# American Institute of Mathematical Sciences

January  2012, 8(1): 1-17. doi: 10.3934/jimo.2012.8.1

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

 1 Department of Industrial Engineering and Management, National Chiao Tung University, Hsingchu 30050, Taiwan 2 Department of Business Administration, Asia University, Wufeng, Taichung 41354, Taiwan 3 Department of Applied Statistics, National Taichung Institute of Technology, Taichung 404, Taiwan 4 Department of Applied Mathematics, National Chung-Hsing University, Taichung 402, Taiwan

Received  December 2009 Revised  June 2011 Published  November 2011

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.
Citation: Chia-Huang Wu, Kuo-Hsiung Wang, Jau-Chuan Ke, Jyh-Bin Ke. A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations. Journal of Industrial & Management Optimization, 2012, 8 (1) : 1-17. doi: 10.3934/jimo.2012.8.1
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