ≤ S(y, t) ≤
Sn n=1 d (y,t) Nm
≤
S(y,t) d (y,t) Nm
≤
PNmd (y,t)+1 n=1 d (y,t)+1 PNm
Sn
d (y,t)+1 Sn Nm n=1 d (y,t)+1 d (y,t) . Nm Nm
From the strong law of large numbers, S(y, t) = µ−1 . d (y, t) t→∞ Nm lim
(7)
Then, we can derive R(y, m) as follows: d µS(y, t) yNm (y, t) = y lim . t→∞ t→∞ t t
R(y, m) = lim
Since we know the last limit exists, define π0 (y, m) = lim S(y, t)/t, which is the long run proportion of time that the system is empty; hence 1 − π0 (y, m) is the long run proportion of time that the server is working, and we have our second expression R(y, m) = yµ[1 − π0 (y, m)].
5
(8)
Existence of Optimal Prices and Infinite Capacity vs. Finite Capacity Systems
In this section, we will be concerned with the existence of optimal prices and we will give an ordering result for the optimal price sets for finite and infinite capacity systems. As it will be shown later in this section, optimal prices may not exist in some cases. However, the assumptions on F given in Section 3 ensure that optimal prices exist for the infinite capacity systems. For the finite capacity systems, the optimal prices exist if BN (λ(y), m) (or equivalently π0 (y, m)) can be shown to be a continuous function of y. 9
Proposition 5.1 states that for the infinite capacity systems, the optimal mark-ups exist, and under a certain assumption any optimal mark-up for the infinite capacity system with customers paying at arrivals is a lower bound on any optimal mark-up for any finite capacity system. We first describe the assumption we need. Assumption A5.1 BN (λ(y), m) is a strictly increasing function of λ(y). Recall that setting a mark-up y basically causes potential customers to be deleted with probability 1 − λ(y)/Λ. Thus, the assumption states that deleting fewer customers strictly increases the long run fraction of customers blocked. We believe that this assumption holds under quite general conditions. One can easily show that it holds for Markovian systems. Let Ym∗ be the set of optimal mark-ups when the system capacity is m < ∞, Ya∗ be the set of optimal mark-ups when the capacity is infinite and customers pay at arrivals, and Yd∗ be the set of optimal mark-ups when the capacity is infinite and customers pay at departures. Proposition 5.1 For the infinite capacity systems, optimal mark-ups exist (Ya∗ 6= ∅ and Yd∗ 6= ∅). Furthermore, if optimal mark-up also exists for a system with capacity m < ∞ (Ym∗ 6= ∅), then under A5.1, any optimal markup for the m capacity system is at least as large as any optimal mark-up for the infinite capacity system under the assumption that customers pay at arrivals (sup{y : y ∈ Ya∗ } ≤ inf{y : y ∈ Ym∗ }). Proposition 5.1 suggests that even if it is not possible to find an optimal price for the finite capacity system, the facility manager can at least find a lower bound on any of the optimal prices by solving the infinite capacity problem. In many cases, finding optimal price sets might be quite difficult, if not impossible. This is especially the case for finite capacity systems since infinite capacity systems have simpler objective functions. Therefore, one would be interested in some bounds on the optimal prices, which would at least give an interval that contains the optimal price set. The proof of Proposition 5.1 uses the assumption that F has a finite mean to prove the existence of the optimal price for the infinite capacity systems. If F has an infinite mean, then the optimal price either exists and is finite, or it does not exist. As an example, consider the distribution 10
1 function F (y) = 1 − y+ε for 0 < 1 − ε ≤ y < ∞, which has an infinite mean. It can be shown that if ε < 0, then ya∗ = 1 − ε. On the other hand, if ε > 0, then the optimal mark-up does not exist, since for any mark-up y, there is always a better mark-up which is greater than y and y = ∞ is not optimal. Note that for ε = 0, any y ∈ [1, ∞) is optimal.
6
Optimal Price Expressions
In this section, we are interested in deriving conditions and expressions for unique optimal prices. If there is a unique optimal price, then the sets Ym∗ , Ya∗ and Yd∗ each have a unique element and these unique elements will be ∗ denoted as ym , ya∗ and yd∗ , respectively. To ensure uniqueness, we will make an assumption on the function e(y) = yr(y) for y ∈ [α, β). Assumption A6.1 e(y) is strictly increasing for y ∈ [inf {y : e(y) ≥ 1}, β). It turns out that the function −e(y) = −yr(y) is the mark-up elasticity of the demand λ(y) = ΛF ([y, ∞)). To be more precise, e(y) =
λ(y+∆y)−λ(y) λ(y) − lim ∆y ∆y→0 y
.
Hence, having e(y) increasing is equivalent to having the mark-up elasticity function increasing. Then, assumption A6.1 is equivalent to the assumption that mark-up elasticity is increasing over the range of mark-ups for which demand is elastic. (We say that demand is elastic for a certain mark-up if the mark-up elasticity for that mark-up is at least 1.) Under A6.1 then, demand is inelastic for low mark-ups and elastic for higher mark-ups. Assumption A6.1 is satisfied by many widely used continuous distribution functions. It is obviously satisfied for distributions with a non-decreasing failure rate r(y) but also for many other distributions due to the factor y in yr(y) (e.g. uniform, exponential, and Weibull distributions). Let y o = sup{y : ρ(y) = 1}. Note that if Λ/µ < 1 then y o = −∞. On the other hand, if Λ/µ ≥ 1, then y o lies in the support of F .
11
Proposition 6.1 (i) Under A6.1, Ra (y, ∞) and Rd (y, ∞) both have unique optimal mark-ups. That is Ya∗ = {ya∗ } and Yd∗ = {yd∗ }, where ya∗ = inf[y ∗ : e(y) ≥ 1], ya if Λ/µ < 1 yd∗ = . ∗ o max{ya , y } if Λ/µ ≥ 1 For Λ/µ ≥ 1 max{ya∗ , y o } = y o if e(y o ) ≥ 1 (or if ρ(ya∗ ) ≥ 1); otherwise, max{ya∗ , y o } = ya∗ . (ii) Suppose that R(y, m) has a unique local maximum and π0 (y, m) is dif∗ ferentiable in y. Then, the optimal mark-up, ym can be defined as ∗ = inf[y : ym
yπ00 (y, m) ≥ 1] 1 − π0 (y, m)
where π00 (y, m) is the derivative of π0 (y, m) w.r.t. y. (iii) Suppose that interarrival and service times have exponential distributions. Then, under A6.1, R(y, m) has a unique optimal mark-up; that ∗ is, Ym∗ = {ym }. Furthermore, ∗ ym = inf[y : e(y)γm (y) ≥ 1]
where ( γm (y) =
1+m(ρ(y))m+1 −(m+1)(ρ(y))m (1−(ρ(y))m+1 )(1−(ρ(y))m ) 1 2
if ρ(y) 6= 1 . if ρ(y) = 1
We have used Assumption A6.1 for proving uniqueness. Several other assumptions have appeared in the literature and been used for the same purpose. These various assumptions are compared in Ziya, Ayhan and Foley (2002). Note that Lariviere and Porteus (2001) make an assumption which is mathematically equivalent to A6.1 although they are interested in a completely different model. They also use the assumption to ensure that a function in the form of (4) is unimodal. In all three cases in Proposition 6.1, when demand function is properly defined, the optimal price can be interpreted as the smallest price such that 12
the mark-up elasticity of demand is greater than or equal to 1. These results are similar to the microeconomics results where under suitable conditions a profit maximizing monopolist should set the price so that the price elasticity of demand is one. It can be easily shown that for a given price y, expected revenue for each customer is yF [y, ∞). This expression is equal to Ra (y, ∞)/Λ. Therefore, since ya∗ maximizes Ra (y, ∞), it also maximizes expected revenue for each customer.
7
Should customers who are willing to pay more be charged more?
Although the answer to the question posed in the section heading might appear to be yes, we will show that it is not always true. This section contains an example of two identical systems A and B except that the nth arrival to B is always willing to pay more than the corresponding arrival to A. Even though it might appear that higher prices should be set in System B, we will give an example where A should have the higher price. First, if FB (y) ≤ FA (y) for all y ∈ (−∞, ∞) then FA is smaller than FB with respect to stochastic dominance, i.e. FA ≤st FB (see Shaked and Shanthikumar 1994). y for y ∈ [0, 2.2] and FB (y) = ln(y) for y ∈ Example: Let FA (y) = 2.2 [1, e]. One can check to see that FA ≤st FB , and the corresponding functions eA (y) and eB (y) are strictly increasing, which implies that A6.1 holds. Let U1 , U2 , · · · be iid uniform (0, 1) random variables, and suppose that the nth customer to A is willing to pay FA−1 (Un ) while the nth customer to B is willing to pay FB−1 (Un ). Since FB−1 (Un ) > FA−1 (Un ), the nth customer to B is willing to pay more than the nth customer to A. However, using Proposition 6.1, it can be easily shown that in System A ya∗ = 1.1 whereas in System B ya∗ = 1. If we turn our attention to the optimal price when customers pay at departure, the relationship between yd∗ for the two systems depends on the value of Λ/µ. There exists some κ such that if Λ/µ > ( ( ΛA /µA and A6.1 holds. Then, yd∗ in System A is less than or equal to yd∗ in System B. Furthermore, if interarrival and service times are exponential, the same holds ∗ for ym for m < ∞.
9
Optimal Prices and System Capacity
∗ In this section, we first consider the optimal mark-up ym as m goes to infinity. Then, we will investigate more closely how the optimal mark-up changes as the capacity m increases or decreases. Proposition 9.1 given below states that under certain assumptions the optimal mark-up for the finite capacity system converges to the optimal markup for the infinite capacity system when customers pay at departure times. First, we will state a new assumption which will be needed for Proposition 9.1. The assumption is
Assumption A9.1 limm→∞ π0 (y, m) = 0 for ρ(y) ≥ 1. We believe that A9.1 holds under quite general conditions if not always. One might be more skeptical whether it holds for ρ(y) = 1. However, using the two equivalent expressions of R(y, m) one can show that for ρ(y) = 1, limm→∞ π0 (y, m) = limm→∞ BN (λ(y), m). Then, if one limit is strictly positive so is the other and this is a good reason to believe that A9.1 holds under quite general conditions. Note that it can be easily shown that A9.1 holds if interarrival and service times are exponentially distributed. We now give two lemmas. These lemmas are used in the proof of Proposition 9.1 but they are given here since they also have some significance by themselves. Lemma 9.1 For m ≥ 1, BN (λ(y), m + 1) ≤ BN (λ(y), m), π0 (y, m + 1) ≤ π0 (y, m) and therefore R(y, m + 1) ≥ R(y, m) for any y. 15
Lemma 9.2 If A5.1 and A9.1 hold, R(y, m) converges to Rd (y, ∞) as m converges to infinity. Lemma 9.1 states the intuitive result that for any mark-up y, it is always better to have higher capacity and Lemma 9.2 states that objective function for the finite capacity system converges to the objective function for the infinite capacity system under the assumption that customers pay at departure times. Using these lemmas, we prove the following proposition. ∗ Proposition 9.1 Let {ym } for m = 1, 2, · · · be a sequence of optimal mark∗ ups for R(y, m). Then, under A5.1, A6.1 and A9.1, ym converges to yd∗ as m converges to ∞. ∗ From Propositions 6.1 and 9.1, one can see that if ρ(ya∗ ) ≤ 1 then ym ∗ ∗ ∗ o converges to ya , on the other hand if ρ(ya ) ≥ 1 then ym converges to y . In other words, if the optimal mark-up for the infinite capacity system under the assumption of payments at arrivals is such that the system is stable, then ∗ ym converges to that optimal mark-up. Otherwise, it converges to a mark-up which makes the realized traffic intensity equal to 1. Proposition 9.1 can be useful to approximate the optimal mark-ups for finite capacity systems in cases where it is difficult or impossible to find them. Obviously, this approximation will be better if the capacity is large. We now look at more closely how the optimal mark-up for the finite capacity system changes with the capacity. For this result we assume that interarrival and service times are exponentially distributed. It turns out that ∗ ym is either monotone non-decreasing or monotone non-increasing depending on the value of Λ/µ. This monotonicity result is given in Corollary 9.1 along with other ordering relations. However, first we give the following propositions, which immediately imply (along with the previous propositions) Corollary 9.1. The following proposition gives another bound on the optimal mark-up for finite capacity systems. Depending on the value of Λ/µ, this bound is either a lower or an upper bound (or it is the optimal mark-up).
Proposition 9.2 Let
c
y =
inf [y : e(y) ≥ 2] if there exists y < ∞ s.t. e(y) ≥ 2 , ∞ otherwise 16
and ρc = F ([yc1,∞)) . Suppose that interarrival and service times are exponentially distributed. Then, under A6.1, we have: ∗ (i) If Λ/µ > ρc then y c ≤ ym . ∗ (ii) If Λ/µ < ρc then ym ≤ yc. ∗ (iii) If Λ/µ = ρc , then ym = yc.
Next, Proposition 9.3 states that traffic intensity for the optimal markup determines whether the optimal mark-up would increase or decrease as a result of an increase in the capacity. Proposition 9.3 Let m, n < ∞ and n > m. Suppose that interarrival and service times are exponentially distributed and A6.1 holds. Then: ∗ ∗ (i) If ρ(ym ) ≥ 1, then yn∗ ≥ ym . ∗ ∗ (ii) If ρ(ym ) ≤ 1, then yn∗ ≤ ym . ∗ Proposition 9.4 states that y o is either a lower or upper bound on ym depending on the value of e(y o ), the mark-up elasticity for the mark-up y o .
Proposition 9.4 Suppose that interarrival and service times are exponen∗ tially distributed and A6.1 holds. If e(y o ) ≥ ( ρc then ya∗ ≤ y c ≤ y1∗ ≤ y2∗ ≤ · · · ≤ ym ∗ ∗ (iii) If Λ/µ < ρc then ya∗ ≤ yd∗ ≤ · · · ≤ ym+1 ≤ ym ≤ · · · ≤ y2∗ ≤ y1∗ ≤ y c . ∗ Furthermore, if ρc > Λ/µ ≥ 1, then y o ≤ ym for m < ∞.
17
In part (i), y c = y o follows from the fact that A6.1 implies that F is strictly increasing for y ∈ [ya∗ , β) which in turn implies that y o is the only mark-up for which ρ(y) = 1. As it can be seen from Corollary 9.1, the ordering relations depend on whether the value of Λ/µ is equal to, greater than or smaller than ρc . Because of this property we call ρc the critical traffic intensity and corresponding mark-up y c the critical mark-up. Note that ρc and y c only depend on the willingness-to-pay distribution function. Part (i) states that if Λ/µ = ρc , the optimal mark-up does not change as the capacity changes and it is equal to y c . Parts (ii) and (iii) give the ordering relations and monotonicity results, which are dependent on the value of Λ/µ. If Λ/µ is greater than the critical ∗ ∗ is non-decreasing in m and y c is a lower bound on ym . If Λ/µ value, then ym ∗ c is less than the critical value, then ym is non-increasing and y is an upper bound. As it has already been stated in Proposition 5.1, ya∗ is always a lower bound. The result that yd∗ is either a lower or an upper bound immediately ∗ follows from Proposition 9.1 once we know the monotonicity result for ym (it can easily be shown that A5.1 and A9.1 hold if interarrival and service times are exponentially distributed). ∗ Intuitively, having monotone non-increasing ym seems to make more sense. When the capacity is small - which means in a sense that the resource is scarce - one would expect high mark-ups and also would expect the optimal markup to drop as this resource becomes more and more available. However, as Corollary 9.1 suggests, if Λ/µ > ρc , this behavior turns to be completely op∗ )≥1 posite. To see why this is so, first note that if Λ/µ > ρc , we have ρ(ym for all m < ∞. This means that, even though the optimal mark-up increases ∗ ) decreases), still for any m, customer arrival rate as m increases (hence ρ(ym ∗ for mark-up ym is more than the service rate. However, with such high demand, for small capacity, another concern of the facility manager is (besides taking advantage of the scarce resource) how to set the price so that the server is rarely idle. When the capacity is large, the idle time of the server is almost zero, because there are almost always waiting customers. However, if the capacity is small (consider the extreme case when m = 1) there is much more chance for the server to be idle. Hence, by decreasing the mark-up as capacity decreases, the facility manager increases the arrival rate to the system and decreases the idle time of the server. Interestingly, if Λ/µ < ρc , which means if demand is relatively small, the facility manager is more interested in utilizing the scarce resource rather than reducing the idle time of the server. 18
10
Conclusions
In this paper, we have studied optimal pricing decisions for a service facility modeled as a single server queueing system. This system can also be viewed as a job shop operating under make-to-order policy. We have considered systems both with finite and infinite waiting area capacity. This restriction on the number of waiting customers is not necessarily a physical constraint. The facility manager may also choose to set a limit on the number of waiting customers (or outstanding orders) to help achieve a certain service level. Our focus has been on the optimal pricing policies which maximize service provider’s long run average profit and how these optimal prices change as a result of changes in customers’ willingness-to-pay or changes in system parameters such as service speed and waiting area capacity. In general, finite capacity systems are more difficult to analyze compared to infinite capacity systems. The blocking term in the objective function for finite capacity systems makes it more difficult to obtain optimal prices. However, more importantly, we do not have an expression for the blocking probabilities for general arrival and service processes. We provided two results on how the optimal prices for infinite capacity systems and optimal prices for finite capacity systems are related. We first showed that optimal prices for the infinite capacity system – when customers pay at arrival times – are lower bounds on the optimal prices for finite capacity systems. For the second result, we proved that as the capacity converges to infinity, the optimal price for the finite capacity system converges to the optimal price for the infinite capacity system under the assumption that customers pay at departure times. These two results can be used to make more informed pricing decisions for finite capacity systems in cases where it is not possible to determine optimal prices. An important result in our paper is on the relationship between the optimal price and waiting area capacity. Considering the fact that optimal prices for the infinite capacity system is a lower bound on the optimal prices for finite capacity systems, one might be tempted to believe that the optimal price is decreasing in the capacity. However, we showed that this is not correct. It turns out that the nature of this relationship depends on the value of other system parameters and customer demand. Under exponential interarrival and service times, we gave necessary and sufficient conditions for the optimal price to be non-increasing, constant or non-decreasing in capacity. It turns out that there exists a critical value for the traffic intensity for which 19
the optimal price is the same for all finite capacity systems. If traffic intensity is larger than this critical value, optimal price increases with the capacity; if it is smaller than the critical value, then optimal price decreases. We also investigated how the optimal prices change in response to changes in the demand function. If there is an upwards shift in the demand function – meaning that for each price there are more customers willing to pay – does this mean that the optimal price should also be higher? It turns out that if the customer arrival rate to the system remains constant, in other words if the shift in demand is a result of the increase in the existing customers’ willingness-to-pay, then the answer is no. However, if the increase is caused by the increase in the customer arrival rate to the system, the answer is yes.
A
Appendix
In this section, we provide the proofs of the results given in the previous sections and also give the lemmas which are needed to prove some of our results. The proof of the following lemma is omitted since it is straightforward but requires tedious algebra. Lemma A.1 Let m ≥ 1 be a finite integer. Then: (i)
1+mz m+1 −(m+1)z m (1−z m )(1−z m+1 )
(ii) (iii)
is a decreasing function of z for z > 0.
1+(m+1)z m+2 −(m+2)z m+1 (1−z m+1 )(1−z m+2 ) 1+(m+1)z m+2 −(m+2)z m+1 (1−z m+1 )(1−z m+2 )
− −
1+mz m+1 −(m+1)z m (1−z m )(1−z m+1 ) 1+mz m+1 −(m+1)z m (1−z m )(1−z m+1 )
< 0 for z > 1. > 0 for 0 < z < 1.
The following corollary follows from Lemma A.1. Part (i) follows from Lemma A.1(i) and part (ii) follows from Lemma A.1(ii) and from Lemma A.1(iii). Corollary A.1 Let γm (y) be defined as in Proposition 6.1 (iii). Then, we have (i) γm (y) is strictly decreasing in ρ(y) and non-decreasing in y, (ii) If ρ(y) > ()(=)γm (y). 20
Proof of Proposition 5.1: We first prove that Ya∗ 6= ∅. Since F is assumed to be absolutely continuous and Λ > 0, yλ(y) is a continuous non-negative function of y. Since α < β, there exists α < y < β such that yλ(y) > 0. If β < ∞, we have βλ(β) = 0. If β = ∞, it follows from the finite mean assumption on F that limy→∞ yλ(y) = 0 (see Chung 1974). Since the function yλ(y) is continuous, we conclude that there exists a finite value of y for which the function yλ(y) is maximized. Hence, Ya∗ 6= ∅. If Λ/µ < 1, then it follows immediately from (5) that Yd∗ 6= ∅. If Λ/µ ≥ 1, then we have the same result from (5) and the fact that y o < ∞. ∗ Suppose that Ym∗ 6= ∅. Let ya∗ ∈ Ya∗ and ym ∈ Ym∗ . It is sufficient to show ∗ ∗ ∗ that if ya∗ 6= ym , then ya∗ < ym . Since ym is optimal for the m-capacity system, ∗ R(ym , m) ≥ R(ya∗ , m).
Using (6), we get ∗ ∗ ym λ(ym 1 − BN (λ(ya∗ ), m) ) ≥ . ∗ ), m) ya∗ λ(ya∗ ) 1 − BN (λ(ym
Note that the numerator and denominator on the left hand side are the expected long run average reward expressions for the infinite capacity system ∗ (under ‘payments at arrivals’) with mark-ups ym and ya∗ , respectively. Then, since ya∗ is optimal for the infinite capacity system, ∗ ∗ ym λ(ym ) ≤ 1. ∗ ∗ ya λ(ya )
This implies
1 − BN (λ(ya∗ ), m) ≤ 1. ∗ ), m) 1 − BN (λ(ym
Therefore, ∗ BN (λ(ya∗ ), m) ≥ BN (λ(ym ), m).
(9)
∗ ∗ Since we assume that ya∗ 6= ym , it follows that λ(ya∗ ) 6= λ(ym ). (Having ∗ ∗ ∗ = λ(ym ) would contradict the fact that both ya and ym are optimal mark-ups for Ra (y, ∞) and R(y, m), respectively.) Then, using A5.1, ∗ we also have BN (λ(ya∗ ), m) 6= BN (λ(ym ), m) and from (9) we conclude that ∗ ∗ BN (λ(ya ), m) > BN (λ(ym ), m). Finally, using A5.1 once again, we conclude ∗ that ya∗ < ym .
λ(ya∗ )
21
Hence, sup{y : y ∈ Ya∗ } ≤ inf{y : y ∈ Ym∗ } for all m < ∞.
2
Proof of Proposition 6.1: (y,∞) (i) Note that if Ra (y, ∞) is differentiable at y ∈ [α, β), then dRady > ()1. We know from the proof of Proposition 5.1 that there exists α < y < β for which Ra (y, ∞) is strictly positive. We also know that if β < ∞ then Ra (β, ∞) = 0; if β = ∞, then limy→β Ra (y, ∞) = 0. It follows that, since Ra (y, ∞) is continuous, there exists y ∈ (α, β) such that e(y) ≥ 1. Then, since e(y) is increasing for y > inf[y : e(y) ≥ 1] , Ra (y, ∞) is decreasing for any y > inf[y : e(y) ≥ 1] for which Ra (y, ∞) is differentiable. Similarly, Ra (y, ∞) is increasing for y < inf[y : e(y) ≥ 1]. Finally, since F is absolutely continuous, Ra (y, ∞) is differentiable a.e., continuous, and we conclude that ya∗ = inf[y : e(y) ≥ 1]. If Λ/µ < 1, then ρ(y) < 1 for all y and by (5) Rd (y, ∞) = Ra (y, ∞). Thus, yd∗ = ya∗ . Suppose that Λ/µ ≥ 1. Then, there exists y such that ρ(y) = 1 and we can rewrite (5) as yµ if y ≤ y o Rd (y, ∞) = . (10) yλ(y) if y > y o From the definition of y o and continuity of F (y), we have λ(y o ) = µ and thus Rd (y, ∞) is continuous. Note that it follows immediately from (10) that the best mark-up in the set {y : y ≤ y o } is y o . If there exists ε > 0 such that e(y o + ε) < 1, then using the arguments of part (i), we conclude that the best mark-up in the set {y : y > y o } is ya∗ and ya∗ λ(ya∗ ) ≥ y o λ(y o ) = y o µ. Thus, yd∗ = ya∗ = max{ya∗ , y o }. If there exists no ε > 0 such that e(y o + ε) < 1, then once again using the arguments of part (i), we conclude that the best mark-up in the set {y : y > y o } is no better than y o and ya∗ ≤ y o . Hence, yd∗ = y o = max{ya∗ , y o }. Using similar arguments, we can also show that if y o r(y o ) ≥ ( ()1. 1 − π0 (y, m) 22
(11)
Since R(y, m) is assumed to have a unique local maximum, it follows that yπ 0 (y,m) ∗ ym = inf[y : 1−π00 (y,m) ≥ 1]. (iii) Since the system is an M/M/1/m queueing system, we have an explicit expression for π0 (y, m), ( 1−ρ(y) if ρ(y) 6= 1 m+1 1−ρ(y) . π0 (y, m) = 1 if ρ(y) = 1 m+1 Suppose that R(y, m) is differentiable at y. Then, we know from the proof π 0 (y,m) of part (ii) that dR(y,m) > ()1. It can be shown that dy π 0 (y,m)
y 1−π0 0 (y,m) = yr(y)γm (y) = e(y)γm (y). It can also be shown that π0 (y, m) is differentiable a.e. and continuous. Thus, R(y, m) is also differentiable a.e. ∗ and continuous. We know from Proposition 5.1 that ym ∈ [inf{y : e(y) ≥ 1}, β) (if interarrival and service times are exponential, A5.1 holds) and from assumption A6.1, we know that e(y) is strictly increasing in the same interval. Finally, from Corollary A.1 (i), we conclude that e(y)γm (y) is also strictly increasing over [inf{y : e(y) ≥ 1}, β). Hence, R(y, m) has a unique local maximum (and therefore a unique global maximum) and we conclude that ∗ ym = inf[y : e(y)γm (y) ≥ 1].
2 Proof of Proposition 7.1: Note that in this proof, we add A and B as subscripts to our original notation to indicate systems A and B. Now, there are three different settings. (i) Suppose that the capacity is infinite and customers pay at arrivals. ∗ Then, yaB = inf {y : eB (y) ≥ 1} and therefore, from assumption A6.1, ∗ ∗ eB (yaB + ε) ≥ 1 for any ε > 0 such that (yaB + ε) < β. Since FB ≥hr FA , ∗ ∗ rB (y) ≤ rA (y). This implies that eA (yaB + ε) ≥ 1. Finally, since yaA = ∗ ∗ inf {y : eA (y) ≥ 1} we conclude that yaA ≤ yaB . (ii) Suppose that the capacity is infinite and customers pay at departures. If Λ/µ < 1, then the result immediately follows from part (i) and Proposition 6.1 (i). If Λ/µ ≥ 1, using Proposition 6.1 (i), it is again sufficient to show that yBo ≥ y o . However, this immediately follows from the definitions of yBo and y o , and using the fact that FB ≥hr FA implies that FB ≥st FA . (iii) Suppose that interarrival and service times are exponentially dis∗ tributed and there is a finite capacity, m. Then, ymB = inf [y : eB (y)γmB (y) ≥ 23
∗ 1]. Let ε > 0 be such that ymB + ε < β. Then, from assumption A6.1 and Corollary A.1 (i) (note that interval [inf {y : eA (y) ≥ 1}, β) contains ∗ ∗ ∗ ymA by Proposition 5.1), eB (ymB + ε)γmB (ymB + ε) ≥ 1. Since FB ≥hr FA , rB (y) ≤ rA (y). Moreover, ρB (y) ≥ ρA (y) since FB ≥hr FA implies FB ≥st FA . ∗ ∗ Then, from Corollary A.1 (ii), it follows that eA (ymB + ε)γmA (ymB + ε) ≥ 1. ∗ ∗ ∗ Finally, since ymA = inf [y : eA (y)γmA (y) ≥ 1], we conclude that ymA ≤ ymB .
2 Proof of Proposition 8.1: Note that in this proof, we add A and B as subscripts to our original notation to indicate systems A and B. Suppose that there is infinite capacity and customers pay at departures. Since ΛB /µB > ΛA /µA , we have yBo > y o . We know from Proposition 6.1 (i) ∗ ∗ ∗ ∗ that yaB = yaA . Then, it follows from Proposition 6.1 (ii) that ydA ≤ ydB . Suppose that interarrival and service times are exponentially distributed ∗ and there is a finite capacity, m. Then, ymB = inf [y : e(y)γmB (y) ≥ 1]. Let ∗ ε > 0 be such that ymB + ε < β. Then, from A6.1 and Corollary A.1 (i) ∗ (note that interval [inf {y : e(y) ≥ 1}, β) contains ymA by Proposition 5.1), ∗ ∗ e(ymB +ε)γmB (ymB +ε) ≥ 1 . Since ρB (y) > ρA (y), from Corollary A.1 (ii), it ∗ ∗ ∗ = inf [y : e(y)γmA (y) ≥ + ε) ≥ 1. Since ymA + ε)γmA (ymB follows that e(ymB ∗ ∗ 2 1], we conclude that ymA ≤ ymB . Proof of Proposition 9.1: It can be shown that R(y, m) is bounded using the fact that R(y, m) ≤ Ra (y, ∞) and the finite mean assumption for F (y) implies that lim sup Ra (y, ∞) = 0. y→∞
From Lemma 9.1, we also know that R(y, m) is non-decreasing in m. Then, we have (see Bartle and Sherbert 1992 and Fischer 1983): lim (sup R(y, m)) = sup( lim R(y, m)).
m→∞
y
m→∞
y
Since we assume that A9.1 holds, we know from Lemma 9.2 that lim R(y, m) = Rd (y, ∞).
m→∞
Then, we have lim (sup R(y, m)) = sup(Rd (y, ∞)),
m→∞
y
y
24
which can also be written as (since we assume the existence of optimal markups) ∗ lim R(ym , m) = Rd (yd∗ , ∞). (12) m→∞
∗ Now, suppose for contradiction that ym does not converge to yd∗ . Then there exists a neighborhood of yd∗ , V , such that if n is any natural number, then there is a natural number k = k(n) ≥ n such that yk∗ ∈ / V (see Bartle ∗ ∗ 1976). Let V = {y : yd − δ1 < y < yd + δ2 } where δ1 > 0 and δ2 > 0. Then,
R(yk∗ , k) = sup R(y, k) ≤ sup Rd (y, ∞) = max(Rd (yd∗ − δ1 ), Rd (yd∗ + δ2 )) {y:y ∈V / }
{y:y ∈V / }
where the last equality follows from the fact that under assumption A6.1, Rd (y, ∞) has a unique local maximum (it is either unimodal or decreasing) and yd∗ is the unique optimal solution. Let ψ > 0 be defined such that Rd (yd∗ , ∞)−max(Rd (yd∗ −δ1 ), Rd (yd∗ +δ2 )) = ψ. Then, Rd (yd∗ , ∞) − R(yk∗ , k) ≥ Rd (yd∗ , ∞) − max(Rd (yd∗ − δ1 ), Rd (yd∗ + δ2 )) = ψ > 0. Let 0 < ε < ψ, then we know from (12) that there exists a natural number N (ε) such that for m ≥ N (ε) we have ∗ 0 ≤ Rd (yd∗ , ∞) − R(ym , m) < ε.
However, we also know that there exists k ≥ N (ε) such that Rd (yd∗ , ∞) − R(yk∗ , k) ≥ ψ ∗ which is a contradiction. Hence, ym converges to yd∗ .
2
Proof of Proposition 9.2: First, suppose that there exists no y such that e(y) ≥ 2. In such a case, c y = ∞ and ρc = ∞. Then, since Λ is finite and µ is strictly positive Λ/µ < ρc . Hence, we conclude that if there is no y such that e(y) ≥ 2, then Λ/µ < ρc ; or, if Λ/µ ≥ ρc , then there exists y such that e(y) ≥ 2. This means that in parts (i) and (iii) below, there exists y such that e(y) ≥ 2. (i) If Λ/µ ≥ ρc , we have ρ(y c ) ≥ 1. Then, from Corollary A.1 (i), γm (y c ) ≤ 1 . We can also write γm (y c − ε) ≤ 12 for any ε > 0 such that y c − ε ≥ α. From 2 25
the definition of y c , we have e(y c − ε) < 2. Thus, e(y c − ε)γm (y c − ε) < 1. ∗ Hence, it follows from Proposition 6.1 (iii) that y c ≤ ym . (ii) Suppose that Λ/µ ≤ ρc . If there exists no y such that e(y) ≥ 2, then y c = ∞ and the result immediately follows. Now, suppose that there exists such y and y c < ∞. Since Λ/µ ≤ ρc , we have ρ(y c ) ≤ 1. Then, from Corollary A.1 (i), γm (y c ) ≥ 21 . We can also write γm (y c + ε) ≥ 12 for any ε > 0 such that y c + ε < β. Since there exists y such that e(y) ≥ 2, from the definition of y c and A6.1, we have e(y c +ε) ≥ 2. Then, e(y c +ε)γm (y c +ε) ≥ 1 ∗ ≤ yc. together with Proposition 6.1 (iii) implies that ym (iii) It follows from parts (i) and (ii) above. 2 Proof of Proposition 9.3: ∗ (i) Suppose for contradiction that yn∗ < ym . Then there exists δ > 0 such ∗ ∗ that yn +δ < ym . By Proposition 6.1 (iii), we can write e(yn∗ +δ)γm (yn∗ +δ) < 1. ∗ Since ρ(ym ) ≥ 1, it is also true that ρ(yn∗ + δ) ≥ 1 and from Corollary A.1 (ii), we have e(yn∗ + δ)γn (yn∗ + δ) ≤ e(yn∗ + δ)γm (yn∗ + δ) < 1. This is a contradiction to the optimality of yn∗ , since yn∗ satisfies yn∗ = inf [y : e(y)γn (y) ≥ 1]. ∗ Hence, yn∗ ≥ ym . ∗ ∗ (ii)Let ε > 0 be such that ym + ε < β. Since ρ(ym ) ≤ 1, we also have ∗ ∗ ∗ +ε). Hence, ρ(ym +ε) ≤ 1. Then from Corollary A.1 (ii), γn (ym +ε) ≥ γm (ym we have ∗ ∗ ∗ ∗ e(ym + ε)γn (ym + ε) ≥ e(ym + ε)γm (ym + ε). ∗ ∗ From Proposition 6.1 (iii) and assumption A6.1, we have e(ym + ε)γm (ym + ∗ ∗ ε) ≥ 1. This implies that e(ym + ε)γn (ym + ε) ≥ 1 and we conclude that ∗ ∗ yn∗ ≤ ym + ε. Taking the limit as ε approaches zero, we find yn∗ ≤ ym for all m < ∞. 2
Proof of Proposition 9.4: If e(y o ) ≥ 2, then since γm (y o ) = 21 we have e(y o )γm (y o ) ≥ 1. Then, from ∗ Proposition 6.1 (iii), ym ≤ y o . Similarly, if e(y o ) < 2, then e(y o )γm (y o ) < 1 ∗ and by Proposition 6.1 (iii), we conclude that ym ≥ yo. 2 Proof of Lemma 9.1:
26
Proof is based on a coupling argument. Consider two systems, system 1 and system 2. Suppose that system 1 has a capacity of m and system 2 has a capacity of m + 1. We will analyze these two systems along the same sample path so that T1n = T2n = T n for all n and S1n = S2n for all n where Tin denotes the arrival time of the nth customer to system i who is willing to pay y but who may not be able to enter the system due to capacity and Sin denotes the nth service time for system i. We define the following: Ai (t): Number of customers accepted to system i until (and including) time t. Di (t): Number of customers departed from system i until (and including) time t. Ii (t): Idle time for server (system) i until time t. We will show that for t ≥ 0, A1 (t) ≤ A2 (t), D1 (t) ≤ D2 (t), I1 (t) ≥ I2 (t).
(13)
First, note that A1 (t) = A2 (t) = D1 (t) = D2 (t) = 0 and I1 (t) = I2 (t) = t for t < T 1 . Thus, (13) holds for t < T 1 . Let n be any integer and suppose that (13) holds for t < T n . Now, to show that (13) holds for all t ≥ 0, using the fact that the process repeats itself and the fact that (13) holds for t < T 1 , it is sufficient to prove that (13) holds for t < T n+1 . We will investigate two cases separately. (i) Suppose that A1 (T n ) ≤ A2 (T n ). Since for T n < t < T n+1 no new arrivals will occur, we immediately have A1 (t) ≤ A2 (t) for T n ≤ t < T n+1 . Also, since server 2 is already ahead of server 1 for t < T n (i.e. D1 (t) ≤ D2 (t) for t < T n ) and since A1 (t) ≤ A2 (t) for T n ≤ t < T n+1 , it is not possible for server 1 to get ahead of server 2 during the time interval [T n , T n+1 ). Hence, we have D1 (t) ≤ D2 (t) for T n ≤ t < T n+1 . This also implies that I1 (t) ≥ I2 (t) for T n ≤ t < T n+1 . (ii) Suppose that A1 (T n ) > A2 (T n ). We will show that this is not possible. In the following discussion, recall that we assume that the probability of having an arrival and service completion at the same time is zero. Let A1 (T n ) = k. Since A1 (T n ) > A2 (T n ) and A1 (t) ≤ A2 (t) for t < T n , we conclude that at time T n , system 2 is full and customer at the end of the
27
queue is customer k − 1. Then, D2 (T n ) = k − 2 − m. On the other hand, since customer k joins system 1 at time T n , we have k −m ≤ D1 (T n ) ≤ k −1. Then, we conclude that D1 (T n ) − D2 (T n ) ≥ 2. This is a contradiction since we have D1 (t) ≤ D2 (t) for t < T n and there is zero probability that an arrival and service completion occurs at the same time. Thus, we conclude that (13) holds for all t ≥ 0. The fact that A1 (t) ≤ A2 (t) implies that BN (λ(y), m + 1) ≤ BN (λ(y), m) and the fact that I1 (t) ≥ I2 (t) implies that π0 (y, m + 1) ≤ π0 (y, m) for m ≥ 1. Finally, R(y, m + 1) ≥ R(y, m) follows from (6) and (8). 2 Proof of Lemma 9.2: Let y¯ be such that ρ(¯ y ) = 1. Using (6) and (8), it can easily be shown that BN (λ(¯ y ), m) = π0 (¯ y , m). From A9.1, we know that limm→∞ π0 (¯ y , m) = 0. Then, we also have lim BN (λ(¯ y ), m) = 0.
m→∞
From A5.1, we have BN (λ(y), m) ≤ BN (λ(¯ y ), m) for ρ(y) < 1 implying that limm→∞ BN (λ(y), m) = 0 for ρ(y) ≤ 1. Then, using (6), we have limm→∞ R(y, m) = yλ(y) for ρ(y) ≤ 1. Also, using (8) and A9.1, we conclude that limm→∞ R(y, m) = yµ for ρ(y) ≥ 1. Hence, limm→∞ R(y, m) = Rd (y, m). 2
References Bartle R.G. and D.R. Sherbert. 1992. Introduction to Real Analysis. John Wiley & Sons, 44. Bartle R.G. 1976. The Elements of Real Analysis. John Wiley & Sons, 99. Chung K.L. 1974. A Course in Probability Theory. Academic Press, 49. Dewan S. and H.Mendelson. 1990. “User Delay Costs and Internal Pricing for a Service Facility.” Management Science. 36 1502–1517. Edelson N.M. and D.K. Hildebrand. 1975. “Congestion Tolls for Poisson Queueing Processes.” Econometrica. 43 81–92. 28
Fischer E. 1983. Intermediate Real Analysis. Springer-Verlag, 102. Franken P., D. Konig, U. Arndt, V.Schmidt. 1981. Queues and Point Processes. Akademie - Verlag, Berlin, 112. Ha A.Y. 1998. “Incentive-Compatible Pricing for a Service Facility with Joint Production and Congestion Externalities.” Management Science. 44 1623–1636. Johansen S.G. 1994. “Optimal Prices of an M/G/1 Jobshop.” Operations Research. 42 765–774. Knudsen N.C. 1972. “Individual and Social Optimization in a Multiserver Queue with a General Cost-Benefit Structure.” Econometrica. 40 515– 528. Lariviere M.A. and E.L. Porteus. 2001. “Selling to the Newsvendor: An Analysis of Price-Only Contracts.” Manufacturing and Service Operations Management 3 293–305. Larsen C. 1998. “Investigating Sensitivity and the Impact of Information on Pricing Decisions in an M/M/1/∞ Queueing Model.” International Journal of Production Economics. 56–57 365–377. Lippman S.A. 1975. “Applying a New Device in the Optimization of Exponential Queueing Systems.” Operations Research. 23 687–708. Lippman S.A. and S.Stidham. 1977. “Individual versus Social Optimization in Exponential Congestion Systems.” Operations Research. 25 233– 247. Low D.W. 1974. “Optimal Dynamic Pricing Policies for an M/M/s Queue.” Operations Research. 22 545–561. Low D.W. 1974. “Optimal Pricing for an Unbounded Queue.” IBM Journal of Research and Development. 18 290-302. Mendelson H. 1985. “Pricing Computer Services: Queueing Effects.” Communications of the ACM. 28 312–321. Mendelson H. and S. Whang. 1990. “Optimal Incentive-Compatible Priority Pricing for the M/M/1 Queue.” Operations Research. 38 870–883. 29
Naor P. 1969. “The Regulation of Queue Size by Levying Tolls.” Econometrica. 37 15–24. Paschalidis I.C. and J.N. Tsitsiklis. 2000. “Congestion-Dependent Pricing of Network Services.” IEEE/ACM Transactions on Networking. 8 171–184. Shaked M. and J.G. Shanthikumar. 1994. Stochastic Orders and Their Applications. Academic Press, Inc. Sigman K. 1995. Stationary Marked Point Processes – An Intuitive Approach. Chapman & Hall, 134. Stidham S. 1992. “Pricing and Capacity Decisions for a Service Facility: Stability and Multiple Local Optima.” Management Science. 38 1121– 1139. Yechiali U. 1971. “On Optimal Balking Rules and Toll Charges in the G/M/1 Queueing Process.” Operations Research. 19 349–370. Ziya, S., H. Ayhan and R. D. Foley. 2002. “On Assumptions Ensuring a Strictly Unimodal Revenue Function.” Working Paper. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta GA.
30
