Semigroup Methods for the M/G/1 Queueing Model with Working Vacation and Vacation Interruption

Ehmet Kasim

doi:10.5539/jmr.v8n5p56

Abstract

By using the strong continuous semigroup theory of linear operators we prove that the M/G/1 queueing model with working vacation and vacation interruption has a unique positive time dependent solution which satisfies probability conditions. When the both service completion rate in a working vacation period and in a regular busy period are constant, by investigating the spectral properties of an operator corresponding to the model we obtain that the time-dependent solution of the model strongly converges to its steady-state solution.

Highlights

According to (Zhang & Hou, 2010), the M/G/1 queueing system with working vacation and vacation interruption can be described by the following system of partial differential equations: d p0,0(t) dt = −λ p0,0 (t) + ∫∞μ0(x)p1,0(x, t)dx μ1(x)p1,1(x, t)dx, ∂p1,0(x, t) ∂t ∂p1,0(x, t) ∂x −[λ θμ0(x)]p1,0(x, t)
By using the strong continuous semigroup theory of linear operators we prove that the M/G/1 queueing model with working vacation and vacation interruption has a unique positive time-dependent solution which satisfies probability conditions
When the both service completion rate in a working vacation period and in a regular busy period are constant, by investigating the spectral properties of an operator corresponding to the model we obtain that the time-dependent solution of the model strongly converges to its steady-state solution

Summary

Introduction

By using the strong continuous semigroup theory of linear operators we prove that the M/G/1 queueing model with working vacation and vacation interruption has a unique positive time-dependent solution which satisfies probability conditions. When the both service completion rate in a working vacation period and in a regular busy period are constant, by investigating the spectral properties of an operator corresponding to the model we obtain that the time-dependent solution of the model strongly converges to its steady-state solution. (Servi & Finn, 2002) first studied the M/M/1 queueing system with multiple working vacation and obtained the transform formulae for the distribution of the number of customers in the system and the sojourn time in a steady state.

Results

Conclusion