Maintenance in Single‐Server Queues: A Game‐Theoretic Approach

Najeeb Al-Matar,Jewgeni H Dshalalow

doi:10.1155/2009/857871

Abstract

We use antagonistic stochastic games and fluctuation analysis to examine a single‐server queue with bulk input and secondary work during server′s multiple vacations. When the buffer contents become exhausted the server leaves the system to perform some diagnostic service of a minimum of L jobs clustered in packets of random sizes (event A). The server is not supposed to stay longer than T units of time (event B). The server returns to the system when A or B occurs, whichever comes first. On the other hand, he may not break service of a packet in a middle even if A or B occurs. Furthermore, the server waits for batches of customers to arrive if upon his return the queue is still empty. We obtain a compact and explicit form functional for the queueing process in equilibrium.

Highlights

In this paper we use a game-theoretic analysis for a single-server queue with maintenance
In our case, the server is vacant from the system not just to perform some unspecified maintenance as it is the case in most systems with server vacations, but to render a random quantity of jobs. This maintenance can be associated with a semiroutine diagnostic, as computer servers render from time to time
Unlike most of queues with vacations, in which the vacationing time is “anonymous,” our server takes on real jobs

Summary

Introduction

In this paper we use a game-theoretic analysis for a single-server queue with maintenance. Jobs are included in packets of random quantities as it is the case in computer networks and telecommunications , so that by no means can the server break a packet with unfinished jobs and return to the system even if the total amount of jobs crosses L or his time spent with maintenance expires beyond T. The paper concludes with a summary of the results obtained and the discussion of the upcoming work on continuous time parameter queueing process and an extended global control under the revival of rendered jobs during the maintenance period

A Relationship to the Existing Literature

Formalism of the Model and the Main Past Results

A Special Case of the Maintenance Process

Phase 2

The System on a Busy Period and in Equilibrium