Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T18:19:03.373Z Has data issue: false hasContentIssue false
  • English
  • Français

The Busy Period of an M/G/1 Queue with Customer Impatience

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Onno Boxma ,
David Perry ,
Wolfgang Stadje  and
Shelley Zacks
Onno Boxma*
Affiliation:
EURANDOM and Eindhoven University of Technology
David Perry*
Affiliation:
University of Haifa
Wolfgang Stadje*
Affiliation:
University of Osnabrück
Shelley Zacks*
Affiliation:
Binghamton University
*
∗Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, HG 9.14, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
∗∗Postal address: Department of Statistics, University of Haifa, Haifa 31909, Israel. Email address: [email protected]
∗∗∗Postal address: Department of Mathematics and Computer Science, University of Osnabrück, 49069 Osnabrück, Germany. Email address: [email protected]
∗∗∗∗Postal address: Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider an M/G/1 queue in which an arriving customer does not enter the system whenever its virtual waiting time, i.e. the amount of work seen upon arrival, is larger than a certain random patience time. We determine the busy period distribution for various choices of the patience time distribution. The main cases under consideration are exponential patience and a discrete patience distribution.

Keywords

Busy periodM/G/1 queueimpatiencevirtual waiting time

MSC classification

Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60J27: Continuous-time Markov processes on discrete state spaces 60J75: Jump processes
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 1 , March 2010 , pp. 130 - 145
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Baccelli, F. and Hébuterne, G. (1981). On queues with impatient customers. In Performance '81 (Amsterdam, 1981), ed. Kylstra, F. J., North Holland, Amsterdam, pp. 159–179.Google Scholar
[2] Baccelli, F., Boyer, P. and Hébuterne, G. (1984). Single-server queues with impatient customers. Adv. Appl. Prob. 16, 887–905.CrossRefGoogle Scholar
[3] Barrer, D. Y. (1957). Queuing with impatient customers and ordered service. Operat. Res. 5, 650–656.CrossRefGoogle Scholar
[4] Borovkov, K. and Burq, Z. (2001). Kendall's identity for the first crossing time revisited. Electron. Commun. Prob. 6, 91–94.Google Scholar
[5] Finch, P. D. (1960). Deterministic customer impatience in the queueing system GI/M/1. Biometrika 47, 45–52.Google Scholar
[6] Kaspi, H. and Perry, D. (1983). Inventory systems of perishable commodities. Adv. Appl. Prob. 15, 674–685.Google Scholar
[7] Perry, D. and Asmussen, S. (1995). Rejection rules in the M/G/1 queue. Queueing Systems 19, 105–130.Google Scholar
[8] Perry, D., Stadje, W. and Zacks, S. (2000). Busy period analysis for M/G/1 and G/M/1 type queues with restricted accessibility. Operat. Res. Lett. 27, 163–174.Google Scholar
[9] Perry, D., Stadje, W. and Zacks, S. (2001). The M/G/1 queue with finite workload capacity. Queueing Systems 39, 7–22.CrossRefGoogle Scholar
[10] Ross, S. M. (1996). Stochastic Processes, 2nd edn. John Wiley, New York.Google Scholar
[11] Stadje, W. and Zacks, S. (2003). Upper first-exit times of compound Poisson processes revisited. Prob. Eng. Inf. Sci. 17, 459–465.Google Scholar
[12] Stanford, R. E. (1979). Reneging phenomenon in single channel queues. Math. Operat. Res. 4, 162–178.Google Scholar
[13] Stanford, R. E. (1990). On queues with impatience. Adv. Appl. Prob. 22, 768–769.Google Scholar
[14] Subba Rao, S. (1967). Queueing models with balking and reneging. Ann. Inst. Statist. Math. 19, 55–71.Google Scholar
[15] Subba Rao, S. (1967/1968). Queuing with balking and reneging in MG1 systems. Metrika 12, 173–188.Google Scholar
[16] Widder, D. V. (1946). The Laplace Transform. Princeton University Press.Google Scholar