# 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 and Management Optimization, 2012, 8 (1) : 1-17. doi: 10.3934/jimo.2012.8.1
##### References:
 [1] Y. Baba, Analysis of a GI/M/1 queue with multiple working vacations, Operations Research Letters, 33 (2005), 201-209. doi: 10.1016/j.orl.2004.05.006. [2] A. D. Banik, U. C. Gupta and S. S. Pathak, On the GI/M/1/N queue with multiple working vacations-analytic analysis and computation, Applied Mathematical Modelling, 31 (2006), 1701-1710. doi: 10.1016/j.apm.2006.05.010. [3] R. L. Burden and J. Douglas, "Numerical Analysis,'' 7th Edition, Brooks/Cole, USA, 2001. [4] U. Chatterjee and S. P. Mukherjee, GI/M/1 queue with server vacations, Journal of the Operational Research Society, 41 (1990), 83-87. [5] E. K. P. Chong and S. H. Zak, "An Introduction to Optimization,'' 2nd Edition, Wiley, New York, 2001. [6] B. T. Doshi, Queueing systems with vacations-a survey, Queueing Systems Theory Appl., 1 (1986), 29-66. doi: 10.1007/BF01149327. [7] S. W. Fuhrmann and R. B. Cooper, Stochastic decompositions in the M/G/1 queue with generalized vacations, Operations Research, 33 (1985), 1117-1129. doi: 10.1287/opre.33.5.1117. [8] F. Karaesmen and S. M. Gupta, The finite capacity GI/M/1 queue with server vacations, Journal of the Operational Research Society, 47 (1996), 817-828. [9] T. Lee, The M/G/1/N queue with vacation and exhaustive service discipline, Operations Research, 32 (1984), 774-784. doi: 10.1287/opre.32.4.774. [10] J.-H. Li, W.-Y. Liu and N.-S. Tian, Discrete time GI/Geo/1 queue with multiple working vacations Queueing Systems, 56 (2007), 53-63. doi: 10.1007/s11134-007-9030-0. [11] C.-H. Lin and J.-C. Ke, Multi-server system with single working vacation, Applied Mathematical Modelling, 33 (2009), 2967-2977. doi: 10.1016/j.apm.2008.10.006. [12] W.-Y. Liu, X.-L. Xu and N.-S. Tian, Stochastic decomposition in the M/M/1 queue with working vacations, Operations Research Letters, 35 (2007), 595-600. doi: 10.1016/j.orl.2006.12.007. [13] M. F. Neuts, "Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach,'' Johns Hopkins Series in the Mathematical Sciences, 2, Johns Hopkins University Press, Baltimore, Md., 1981. [14] L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (M/M/1/WV), Performance Evaluation, 50 (2002), 41-52. doi: 10.1016/S0166-5316(02)00057-3. [15] H. Takagi, "Queueing Analysis: A Foundation of Performance Evaluation," Vol. 1, Vacation and Priority Systems, Part 1, North-Holland Publishing Co., Amsterdam, 1991. [16] N. Tian, D. Zhang and C. Cao, The GI/M/1 queue with exponential vacations, Queueing Systems Theory Appl., 5 (1989), 331-344. doi: 10.1007/BF01225323. [17] J. A. White, J. W. Schmidt and G. K. Benett, "Analysis of Queueing System,'' Operations Research and Industrial Engineering, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. [18] D. A. Wu and H. Takagi, M/G/1 queue with multiple working vacations, Performance Evaluation, 63 (2006), 654-681. doi: 10.1016/j.peva.2005.05.005.

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##### References:
 [1] Y. Baba, Analysis of a GI/M/1 queue with multiple working vacations, Operations Research Letters, 33 (2005), 201-209. doi: 10.1016/j.orl.2004.05.006. [2] A. D. Banik, U. C. Gupta and S. S. Pathak, On the GI/M/1/N queue with multiple working vacations-analytic analysis and computation, Applied Mathematical Modelling, 31 (2006), 1701-1710. doi: 10.1016/j.apm.2006.05.010. [3] R. L. Burden and J. Douglas, "Numerical Analysis,'' 7th Edition, Brooks/Cole, USA, 2001. [4] U. Chatterjee and S. P. Mukherjee, GI/M/1 queue with server vacations, Journal of the Operational Research Society, 41 (1990), 83-87. [5] E. K. P. Chong and S. H. Zak, "An Introduction to Optimization,'' 2nd Edition, Wiley, New York, 2001. [6] B. T. Doshi, Queueing systems with vacations-a survey, Queueing Systems Theory Appl., 1 (1986), 29-66. doi: 10.1007/BF01149327. [7] S. W. Fuhrmann and R. B. Cooper, Stochastic decompositions in the M/G/1 queue with generalized vacations, Operations Research, 33 (1985), 1117-1129. doi: 10.1287/opre.33.5.1117. [8] F. Karaesmen and S. M. Gupta, The finite capacity GI/M/1 queue with server vacations, Journal of the Operational Research Society, 47 (1996), 817-828. [9] T. Lee, The M/G/1/N queue with vacation and exhaustive service discipline, Operations Research, 32 (1984), 774-784. doi: 10.1287/opre.32.4.774. [10] J.-H. Li, W.-Y. Liu and N.-S. Tian, Discrete time GI/Geo/1 queue with multiple working vacations Queueing Systems, 56 (2007), 53-63. doi: 10.1007/s11134-007-9030-0. [11] C.-H. Lin and J.-C. Ke, Multi-server system with single working vacation, Applied Mathematical Modelling, 33 (2009), 2967-2977. doi: 10.1016/j.apm.2008.10.006. [12] W.-Y. Liu, X.-L. Xu and N.-S. Tian, Stochastic decomposition in the M/M/1 queue with working vacations, Operations Research Letters, 35 (2007), 595-600. doi: 10.1016/j.orl.2006.12.007. [13] M. F. Neuts, "Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach,'' Johns Hopkins Series in the Mathematical Sciences, 2, Johns Hopkins University Press, Baltimore, Md., 1981. [14] L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (M/M/1/WV), Performance Evaluation, 50 (2002), 41-52. doi: 10.1016/S0166-5316(02)00057-3. [15] H. Takagi, "Queueing Analysis: A Foundation of Performance Evaluation," Vol. 1, Vacation and Priority Systems, Part 1, North-Holland Publishing Co., Amsterdam, 1991. [16] N. Tian, D. Zhang and C. Cao, The GI/M/1 queue with exponential vacations, Queueing Systems Theory Appl., 5 (1989), 331-344. doi: 10.1007/BF01225323. [17] J. A. White, J. W. Schmidt and G. K. Benett, "Analysis of Queueing System,'' Operations Research and Industrial Engineering, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. [18] D. A. Wu and H. Takagi, M/G/1 queue with multiple working vacations, Performance Evaluation, 63 (2006), 654-681. doi: 10.1016/j.peva.2005.05.005.
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