SERVICE RATE DECISIONS
6. Co-production
Ha [62, 63] considered an unobservable system where each customer chooses a service rate. The choice reflects the amount of service he requests, and affects his utility from service. Cachon and Harker [31] (see Section 7) made a similar assumption in their model. They refer to it as an engagement of customers inco-production or outsourcing of service to customers. Note that a high rate of service means that less service is given by the server, whereas longer service is associated with higher value to the customer.
The decrease in utility that a customer obtains from faster service is reflected in a cost functionh(µ) incurred by a customer if he chooses ser- vice rateµ. The function h(µ) is continuously differentiable, monotone increasing, and strictly convex.
Customers ignore the externalities of their choice of a service rate on the delays incurred by others and choose a rate which is smaller than the service rate a social planner would choose.5 Ha derived pricing schemes under which customers make decisions that are compatible with social optimality.
Ha’s approach distinguishes between two types of externalities in- volved with the decision process of co-production models. Service exter- nalities are associated with the individual’s optimization of his service requirement. Specifically, the longer a customer’s service, the more wait- ing time is added to others. When choosing a service rate, the customer ignores these externalities and therefore his choice tends to be too small from a social point of view. Admission externalities are involved with the increase in arrival rate caused when more customers decide to join. We note that this distinction is mainly semantic: The two types of ex- ternalities are caused by the same action, that of joining the queue in order to obtain service.
6.1.
Single class FCFS model
The first model of Ha [62] assumes a GI/GI/1 FCFS queue with customers who are identical in all parameters except for their willingness to join the system at a given value of the expected full price. Note that the full price in this model includes the costh(µ) (see (8.11)).
5The following illustrative situation is described by Schelling [153]. An accident occurs in an
freeway. Drivers in the opposite lane slow down to watch, creating long lines of cars behind them. Eventually, many commuters spend ten minutes extra driving for a ten-second look. When they get to the scene, the ten minutes’ delay is a sunk cost, and they pay the extra ten seconds for their own sightseeing. As a collective body, the drivers might vote to maintain speed, each foregoing a ten-second look and saving ten minutes on the freeway.
168 TO QUEUE OR NOT TO QUEUE LetW(λ, µ) denote the expected waiting time and letWq(λ, µ) denote
the expected queueing time, given the rates λ and µ.6 A customer’s
choice of service rate does not affect his queueing time. Therefore, his decision amounts to selecting a service rateµewhich minimizes Cµ+h(µ)
over µ >0, andµe satisfies
h′(µe) =
C µ2
e
.
Note that µe is not a function of λ. In contrast, the social objective
(given that the arrival rate isλ) is to setµto minimizeCW(λ, µ)+h(µ). Given the assumption on the function h(µ), the optimal service rate,
µ∗(λ), is uniquely determined by the first-order condition
h′(µ∗(λ)) =−C∂W(λ, µ∗(λ))
∂µ . (8.8)
By the convexity of W(λ, µ) in µ, and the assumptions on the cost functionh:
µ∗(λ) increases inλ;
µ∗(λ)≥µe forλ≥0.
To induce the socially optimal behavior in equilibrium, Ha suggested a price scheme which consists of both a fixed admission fee α and a variable service feeβ proportional to the realized time of service. (If it is possible to observe the service rate chosen by a customer, the variable part of the price can be made proportional to the expected service time.) The goal of these fees is to attain equilibrium with the optimal arrival and service rates.
We first determine the variable cost, β. The customer’s choice of µ
minimizes h(µ) + C+µβ. This is a strictly convex function ofµ and the unique minimizer satisfies
h′(µ) = C+β
µ2 . (8.9)
Since the expected number of customers in the system, L(λ, µ), is a function ofλand µonly through λµ,λ∂L∂λ+µ∂L∂µ = 0.WithL=λW this becomes
W +λ∂W ∂λ +µ
∂W
∂µ = 0. (8.10)
6A more formal approach would require to define the expected waiting time for any strategy
profile selected by the customers but these definitions are sufficient since we only consider symmetric profiles.
Service rate decisions 169 Let λ∗ and µ∗ be the arrival and service rates which jointly maximize social welfare. By substituting (8.9) and (8.10) into (8.8) we obtain the following theorem:
Theorem 8.2 Suppose that λ=λ∗ and that the service charge per unit of actual service is β=µ∗C Wq(λ∗, µ∗) +λ∗ ∂W(λ∗, µ∗) ∂λ .
Then the resulting equilibrium service rate equals µ∗.
We now determine the optimal admission fee, α. The social objective is to maximize V(λ)−CλW(λ, µ)−λh(µ). This function is strictly concave inλand hence its unique maximizer is determined by the first- order condition
V′(λ) =CW(λ, µ) +Cλ∂W(λ, µ)
∂λ +h(µ).
In equilibrium, the arrival rate is such that V′(λ) equals the expected full price:
α+β
µ +CW(λ, µ) +h(µ). (8.11)
Thus we obtain the following theorem:
Theorem 8.3 Suppose that an admission fee ofα=−CWq(λ∗, µ∗)and a variable fee of β as stated in Theorem 8.2 are imposed. Then the socially optimal solution (λ∗, µ∗) defines an equilibrium.
6.2.
Multi-class extensions
Ha [63] extended the model of [62], assuming that the demand pro- cess consists ofm classes that differ by their aggregate utility functions
Vi(λi), cost functions hi(µi), and time values Ci, i= 1, . . . , m. We will
assume that class identities are unobservable.7 It is assumed that for each classi, the values of Vi′(0) and hi(0) are sufficiently large to guar-
antee interior solutions. Let λ = (λ1, . . . , λm) and µ = (µ1, . . . , µm).
The social objective is to maximize
m X i=1 h Vi(λi)−λiCiWi(λ, µ)−λihi(µi) i (8.12)
170 TO QUEUE OR NOT TO QUEUE with respect toλand µ, whereWi(λ, µ) is the expected waiting time of
ani-customer given the rates λandµ.
Letλ∗ andµ∗ be the vectors of arrival and service rates which jointly maximize (8.12). The first-order conditions are fori= 1, . . . , m,
m X j=1 Cjλ∗j ∂Wj(λ∗, µ∗) ∂µi +λ∗ih′i(µ∗i) = 0, (8.13) and Vi′(λ∗i) =CiWi(λ∗, µ∗) + m X j=1 Cjλ∗j ∂Wj(λ∗, µ∗) ∂λi +hi(µ∗i). (8.14)
Let tbe an observable measure associated with a customer’s service rate. Specifically, Ha considered two service disciplines, FCFS in whicht
is the time in service, and EPS in whichtis the time in the system. Let
τ be a random variable denoting the realization of t. Suppose that the server charges customers a pricep(t). LetWi(λ, µ, µ) andE[p(τi)|λ, µ, µ]
denote the expected waiting time and price, respectively, incurred by an
i-customer who chose a service rate µ. The objective of an i-customer is to choose aµvalue that minimizes
E[p(τi)|λ, µ, µ] +CiWi(λ, µ, µ) +hi(µ).
The first-order conditions are that fori= 1, . . . , m,
Ci ∂Wi(λ, µ, µ) ∂µ + ∂E(p(τi)|λ, µ, µ) ∂µ +h ′ i(µ) = 0. (8.15)
In a symmetric equilibrium, (8.15) holds withµ=µi fori= 1, . . . , m.
The arrival ratesλi are determined by
Vi′(λi) =E[p(τi)|λ, µ, µi] +CiWi(λ, µ) +h(µi). (8.16)
Comparing (8.13), (8.14), (8.15), and (8.16), the price functionp(t) will induce optimal arrival and service rates in equilibrium if fori= 1, . . . , m,
∂E[p(τi)|λ∗, µ∗, µ∗i] ∂µi = 1 λ∗ i m X j=1 Cjλ∗j ∂Wj(λ∗, µ∗) ∂µi − Ci ∂Wi(λ∗, µ∗, µ∗i) ∂µi ≡ Es i, (8.17) and E[p(τi)|λ∗, µ∗] = m X j=1 Cjλ∗j ∂Wj(λ∗, µ∗) ∂λi ≡ Eia. (8.18)
Service rate decisions 171 Ha interpreted Eis and Eia as service and admission externalities, re- spectively.
Ha applied (8.17) and (8.18) to the following two models:
Consider anM/G/ssystem with egalitarian processor sharing (EPS): a server is dedicated to a customer if the number of customers does not exceed s; otherwise, the service capacity is allocated equally among the customers in the system.
Important features of the EPS model are:
– The service externalities are identical across all classes, that is,
Es
i =Es fori= 1, . . . , m.
– The admission externalities equal the service externalities, that is,Eia= Eµ∗s
i, fori= 1, . . . , m.
Consequently, a single undifferentiated price per unit of time in the system can be applied to optimally regulate customers’ behavior in this model of heterogeneous customer classes. For some constant
β,p(t) =βtinduces an equilibrium in which customers in each class make the systemwide optimal admission decisions and those who join are induced to select the optimal service requirement intended for their class.
Consider anM/G/1 FCFS system. In this case, Ha proved that there are constants β and γ such that the pricing functionp(t) =βt+γt2
induces the optimal behavior in equilibrium.