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January  2014, 10(1): 89-112. doi: 10.3934/jimo.2014.10.89

## Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy

 1 Department of Statistics, College of Sciences, Yanshan University, Qinhuangdao 066004 2 Department of Intelligence and Informatics, Konan University, 8-9-1 Okamoto, Kobe 658-8501 3 Department of Telecommunications, Budapest University of Technology and Economics, Budapest 4 College of Science, Yanshan University, Qinhuangdao 066004, China

Received  September 2012 Revised  June 2013 Published  October 2013

In this paper, we consider an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy, where we examine the case that customer impatience is due to the servers' vacation. Whenever a system becomes empty, the server takes a vacation. However, the server is allowed to take a maximum number $K$ of vacations if the system remains empty after the end of a vacation. This vacation policy includes both a single vacation and multiple vacations as special cases. We derive the probability generating functions of the steady-state probabilities and obtain the closed-form expressions of the system sizes when the server is in different states. We further make comparisons between the mean system sizes under the variant vacation policy and the mean system sizes under the single vacation policy or the multiple vacation policy. In addition, we obtain the closed-form expressions for other important performance measures and discuss their monotonicity with respect $K$. Finally, we present some numerical results to show the effects of some parameters on some performance measures.
Citation: Dequan Yue, Wuyi Yue, Zsolt Saffer, Xiaohong Chen. Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy. Journal of Industrial & Management Optimization, 2014, 10 (1) : 89-112. doi: 10.3934/jimo.2014.10.89
##### References:
 [1] E. Altman and U. Yechiali, Analysis of customers' impatience in queues with server vacations,, Queueing Systems, 52 (2006), 261. doi: 10.1007/s11134-006-6134-x. [2] E. Altman and U. Yechiali, Infinite-server queues with systems' additional task and impatient customers,, Probability in the Engineering and Informational Sciences, 22 (2008), 477. doi: 10.1007/978-1-4020-8741-7_57. [3] A. D. Banik, The infinite-buffer single server queue with a variant of multiple vacation policy and batch Markovian arrival process,, Applied Mathematical Modelling, 33 (2009), 3025. doi: 10.1016/j.apm.2008.10.021. [4] S. Benjaafar, J. Gayon and S. Tepe, Optimal control of a production-inventory system with customer impatience,, Operations Research Letters, 38 (2010), 267. doi: 10.1016/j.orl.2010.03.008. [5] T. Bonald and J. Roberts, Performance modeling of elastic traffic in overload,, ACM Sigmetrics Performance Evaluation Review, 29 (2001), 342. [6] B. Doshi, Single server queues with vacation: A survey,, Queueing Systems, 1 (1986), 29. [7] S. Economou and S. Kapodistria, Synchronized abandonments in a single server unreilable queue,, European Journal of Operational Research, 203 (2010), 143. [8] N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: Tutotial, review, and research prospects,, Manufacturing and Service Operations Management, 5 (2003), 79. doi: 10.1287/msom.5.2.79.16071. [9] J. C. Ke, Operating characteristic analysis on the $M^{[X]}$/G/1 system with a variant vacation policy and balking,, Applied Mathematical Modelling, 31 (2007), 1321. doi: 10.1016/j.apm.2006.02.012. [10] J. C. Ke and F. M. Chang, Modified vacation policy for M/G/1 retrial queue with balking and feedback,, Computer & Industrial Engineering, 57 (2009), 433. doi: 10.1016/j.cie.2009.01.002. [11] J. C. Ke, H. I. Huang and Y. K. Chu, Batch arrival queue with N-policy and at most J vacations,, Applied Mathematical Modelling, 34 (2010), 451. doi: 10.1016/j.apm.2009.06.003. [12] N. Perel and U. Yechiali, Queues with slow servers and impatient customers,, European Journal of Operational Research, 201 (2010), 247. doi: 10.1016/j.ejor.2009.02.024. [13] H. Takagi, "Queueing Analysis, A Foundation of Performance Evaluation, Volume 1: Vacation and Priority Systems,", Part 1. North-Holland Publishing Co., (1991). [14] N. Tian and Z. G. Zhang, "Vacation Queueing Models: Theory and Applications,", New York: Springer, (2006). [15] T. Y. Wang, J. C. Ke and F. M. Chang, On the discrete-time Geo/G/1 queue with randomized vacations and at most $J$ vacations,, Applied Mathematical Modelling, 35 (2011), 2297. doi: 10.1016/j.apm.2010.11.021. [16] U. Yechiali, Queues with system disasters and impatient customers when system is down,, Queueing Systems, 56 (2007), 195. doi: 10.1007/s11134-007-9031-z. [17] D. Yue, W. Yue and G. Xu, Analysis of customers' impatience in an M/M/1 queue with and working vacations,, Journal of Industrial and Management Optimization, 8 (2012), 895. doi: 10.3934/jimo.2012.8.895. [18] Z. G. Zhang and N. Tian, Discrete time Geo/G/1 queue with multiple adaptive vacations,, Queueing Systems, 38 (2001), 419. doi: 10.1023/A:1010947911863.

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##### References:
 [1] E. Altman and U. Yechiali, Analysis of customers' impatience in queues with server vacations,, Queueing Systems, 52 (2006), 261. doi: 10.1007/s11134-006-6134-x. [2] E. Altman and U. Yechiali, Infinite-server queues with systems' additional task and impatient customers,, Probability in the Engineering and Informational Sciences, 22 (2008), 477. doi: 10.1007/978-1-4020-8741-7_57. [3] A. D. Banik, The infinite-buffer single server queue with a variant of multiple vacation policy and batch Markovian arrival process,, Applied Mathematical Modelling, 33 (2009), 3025. doi: 10.1016/j.apm.2008.10.021. [4] S. Benjaafar, J. Gayon and S. Tepe, Optimal control of a production-inventory system with customer impatience,, Operations Research Letters, 38 (2010), 267. doi: 10.1016/j.orl.2010.03.008. [5] T. Bonald and J. Roberts, Performance modeling of elastic traffic in overload,, ACM Sigmetrics Performance Evaluation Review, 29 (2001), 342. [6] B. Doshi, Single server queues with vacation: A survey,, Queueing Systems, 1 (1986), 29. [7] S. Economou and S. Kapodistria, Synchronized abandonments in a single server unreilable queue,, European Journal of Operational Research, 203 (2010), 143. [8] N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: Tutotial, review, and research prospects,, Manufacturing and Service Operations Management, 5 (2003), 79. doi: 10.1287/msom.5.2.79.16071. [9] J. C. Ke, Operating characteristic analysis on the $M^{[X]}$/G/1 system with a variant vacation policy and balking,, Applied Mathematical Modelling, 31 (2007), 1321. doi: 10.1016/j.apm.2006.02.012. [10] J. C. Ke and F. M. Chang, Modified vacation policy for M/G/1 retrial queue with balking and feedback,, Computer & Industrial Engineering, 57 (2009), 433. doi: 10.1016/j.cie.2009.01.002. [11] J. C. Ke, H. I. Huang and Y. K. Chu, Batch arrival queue with N-policy and at most J vacations,, Applied Mathematical Modelling, 34 (2010), 451. doi: 10.1016/j.apm.2009.06.003. [12] N. Perel and U. Yechiali, Queues with slow servers and impatient customers,, European Journal of Operational Research, 201 (2010), 247. doi: 10.1016/j.ejor.2009.02.024. [13] H. Takagi, "Queueing Analysis, A Foundation of Performance Evaluation, Volume 1: Vacation and Priority Systems,", Part 1. North-Holland Publishing Co., (1991). [14] N. Tian and Z. G. Zhang, "Vacation Queueing Models: Theory and Applications,", New York: Springer, (2006). [15] T. Y. Wang, J. C. Ke and F. M. Chang, On the discrete-time Geo/G/1 queue with randomized vacations and at most $J$ vacations,, Applied Mathematical Modelling, 35 (2011), 2297. doi: 10.1016/j.apm.2010.11.021. [16] U. Yechiali, Queues with system disasters and impatient customers when system is down,, Queueing Systems, 56 (2007), 195. doi: 10.1007/s11134-007-9031-z. [17] D. Yue, W. Yue and G. Xu, Analysis of customers' impatience in an M/M/1 queue with and working vacations,, Journal of Industrial and Management Optimization, 8 (2012), 895. doi: 10.3934/jimo.2012.8.895. [18] Z. G. Zhang and N. Tian, Discrete time Geo/G/1 queue with multiple adaptive vacations,, Queueing Systems, 38 (2001), 419. doi: 10.1023/A:1010947911863.
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