×

zbMATH — the first resource for mathematics

Queues with system disasters and impatient customers when system is down. (English) Zbl 1124.60076
Summary: Consider a system operating as an M/M/\(c\) queue, where \(c =1\), \(1< c <\infty\), or \(c =\infty\). The system as a whole suffers occasionally a disastrous breakdown, upon which all present customers (waiting and served) are cleared from the system and lost. A repair process then starts immediately. When the system is down, inoperative, and undergoing a repair process, new arrivals become impatient: each individual customer, upon arrival, activates a random-duration timer. If the timer expires before the system is repaired, the customer abandons the queue never to return. We analyze this model and derive various quality of service measures: mean sojourn time of a served customer; proportion of customers served; rate of lost customers due to disasters; and rate of abandonments due to impatience.

MSC:
60K25 Queueing theory (aspects of probability theory)
60K37 Processes in random environments
90B22 Queues and service in operations research
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Altman, E., Yechiali, U.: Analysis of customers’ impatience in queues with server vacations. Queueing Syst. 52, 261–279 (2006) · Zbl 1114.90015 · doi:10.1007/s11134-006-6134-x
[2] Altman, E., Yechiali, U.: Infinite-server queues with system’s additional tasks and impatience customers. Technical Report, Tel Aviv University, April 2005 · Zbl 1228.60096
[3] Avi-Itzhak, B., Naor, P.: Some queueing problems with the service station subject to breakdowns. Oper. Res. 11, 303–320 (1963) · Zbl 0114.34202 · doi:10.1287/opre.11.3.303
[4] Baccelli, F., Boyer, P., Hebuterne, G.: Single-server queues with impatient customers. Adv. Appl. Probab. 16, 887–905 (1984) · Zbl 0549.60091 · doi:10.2307/1427345
[5] Baykal-Gursoy, M., Xiao, W.: Stochastic decomposition in M/M/queues with Markov-modulated service rates. Queueing Syst. 48, 75–88 (2004) · Zbl 1059.60093 · doi:10.1023/B:QUES.0000039888.52119.1d
[6] Baykal-Gursoy, M., Xiao, W., Ozbay, K.: Modeling traffic flow interrupted by incidents, http://coewww.rutgers.edu/ie/research/working-paper/paper-2004-005.pdf · Zbl 1159.90014
[7] Daley, D.J.: General customer impatience in the queue GI/G/1. J. Appl. Probab. 2, 186–205 (1965) · Zbl 0134.14402 · doi:10.2307/3211884
[8] D’Auria, B.: Stochastic decomposition of the M/G/queue in random environment. Report 2005-045, Eurandom, The Netherlands
[9] Gans, N., Koole, G., Mandelbaum, A.: Telephone call centers: tutorial, review, and research prospects. Manuf. Serv. Oper. Manag. 5, 79–141 (2003) · doi:10.1287/msom.5.2.79.16071
[10] Gaver, D.P.: A waiting line with interrupted service, including priorities. J. Roy. Stat. Soc. Ser. B 24, 73–90 (1962) · Zbl 0108.31403
[11] Gupta, V., Scheller-Wolf, A., Harchol-Balter, M., Yechiali, U.: Fundamental characteristics of queues with fluctuating load. Proceedings of ACM SIGMETRIC, 2006
[12] Martin, S.P., Mitrani, I.: Job transfers between unreliable servers. 2nd Madrid Conference on Queueing Theory, July 2006
[13] Polyanin, A.D., Zaitsev, V.F.: Handbook of Exact Solutions for Ordinary Differential Equations. Chapman & Hall, Boca Raton (2003) · Zbl 1015.34001
[14] Takacs, L.: A single-server queue with limited virtual waiting time. J. Appl. Probab. 11, 612–617 (1974) · Zbl 0303.60098 · doi:10.2307/3212710
[15] Towsley, D., Tripathi, S.K.: A single server priority queue with server failures and queue flushing. Oper. Res. Lett. 10, 362–363 (1991) · Zbl 0737.60086
[16] Yang, W.S., Kim, J.D., Chae, K.C.: Analysis of M/G/1 stochastic clearing systems. Stoch. Anal. Appl. 20 (2002) · Zbl 1019.60087
[17] Yechiali, U., Naor, P.: Queuing problems with heterogeneous arrivals and service. Oper. Res. 19, 722–734 (1971) · Zbl 0226.60107 · doi:10.1287/opre.19.3.722
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.