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The queue with impatience: construction of the stationary workload under FIFO

Part of: Probability theory on algebraic and topological structures Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Pascal Moyal
Pascal Moyal*
Affiliation:
Université de Technologie de Compiègne
*
∗Postal address: Laboratoire de Mathématiques Appliquées de Compiègne, Université de Technologie de Compiègne, Département Génie Informatique, Centre de Recherches de Royallieu, BP 20 529, 60 205 Compiegne Cedex, France. Email address: [email protected]
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Abstract

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In this paper we study the stability of queueing systems with impatient customers and a single server operating under a FIFO (first-in-first-out) discipline. We first give a sufficient condition for the existence of a stationary workload in the case of impatience until the beginning of service. We then provide a weaker condition of existence on an enriched probability space using the theory of Anantharam et al. (1997), (1999). The case of impatience until the end of service is also investigated.

Keywords

Stochastic recursionstationary solutionqueues with impatiencerenovative eventenrichment of probability space

MSC classification

Secondary: 60F17: Functional limit theorems; invariance principles 60K25: Queueing theory 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case)
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 498 - 512
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Anantharam, V. and Konstantopoulos, T. (1997). Stationary solutions of stochastic recursions describing discrete event systems. Stoch. Process. Appl. 68, 181–194.CrossRefGoogle Scholar
[2] Anantharam, V. and Konstantopoulos, T. (1999). A correction and some additional remarks on: stationary solutions of stochastic recursions describing discrete event systems. Stoch. Process. Appl. 80, 271–278.Google Scholar
[3] Baccelli, F. and Brémaud, P. (2002). Elements of Queueing Theory, 2nd edn. Springer, Berlin.Google Scholar
[4] Baccelli, F. and Hébuterne, G. (1981). On queues with impatient customers. In Performance '81 (Amsterdam, 1981), North-Holland, Amsterdam, pp. 159–179.Google Scholar
[5] Baccelli, F., Boyer, P. and Hébuterne, G. (1984). Single-server queues with impatient customers. Adv. Appl. Prob. 16, 887–905.Google Scholar
[6] Barrer, D. Y. (1957). Queuing with impatient customers and ordered service. Operat. Res. 5, 650–656.CrossRefGoogle Scholar
[7] Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory. Springer, New York.CrossRefGoogle Scholar
[8] Borovkov, A. A. and Foss, S. G. (1992). Stochastically recursive sequences and their generalizations. Siberian Adv. Math. 2, 16–81.Google Scholar
[9] Flipo, D. (1983). Files d'attente à rejet à un serveur. Application au cas indépendant. C. R. Acad. Sci. Paris 297, 365–367.Google Scholar
[10] Loynes, R. M. (1962). The stability of a queue with non-independent interarrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
[11] Moyal, P. (2008). Stability of a processor-sharing queue with varying throughput. J. Appl. Prob. 45, 953–962.Google Scholar
[12] Moyal, P. (2009). Weak solutions of stochastic recursions: an explicit construction. Preprint. Available at http://arxiv.org/abs/0904.3240v1.Google Scholar
[13] Neveu, J. (1984). Construction de files d'attente stationnaires. In Modeling and Performance Evaluation Methodology (Lecture Notes Control Inf. Sci. 60), Springer, Berlin, pp. 29–41.Google Scholar