Study on the Queue-Length Distribution inGeo/G(MWV)/1/NQueue with Working Vacations

Chuanyi Luo,Yinghui Tang,Jiangyi Li

doi:10.1155/2015/160697

Abstract

This paper analyzes a finite buffer size discrete-timeGeo/G/1/Nqueue with multiple working vacations and different input rate. Using supplementary variable technique and embedded Markov chain method, the queue-length distribution solution in the form of formula at arbitrary epoch is obtained. Some performance measures associated with operating cost are also discussed based on the obtained queue-length distribution. Then, several numerical experiments follow to demonstrate the effectiveness of the obtained formulae. Finally, a state-dependent operating cost function is constructed to model an express logistics service center. Regarding the service rate during working vacation as a control variable, the optimization analysis on the cost function is carried out by using parabolic method.

Highlights

Discrete-time queues with classical vacation policies have been explored more in depth during the last few decades due to their widespread application in telecommunication system, electronic information network, production system, and so on
On account of the introduction mentioned above, we study the finite buffer Geo/G/1 queue with working vacations and different arrival rates, denoted by Geoλ1,λ2 /G(MWV)/ 1/N
The length of the service time for a customer during working vacation period, denoted by χ(V), is a random variable following another arbitrary distribution with probability mass function (PMF) gj(V) = P{χi(V) = j}, j ≥ 1 and probability generating function (PGF) G(V)(z) = ∑∞ j=1 gj(V)zj and finite mean service time E[χ(V)] = α2 > α1

Summary

Introduction

Discrete-time queues with classical vacation policies have been explored more in depth during the last few decades due to their widespread application in telecommunication system, electronic information network, production system, and so on (see Takagi [1], Tian et al [2], Alfa [3], and the references therein). On the basis of working-vacation Geo/G/1 queue, Goswami and Selvaraju [11] extended the working-vacation policy to queueing model with Markovian arrival process and general phase-type distributed service time. These mentioned research works all concentrated on working-vacation queueing models with infinite buffer size; the finite buffer size counterparts received little attention. The existing research topics with finite buffer size and working vacations mentioned above all concentrate on GI/Geo/1/N queueing system and its different varieties. On account of the introduction mentioned above, we study the finite buffer Geo/G/1 queue with working vacations and different arrival rates, denoted by Geoλ1,λ2 /G(MWV)/ 1/N.

Model Description

The Queue-Length Solutions in the Form of Formula

Performance Evaluation

Discussion of Numerical Performance Characteristics

A Numerical Example

Conclusions