Regulating an observable M/M/1 queue

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Abstract

Naor (1969) was the first to observe that in observable queues, customers’ joining behavior may not coincide with the socially optimal one. The question then is how to regulate the system such that customers, while minding their own utility, will decide to join only when it is advised by society. After reviewing some existing mechanisms (some involving money transfers and some not), we suggest novel ones that do not involve money transfers and possess some advantages over the existing ones.

Introduction

Naor  [10] was the first to observe that queues call for regulation: left to themselves, customers join a queue at a rate that is greater than is socially desired. The model he used is as follows. In a single-server queue there exists a Poisson arrival stream of customers at rate λ. Service times follow an exponential distribution at rate μ. All customers value service by R and suffer a cost of C per unit of time in the system (service inclusive). The default service regime is First Come First Served (FCFS). Upon arrival, customers inspect the queue length and decide whether to join or not. Customers are risk neutral, and hence wish to maximize their expected individual monetary utility (assume without loss of generality that not joining comes with a zero utility). The customers’ optimal strategy here is trivial: join if and only if the number in the system (inclusive of themselves), n, is less than or equal to ne, where ne=Rμ/C.

Next Naor assumed that all costs and rewards go to a single entity (to be called “society”) instead of to the individuals themselves, and that society can enforce its (socially) optimal join-do-not-join entry policy. This leads to another threshold-based policy: join if and only if the number in the system upon arrival is less than or equal to ns for some nsne. Naor showed that ns is the largest integer n obeying n(1ρ)ρ(1ρn)(1ρ)2RμC, where ρ=λ/μ. Note that it is not assumed that ρ<1 and in the case where ρ=1 the right-hand side of the inequality above is derived by continuity and equals (n+1)n/2.

The reason behind the fact that nsne is that when selfish customers put all costs and rewards in their equation, they ignore the negative externalities associated with joining a queue. These externalities take the form of making others (who join later) wait longer than they would have to do otherwise. Society minds these externalities, and hence the social utility associated with an individual joining customer is less than her individual utility. Therefore, joining may but need not be socially beneficial only if it is also worthy for the joiner herself; that is, the observed queue length is less than or equal to ne.

All the above call for regulation, i.e., a set of rules, administrated by a central planner, under which the equilibrium behavior coincides with the socially optimal behavior. The purpose of this paper is to review some regulation mechanisms from the existing literature and suggest new ones.

The basic idea behind the new mechanisms suggested here is that customers should be treated differently, depending on the queue length upon their arrival. Those who arrive when the queue length is shorter than ns are welcome by society, and should get an incentive to join, while all others are not welcome, and hence should get an incentive to balk. We would like to point out that this discrimination is not conceptually different from the situation under an unregulated system, where customers’ utility is queue length dependent.

In Section  2 we review the existing and the new mechanisms. A succinct comparison of the mechanisms in this paper, with respect to some properties to be defined next, is given in Section  2.7. Finally, in Section  3, we briefly describe the unobservable version of this model as well as some regulation mechanisms that are applicable in that case.

Section snippets

Regulation schemes

The next question looked at by Naor is how to make customers adopt the socially optimal policy while they, from their point of view, maximizing only their own individual utility (and without forcing them to do so).

Unobservable queue

Our final comment here concerns the counterpart problem when customers have to make up their mind regarding whether to join or not without inspecting the queue length first. This is Edleson and Hildebrand’s  [3] unobservable model. Assuming C/μRC/(1ρ), they show that customers’ (Nash) equilibrium joining probability equals pe=(μC/R)/λ while the socially optimal joining probability is ps=(μCμ/R)λ. For the same reasons as those outlined in the observable version, ps<pe (a fact that can also

Acknowledgment

This research was partly supported by Israel Science Foundation Grant Nos. 1319/11 and 515/15.

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