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The stability of storage models with shot noise input

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Robert B. Lund
Robert B. Lund*
Affiliation:
The University of Georgia
*
∗Postal address: Department of Statistics, The University of Georgia, Athens, GA 30602–1952, USA.
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Abstract

We examine the existence of limiting behavior, or stability, for storage models with shot noise input and general release rules. The shot noise feature of the input process allows the individual inputs to gradually enter the store.

We first show that a store under the unit release rule is stable if and only if the traffic intensity is less than one; this extends the classic result of Prabhu (1980) to the case of shot noise input. The stability of the unit release rule store and various stochastic orderings are then used to derive a sufficient condition for a store with a general release rule to be stable. Finally, we show that when restricted to a compact state space, our storage model is always stable.

An important component of the paper is the methodology employed: coupling and stochastic monotonicity play a key role in analyzing the non-Markov processes encountered.

Keywords

STORAGE MODELSHOT NOISECOUPLINGSTOCHASTIC MONOTONICITYCOMPOUND POISSON PROCESSSTORAGE EQUATION

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J25: Continuous-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 3 , September 1996 , pp. 830 - 839
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
[2] Brockwell, P. J., Resnick, S. I. and Tweedie, R. L. (1982) Storage processes with general release rule and additive inputs. Adv. Appl. Prob. 14, 392–433.Google Scholar
[3] Doney, R. A. and O'Brien, G. L. (1991) Loud shot noise. Ann. Appl. Prob. 1, 88–103.Google Scholar
[4] Harrison, J. M. and Resnick, S. I. (1976) The stationary distribution and first exit probabilities of a storage process with general release rule. Math. Operat. Res. 1, 347–358.CrossRefGoogle Scholar
[5] Lindvall, T. (1992) Lectures on the Coupling Method. Wiley, New York.Google Scholar
[6] Loynes, R. M. (1962) The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
[7] Lund, R. B. (1993) Some limiting and convergence rate results in the theory of dams. Ph. D. dissertation. The University of North Carolina at Chapel Hill.Google Scholar
[8] Lund, R. B. (1994) A dam with seasonal input. J. Appl. Prob. 31, 526–541.CrossRefGoogle Scholar
[9] Mccormick, W. P. and Homble, P. (1995) Weak limit results for the extremes of a class of shot noise processes. J. Appl. Prob. 32, 707–727.Google Scholar
[10] Moran, P. A. P. (1967) Dams in series with continuous release. J. Appl. Prob. 4, 380–388.CrossRefGoogle Scholar
[11] Prabhu, N. U. (1980) Stochastic Storage Processes. Springer, New York.Google Scholar
[12] Sigman, K. and Yamazaki, G. (1992) Fluid models with burst arrivals: a sample path analysis. Prob. Eng. Inf. Sci. 6, 17–27.CrossRefGoogle Scholar