Skip to main content Accessibility help
×
Home

A Lévy Process Reflected at a Poisson Age Process

  • Offer Kella (a1), Onno Boxma (a2) and Michel Mandjes (a3)

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      A Lévy Process Reflected at a Poisson Age Process
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      A Lévy Process Reflected at a Poisson Age Process
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      A Lévy Process Reflected at a Poisson Age Process
      Available formats
      ×

Copyright

Corresponding author

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

References

Hide All
[1] Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8, 354374.
[2] Asmussen, S. (2003). Applied Probability and Queues (Appl. Math. (New York) 51), 2nd edn. Springer, New York.
[3] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.
[4] Bertoin, J. and Doney, R. A. (1994). Cramér's estimate for Lévy processes. Statist. Prob. Lett. 21, 363365.
[5] Dieker, A. B. (2005). Applications of factorization embeddings for Lévy processes. Submitted. Available at http://www.cwi.nl/∼ton.
[6] Kella, O. and Whitt, W. (1991). Queues with server vacations and Lévy processes with secondary Jump input. Ann. Appl. Prob. 1, 104117.
[7] Kella, O. and Whitt, W. (1992). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29,{} 396403.
[8] Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Prob. 41, 407424.

Keywords

MSC classification

A Lévy Process Reflected at a Poisson Age Process

  • Offer Kella (a1), Onno Boxma (a2) and Michel Mandjes (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed