# American Institute of Mathematical Sciences

July  2011, 7(3): 593-606. doi: 10.3934/jimo.2011.7.593

## Tail asymptotics for waiting time distribution of an M/M/s queue with general impatient time

 1 Department of Distribution and Information Engineering, Hiroshima National College of Maritime Technology, Osakikamijima-Town, 725-0231, Japan 2 Department of Information Network and Communication, Kanagawa Institute of Technology, Atsugi-City, 243-0292, Japan 3 Department of Computer Science, Gunma University, Kiryu-City, 376-8515, Japan 4 Department of Information Sciences, Tokyo University of Science, Noda-City, 278-8510, Japan

Received  September 2010 Revised  May 2011 Published  June 2011

In this paper, we consider an $M/M/s$ queueing model where customers may abandon waiting for service and leave the system without receiving their services. We assume that impatient time on waiting for each customer is an independent and identically distributed nonnegative random variable with a general distribution where the probability distribution is light-tailed and unbounded. The main objective of this paper is to provide an approximation for the waiting time distribution in an analytically tractable form. To this end, we obtain the tail asymptotics of the waiting time distributions of served and impatient customers. By using the tail asymptotics, we show that the fairly good approximations of the waiting time distributions can be obtained in asymptotic region with low numerical complexity.
Citation: Yutaka Sakuma, Atsushi Inoie, Ken’ichi Kawanishi, Masakiyo Miyazawa. Tail asymptotics for waiting time distribution of an M/M/s queue with general impatient time. Journal of Industrial & Management Optimization, 2011, 7 (3) : 593-606. doi: 10.3934/jimo.2011.7.593
##### References:
 [1] S. Asmussen, "Applied Probability and Queues," 2nd ed., Applications of Mathematics (New York), 51, Springer-Verlag, New York, 2003.  Google Scholar [2] 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.  Google Scholar [3] F. Baccelli and G. Hebuterne, On queues with impatient customers, in "Performance '81" (ed. F. J. Kylstra), North-Holland, Amsterdam-New York, 32 (1981), 159-179.  Google Scholar [4] D. Y. Barrer, Queueing with impatient customers and indifferent clerks, Operations Research, 4 (1957), 644-649. Google Scholar [5] D. Y. Barrer, Queueing with impatient customers and ordered service, Operations Research, 4 (1957), 650-656. doi: 10.1287/opre.5.5.650.  Google Scholar [6] A. Brandt and M. Brandt, On the $M(n)$/$M(n)$/$s$ queue with impatient calls, Performance Evaluation, 35 (1999), 1-18. doi: 10.1016/S0166-5316(98)00042-X.  Google Scholar [7] A. Brandt and M. Brandt, Asymptotic results and a Markovian approximation for the $M(n)$/$M(n)$/$s+GI$ system, Queueing Systems, 41 (2002), 73-94. doi: 10.1023/A:1015781818360.  Google Scholar [8] L. Brown, N. Gans, A. Mandelbaum, A. Sakov, H. Shen, S. Zeltyn and L. Zhao, Statistical analysis of a telephone call center: A queueing-science perspective, Journal of the American Statistical Association, 100 (2005), 36-50. doi: 10.1198/016214504000001808.  Google Scholar [9] B. D. Choi and B. Kim, $MAP$/$M$/$c$ queue with constant impatient time, Mathematics of Operations Research, 29 (2004), 309-325. doi: 10.1287/moor.1030.0081.  Google Scholar [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.  Google Scholar [11] A. G. de Kok and H. C. Tijms, A queueing system with impatient customers, Journal of Applied Probability, 22 (1985), 688-696. doi: 10.2307/3213871.  Google Scholar [12] G. Evans, "Practical Numerical Analysis," John Wiley & Sons, 1996. Google Scholar [13] P. D. Finch, Deterministic customer impatience in the queueing system $GI$/$M$/$1$, Biometrika, 47 (1960), 45-52.  Google Scholar [14] N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: Tutorial, review, and research prospects, Manufacturing and Service Operations Management, 5 (2003), 79-141. doi: 10.1287/msom.5.2.79.16071.  Google Scholar [15] O. Garnett, A. Mandelbaum and M. Reiman, Designing a call center with impatient customers, Manufacturing & Service Operations Management, 4 (2002), 208-227. doi: 10.1287/msom.4.3.208.7753.  Google Scholar [16] R. B. Haugen and E. Skogan, Queueing systems with stochastic time out, IEEE Transactions on Communications, 28 (1980), 1984-1989. doi: 10.1109/TCOM.1980.1094632.  Google Scholar [17] G. Latouche and V. Ramaswami, "Introduction to Matrix Analytic Methods in Stochastic Modeling," American Statistical Association and the Society for Industrial and Applied Mathematics, Philadelphia, American Statistical Association, Alexandria, VA, 1999. doi: 10.1137/1.9780898719734.  Google Scholar [18] A. Movaghar, On queueing with customer impatience until the beginning of service, Queueing Systems Theory Appl., 29 (1998), 337-350. doi: 10.1023/A:1019196416987.  Google Scholar [19] C. Palm, Methods of judging the annoyance caused by congestion, Tele (English ed.), 2 (1953), 1-20. Google Scholar [20] R. E. Stanford, Reneging phenomena in single server queues, Mathematics of Operations Research, 4 (1979), 162-178. doi: 10.1287/moor.4.2.162.  Google Scholar [21] W. Xiong, D. Jagerman and T. Altiok, $M$/$G$/$1$ queue with deterministic reneging times, Performance Evaluation, 65 (2008), 308-316. doi: 10.1016/j.peva.2007.07.003.  Google Scholar [22] S. Zeltyn and A. Mandelbaum, Call centers with impatient customers: Many-server asymptotics of the M/M/$n$ + G queue, Queueing Systems, 51 (2005), 361-402. doi: 10.1007/s11134-005-3699-8.  Google Scholar

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##### References:
 [1] S. Asmussen, "Applied Probability and Queues," 2nd ed., Applications of Mathematics (New York), 51, Springer-Verlag, New York, 2003.  Google Scholar [2] 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.  Google Scholar [3] F. Baccelli and G. Hebuterne, On queues with impatient customers, in "Performance '81" (ed. F. J. Kylstra), North-Holland, Amsterdam-New York, 32 (1981), 159-179.  Google Scholar [4] D. Y. Barrer, Queueing with impatient customers and indifferent clerks, Operations Research, 4 (1957), 644-649. Google Scholar [5] D. Y. Barrer, Queueing with impatient customers and ordered service, Operations Research, 4 (1957), 650-656. doi: 10.1287/opre.5.5.650.  Google Scholar [6] A. Brandt and M. Brandt, On the $M(n)$/$M(n)$/$s$ queue with impatient calls, Performance Evaluation, 35 (1999), 1-18. doi: 10.1016/S0166-5316(98)00042-X.  Google Scholar [7] A. Brandt and M. Brandt, Asymptotic results and a Markovian approximation for the $M(n)$/$M(n)$/$s+GI$ system, Queueing Systems, 41 (2002), 73-94. doi: 10.1023/A:1015781818360.  Google Scholar [8] L. Brown, N. Gans, A. Mandelbaum, A. Sakov, H. Shen, S. Zeltyn and L. Zhao, Statistical analysis of a telephone call center: A queueing-science perspective, Journal of the American Statistical Association, 100 (2005), 36-50. doi: 10.1198/016214504000001808.  Google Scholar [9] B. D. Choi and B. Kim, $MAP$/$M$/$c$ queue with constant impatient time, Mathematics of Operations Research, 29 (2004), 309-325. doi: 10.1287/moor.1030.0081.  Google Scholar [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.  Google Scholar [11] A. G. de Kok and H. C. Tijms, A queueing system with impatient customers, Journal of Applied Probability, 22 (1985), 688-696. doi: 10.2307/3213871.  Google Scholar [12] G. Evans, "Practical Numerical Analysis," John Wiley & Sons, 1996. Google Scholar [13] P. D. Finch, Deterministic customer impatience in the queueing system $GI$/$M$/$1$, Biometrika, 47 (1960), 45-52.  Google Scholar [14] N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: Tutorial, review, and research prospects, Manufacturing and Service Operations Management, 5 (2003), 79-141. doi: 10.1287/msom.5.2.79.16071.  Google Scholar [15] O. Garnett, A. Mandelbaum and M. Reiman, Designing a call center with impatient customers, Manufacturing & Service Operations Management, 4 (2002), 208-227. doi: 10.1287/msom.4.3.208.7753.  Google Scholar [16] R. B. Haugen and E. Skogan, Queueing systems with stochastic time out, IEEE Transactions on Communications, 28 (1980), 1984-1989. doi: 10.1109/TCOM.1980.1094632.  Google Scholar [17] G. Latouche and V. Ramaswami, "Introduction to Matrix Analytic Methods in Stochastic Modeling," American Statistical Association and the Society for Industrial and Applied Mathematics, Philadelphia, American Statistical Association, Alexandria, VA, 1999. doi: 10.1137/1.9780898719734.  Google Scholar [18] A. Movaghar, On queueing with customer impatience until the beginning of service, Queueing Systems Theory Appl., 29 (1998), 337-350. doi: 10.1023/A:1019196416987.  Google Scholar [19] C. Palm, Methods of judging the annoyance caused by congestion, Tele (English ed.), 2 (1953), 1-20. Google Scholar [20] R. E. Stanford, Reneging phenomena in single server queues, Mathematics of Operations Research, 4 (1979), 162-178. doi: 10.1287/moor.4.2.162.  Google Scholar [21] W. Xiong, D. Jagerman and T. Altiok, $M$/$G$/$1$ queue with deterministic reneging times, Performance Evaluation, 65 (2008), 308-316. doi: 10.1016/j.peva.2007.07.003.  Google Scholar [22] S. Zeltyn and A. Mandelbaum, Call centers with impatient customers: Many-server asymptotics of the M/M/$n$ + G queue, Queueing Systems, 51 (2005), 361-402. doi: 10.1007/s11134-005-3699-8.  Google Scholar
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