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When the service discipline in an observable queue is not FCFS, the decision of whether or not to join depends not only on the state of the queue but also on the strategy adopted by future arrivals. In Section 2
15See [33] for a criticism of the model of [41].
16There is no waiting in queue and therefore the value of service is compared to the price
and not to a full price.
17Note that A
s−1(T∗) is not the maximum rate of social gains in thes−1 servers model,
Observable queues 37 we described the consequences of a LCFS-PR discipline when reneging is allowed. In this section we describe several models where reneging is forbidden. The equilibrium strategies in such models are different in nature from those obtained when reneging is allowed.
6.1.
LCFS
Tilt and Balachandran [168] considered aGI/M/sLCFS model (with- out preemption) where reneging is forbidden, but balking is allowed. Hassin and Haviv [73] considered the same model with emphasis on sub- game perfect solutions. Consider first aGI/M/s/(N +s) LCFS model, that is, where the number of customers in the system is at mostN+s. A customer who observesN customers in the queue balks. Since there is no preemption in the model, a customer who observes a free server joins if and only ifR≥ Cµ.
Tilt and Balachandran allowed heterogeneous service and time values, and admission fees that depend on the customer’s type. They showed how to compute an SPE when the queue length is bounded. We will consider below the case of homogeneous customers.
Suppose that a customer arrives when all the servers are busy and there are i customers in the queue. Recall that i = N means that the customer must balk, so assume that 0 ≤ i < N. Note that of all possible values of iin this range, the preferred one for a new customer isi=N −1. The reason is that theseN −1 customers do not impose any waiting time on the new customer, since they are guaranteed to stay behind him as long as he is in the queue. Moreover, the customer is not concerned with future arrivals, as they are forced to balk as long as the new customer is still in the queue.
Theforced joining strategyprescribes joining the system as long as this is possible. Let Qn be the expected queueing time of a customer who
observes n,n= 1, . . . , N, vacant positions upon arrival, when all other customers adopt the forced joining strategy. Tilt and Balachandran showed that for n= 1, . . . , N
Qn= 1 sµ n−1 X j=0 λ sµ j .
Clearly,Qn is monotone increasing inn: a higher value ofnmeans that
more future arrivals are expected to overtake the new customer. The expected net benefit of such a customer, if he joins, is
Rn=R−C
1
µ+Qn
38 TO QUEUE OR NOT TO QUEUE Suppose the model’s parameters are set so that for some k, 1≤k≤ N −1, Rk > 0 but Rk+1 <0. Then, the best response for a customer
who observes staten(when all others using the forced joining strategy) is to join when 1≤n≤k and balk otherwise. Remove now the forced joining assumption. Still by induction we conclude that under an SPE, customers join when n ≤ k and balk when n = k+ 1. The fact that customers do not join whenn=k+ 1 makes it worthwhile to join when
n=k+ 2. Continuing with this line of reasoning we conclude that under the SPE strategy, customers balk if n=ik+ 1 for somei≤1 and join otherwise.
There are other pure equilibrium strategies. An example for such a strategy whenN >2k+ 1 is as follows: join if and only if the number of customers in the queue is smaller thenj for some j < k. The strategy does not prescribe an optimal response when the number of customers observed upon arrival isi,N−k≤i≤N−1: joining is certainly better. Yet, it is still an equilibrium since these states are transient under this strategy.
Consider now the model with N = ∞. Assume again that Rk > 0
and Rk+1 < 0. The above-mentioned properties are still valid for an
equilibrium. However, in this case there are more pure subgame perfect equilibria. An SPE prescribes for some l∈ {0, . . . , k−1} joining in all states except for those whose index is (k+ 2)i+lfor some integer i >0. For example, when λ is sufficiently large so that A0 > 0 but A1 < 0,
there are two SPE solutions. One prescribes joining if and only if the queue length has an odd value, the other if and only if it is even.
6.2.
EPS and random queues
Altman and Shimkin [10] considered a system of observable egalitar- ian processor sharing (EPS) where reneging is forbidden. Customers decide whether to join the queue after observing the number of cus- tomers already there. As in the LCFS model, customers are affected by the strategies adopted by future customers. This is an ATC situation, and Altman and Shimkin showed how to compute the unique (pure or mixed) threshold equilibrium strategy. This strategy is also the unique SPE.18
In a similar model, service is granted in random order. In fact, due to the memoryless service process, the two models coincide with respect to the decision problem posed here if the decision about whose service was completed is done randomly among the customers in the queue
18An extension where customers differ in their expected service time is considered in Ben-
Observable queues 39 afterservice completions. When customerscommenceservice in random order, the threshold can be computed in a similar way.
A variation of the model, where reneging is allowed, leads to a different model which is analyzed in detail in§5.1.