Métodos Simples Estáticos (Basados en el Balance)

This section describes long-run decisions in models where the cus- tomers are heterogeneous, having different time or service values, but this private information is not available to the server who cannot use it to discriminate among customers.

4.1.

Heterogeneous service values

The model of Whang [173] resembles the unobservable model of _§3.3, except that the queue manager and the customers do not know the marginal value function V′_{(λ). Thus, the service value of a customer is} his private information.

Whang modeled the situation as a non-cooperative game. The game proceeds in several stages:

Customers anonymously report their service values. The meaning of anonymity is that these reports cannot be used to discriminate among customers. Denote the resulting reported marginal value function by

m(λ).

Using these reports, the server makes the long-run decision of choos- ing a capacityµ(m). This comes at a linear cost c(µ) =bµ.

The actual function V′ is realized and observed and the server uses it for the short-run decision of the service feep(V′_{, µ). It is assumed} that the server can commit to fixed rulesµ(m) andp(V′_{, µ).}

164 TO QUEUE OR NOT TO QUEUE c1 c2 µ µ1 ΛR S S=Z Z µ0 µ2

Figure 8.1. Optimal service rates in a system with bribery

Lastly, the equilibrium arrival rate, given µand p, is obtained, as in §3.3.

Whang observed that those customers who choose to arrive have the same objective with respect to the server’s long- and short-run decisions, namely, they all wish to minimizeCW(λ, µ)+p. Therefore, it is assumed that customers cooperate in their reports so that a customer with service valueR reports a value σ(R) so that the resulting rate,m, solves

min

m [CW(λ, µ(m)) +p(V

′_{, µ(m))]} subject to the equilibrium condition

V′(λ) =CW(λ, µ(m)) +p(V′, µ(m)).

The ruleσ may depend on the customers’ service value, but not on the unknown functionV′_.

Service rate decisions 165 Whang observed that if according to the rule µ(m) higher reports lead to a higher service rate, then customers will report increasingly high values so that a higher than social optimal µ is set by the server. Therefore, an optimal rule for the server is not monotone.

A strategy (µ(m), p(V′_{, µ)) that achieves a solution which is socially} optimal under full information is said to be a full-information-efficient strategy. Whang’s main claim is that if W = W(ρ), then the follow- ing strategy is full-information-efficient and motivates each customer to reveal his true service value:

On receiving the reports m, the manager should solve the system’s problem assuming that the reports are true, and fully allocate the capacity costbµ∗_{(m) to the customers.}

Note that the assumption W = W(ρ) does not hold in most queueing models, in particularM/M/1.

4.2.

Heterogeneous time values

Balachandran and Radhakrishnan [19] analyzed a model of class de- cision with asymmetric information. In this model, the demand for service consists of a finite number of classes of customers each of which with given rates of demand. The cost of operating the server is a con- vex monotone increasing functionc(µ). The time values,Ci, are private

information of the controllers of the classes. The sequence of events is as follows:

The server announces a rule by which the server’s capacity is deter- mined and the operating costs are divided among the classes. Classes report their time values, C_ir (that may be different from the true values, Ci).

The server decides on the service rate µ.

Classes pay for the operating costs according to the rule.

Balachandran and Radhakrishnan observed that if the server deter- mines the overall optimal µ assuming that the time values are as re- ported, then classes have incentives to overstate their time values and by doing so induce higher service rates and shorter waiting costs. It is assumed therefore that the server can obtain measures, Xi, of the

true values Ci. These measures are required to be unbiased estimators

for Ci (that is, E(Xi) = Ci). Balachandran and Radhakrishnan show

166 TO QUEUE OR NOT TO QUEUE the operating costs so that classes are induced to report their true time values.

It should be emphasized that the purpose of the pricing scheme here is not to control the arrival rates (these are considered to be fixed) but to induce truthful reports of the time costs in order to determine optimal service capacity.

5.

Observable vs. unobservable queues

Hassin [66] compared the social welfare under the profit-maximizing number of single server facilities, in the basic models of Naor [133] and Edelson and Hildebrand [47]. Let c denote the cost per unit of time associated with operating a facility, regardless of its utilization. Assume that the potential arrival rate is large and that the queue organizer determines the optimal arrival rate to each facility. If the gain from a facility that serves an arrival process with rate λ is Z(λ), then the optimal arrival rate per facility maximizes Z(λ_λ)−c, that is, maximizes gain per unit of arrival rate.

Suppose that the service manager can choose between revealing the queue length to the customers and operating an observable queue, or concealing this information and operating an unobservable queue (_§3.2). Hassin showed that social welfare may, in some cases, be increased by motivating the profit maximizing firm to reveal the queue length when it otherwise prefers to conceal it. On the other hand, it never pays, from the social point of view, to induce the firm to conceal the queue length when it is willing to reveal it. To see this let the operating costcbe such that the firm prefers to reveal the queue length. Let λ1 be the rate it

chooses in this case, and letλ2 be the rate it would choose had it been

impossible for it to reveal this information. Then,

SO−c λ1 ≥ ZO−c λ1 > ZU−c λ2 = SU−c λ2 ,

where Z denotes profit,S denotes social welfare, O denotes observable queues and U denotes unobservable queues. The first inequality holds sinceSO ≥ ZO, the second since the firm prefers an observable queue.

Thus, social welfare decreases when the firm operates an unobservable queue at rate ofλ2 .

Service rate decisions 167