Analysis of customers' impatience in an M/M/1 queue with working vacations

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October 2012, 8(4): 895-908. doi: 10.3934/jimo.2012.8.895

Analysis of customers' impatience in an M/M/1 queue with working vacations

Dequan Yue ^1, , Wuyi Yue ^2, and Gang Xu ^1,

1.	Department of Statistics, College of Sciences, Yanshan University, Qinhuangdao 066004, China
2.	Department of Intelligence and Informatics, Konan University, 8-9-1 Okamoto, Kobe 658-8501

Received September 2011 Revised July 2012 Published September 2012

Abstract
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In this paper, we analyze an M/M/1 queueing system with working vacations and impatient customers. We examine the case that the customers' impatience is due to a working vacation. During a working vacation, customers are served at a slower than usual service rate and are likely to become impatient. Whenever a customer arrives in the system and realizes that the server is on vacation, the customer activates an ``impatience timer" which is exponentially distributed. If a customer's service has not been completed before the customer's timer expires, the customer leaves the queue, never to return. By analyzing this model, we derive the probability generating functions of the number of customers in the system when the server is in a service period and a working vacation, respectively. We further obtain the closed-form expressions for various performance measures, including the mean system size, the mean sojourn time of a customer served, the proportion of customers served and the rate of abandonment due to impatience. Finally, we present some numerical results to demonstrate effects of some parameters on these performance measures of the system.

Keywords: probability generating functions, impatience, mean system sizes, Queueing systems, working vacations, sojourn times..

Mathematics Subject Classification: Primary: 60K25; Secondary: 90B2.

Citation: Dequan Yue, Wuyi Yue, Gang Xu. Analysis of customers' impatience in an M/M/1 queue with working vacations. Journal of Industrial and Management Optimization, 2012, 8 (4) : 895-908. doi: 10.3934/jimo.2012.8.895

References:

[1]	R. O. Al-Seedy, S. A. El-Shehawy, A. A. El-Sherbiny and S. I. Ammar, Transient solution of the M/M/c queue with balking and reneging, Computers and Mathematics with Applications, 57 (2009), 1280-1285. doi: 10.1016/j.camwa.2009.01.017.
[2]	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.
[3]	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: 10.1017/S0269964808000296.
[4]	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.
[5]	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.
[6]	A. D. Banik, U. C. Gupta and S. S. Pathak, On the GI/M/1/N queue with multiple vacation-analytic analysis and computation, Applied Mathematical Modelling, 31 (2007), 1701-1710. doi: 10.1016/j.apm.2006.05.010.
[7]	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.
[8]	T. Bonald and J. Roberts, Performance modeling of elastic traffic in overload, ACM Sigmetrics Performance Evaluation Review, 29 (2001), 342-343. doi: 10.1145/384268.378845.
[9]	O. J. Boxma and P. R. de Waal, Multiserver queues with impatient customers, in "The Fundamental Role of Teletraffic in the Evolution of Telecommunications Networks" (eds. J. Labetouille and J. W. Roberts), North-Holland, Amsterdam, (1994), 743-756.
[10]	D. J. Daley, General customer impatience in the queue GI/G/1, Journal of Applied Probability, 2 (1965), 186-205.
[11]	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.
[12]	N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: Tutorial, review, and research prospects, Manufacturing and Service Operations Management, 5 (2003), 79-141.
[13]	E. R. Obert, Reneging phenomenon of single channel queues, Mathematics of Operations Research, 4 (1979), 162-178.
[14]	C. Palm, Methods of judging the annoyance caused by congestion, Tele., 4 (1953), 189-208.
[15]	N. Perel and U. Yechiali, Queues with slow servers and impatient customers, European Journal of Operational Research, 201 (2010), 247-258. doi: 10.1016/j.ejor.2009.02.024.
[16]	Y. Sakuma, A. Inoie, K. Kawanishi and M. Miyazawa, Tail asymptotics for waiting time distribution of an M/M/$s$ queue with general impatient time, Journal of Industrial and Management Optimization, 7 (2011), 593-606.
[17]	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.
[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]	B. Van Houdt, R. B. Lenin and C. Blonia, Delay distribution of (im)patient customers in a discrete time D-MAP/PH/1 queue with age-dependent service times, Queueing Systems, 45 (2003), 59-73. doi: 10.1023/A:1025695818046.
[20]	D. 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.
[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" (eds. W. Yue et al.), Springer-Verlag, New York, (2009), 165-180.
[23]	D. Yue and W. Yue, Block-partioning matrix solution of M/M/R/N queueing system with balking, reneging and server breakdowns, Journal of Industrial and Management Optimization, 5 (2009), 417-430.
[24]	M. Zhang and Z. Hou, Performance analysis of MAP/G/1 queue with working vacations and vacation interruption, Applied Mathematical Modelling, 35 (2011), 1551-1560. doi: 10.1016/j.apm.2010.09.031.

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References:

[1]	R. O. Al-Seedy, S. A. El-Shehawy, A. A. El-Sherbiny and S. I. Ammar, Transient solution of the M/M/c queue with balking and reneging, Computers and Mathematics with Applications, 57 (2009), 1280-1285. doi: 10.1016/j.camwa.2009.01.017.
[2]	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.
[3]	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: 10.1017/S0269964808000296.
[4]	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.
[5]	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.
[6]	A. D. Banik, U. C. Gupta and S. S. Pathak, On the GI/M/1/N queue with multiple vacation-analytic analysis and computation, Applied Mathematical Modelling, 31 (2007), 1701-1710. doi: 10.1016/j.apm.2006.05.010.
[7]	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.
[8]	T. Bonald and J. Roberts, Performance modeling of elastic traffic in overload, ACM Sigmetrics Performance Evaluation Review, 29 (2001), 342-343. doi: 10.1145/384268.378845.
[9]	O. J. Boxma and P. R. de Waal, Multiserver queues with impatient customers, in "The Fundamental Role of Teletraffic in the Evolution of Telecommunications Networks" (eds. J. Labetouille and J. W. Roberts), North-Holland, Amsterdam, (1994), 743-756.
[10]	D. J. Daley, General customer impatience in the queue GI/G/1, Journal of Applied Probability, 2 (1965), 186-205.
[11]	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.
[12]	N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: Tutorial, review, and research prospects, Manufacturing and Service Operations Management, 5 (2003), 79-141.
[13]	E. R. Obert, Reneging phenomenon of single channel queues, Mathematics of Operations Research, 4 (1979), 162-178.
[14]	C. Palm, Methods of judging the annoyance caused by congestion, Tele., 4 (1953), 189-208.
[15]	N. Perel and U. Yechiali, Queues with slow servers and impatient customers, European Journal of Operational Research, 201 (2010), 247-258. doi: 10.1016/j.ejor.2009.02.024.
[16]	Y. Sakuma, A. Inoie, K. Kawanishi and M. Miyazawa, Tail asymptotics for waiting time distribution of an M/M/$s$ queue with general impatient time, Journal of Industrial and Management Optimization, 7 (2011), 593-606.
[17]	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.
[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]	B. Van Houdt, R. B. Lenin and C. Blonia, Delay distribution of (im)patient customers in a discrete time D-MAP/PH/1 queue with age-dependent service times, Queueing Systems, 45 (2003), 59-73. doi: 10.1023/A:1025695818046.
[20]	D. 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.
[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" (eds. W. Yue et al.), Springer-Verlag, New York, (2009), 165-180.
[23]	D. Yue and W. Yue, Block-partioning matrix solution of M/M/R/N queueing system with balking, reneging and server breakdowns, Journal of Industrial and Management Optimization, 5 (2009), 417-430.
[24]	M. Zhang and Z. Hou, Performance analysis of MAP/G/1 queue with working vacations and vacation interruption, Applied Mathematical Modelling, 35 (2011), 1551-1560. doi: 10.1016/j.apm.2010.09.031.

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