American Institute of Mathematical Sciences

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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

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 & 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. doi: 10.1016/j.camwa.2009.01.017. Google Scholar

[2]

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. Google Scholar

[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. doi: 10.1017/S0269964808000296. Google Scholar

[4]

Y. Baba, Analysis of a GI/M/1 queue with multiple working vacations,, Operations Research Letters, 33 (2005), 201. doi: 10.1016/j.orl.2004.05.006. Google Scholar

[5]

F. Baccelli, P. Boyer and G. Hebuterne, Single-server queues with impatient customers,, Advances in Applied Probability, 16 (1984), 887. doi: 10.2307/1427345. Google Scholar

[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. doi: 10.1016/j.apm.2006.05.010. Google Scholar

[7]

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. Google Scholar

[8]

T. Bonald and J. Roberts, Performance modeling of elastic traffic in overload,, ACM Sigmetrics Performance Evaluation Review, 29 (2001), 342. doi: 10.1145/384268.378845. Google Scholar

[9]

O. J. Boxma and P. R. de Waal, Multiserver queues with impatient customers,, in, (1994), 743. Google Scholar

[10]

D. J. Daley, General customer impatience in the queue GI/G/1,, Journal of Applied Probability, 2 (1965), 186. Google Scholar

[11]

S. Economou and S. Kapodistria, Synchronized abandonments in a single server unreliable queue,, European Journal of Operational Research, 203 (2010), 143. doi: 10.1016/j.ejor.2009.07.014. Google Scholar

[12]

N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: Tutorial, review, and research prospects,, Manufacturing and Service Operations Management, 5 (2003), 79. Google Scholar

[13]

E. R. Obert, Reneging phenomenon of single channel queues,, Mathematics of Operations Research, 4 (1979), 162. Google Scholar

[14]

C. Palm, Methods of judging the annoyance caused by congestion,, Tele., 4 (1953), 189. Google Scholar

[15]

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. Google Scholar

[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. Google Scholar

[17]

L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (M/M/1/WV),, Performance Evaluation, 50 (2002), 41. doi: 10.1016/S0166-5316(02)00057-3. Google Scholar

[18]

L. Takacs, A single-server queue with limited virtual waiting time,, Journal of Applied Probability, 11 (1974), 612. doi: 10.2307/3212710. Google Scholar

[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. doi: 10.1023/A:1025695818046. Google Scholar

[20]

D. Wu and H. Takagi, M/G/1 queue with multiple working vacations,, Performance Evaluation, 63 (2006), 654. doi: 10.1016/j.peva.2005.05.005. Google Scholar

[21]

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. Google Scholar

[22]

D. Yue and W. Yue, Analysis of M/M/$c$/N queueing system with balking, reneging and synchronous vacations,, in, (2009), 165. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.apm.2010.09.031. Google Scholar

show all references

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. doi: 10.1016/j.camwa.2009.01.017. Google Scholar

[2]

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. Google Scholar

[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. doi: 10.1017/S0269964808000296. Google Scholar

[4]

Y. Baba, Analysis of a GI/M/1 queue with multiple working vacations,, Operations Research Letters, 33 (2005), 201. doi: 10.1016/j.orl.2004.05.006. Google Scholar

[5]

F. Baccelli, P. Boyer and G. Hebuterne, Single-server queues with impatient customers,, Advances in Applied Probability, 16 (1984), 887. doi: 10.2307/1427345. Google Scholar

[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. doi: 10.1016/j.apm.2006.05.010. Google Scholar

[7]

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. Google Scholar

[8]

T. Bonald and J. Roberts, Performance modeling of elastic traffic in overload,, ACM Sigmetrics Performance Evaluation Review, 29 (2001), 342. doi: 10.1145/384268.378845. Google Scholar

[9]

O. J. Boxma and P. R. de Waal, Multiserver queues with impatient customers,, in, (1994), 743. Google Scholar

[10]

D. J. Daley, General customer impatience in the queue GI/G/1,, Journal of Applied Probability, 2 (1965), 186. Google Scholar

[11]

S. Economou and S. Kapodistria, Synchronized abandonments in a single server unreliable queue,, European Journal of Operational Research, 203 (2010), 143. doi: 10.1016/j.ejor.2009.07.014. Google Scholar

[12]

N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: Tutorial, review, and research prospects,, Manufacturing and Service Operations Management, 5 (2003), 79. Google Scholar

[13]

E. R. Obert, Reneging phenomenon of single channel queues,, Mathematics of Operations Research, 4 (1979), 162. Google Scholar

[14]

C. Palm, Methods of judging the annoyance caused by congestion,, Tele., 4 (1953), 189. Google Scholar

[15]

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. Google Scholar

[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. Google Scholar

[17]

L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (M/M/1/WV),, Performance Evaluation, 50 (2002), 41. doi: 10.1016/S0166-5316(02)00057-3. Google Scholar

[18]

L. Takacs, A single-server queue with limited virtual waiting time,, Journal of Applied Probability, 11 (1974), 612. doi: 10.2307/3212710. Google Scholar

[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. doi: 10.1023/A:1025695818046. Google Scholar

[20]

D. Wu and H. Takagi, M/G/1 queue with multiple working vacations,, Performance Evaluation, 63 (2006), 654. doi: 10.1016/j.peva.2005.05.005. Google Scholar

[21]

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. Google Scholar

[22]

D. Yue and W. Yue, Analysis of M/M/$c$/N queueing system with balking, reneging and synchronous vacations,, in, (2009), 165. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.apm.2010.09.031. Google Scholar

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