Dynamic Pricing to Control Loss Systems with Quality of Service Targets

Transcription

1 Dynamic Pricing to Control Loss Systems with Quality of Service Targets Robert C. Hampshire Carnegie Mellon University Pittsburgh, PA U.S.A Qiong Wang Bell Laboratories Murray Hill, NJ U.S.A December 19, 26 William A. Massey Princeton University Princeton, NJ U.S.A Abstract A wide range of real time services can be modeled as loss systems. Numerous examples arise in the service industry including: agent staffing for call centers, provisioning bandwidth for private line services, making rooms available for hotel reservations and congestion pricing for parking spaces. In general these services charge arriving customers a price for admission and customers expect a specific quality of service level. Given that arriving customers make their decision to join the system based on the current service price, the manager may use price as a mechanism to control the utilization of the system. A major objective for the manager is then to find a pricing policy that maximizes total revenue while meeting the quality of service targets desired by the customers. For systems with growing demand and service capacity, we provide a dynamic pricing algorithm. A key feature of our solution is congestion pricing. Our solution uses demand forecasts to anticipate future service congestion and set the present price accordingly. The optimal price is a function of the number of customers in the system and a future congestion tax. We also quantify revenue tradeoffs between service capacity expansion and quality of service changes. Keywords: Dynamic pricing, Erlang loss model, infinite server queue, calculus of variations, Lagrangian, Hamiltonian, Pontryagin principle, revenue management. Submitted to a Special Issue of Probability in the Engineering and Informational Sciences on the Analysis and Control of Queues in Manufacturing and Service Systems 1

2 Contents 1 Introduction 2 2 The Carried Load Model and Pricing Problem Traffic and Offered Load Models The Carried Load Service Model and the Pricing Problem Analyzing the Pricing Problem Reduction to a Constrained Optimal Control Problem Lagrangian and Hamiltonian Dynamics Sensitivity Analysis Derivation of the Algorithm Numerical Example The Base Case Comparing with Myopic Pricing Policy The Effect of the Price Elasticity of Demand Conclusions and Future Work 2 6 Appendix: Properties of the ψ function 23 1 Introduction Many finite capacity service systems can be formulated as a loss system, where arriving customers are rejected (i.e., lost ) at times when the system capacity is full. This is in contrast to a delay system where the customers are put into a queue or a waiting line until service capacity becomes available. Examples of loss systems include private line communication services, call centers, hotels, and parking garages. A given system may also evolve from a delay system to a loss system as the nature of its offered services changes. Consider, for example, the underlying packet-switched data network of the Internet. It fits the description of a delay system when the network only offers the so-called best-effort service, where admission is open to all consumers and excess traffic (packets) is queued in the buffer. Nevertheless, for many recent real time applications, such as voice over IP (VOIP) and internet video, the network has to offer guaranteed service where consumers are rejected if there is not enough capacity to handle their demands. For such situations, a loss system formulation becomes appropriate. As is the case with many service facilities, customer arrivals at a loss system depend on the price of the service. Moreover, pricing serves the following 3 functions: A device for revenue collection An instrument for admission control A control mechanism to attain a given quality of service level. 2

3 Note that admission control is the only way that the service manager can reduce the system load. Moreover, the quality of service level is a measure of satisfaction for the customer. For loss systems, the primary measure of the quality of service is the blocking rate, or the probability that an arriving customer is denied service. For some cases, quality of service is a contractual agreement between the customer and the service provider (e.g. leased lines and call centers). For other services, the quality of service is a target which the provider aspires to meet (e.g. parking garages and hotels). Even in the latter case, missing a quality of service target results in customer dissatisfaction and hurts the provider s business in the long run. Therefore, a sensible pricing policy should not only strive to maximize the provider s immediate revenue, but also to satisfy the quality of service target, which is a property that we require for the pricing policy developed in the paper. Early work on combining queuing and pricing, with applications to information systems, concentrate on the delay model (see Dewan and Mendelson [3], Mendelson [14], Stidham [17], and Westland [18]). This remains to be the case in the early discussions about Internet pricing (see Bailey and Knight [1] and Mackie-Mason and Varian [13]). Pricing of a loss system that offers guaranteed services has been proposed in Wang, Peha and Sirbu [2] and studied in great detail in Courcoubetis et al. [2], Lanning et al.[11], and Maglaras and Zeevi [12]. Developing a revenue-maximizing pricing policy that accommodates time-varying demands and satisfies the required QoS constraint remains an open problem, even for the classical blocking model. The main difficulty is integrating the blocking rate calculation into the price optimization, especially in cases when customer arrivals are non-stationary. Our paper addresses this issue. We consider a growing service where the demand and capacity are large. In this setting we develop a dynamic pricing solution that maximizes revenue while meeting the blocking rate target. The policy explicitly incorporates the non-stationarity of demand and is forwardlooking by setting the price in anticipation of future congestion. There are several key contributions of our work. First, using the ψ function, the inverse of the hazard rate function for a normal distribution, and the modified offered load approximation, we convert the complex blocking rate constraint for the time varying finite capacity system into a simple threshold constraint for the mean of a time varying infinite capacity system. The use of the ψ function improves the approximation of the blocking rate compared to a previous approach that used the tail distribution of the normal distribution (see Lanning, Massey, Rider and Wang [11]). An additional contribution of this work is that the blocking rate approximation allows us to formulate the pricing problem as a deterministic optimal control problem. Insights from both the Lagrangian and Hamiltonian perspectives are used to solve this problem. We find that the optimal price is not a function of the number of customers in the system. Instead the optimal price is uniquely determined by the number customers in the system and the opportunity cost per customer which captures the effects of future congestion. Numerical algorithms are presented to solve the control problem. The case of bounded elastic demand is presented in detail. Finally, we derive sensitivity results that provide managerial insights for quantifying the revenue tradeoff between expanding capacity and increasing the blocking rate. Additionally, several new properties of the ψ function are developed and used to further our analysis of these managerial design choices. 3

4 In the section that follows we introduce a carried load model of the service and formulate the pricing problem. The tools of calculus of variations, Lagrangian and Hamiltonian mechanics are employed in Section 3 to produce an approximate solution to the carried load pricing problem. In Section 4 the case of bounded elastic demand is treated with a numerical example. 2 The Carried Load Model and Pricing Problem A traffic and offered load model are introduced for customer arrivals and resource demand processes.they form the foundation for the analytical tools needed to address our ultimate loss model, the carried load (blocking) system. 2.1 Traffic and Offered Load Models We assume that customers arrive to the service system according to a non-homogeneous Poisson process, { A(t) t } with rate function { λ(t) t } where for all non-negative integers n, we have P {A(t) = n} = 1 n! ( t n ( λ(s) ds) exp t ) λ(s) ds. (2.1) The service system informs the arriving customers of their price of admission. We assume that each arriving customer assigns a utility value to the service. This value is private and hence unknown to the service manager. From the manager s perspective each arriving customer has random, identically distributed utility values that are mutually independent. If the current service price is below an arriving customer s utility value, then the customer accepts that price and joins the system. If the current service price is above an arriving customer s utility value, then that customer rejects the price and does not join the system. This process represents a series of auctions for admission into the system. This auction procedure produces a thinned Poisson process with rate function λ t (π t ) where π t is the service admission price at time t. We assume that the non-homogeneous Poisson process with this rate function is our traffic model for customers requesting this service. We can now formally state the traffic pricing objective. Optimization Problem 2.1 (Traffic Pricing) Find π { π t t T }, a dynamic pricing policy, such that we maximize π t λ(t, π t ) dt. (2.2) What is a calculus of variation problem (see Ewing [7]) reduces here to a static optimization problem at each time t or π t λ(t, π t ) = max z λ(t, z). (2.3) z We can restrict z to being positive since z λ(t, z) is always a negative number when z is negative. When λ is smooth or differentiable, then π must solve the equation π π t λ(t, π t ) = λ(t, π t ) + π t λ π (t, π t) = 4 or π t λ(t, π t ) λ π (t, π t) = 1. (2.4)

5 The last statement says that pricing policy solution to the traffic pricing problem has the property that the percentage increase in price is matched by the percentage decrease in demand. The traffic price corresponds to the optimal price assuming infinite capacity. The offered load model represents the demand for the resources of a system with infinite capacity, and corresponds to the infinite server queue, { Q (t) t }. Given our traffic model, the number of resources requested at any given time has a Poisson distribution with mean q(t) E [Q (t)], P {Q (t) = i} = e q(t) q(t) i, (2.5) i! whenever Q () has a Poisson distribution (which includes Q () = ). If the customer service times are exponential with rate µ then, d dt q(t) = λ(t, π t) µ q(t). (2.6) The work of Eick, Massey and Whitt [5] explores the dynamic properties of infinite server queues with non-homogenous Poisson input. The solution of the offered load model provides a basis for the analysis of the loss system in the next section. 2.2 The Carried Load Service Model and the Pricing Problem The carried load model has finite capacity and arriving customers are denied service if all system resources are occupied. Throughout this paper, the term loss system is used interchangeably with carried load model. Hampshire, Massey, Mitra and Wang [8] explore the connection between the offered load model and loss system in detail. We present a brief summary of this relationship. Let {Q C (t) t } be the number of customers in a M t /M t /C/C queue. The fundamental connection between our offered load and carried load models is given by the modified offered load approximation due to Jagerman [9] P (Q C (t) = C) β C (q(t)) = P (Q (t) = C Q (t) C). (2.7) The formula given by β C ( ) is the Erlang blocking formula (see Erlang [6]). Observe that this approximation is exact when the Poisson arrival rate is constant and both systems are in steady state equilibrium. Estimates on the error of this approximation can be found in Massey and Whitt [15]. Due to the Poisson distribution of the M t /G/ queue, it is natural to use the square root staffing representation used in Jennings et al. [1] of the service capacity C = q + x q, (2.8) since the square of the mean for a Poisson random variable is simply its standard deviation. Inspired by growing a business to match a corresponding growth in customer demand, we scale the arrival rate by η >, and hence the offered load q. Using the square root staffing rule, Jagerman [9] shows that lim η β ηq+x ηq (ηq) = 1 φ(x) η q Φ(x). (2.9) 5

6 This asymptotic result says that if a manager wants to keep the blocking probability of the system below ɛ, this can be approximated by the relation 1 q φ(x) Φ(x) ɛ. (2.1) Now we define ψ to be the inverse of the Gaussian hazard rate function or φ(ψ(y)) Φ(ψ(y)) = y (2.11) for all y >. This function was first introduced in Hampshire et al. [8]. Some new properties of ψ are presented in the Appendix of this paper. One of the properties of ψ is that it is a decreasing function, so we have 1 φ(x) q Φ(x) ɛ x ψ (ɛ q). (2.12) This suggests that in terms of provisioning for a QoS level of ɛ, the smallest effective value for x is x = ψ ( ɛ q ). Given C and ɛ, we can show that there exists a unique positive value θ such that ( C = θ + ψ ɛ θ) θ. (2.13) We refer to this constant θ as the critical offered load since, asymptotically, it is the largest offered load that has a blocking probability less than ɛ for C channels. Now we introduce the notion of carried load revenue. The problem faced by the service manager is to set a pricing policy that maximizes carried load revenue while satisfying the blocking probability constraints for the carried load system. The carried load pricing problem reduces to a search among all pricing policies π { π t t T } that yield a customer arrival rate function λ { λ(t, π t ) t T } that solves the following optimization problem. Optimization Problem 2.2 (Carried Load Pricing) Find a dynamic pricing policy π such that we maximize π t λ(t, π t ) P {Q C (t) < C} dt subject to max P {Q C(t) = C} ɛ. (2.14) t T Figure 1 presents a flowchart representation of this optimization problem. In what follows, we present the constrained offered load pricing algorithm that solves the carried load pricing problem under the blocking rate approximation. 3 Analyzing the Pricing Problem In this section we transform the optimization problem of carried load dynamic pricing optimization into an optimal control problem for the mean offered load. The key step that facilitates this transformation is approximating the blocking probability for a loss system 6

7 C ε { ( ) } T P QC t = C ε t: t T ( π ) { } max π λ t, P Q ( t) < C dt t λ( t, x) t µ C π t Figure 1: The carried load pricing problem by a non-linear function of the mean offered load. Once the constrained optimal control problem is derived, we use the tools of calculus of variations to derive necessary conditions of optimality. In addition, we also explore the parameter sensitivity of the total optimal revenue. We obtain a new formula that shows the tradeoff between increasing the optimal revenue by either increasing capacity or increasing the blocking rate. 3.1 Reduction to a Constrained Optimal Control Problem Our set of feasible pricing policies can be restricted to those where, All such policies yield the following set of inequalities max P {Q C(t) = C} ɛ. (3.1) t T (1 ɛ) π t λ(t, π t ) dt π t λ(t, π t ) P {Q C (t) < C} dt π t λ(t, π t ) dt, (3.2) which gives us the inequality / π t λ(t, π t ) dt π t λ(t, π t ) P {Q C (t) < C} dt π t λ(t, π t ) dt ɛ (3.3) Thus we see that for all pricing policies where the probability of blocking is always less than ɛ, the relative error between the corresponding carried and offered load revenues is also less than ɛ. This means that for small ɛ a reasonable substitution for the pricing problem is to maximize the offered load revenue given by some pricing policy with the quality of service constraint or maximize π t λ(t, π t ) dt subject to max P {Q C(t) = C} ɛ. (3.4) t T 7

8 C ε max π λ, l t T ε ( q ) C t: t T t λ( t, x) ( t π ) t dt π t µ qt = λ ( t,π t ) µ qt Figure 2: The constrained offered load pricing problem. Combining the modified offered load approximation with the hazard rate limit result of Jagerman [9], we can replace our quality of service constraint Equation 3.1 with the offered load constraint of max q(t) θ, where C = θ + ψ( ɛ θ) θ. (3.5) t T Therefore, we can reduce our analysis of the pricing problem to the following constrained optimization problem. Optimization Problem 3.1 (Constrained Offered Load Pricing) Find a dynamic pricing policy π such that we maximize π t λ(t, π t ) dt subject to max q(t) θ, (3.6) t T where d dt q(t) = λ(t, π t) µ q(t). (3.7) Figure 2 presents a flowchart representation of this optimization problem. 3.2 Lagrangian and Hamiltonian Dynamics We can reformulate the above constrained optimal control problem as a classical physics problem of Lagrangian mechanics with the necessary auxiliary variables. Optimization Problem 3.2 (Constrained Offered Load Revenue) Find a dynamic pricing policy π such that maximize L(p, q, π, x, y) dt, (3.8) 8

9 where L(p, q, π, x, y) = π λ(π) + p ( q λ(π) + µ q) + x (C l ɛ (q) y 2). (3.9) The terms q and π are the state variables. The terms p and x are the Lagrange multipliers. Finally, y is the slack variable for x. The optimized integral for the total revenue Equation (3.8) is called the action for the system in classical mechanics. In operations research, it is referred to as the Bellman value function. Revenue plays the role of value here and the integrand L(p, q, π, x, y) for the revenue rate as given by (3.9) is the Lagrangian. The Lagrange multiplier p (opportunity cost per customer) is dual to the state variable q (offered load). Using the Euler-Lagrange equations, the optimal p and q must satisfy the following two differential equations ṗ = µ p x l ɛ(q) and q = λ(π) µ q. (3.1) These equations also play the role of Hamilton s equations. The variable q corresponds to the position variable, where the initial condition q is given. The variable p corresponds to the momentum variable. It is defined formally as p L (q, q). (3.11) q The Euler-Lagrange equations for q can now be written compactly as ṗ = L (q, q). (3.12) q Moreover, the terminal condition p T = is given when only the initial condition for q is specified. Combining these two results, we can write a general formula for an optimal p, namely L p t = (q, q) ds. (3.13) t q Since the Lagrangian equals the revenue rate for this queueing system, its partial derivative with respect to the q equals the marginal revenue rate per customer. This means that p t equals the opportunity cost per customer arriving at time t. Applying the Pontryagin principle (see Pontryagin et al. [16]) gives us the following optimization relations for π: (π p) λ(π) = { maxz (z p) λ(z) if y >, max z:λ(z) µ θ (z p) λ(z) if y =. (3.14) The last condition follows from the fact that y = if and only if q is at the constraint boundary where its time derivative cannot be positive. Observe that if our demand function is a strictly decreasing function of the price for service, then it follows from (3.14) that p < π always holds. Finally, our queueing state variable q, multiplier x, and its slack variable y must satisfy the following constraint and complementarity equations: C l ɛ (q) = y 2 and xy =. (3.15) 9

10 Applying the Pontryagin principle, we must then have x to satisfy the relation x y 2 = max x <z< z2 =. (3.16) Theorem 3.3 Let Σ be a random service time that is exponentially distributed with mean 1/µ. We can rewrite the optimal solutions for p and q as [ t ] [ t+σ ] q t = q P {Σ > t} + E λ s ds and p t = l ɛ(θ) E x s ds, (3.17) t Σ where we use the convention that λ s = for all s < and x s = for all s > T. Moreover, p has the following three properties: 1. The multiplier p is always non-negative. 2. The multiplier p stays equal to zero after it first hits that value. 3. If the multiplier p is zero after any time t < T, then so is x. Let us define l ɛ(θ) to be the marginal critical channel (bandwidth) per customer given the critical offered load θ. We can then interpret x to be the marginal optimal revenue rate per channel and p to be the opportunity cost per customer. The complementary condition xy = means that the marginal optimal revenue per channel is zero whenever there is no congestion or y >. This probabilistic formulation for p and q gives us additional insight into their dynamical behavior. If we think of q as looking backward into the past as it moves forward in time, then p looks forward into the future as it moves backward in time. What p is anticipating is future levels of congestion in the system. This is precisely the mechanism that is needed for congestion pricing. 3.3 Sensitivity Analysis The optimal, constrained offered load revenue can be written as R(T ) L(p, q, π, x, y) dt = t π λ(π) dt. (3.18) Now we summarize the sensitivity of the optimal revenue to various parameters of the carried load system. Theorem 3.4 The optimal marginal revenue per channel is given by the formula dr dc (T ) = The optimal marginal revenue per quality of service level ɛ is given by dr dɛ (T ) = x dt. (3.19) θ ɛ (C θ (1 ɛ)) dr (T ). (3.2) dc 1

11 Moreover, the optimal marginal revenue due to productivity equals dr T dµ (T ) = pq dt. (3.21) Proof: The envelope theorem (see Dixit [4]) leads to and d T L(p, q, π, x, y) dt = dc d T L(p, q, π, x, y) dt = dɛ This completes the proof. d T L(p, q, π, x, y) dt = dµ T L(p, q, π, x, y) dt = x dt, (3.22) C ɛ L(p, q, π, x, y) dt = d dɛ l ɛ(θ) x dt, (3.23) T L(p, q, π, x, y) dt = pq dt. (3.24) µ Let dr/dc equal the change in revenue due to an increase in the capacity and dr/dɛ equal the change in revenue due to an incremental increase in the QoS target ɛ. Our analysis suggests a formula that quantifies the tradeoff between these two quantities dr dɛ (T ) = θ ɛ (C θ (1 ɛ)) dr (T ). (3.25) dc We convert this equation into an expression for the the relative change in revenue, dr R / dɛ ɛ = θ C (C θ (1 ɛ)) dr R / dc C. (3.26) This equation suggests a tradeoff for making a percentage change in the optimal revenue between the percentage change in ɛ and the percentage change in capacity C where Γ (C, ɛ) ɛ ɛ C (C θ (1 ɛ)) θ C Γ (C, ɛ) C ( ) = θ + ψ(ɛ θ) (3.27) ( ɛ θ + ψ(ɛ ) θ). (3.28) We call Γ (C, ɛ) the QoS-Capacity efficiency ratio. It implies that the percentage increase in optimal revenue obtained by a percentage increase in capacity can also be obtained by a percentage increase in the QoS level ɛ that is a factor of Γ (C, ɛ) times the capacity percentage increase. Proposition 3.5 As the capacity increases the QoS-Capacity efficiency ratio Γ converges to lim Γ (C, ɛ) = 1 1. (3.29) C ɛ 11

12 Proof: Recalling that ( C = θ + ψ ɛ ) θ ɛ θ, we show in Corollary 6.2 that l ɛ (x) is an increasing function of the critical offered load θ. Therefore C implies that θ. Rewriting Equation 3.29, we have Γ (C, ɛ) = 1 ɛ ( ɛ ( θ ɛ ( θ + ψ ɛ ))) ( θ + ɛ θ ( ( ɛ ( θ + ψ ɛ ))) ψ ɛ ) θ θ ɛ. (3.3) θ The first term goes to 1/ɛ as θ by statement 2 of Corollary 6.2. The second term goes to 1 by statements 1 and 2 of Corollary 6.2. See the Appendix for the statement and proof of Corollary 6.2. These results support the manager s decision of choosing the capacity level and setting the QoS target by quantifying the revenue tradeoff between the two approaches. 3.4 Derivation of the Algorithm Now we derive the constrained offered load pricing algorithm. Given the differential equations (3.1) for p and q, we can numerically integrate them as an autonomous system if we can compute π and x from a given pair of p and q. First, we assume that λ(t, ) is a smooth, decreasing, invertible function of the price. Moreover, we assume that for all t, the equation λ(t, z) + (z p t ) λ (t, z) = (3.31) π has a unique solution in z that we call ˆπ t, where we also have (ˆπ p) λ(ˆπ) = max(z p) λ(z). (3.32) z Whenever y >, it follows that the optimal price π = ˆπ. We cannot use this solution for π whenever y = and µ θ < λ(ˆπ), since we now have q = θ. Since q, we must have λ(π) µ θ < λ(ˆπ) or ˆπ < λ 1 (µ θ) π. (3.33) As a function of z, we are assuming that (z p) λ(z) has a unique critical point at ˆπ. Hence, this function must be a strictly decreasing function on the interval [ˆπ, ). This means that max (z p) λ(z) = max (z p) λ(z) = ( λ 1 (µ θ) p ) µ θ (3.34) z:λ(z) µ θ z:λ(z)=µ θ and so π = λ 1 (µ θ). To obtain the non-zero expression for x, first observe that the sensitivity relation (3.19) holds for any initial time t where t < T, we also have d π λ(π) = x. (3.35) dc 12

13 Now observe that whenever we have x >, it follows that y =. This means that our system is congested or π = λ 1 (µ θ) = λ 1 (µ l 1 ɛ (C)). From this formula, we then obtain x = d dc λ 1 (µ θ) µ θ = d { λ 1 (µ l 1 ε (C)) µ l 1 ɛ (C) } { dc } d = dc λ 1 (µ l 1 ε (C)) µ l 1 ɛ (C) + { } d = dc λ 1 (µ l 1 ε (C)) µ θ + [ ] µ θ = λ λ 1 (µ l 1 ɛ (C)) + π d µ ɛ [ ] µ θ = λ (π) + π µ l ɛ l 1 ɛ (C) [ ] µ θ = λ (π) + π µ l ɛ(θ). { π µ dc l 1 { λ 1 (µ l 1 d dc l 1 ɛ (C) ɛ (C)) µ } (C) d dc l 1 ε Figure 3 presents a flowchart that summarizes our dynamic pricing algorithm. 4 Numerical Example } (C) In this section, we demonstrate the application of our optimal pricing approach through a numerical case study. The first and most important step is to define the demand function. We assume that the user arrival rate has the following bounded elastic demand function λ(π) = γ(t) (α + βπ) σ α >, β > and σ > 1. (4.1) This demand model differs from the standard constant elasticity demand (CED) function, λ(π) = βπ σ, in two aspects. First, the CED function suffers from an anomaly that the revenue rate πλ(π) = βπ 1 σ can grow without bound as the price π approaches. With Equation (4.1), we avoid this difficulty since max π λ(t, π) = λ(t, ) = γ(t) α σ. Second, we make our demand dynamic by introducing a time-varying scaling function γ(t). More specifically, we assume γ(t) = z [W (2t/T 1) 2 ], z >, W >, (4.2) so that the demand is unimodal for a fixed price π. This means that it reaches its unique peak in the middle of the planning horizon (t = T/2). From Equation (4.1), we have πλ (π) λ(π) = σ βπ α + βπ. 13

14 p q λ( ˆ π ) + ( ˆ π p) λ ( ˆ π ) = 2 y C q ( ) = l ε ˆ π y x = π = πˆ yes y >? no yes θ l ε1 ( C) λ( ˆ π ) µ θ? no π = λ 1 ( µ θ ) µ θ µ x = + π λ ( π ) l ε ( θ ) Figure 3: The constrained offered load pricing algorithm So at any given price, a larger value of σ implies a larger percentage change of demand ( λ/λ) with respect to the percentage change of price ( π/π). Since the ratio involving π is a number between and 1, this means that the percentage change in demand is bounded between σ and. Also, in the limit as the price approaches infinity, the elasticity of demand actually becomes σ. We can use σ as both a bound and an approximate measure of the price elasticity of the demand, which is the original definition of the parameter in the standard CED function. With this demand function, the optimal traffic price, i.e., the price that maximizes the revenue in the absence of a capacity constraint, is α ˆπ = β (σ 1), (4.3) which does not change over time despite demand variation. Whenever q θ the optimal price π equals ˆπ, where p ˆπ = ˆπ + 1 1/σ and (4.4) λ(ˆπ) = (1 1/σ) σ λ(p). (4.5) 14

15 Furthermore when q θ, we have x = so the dynamics of p simply follow the equation ṗ = µ p, and so p(t) = p() e µt. (4.6) Now let t 1 be the first time s where q(s) = θ. The first step of our algorithm is to jointly find t 1 and p(). Step 1 Define δ(t), Now solve for p() and t 1, where δ(t) ( ) 1/σ γ(t). µ θ p() = 1 β (1 1/σ) (δ(t 1) α) e µt 1 (4.7) and θ = q()e µt 1 + ( 1 1 t1 ) σ λ(p() e µs, s) e µ(t1 s) ds. (4.8) The first equation ( 4.7) is equivalent to λ(ˆπ, t 1 ) = µ θ. In turn we write this as ( 1 1 ) λ(p() e µt 1, t) = µ θ. (4.9) σ The second equation follows from having q(t 1 ) = θ. Now for the second step of our algorithm, we find the last time that our constrained offered load equals θ and call it t 2. Step 2 Solve for t 2 γ(t 2 ) = µ θ ( ) σ α. (4.1) 1 1/σ Due to the unimodal structure of the demand there is at most one congestion period. It follows that p(t 2 ) =, which implies that λ(ˆπ, t 2 ) = µ θ. Now we can compute the opportunity cost per customer. Step 3 Solve ṗ = µ p l ɛ(θ) x backwards in time with p(t 2 ) = and setting when t < t 1, x(t) = (δ(t) (1 1/σ) α) µ/(β l ɛ(θ)) when t 1 t < t 2, when t 2 t T. (4.11) Finally, we compute the constrained offered load. Step 4 Solve q = λ(π) µ q forwards in time given q() and ˆπ + p() e µt /(1 1/σ) when t < t 1, π(t) = (δ(t) α) /β when t 1 t < t 2, ˆπ when t 2 t T. (4.12) 15

16 Price 5 4 Congestion Price Traffic Price Time Figure 4: The Dynamics of the Congestion Price (π) to the Traffic Price (ˆπ ). For our numerical cases we set T = 1. In the base case, we have capacity C = 5 and the required blocking rate ɛ =.5 or 5%. The critical carried load for this case is θ = This means that, l ɛ (q) = q + ψ(ɛ q) q l ɛ (θ) = C = 5, if and only if q θ. (4.13) We assume that the customer departure rate is µ =.1. Finally, for the demand parameters, we let α = β =.5, σ = 1.5, z = 1.5 and W = The Base Case We find that the opportunity cost per customer and the price follow the change of demand pattern that is parameterized by γ(t). In initial periods, the demand growth induces an increase in both the price and the opportunity cost per customer. After the demand reaches its peak and subsides, the two variables decline over time. Near the end of the planning horizon when the demand drops to the level at which the QoS constraint becomes nonbinding, the opportunity cost per customer becomes and the price converges to the optimal traffic price (Equation4.3). The optimal price π is shown in Figure 4 along this the traffic price ˆπ. The onset of congestion (t 1 = 22.5) and the end of congestion (t 2 = 95.5) are also labeled. Starting from q =, the constrained offered load q t grows until it reaches the critical offered load θ ( see Figure 6). It stays at the critical offered load until λ(t, π(t)) < µ θ once 16

17 Cost Time Figure 5: Opportunity Cost per Customer Dynamics (p) the constrained offered load starts to fall off the boundary. This happens at the same point where the opportunity cost per customer becomes and the price converges to the optimal offered load price. The arrival rate grows and reaches its peaks before the constrained offered load meets its upper limit. The pattern indicates that the price increase first lags behind demand growth and then exceeds the latter, during the periods when QoS constraint is not binding. Once the constrained offered load reaches the critical offered load, the price is set so that demand is constant and equals µ θ. 4.2 Comparing with Myopic Pricing Policy The opportunity cost per customer p can be viewed as being proportional to a future congestion tax that raises the price π above the optimal traffic price to keep the constrained offered load from exceeding the critical offered load θ. Our approach is forward-looking: it anticipates that users admitted during non-congested periods have a positive probability to contribute to future congestion. The positive opportunity cost per customer accounts for this effect and drives the optimal price dynamics accordingly. Alternatively, one may consider a myopic approach under which the traffic price is optimal when QoS constraint is slack. The price is raised to reduce the demand only when the QoS constraint becomes tight, i.e., when the constrained offered load reaches its critical value θ. In Table 1, we compare these two approaches by plotting their percentage revenue difference 17

18 Offered Load Time Figure 6: Constrained Offered Load Dynamics (q) as defined by R (t) R m (t). (4.14) R (t) Here R (t) and R m (t) are the revenues collected in period t under the optimal and myopic pricing policies respectively. We demonstrate the difference for the first 3 units. Ignoring future QoS constraints, the myopic policy generates more revenue in the initial periods when the system is not congested. However, the policy induces excessive demand causing the constrained offered load to quickly increase. The QoS constraint becomes tight sooner (t = 11.5) than under the optimal pricing policy (t = 22.5). Between these two times, the myopic policy has to charge too high a price (relative to the optimal pricing) thus causes the revenue to fall below the optimal level. Note that when the constrained offered load hits the critical offered load under both policies, the two policies charge the same price and generate the same amount of revenue. Therefore, we only need to compare the total revenue up to that point (i.e., t = 22.5). The total revenue under the myopic policy is 383 and under the optimal policy is 39. The latter dominates the former by 1.9%. We conduct the same comparisons for various values of the service rate µ. The results are shown in Table 1. In all these comparisons, we keep the input rate λ(t) and the output rate C µ constant. This provides a controlled experiment for testing the effects of varying µ. Here, the onset of congestion t 1 always occurs earlier under the myopic policy. We compare the revenue up to the period where the QoS constraint becomes tight under both 18

19 Price Price Offered Load Opportunity Cost Offered Load Figure 7: The Constrained Offered Load(q) and Opportunity Cost per Customer (p) uniquely determine the Optimal Price(π). C µ congestion starting time(t 1 ) total revenue myopic optimal myopic optimal % gain % % % % % % % % % Table 1: A Comparison of Myopic and Optimal Pricing policies. As one might expect, the myopic policy always generates less revenue. We observe that as the average service time increases, the difference diminishes. Intuitively, this trend should be expected since a higher service rate implies a shorter average stay in the system by a user. Consequently, the value of setting the price by considering future congestion decreases as µ increases. 4.3 The Effect of the Price Elasticity of Demand We now discuss the effects of price elasticity. In this subsection we hold the traffic revenue fixed at 51.6 (column 2 times column 3) to provide a controlled scenario for testing the effects of varying σ. Let the elasticity parameter σ to vary from 1.2 to 1.8 at.1 increments, and change the scale function γ(t) to make the traffic revenue the same for all cases. The results of our analysis is shown in Table 2. Table 2 shows that the total revenue declines monotonically as the value of σ increases. Additionally, the traffic price ˆπ is always less than or equal to ˆπ, and the corresponding 19

20 σ ˆπ λ(ˆπ ) congestion period (t 1, t 2 ) % increase over traffic revenue demand carried (31.7, 87.8) 223.8% 42.71% (26.7, 92.3) 311.6% 28.48% (24.1, 94.3) 381.7% 21.36% (22.5, 95.5) 437.1% 17.9% (21.2, 96.3) 481.% 14.24% (2.3, 96.9) 515.8% 12.21% (19.6, 97.3) 543.7% 1.68% Table 2: Pricing and Congestion at Different Elasticity Levels optimal arrival rate λ(ˆπ ) is greater than or equal to λ(ˆπ). We find that for higher elasticities σ, the constrained offered load revenue is generated by high demand volume induced by a low price. While for lower elasticities, the constrained offered load revenue is generated by a smaller number of customers paying a higher price. Specifically, we find that the optimal traffic price ˆπ for the case of σ = 1.8 is 1/4 the optimal price traffic ˆπ when σ = 1.2. Furthermore, the optimal traffic demand λ(ˆπ ) at σ = 1.8 is 4 times the optimal demand λ(ˆπ ) when σ = 1.2. As a consequence, everything else being equal, the capacity constraint disproportionately affects the cases associated with higher price elasticity, and thus higher traffic demand. As the table shows, as the σ value grows, the congestion periods become longer, starting earlier and ending later (column 4), the price increases becomes more excessive (column 5), and reduction of demand, reflected by the percentage of the optimal traffic demand carried during congested periods (column 6), is more severe. Since the traffic revenue is the same across all cases, those associated with largest deviations of optimal price and demand will get smaller revenues, just as what we have found in this study. Finally, a manager may consider increasing the relative amount of capacity C or increasing the relative amount of blocking ɛ to increase the relative amount of revenue. The QoS-Capacity efficiency ratio for this problem is Γ (C, ɛ) = To induce the same percentage change in revenue, the percentage change in blocking the probability must be 9.16 times the percentage change in capacity. In the current example, a 1% percent increase in capacity produces the same percentage gain in revenue as a 91.6% percent increase in the probability of blocking ɛ. This implies that increasing C from 5 to 55 induces the same percent increase in revenue as changing the blocking probability ɛ from 5% to 9.58%. Before making a decision, the manager must consider costs of these two actions. The cost of increasing the relative capacity can be calculated directly. While the cost of increasing the relative blocking probability is related to customer satisfaction. 5 Conclusions and Future Work We provide a dynamic pricing algorithm for loss systems that maximize revenue while meeting quality of service targets over a finite time horizon. The modified offered load (MOL) approximation and the ψ function, inverse of the normal hazard rate, are tools that convert 2

21 the QoS blocking constraint into a threshold constraint for the mean of the dynamic infinite server queue. The resulting dynamic optimization problem is analyzed with calculus of variations, and we derive necessary conditions for the optimal pricing policy. Our pricing policy is dynamic in the sense that the price is a function of both the number of customers in the system and time. The optimal pricing policy cannot be implemented by observing only the queue. Figure 7 shows that the relationship between the constrained offered load and the optimal price is not a function. A notion of future congestion is needed. The opportunity cost per customer captures the future congestion tax levied on the system when admitting an additional customer. Using the constrained offered load and the opportunity cost per customer, our solution is able to anticipate future service congestion and set the present price accordingly. Our method can be viewed as state dependent pricing where the state space is expanded to include the opportunity cost per customer. Further, the price is not an increasing function of the constrained offered load. This observation contradicts the heuristic: more congestion, higher price. Our simple example illustrates that during the end of congestion, the price may be lower than during the onset of congestion, even when the number of customers in the system is the same. This is a direct consequence of having a dynamic demand rate and a QoS target constraint. A myopic policy that does not consider future congestion is shown not to be as effective as our approach. The myopic policy considers only the constrained offered load, not the opportunity cost per customer. As the average customer service time increases, our optimal pricing policy displays larger gains over the myopic policy. Finally, the tradeoff between increasing capacity and increasing the QoS target is considered. Increasing capacity would allow more customers to be admitted for service, generating more revenue. While increasing the QoS target ɛ, would increase the critical offered load θ allowing more customers to enter into service. The QoS-Capacity efficiency ratio captures this tradeoff. Additionally, this ratio converges to a limiting value proportional to 1/ɛ as the capacity increases. References [1] Bailey, J. and McKnight, L. Internet Economics, The MIT Press, [2] Courcoubetis, C. A., Dimakis, A., and M. I. Reiman, Providing Band width Guarantees Over a Best-effort Network: Call Admission and Pricing, Proceedings of IEEE INFOCOM 21, pp , April 21. [3] Dewan, S. and Mendelson H., User Delay Cost And Internal Pricing For A Service Facility, Management Science, pp , 199. [4] Dixit, A.K., Optimization in Economic Theory, Oxford University Press, 199. [5] Eick, S., Massey, W. A., and Whitt, W., The Physics of the M(t)/G/ Queue, Operations Research, pp. 4 48,

22 [6] Erlang, A. K., Solutions of Some Problems in the Theory of Probabilities of Significance in Automatic Telephone Exchanges, The Post Office Electrical Engineers Journal; (from the 1917 article in Danish in Elektroteknikeren vol. 13), pp , [7] Ewing, G. M. Calculus of Variations with Applications, Dover Publications Inc., [8] Hampshire M., Massey W. A., Mitra D., and Wang, Q., Provisioning of Bandwidth Sharing and Exchange, Telecommunications Network Design and Economics and Management: Selected Proceedings of the 6th INFORMS Telecommunications Conferences, Kluwer Academic Publishers, Boston/Dordrecht/London, 22, pp [9] Jagerman, D. L., Nonstationary Blocking in Telephone Traffic, Bell System Technical Journal; pp , [1] Jennings, O. B., Massey, W. A., and McCalla, C. Optimal Profit for Leased Lines Services, Proceedings of the 15th International Teletraffic Congress - ITC 15, pp , [11] Lanning, S. G., Massey, W. A., Rider, B. and Q. Wang, Optimal Pricing in Queuing Systems with Quality of Service Constraints, Proceedings of the 16th International Teletraffic Congress, Edinburgh, UK, pp , June [12] Maglaras, C. and Zeevi A., Pricing and Design of Differentiated Services: Approximate Analysis and Structural Insights, Operations Research, Vol. 53, March 25, pp [13] Mackie-Mason, J. F. and Varian, H., Pricing Congestible Network Resources, IEEE Journal on Selected Areas in Communications, pp , [14] Mendelson, H., Pricing Computer Services: Queuing Effects, Communications of the ACM, pp , [15] Massey, W. A. and Whitt, W., An Analysis of the Modified Offered Load Approximation for the Nonstationary Erlang Loss Model, Annals of Applied Probability, pp , [16] Pontryagin, L.S., V.G. Boltyanshii, R.V. Gamkredlidze, and E.F. Mishchenko. The Mathematical Theory of Optimal Processes. John Wiley and Sons, New York, N.Y., [17] S. Stidham, Pricing and Capacity Decisions for A Service Facility: Stability and Multiple Local Optima, Management Science, Vol. 38, No. 8, pp , [18] J. C. Westland, Congestion and Network Externalities in the Short Run Pricing of Information Systems Services, Management Science, vol. 38, no. 6, , July [19] Mendelson, H. and S. Whang, Optimal Incentive-Compatible Priority Pricing for the M/M/1 Queue, Operations Research, vol. 38, no. 5, pp ,

23 [2] Wang, Q., Peha, J. M., and M. A. Sirbu, Optimal Pricing for Integrated Services Networks, Internet Economics, edited by L. W. McKnight and J. P. Bailey, pp , MIT Press, Cambridge, MA, Appendix: Properties of the ψ function The ψ function is introduced in Hampshire, Massey, Mitra and Wang [8], where many of its properties are derived. Below, we state and prove some additional properties that are used in this paper. Theorem 6.1 For all x >, we have > ψ(x) + x 1 x > 1 x 3 + x. (6.1) Proof of Theorem 6.1: First, we show that ψ(x) + x > for all x > and converges to zero as x +. Since φ (y) = y φ(y), we have ψ(x) + x = y + φ(y) Φ(y) = y Φ(y) + φ(y) Φ(y) = y Φ(z) dz Φ(y) >. (6.2) We now have ψ(x) + x > for all positive x. If we set ψ(x) = y, then x + is equivalent to y. We then have lim x + ψ(x) + x =, since by L Hopital s rule y lim ψ(x) + x = lim Φ(z) dz Φ(y) = lim x y Φ(y) y φ(y) = lim φ(y) y y φ(y) = lim 1 =. (6.3) y y Now let h(x) ψ(x) + x 1 x. (6.4) We have lim x + h(x) = and differentiating h gives us h (x) = 1 x(x + ψ(x)) x 2 = h(x) x + ψ(x) + 1 x 2. (6.5) If h(x), then h (x) >. This contradicts h converging to zero as x +, so we must have h(x) < for all x >. Differentiating h a second time gives us h (x) = = h(x) (x + ψ(x)) 2 ( 1 + ) 1 + h (x) x(x + ψ(x)) x + ψ(x) 2 (6.6) x 3 h(x) 2 (x + ψ(x)) + h (x) 3 x + ψ(x) 2 x. (6.7) 3 If we set h (x) =, then by (6.7) we must have h (x) <. Hence extreme points for h are always local maxima. 23

24 If h (x) for some x, then h is locally decreasing. Since h is negative and convergent to zero for large x, then h must have a local minimum point, which is a contradiction. Therefore, h (x) > for all x >, and so This gives us h(x) x + ψ(x) + 1 >. (6.8) x2 h(x) > 1 ( 1 (x + ψ(x)) = h(x) + 1 ). (6.9) x2 x 2 x Finally, when resolve the inequality for h, we obtain which completes the proof. h(x) > Corollary 6.2 The following statements are true and equivalent: 1 x 3 + x, (6.1) 1. The function f(x) = ψ(x) + x is positive, decreasing and convex. 2. The function g(x) = x(ψ(x) + x) is positive, increasing and less than 1. Moreover, if we let then l ɛ is an increasing function with l ɛ(x) > 1 ɛ. l ɛ (x) x + ψ ( ɛ x ) x, (6.11) Proof: The equivalence of the statements follows from the identity Now it remains only to prove the second statement. Differentiating g gives us If we do it a second time, then If g (x) =, then f (x) = (6.12) g(x) g (x) = g(x) x x + x. (6.13) g(x) g (x) = g(x) 1 x 2 g(x) + g (x) x + x g (x) + 1. (6.14) g(x) 2 g (x) = g(x) <, (6.15) x 2 g(x) since Equation 6.1 implies that g(x) < 1. Hence all extreme points of g must be local maxima. If g (x) for some x, then by Equation 6.15 it follows that g must be locally decreasing after x. However, g is positive and converges to 1 by Equation 6.1, so g must have after x a local minimum. This is a contradiction, so g (x) > for all x >. Therefore g is a strictly increasing function and this completes the proof. 24