Analysis of an M/M/1 Queue Using Fixed Order of Search for Arrivals and Service

TL;DR

This paper investigates an M/M/1 queue with a unique search-based discipline for arrivals and service, deriving asymptotic expansions for waiting time moments and comparing variance characteristics to known queue disciplines.

Contribution

It introduces a novel queue discipline involving search for stations, providing asymptotic analysis of waiting times and variance behavior in heavy traffic.

Findings

Asymptotic expansions for all moments of waiting time were derived.

Variance of waiting time resembles that of last-come-first-served discipline.

The discipline's variance behavior differs from first-come-first-served, with a higher pole order.

Abstract

We analyze an M/M/1 queue with a service discipline in which customers, upon arriving when the server is busy, search a sequence of stations for a vacant station at which to wait, and in which the server, upon becoming free when one or more customers are waiting, searches the stations in the same order for a station occupied by a customer to serve. We show how to find complete asymptotic expansions for all the moments of the waiting time in the heavy traffic limit. We show in particular that the variance of the waiting time for this discipline is more similar to that of last-come-first-served (which has a pole of order three as the arrival rate approaches the service rate) than that of first-come-first-served (which has pole of order two).