Finance, Economics, and Mathematics

Chapter 37An Inequality for the Variance of Waiting Time under a General Queueing Discipline

Operations Research, 25 (5) (1977), 879–884.

Abstract

We show that the expected value of any convex function of the waiting time (such as the variance) in a general queuing system under any queuing discipline independent of the service times does not exceed that under the last-come-first-served discipline, and is not less than that under the first-come-first-served discipline.

Introduction

It has been noted (see, for instance, Cohen (1969), Riordan (1962), Vaulot (1954)) that the variance of the waiting time in the and M/G/1 queuing systems under the last-come-first-served (LCFS) discipline exceeds that under service in the order of arrivals, first-come-first-served (FCFS). This observation has been made by comparison of explicit expressions for the variance. Moreover, in the cases when the waiting-time variance has been determined under service in random order (SIRO), it was found to attain an intermediate value (see Riordan 1962). The expected waiting times in all three cases are, of course, equal.

This has led to an interpretation of the FCFS discipline as more “fair” than either the SIRO or LCFS: While the expected waiting time of a given customer is not influenced, the total waiting time of all customers is more equally divided under FCFS than under SIRO, while the LCFS discipline ...