OBSERVABLE QUEUES
2. The LCFS-PR model
Hassin [65] suggested a way to achieve social optimality without im- posing admission fees. This section is devoted to describing this ap- proach and its implications. We adopt Naor’s model with two changes:
Assumption 8 changes as follows:
The service discipline is LCFS-PR, that is, a newly arrived customer joins the system and is immediately served, possibly preempting the service of another customer. Preempted customers join a queue where later arrivals get priority over earlier arrivals. When a preempted customer’s turn to re-enter service comes, his service is resumed from the point of interruption.
Assumption 9 changes as follows:
At any instant, each customer is allowed to renege at no additional cost and never return. The queue is fully observable at any instant, so that a customer can base his decision on the queue length and on his position in it.
Observable queues 25 In a FCFS queue, a new customer is placed at the end of the queue, and therefore imposes no negative externalities on customers already in the system. However, this customer may impose negative externalities on future arrivals. The essence of the discrepancy between individual and social optimization in Naor’s FCFS model lies in the fact that the customer ignores these externalities. Therefore, the individual may join a queue even when his own expected welfare is smaller than the expected reduction in welfare to future customers.
The externalities imposed by a newly arrived customer on those who are presently in the system, are highest if he is assigned to the head of the queue. Under LCFS-PR, every arriving customer is placed at the head of the queue, pushing back those customers who arrived earlier. However, all future arrivals will be placed in front of him and therefore he does not impose any external effects on them.
Hassin observed that LCFS-PR leads to a socially optimal behavior by the customers. The relevant decision that an individual faces iswhen to leave the queue rather than whether to join it. By the memoryless property of the exponential distribution it follows that the distributions of the customers’ residual service are independent of the queue length and of the amount of service each of them has already received. Since the model assumes homogeneous customers, the waiting customers have identical time and service values as well as identical distributions of residual service time. Therefore, when a customer decides to renege there is no other customer behind him. Since everybody present is served prior to the person at the end of the queue, he imposes no externalities, regardless of his action. In other words, his considerations coincide with those of the society, and hence he will reach the same conclusion of whether or not to renege. In particular, his threshold isn∗. (Note that from the social point of view the order of service is irrelevant, so that the socially optimal threshold is the same under the FCFS and LCFS- PR regimes.) In the next section we use this observation in order to determinen∗.7
We now discuss the LCFS-PR model and its implications.
There is a strategic difficulty associated with the LCFS-PR model. A customer whose service has been preempted is motivated to renege and re-enter the system, pretending to be a new arrival. Such behav-
7Remarks 2.1 and 2.2 also apply to the FCFS-PR model. A customer in positionn∗knows
that the same reasoning that led him not to renege earlier leads everyone in front of him not to renege while in positions 1 ton∗−1. Similarly, the equilibrium is not unique. For
example, reneging if and only if there are exactlyn∗customers ahead in the queue is also an
26 TO QUEUE OR NOT TO QUEUE ior contradicts Assumption 10 in Naor’s model and therefore must be prevented administratively.8
The important property of the LCFS-PR model that leads to optimal individual behavior is that the last customer in the queue remains at the end of the queue as long as he is in the system and therefore he imposes no externalities. This property is preserved by any queue discipline with the property that the newly arrived customer is placed anywhere except for at the end of the queue. A particularly appeal- ing policy is to assign a newly arrived customer, whenever the server is busy, to the position before the last. This policy reduces the cus- tomer’s incentive to renege and re-enter as a new arrival. There is, however, another difficulty associated with this solution. Suppose that customer A is now at the end of the queue while B is just one position ahead of him. If A reneges, then B becomes the last one, and all future arrivals will be positioned ahead of him. Thus B may find it beneficial to offer A a payment so that A doesn’t renege. Such side payments must be prevented to preserve optimal behavior. This can be done by concealing the identities of the customers in the queue. The solution just proposed has other advantages over LCFS-PR: (i) Preemption may incur some loss of service and this solution is asso- ciated with fewer preemptions. (ii) Risk averse customers are worse off under the LCFS-PR discipline than under other queue disciplines like FCFS, since LCFS-PR is associated with a larger waiting time variance.9 Under the LCFS-PR rule some customers are continuously
served without waiting while others wait for long periods of time and finally renege without being served. In particular, in a FCFS queue no customer incurs negative utility (assuming that the utility asso- ciated with immediate balking is 0), while this is not the case with LCFS-PR. Assigning new arrivals to the position before last reduces all these drawbacks while maintaining a socially optimal behavior. The model is of course a simplified one. However, the qualitative implications are quite general. It is well known that if customers differ by their characteristics (waiting cost, service distribution, ser- vice value, etc.) social welfare can be increased by proper assignment
8If waiting “at home” is less costly than waiting in the queue it may be socially desirable
that a customer returns to the system after balking or reneging. Models with retrials are discussed in§5.
9Kingman [87] showed that FCFS (respectively, LCFS-PR) minimizes (respectively, maxi-
Observable queues 27 of priorities (this is the subject of §4). A consequence of the above discussion is that:
– Assigning priorities may be beneficial even when the customers are identical!
Suppose that priorities are assigned randomly or according to some irrelevant basis. The customer at the end of the queue will usually have low priority, and may expect most future arrivals to be placed in front of him. This decreases the externalities he imposes and makes his decision of whether to renege closer to the socially optimal one. Olson [136] showed that a LCFS-PR regime can be induced through an appropriate price menu, so that customers receive priority levels based on the amount they paid rather than administratively. Such a pricing system will also achieve social optimality. See also§4.1.3. An LCFS-PR discipline induces optimal customers’ behavior also in more general observable models. For example, consider an M/M/s
system, where the servers may have different service rates. An ar- riving customer starts service at the fastest server. This action may preempt the service of an earlier customer who is then moved to the second fastest server, and so on. A customer at the slowest server may be returned to the queue. It is also possible that the customer at the end of the queue reneges at this stage as a result of the increase in his expected waiting time. As in the single server system (and because of the same reasons), reneging is done in the socially optimal way. Variations of this model with s= 2 were analyzed by Xu [177]. Illustrative descriptions of the LCFS-PR model and its consequences were given by Nalebuff in [132] and Landsburg in [98].