A Lévy process reflected at a poisson age process

Offer Kella*, Onno Boxma, Michel Mandjes

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

20 Scopus citations

Abstract

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.

Original languageAmerican English
Pages (from-to)221-230
Number of pages10
JournalJournal of Applied Probability
Volume43
Issue number1
DOIs
StatePublished - 2006

Keywords

  • Decomposition
  • Lévy process
  • Queue with server vacations
  • Secondary input
  • Storage process

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