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ON AN EQUIVALENCE BETWEEN LOSS RATES AND CYCLE MAXIMA IN QUEUES AND DAMS

Published online by Cambridge University Press:  23 March 2005

René Bekker  and
Bert Zwart
René Bekker
Affiliation:
CWI, 1090 GB Amsterdam, The Netherlands, and, Department of Mathematics & Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands, E-mail: rbekker@win.tue.nl; zwart@win.tue.nl
Bert Zwart
Affiliation:
CWI, 1090 GB Amsterdam, The Netherlands, and, Department of Mathematics & Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands, E-mail: rbekker@win.tue.nl; zwart@win.tue.nl
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Abstract

We consider the loss probability of a customer in a single-server queue with finite buffer and partial rejection and show that it can be identified with the tail distribution of the cycle maximum of the associated infinite-buffer queue. This equivalence is shown to hold for the GI/G/1 queue and for dams with state-dependent release rates. To prove this equivalence, we use a duality for stochastically monotone recursions, developed by Asmussen and Sigman (1996). As an application, we obtain several exact and asymptotic results for the loss probability and extend Takács' formula for the cycle maximum in the M/G/1 queue to dams with variable release rate.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 19 , Issue 2 , April 2005 , pp. 241 - 255
Copyright
© 2005 Cambridge University Press

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References

REFERENCES

Asmussen, S. (2003). Applied probability and queues, 2nd ed. New York: Springer-Verlag.
Asmussen, S. (1998). Extreme value theory for queues via cycle maxima. Extremes 2: 137–168.Google Scholar
Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: Extremal behaviour, stationary distributions and first passage times. Annals of Applied Probability 8: 354–374.Google Scholar
Asmussen, S. & Kella, O. (1996). Rate modulation in dams and ruin problems. Journal of Applied Probability 33: 523–535.Google Scholar
Asmussen, S. & Perry, D. (1992). On cycle maxima, first passage problems and extreme value theory for queues. Stochastic Models 8: 421–458.Google Scholar
Asmussen, S. & Schock Petersen, S. (1988). Ruin probabilities expressed in terms of storage processes. Advances in Applied Probability 20: 913–916.Google Scholar
Asmussen, S. & Sigman, K. (1996). Monotone stochastic recursions and their duals. Probability in the Engineering and Informational Sciences 10: 1–20.Google Scholar
Bekker, R. (2004). Finite-buffer queues with workload-dependent service and arrival rates. SPOR Report 2004-01, Eindhoven University of Technology, The Netherlands.
Browne, S. & Sigman, K. (1992). Work-modulated queues with applications to storage processes. Journal of Applied Probability 29: 699–712.Google Scholar
Cohen, J.W. (1976). Regenerative processes in queueing theory. Berlin: Springer-Verlag.CrossRef
Cohen, J.W. (1982). The single server queue. Amsterdam: North-Holland.
Dette, H., Fill, J.A., Pitman, J., & Studden, W.J. (1997). Wall and Siegmund duality relations for birth and death chains with reflecting barrier. Journal of Theoretical Probability 10: 349–374.Google Scholar
Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997). Modelling extremal events. Berlin: Springer-Verlag.
Harrison, J.M. & Resnick, S.I. (1976). The stationary distribution and first exit probabilities of a storage process with general release rule. Mathematics of Operations Research 1: 347–358.Google Scholar
Hooghiemstra, G. (1987). A path construction for the virtual waiting time of an M/G/1 queue. Statistica Neerlandica 41: 175–181.Google Scholar
Iglehart, D.G. (1972). Extreme values in the GI/G/1 queue. Annals of Mathematical Statistics 43: 627–635.Google Scholar
Lindley, D.V. (1959). Discussion of a paper by C.B. Winsten. Proceedings of the Cambridge Philosophical Society 48: 277–289.Google Scholar
Loynes, R.M. (1965). On a property of the random walks describing simple queues and dams. Journal of the Royal Statistical Society Series B 27: 125–129.Google Scholar
Perry, D. & Stadje, W. (2003). Duality of dams via mountain processes. Operations Research Letters 31: 451–458.Google Scholar
Siegmund, D. (1976). The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes. Annals of Probability 4: 914–924.Google Scholar
Takács, L. (1967). Combinatorial methods in the theory of stochastic processes. New York: Wiley.
Van Ommeren, J.C.W. & De Kok, A.G. (1987). Asymptotic results for buffer systems under heavy load. Probability in the Engineering and Informational Sciences 1: 327–348.Google Scholar
Zwart, A.P. (2000). A fluid queue with a finite buffer and subexponential input. Advances in Applied Probability 32: 221–243.Google Scholar