OBSERVABLE QUEUES
7. Discounting
Chen and Frank [34] generalized Naor’s model assuming that both the customers and the server maximize their expected discounted utility using a common discount rate. Chen and Frank computed the profit and social welfare maximizing pricing schemes for this generalization of Naor’s model.19 We illustrate this approach by describing the customers response to a given admission fee of sizep.
Let γ be the discount rate. Consider a customer who joins at time 0 and denote by θ his service completion time. The expected benefit for this customer is E " e−γθR−p−C Z θ 0 e−γtdt # , (2.14)
where the expectationE is taken with respect toθ.
Assume that the service times are exponentially distributed with a rate µ. If the arriving customer observes n customers already in the system, thenθfollows an Erlang distribution with parametersn+ 1 and
µ. In this case,E[e−γθ] =φn+1 where
φ= µ
µ+γ.
Therefore, the expression in (2.14) equals
φn+1 R+ C γ − p+C γ .
The customer prefers joining to balking if
φn+1 R+ C γ ≥ p+C γ , or n≤logφ p+ C γ R+Cγ ! −1.
40 TO QUEUE OR NOT TO QUEUE
8.
State dependent pricing
The discrepancy between the profit-maximizing price and the welfare- maximizing price in Naor’s model follows from the monopoly’s inability to extract all of the consumer surplus. Therefore, the monopoly’s objec- tive differs from the social one. In§3 we will see that this discrepancy doesn’t exist in the unobservable version of this model. Chen and Frank [34] observed another case where the profit maximizer’s objective co- incides with the social one, namely, when the server is able to adjust the price to the state of the system (and the population of customers is homogeneous). They showed that the profit-maximizing pricing scheme is to charge the maximum possible fee that does not deter customers from joining, as long as the queue length is less than a threshold, and to charge a high fee otherwise. All of the consumer surplus then goes to the server whose strategy (of whether to accept or reject a customer) is therefore socially optimal.
Chen and Frank also considered the case where customers have het- erogeneous service values. It is assumed that these values are not known to the server and hence they cannot be used to discriminate among cus- tomers. In this case, Chen and Frank found that the socially optimal strategy is as in the homogeneous case where the expected service value is used as a common value. Thus, the socially optimal behavior depends on the service distribution only trough its mean value. In particular, there is no loss of generality in assuming an exponential distribution. Moreover, as in the case where the server cannot adjust the fee to the state of the system (Section 5), the profit-maximizing fees tend to be higher than the socially optimal fees.20
The profit-maximizing strategy is socially optimal also when cus- tomers differ in their attributes, as long as the relevant information is available to the server and can be used to determine the admission fee.
Motivated by applications in the mortgage market, Levy, A and Levy, H [106] considered anM/M/1 queue where the server advertises a price
pi (from a given set) whenever there are i customers in the system. It
is assumed that there is a demand function D so that the joining rate associated with pi is λi = D(pi). The novel assumption of the model
is that a customer who leaves at time t pays the price advertised just before t. Thus, upon joining, the customer doesn’t know how much he will have to pay for the service. Levy and Levy proved that under
20Socially, the exact fees are unimportant, as long as they induce joining and balking in the
socially desired states. Hence, by saying that the server charges fees that are too high we mean that the threshold induced by these fees is too low.
Observable queues 41 the profit-maximizing pricing scheme, pi+1 ≥ pi i= 0,1, . . . , and that
the expected profit is higher than in the corresponding system where customers pay the price advertised before their arrival.
We suggest an extension of the model where the joining rates λi are
determined through an equilibrium mechanism. Suppose that the po- tential rate of demand is Λ, and that customers differ by their value of service. Consider a known strategy of the server, consisting of the prices
p0, p1, . . .. The joining ratesλ0, λ1, . . .define an equilibrium if λΛi equals
the probability that the service value of a random customer is at least the expected full pricePi associated with statei. For given joining rates
λ0, λ1, . . . , the expected full pricesPi are computed as follows: let qi,j
be the probability that a customer who joins while the system is in state
ileaves the system in state j (the state just before he leaves is j+ 1), then Pi=C i+ 1 µ + ∞ X j=0 qi,jpj+1.