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April  2016, 12(2): 653-666. doi: 10.3934/jimo.2016.12.653

## Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states

 1 Department of Statistics, College of Sciences, Yanshan University, Qinhuangdao 066004 2 Department of Intelligence and Informatics, Konan University, Kobe 658-8501 3 School of Economics and Management, Yanshan University, Qinhuangdao 066004, China

Received  September 2014 Revised  March 2015 Published  June 2015

We consider an M/M/1 queueing system with vacations and impatient customers. Whenever a customer arrives at the system, it activates an random impatience timer". If the customer's service has not been completed before the customer's impatience timer expires, the customer abandons the queue, and never returns. It is assumed that the impatience timer depends on the server's states. We analyze both multiple and single vacation scenarios and derive the probability generating functions of the number of customers in the system when the server is in vacation period and busy period. Then, we obtain explicit expressions for various performance measures such as the mean system sizes when the server is either on vacation or busy, the proportion of customers served, and the average rate of abandonments due to impatience. We present some numerical results for multiple vacation scenario to show the effects of the parameters of impatience timers on some performance measures. Finally, we show some inequalities on some performances under the single vacation policy and under multiple vacation policy.
Citation: Dequan Yue, Wuyi Yue, Guoxi Zhao. Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states. Journal of Industrial and Management Optimization, 2016, 12 (2) : 653-666. doi: 10.3934/jimo.2016.12.653
##### References:
 [1] E. Altman and U. Yechiali, Analysis of customers' impatience in queues with server vacations, Queueing Systems, 52 (2006), 261-279. 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), 477-493. doi: 10.1007/978-1-4020-8741-7_57. [3] C. J. Ancker and A. V. Gafarian, Some queueing problems with balking and reneging, Operations Research, 11 (1963), 88-100. doi: 10.1287/opre.11.1.88. [4] F. Baccelli, P. Boyer and G. Hebuterne, Single-server queues with impatient customers, Advances in Applied Probability, 16 (1984), 887-905. doi: 10.2307/1427345. [5] F. Baccelli and G. Hebuterne, On queues with impatient customers, in Perforamnce' 81 (F. Kylstra, ed.), North-Holland, Amsterdam, 1981, 159-179. [6] S. Benjaafar, J. Gayon and S. Tepe, Optimal control of a production-inventory system with customer impatience, Operations Research Letters, 38 (2010), 267-272. doi: 10.1016/j.orl.2010.03.008. [7] N. K. Boots and H. Tijms, A multiserver queueing system with impatient customers, Management Science, 45 (1999), 444-448. doi: 10.1287/mnsc.45.3.444. [8] O. J. Boxma and P. R. de Waal, Multiserver queues with impatient customers, ITC, 14 (1994), 743-756. doi: 10.1016/B978-0-444-82031-0.50079-2. [9] S. R. Chakravarthy, A disater queue with Markovian arrivals and impatient customers, Applied Mathematics and Computation, 214 (2009), 48-59. doi: 10.1016/j.amc.2009.03.081. [10] D. J. Daley, General customer impatience in the queue GI/G/1, Journal of Applied Probability, 2 (1965), 186-205. doi: 10.2307/3211884. [11] S. Dimou, A. Economou and D. Fakinos, The single server vacation queueing model with geometric abandonments, Journal of Statistical Planning and Inference, 141 (2011), 2863-2877. doi: 10.1016/j.jspi.2011.03.010. [12] B. Doshi, Single server queues with vacation: A survey, Queueing Systems, 1 (1986), 29-66. [13] S. Economou and S. Kapodistria, Synchronized abandonments in a single server unreliable queue, European Journal of Operational Research, 203 (2010), 143-155. doi: 10.1016/j.ejor.2009.07.014. [14] N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: Tutotial, review, and research prospects, Manufacturing and Service Operations Management, 5 (2003), 79-141. doi: 10.1287/msom.5.2.79.16071. [15] O. Garnett, A. Mandelbaum and M. Reiman, Designing a call center with impatient customers, Manufacturing and Service Operations Management, 4 (2002), 208-227. doi: 10.1287/msom.4.3.208.7753. [16] S. Graves, The application of queueing theory to continous perishable inventory systems, Management Science, 28 (1984), 401-406. [17] Y. W. Shin and T. S. Choo, M/M/s queue with impatient customers and retrials, Applied Mathematical Modelling, 33 (2009), 2596-2606. doi: 10.1016/j.apm.2008.07.018. [18] L. Takacs, A single-server queue with limited virtual waiting time, Journal of Applied Probability, 11 (1974), 612-617. doi: 10.2307/3212710. [19] H. Takagi, Queueing Analysis, A Foundation of Performance Evaluation, Volume 1: Vacation and Priority Systems, Part 1. North-Holland, Amsterdam, 1991. [20] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, Springer, New York, 2006. [21] U. Yechiali, Queues with system disasters and impatient customers when system is down, Queueing Systems, 56 (2007), 195-202. doi: 10.1007/s11134-007-9031-z. [22] D. Yue and W. Yue, Analysis of M/M/c/N queueing system with balking, reneging and synchronous vacations, in Advanced in Queueing Theory and Network Applications (ed. W. Yue etal.), Springer, 2009, 165-180. doi: 10.1007/978-0-387-09703-9_9. [23] D. Yue, W. Yue, Z. Saffer and X. Chen, Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy, Journal of Industrial and Management Optimization, 10 (2014), 89-112. doi: 10.3934/jimo.2014.10.89.

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##### References:
 [1] E. Altman and U. Yechiali, Analysis of customers' impatience in queues with server vacations, Queueing Systems, 52 (2006), 261-279. 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), 477-493. doi: 10.1007/978-1-4020-8741-7_57. [3] C. J. Ancker and A. V. Gafarian, Some queueing problems with balking and reneging, Operations Research, 11 (1963), 88-100. doi: 10.1287/opre.11.1.88. [4] F. Baccelli, P. Boyer and G. Hebuterne, Single-server queues with impatient customers, Advances in Applied Probability, 16 (1984), 887-905. doi: 10.2307/1427345. [5] F. Baccelli and G. Hebuterne, On queues with impatient customers, in Perforamnce' 81 (F. Kylstra, ed.), North-Holland, Amsterdam, 1981, 159-179. [6] S. Benjaafar, J. Gayon and S. Tepe, Optimal control of a production-inventory system with customer impatience, Operations Research Letters, 38 (2010), 267-272. doi: 10.1016/j.orl.2010.03.008. [7] N. K. Boots and H. Tijms, A multiserver queueing system with impatient customers, Management Science, 45 (1999), 444-448. doi: 10.1287/mnsc.45.3.444. [8] O. J. Boxma and P. R. de Waal, Multiserver queues with impatient customers, ITC, 14 (1994), 743-756. doi: 10.1016/B978-0-444-82031-0.50079-2. [9] S. R. Chakravarthy, A disater queue with Markovian arrivals and impatient customers, Applied Mathematics and Computation, 214 (2009), 48-59. doi: 10.1016/j.amc.2009.03.081. [10] D. J. Daley, General customer impatience in the queue GI/G/1, Journal of Applied Probability, 2 (1965), 186-205. doi: 10.2307/3211884. [11] S. Dimou, A. Economou and D. Fakinos, The single server vacation queueing model with geometric abandonments, Journal of Statistical Planning and Inference, 141 (2011), 2863-2877. doi: 10.1016/j.jspi.2011.03.010. [12] B. Doshi, Single server queues with vacation: A survey, Queueing Systems, 1 (1986), 29-66. [13] S. Economou and S. Kapodistria, Synchronized abandonments in a single server unreliable queue, European Journal of Operational Research, 203 (2010), 143-155. doi: 10.1016/j.ejor.2009.07.014. [14] N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: Tutotial, review, and research prospects, Manufacturing and Service Operations Management, 5 (2003), 79-141. doi: 10.1287/msom.5.2.79.16071. [15] O. Garnett, A. Mandelbaum and M. Reiman, Designing a call center with impatient customers, Manufacturing and Service Operations Management, 4 (2002), 208-227. doi: 10.1287/msom.4.3.208.7753. [16] S. Graves, The application of queueing theory to continous perishable inventory systems, Management Science, 28 (1984), 401-406. [17] Y. W. Shin and T. S. Choo, M/M/s queue with impatient customers and retrials, Applied Mathematical Modelling, 33 (2009), 2596-2606. doi: 10.1016/j.apm.2008.07.018. [18] L. Takacs, A single-server queue with limited virtual waiting time, Journal of Applied Probability, 11 (1974), 612-617. doi: 10.2307/3212710. [19] H. Takagi, Queueing Analysis, A Foundation of Performance Evaluation, Volume 1: Vacation and Priority Systems, Part 1. North-Holland, Amsterdam, 1991. [20] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, Springer, New York, 2006. [21] U. Yechiali, Queues with system disasters and impatient customers when system is down, Queueing Systems, 56 (2007), 195-202. doi: 10.1007/s11134-007-9031-z. [22] D. Yue and W. Yue, Analysis of M/M/c/N queueing system with balking, reneging and synchronous vacations, in Advanced in Queueing Theory and Network Applications (ed. W. Yue etal.), Springer, 2009, 165-180. doi: 10.1007/978-0-387-09703-9_9. [23] D. Yue, W. Yue, Z. Saffer and X. Chen, Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy, Journal of Industrial and Management Optimization, 10 (2014), 89-112. doi: 10.3934/jimo.2014.10.89.
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