A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations

,Chia-Huang Wu,Kuo-Hsiung Wang,,Department Of Applied Mathematics, National Chung-Hsing University, Taichung 402 ,Jau-Chuan Ke,,Department Of Business Administration, Asia University, Wufeng, Taichung 41354 ,Jyh-Bin Ke

doi:10.3934/jimo.2012.8.1

Abstract

This paper focuses on an M/M/$s$ queue with multiple working vacations such that the server works with different service rates rather than no service during the vacation period. We show that this is a generalization of an M/M/1 queue with working vacations in the literature. Service times during vacation period, or during service period and vacation times are all exponentially distributed. We obtain the useful formula for the rate matrix $\textbf{R}$ through matrix-geometric method. A cost function is formulated to determine the optimal number of servers subject to the stability conditions. We apply the direct search algorithm and Newton-Quasi algorithm to heuristically find an approximate solution to the constrained optimization problem. Numerical results are provided to illustrate the effectiveness of the computational algorithm.

Highlights

We analyze an M/M/s queue with multiple working vacations such that the server works with variable service rates rather than completely terminates service during a vacation period
In steady-state, we introduce the following notations: P0(n) ≡ probability that there are n customers in the system when all servers are on a working vacation, where n = 0, 1, 2, . . .; P1(n) ≡ probability that there are n customers in the system when there are (s−1) servers on a working vacation, where n = 1, 2, . . .; Pk(n) ≡ probability that there are n customers in the system when there are (s−k) servers on a working vacation, where n = k, k + 1, k + 2, . . ., k = 2, 3, 4, . . . , s
For the multiple-server queue, we assume that there exists a matrix R satisfying the following equation: R2As + RBs + C = 0, where As, Bs, and C are given in (13), (12), (10), respectively

Summary

Introduction

We analyze an M/M/s queue with multiple working vacations such that the server works with variable service rates rather than completely terminates service during a vacation period. Newton-Quasi algorithm; optimization; rate matrix; sensitivity analysis; working vacations.

Results

Conclusion