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A Lévy Process Reflected at a Poisson Age Process

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Offer Kella ,
Onno Boxma  and
Michel Mandjes
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
Onno Boxma*
Affiliation:
EURANDOM and Eindhoven University of Technology
Michel Mandjes*
Affiliation:
CWI and The University of Amsterdam
*
∗ Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email address: offer.kella@huji.ac.il
∗∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: boxma@win.tue.nl
∗∗∗ Postal address: CWI, PO Box 94079, 1090 GB Amsterdam, The Netherlands. Email address: michel.mandjes@cwi.nl
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Abstract

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We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.

Keywords

Lévy processstorage processdecompositionqueue with server vacationssecondary input

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 1 , March 2006 , pp. 221 - 230
Copyright
© Applied Probability Trust 2006 

References

Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8, 354–374.CrossRefGoogle Scholar
Asmussen, S. (2003). Applied Probability and Queues (Appl. Math. (New York) 51), 2nd edn. Springer, New York.Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bertoin, J. and Doney, R. A. (1994). Cramér's estimate for Lévy processes. Statist. Prob. Lett. 21, 363–365.Google Scholar
Dieker, A. B. (2005). Applications of factorization embeddings for Lévy processes. Submitted. Available at http://www.cwi.nl/∼ton.Google Scholar
Kella, O. and Whitt, W. (1991). Queues with server vacations and Lévy processes with secondary Jump input. Ann. Appl. Prob. 1, 104–117.CrossRefGoogle Scholar
Kella, O. and Whitt, W. (1992). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29,{} 396–403.Google Scholar
Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Prob. 41, 407–424.CrossRefGoogle Scholar