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On the exact asymptotics of the busy period in GI/G/1 queues

Part of: Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Zbigniew Palmowski  and
Tomasz Rolski
Zbigniew Palmowski*
Affiliation:
Wrocław University and Utrecht University
Tomasz Rolski*
Affiliation:
Wrocław University
*
∗ Postal address: Mathematical Institute, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
∗ Postal address: Mathematical Institute, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
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Abstract

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In this paper we study the busy period in GI/G/1 work-conserving queues. We give the exact asymptotics of the tail distribution of the busy period under the light tail assumptions. We also study the workload process in the M/G/1 system conditioned to stay positive.

Keywords

GI/G/1 queuebusy periodworkload process conditioned to stay positiveexponential change of measurequasistationary distributionexact asymptoticse

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J35: Transition functions, generators and resolvents
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 3 , September 2006 , pp. 792 - 803
Copyright
Copyright © Applied Probability Trust 2006 

References

Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.Google Scholar
Bertoin, J. and Doney, R. A. (1994). On conditioning a random walk to stay nonnegative. Ann. Prob. 22, 2152–2167.Google Scholar
Chover, J., Ney, P. and Wainger, S. (1973). Functions of probability measures. J. Anal. Math. 26, 255–302.Google Scholar
Cox, D. R. and Smith, W. L. (1961). Queues. Methuen, London.Google Scholar
Davis, M. H. A. (1993). Markov Models and Optimization (Monogr. Statist. Appl. Prob. 49). Chapman and Hall, London.CrossRefGoogle Scholar
Doney, R. A. (1989). On the asymptotic behaviour of first passage times for transient random walk. Prob. Theory Relat. Fields 81, 239–246.Google Scholar
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York.Google Scholar
Iglehart, D. L. (1974). Random walks with negative drift conditioned to stay positive. J. Appl. Prob. 11, 742–751.Google Scholar
Keener, R. W. (1992). Limit theorems for random walks conditioned to stay positive. Ann. Prob. 20, 801–824.Google Scholar
Knight, F. B. (1969). Brownian local times and taboo processes. Trans. Amer. Math. Soc. 143, 173–185.Google Scholar
Kyprianou, A. E. and Palmowski, Z. (2005). A martingale review of some fluctuation theory for spectrally negative Lévy processes. In 38th Seminar on Probability, eds Émery, M. et al., Springer, Berlin, pp. 16–29.Google Scholar
Kyprianou, A. E. and Palmowski, Z. (2006). Quasi-stationary distributions for Lévy processes. To appear in Bernoulli.Google Scholar
Kyprianou, E. K. (1971). On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals. J. Appl. Prob. 8, 494–507.Google Scholar
Mandjes, M. and Zwart, B. (2006). Large deviations of sojourn times in processor sharing queues. Queueing Systems 52, 237–250.Google Scholar
Palmowski, Z. and Rolski, T. (2002). A technique for exponential change of measure for Markov processes. Bernoulli 8, 767–785.Google Scholar
Palmowski, Z. and Rolski, T. (2004). Markov processes conditioned to never exit a subspace of the state space. Prob. Math. Statist. 24, 339–353.Google Scholar
Petrov, V. V. (1965). On the probabilities of large deviations for sums of independent random variables. Theory Prob. Appl. 10, 287–298.Google Scholar
Petrov, V. V. (1972). Summy Nezavisimyk Slucainyh Velicin. Nauka, Moscow.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.Google Scholar