Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T15:18:07.739Z Has data issue: false hasContentIssue false
  • English
  • Français

Scale functions of Lévy processes and busy periods of finite-capacity M/GI/1 queues

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Parijat Dube ,
Fabrice Guillemin  and
Ravi R. Mazumdar
Parijat Dube*
Affiliation:
IBM Research
Fabrice Guillemin*
Affiliation:
France Télécom
Ravi R. Mazumdar*
Affiliation:
Purdue University
*
∗ Postal address: IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA. Email address: pdube@us.ibm.com
∗∗ Postal address: France Télécom, Division R&D, 2 Avenue Pierre Marzin, 22300 Lannion, France. Email address: fabrice.guillemin@francetelecom.com
∗∗∗ Postal address: School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285, USA. Email address: mazum@ecn.purdue.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we use the exit time theory for Lévy processes to derive new closed-form results for the busy period distribution of finite-capacity fluid M/G/1 queues. Based on this result, we then obtain the busy period distribution for finite-capacity queues with on–off inputs when the off times are exponentially distributed.

Keywords

Queuebusy periodLévy processscale functionLaplace transformon–off input

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J50: Boundary theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1145 - 1156
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avram, F., Kyprianou, A. E., and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215–238.Google Scholar
Baccelli, F., and Makowski, A. M. (1989). Dynamic, transient and stationary behavior of the M/G/1 queue via martingales. Ann. Prob. 17, 1691–1699.Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156–169.Google Scholar
Emery, D. J. (1973). Exit problem for a spectrally positive process. Adv. Appl. Prob. 5, 498–520.Google Scholar
Guillemin, F. M., and Mazumdar, R. R. (1995). Excursions of the workload process in G/GI/1 queues. Stoch. Process. Appl. 58, 139–154.Google Scholar
Kinateder, K. K. J., and Lee, E. Y. (2000). A new approach to the busy period of the M/M/1 queue. Queueing Systems 35, 105–115.Google Scholar
Kyprianou, A., and Palmowski, Z. (2004). A martingale review of some fluctuation theory for spectrally negative Lévy processes. To appear in Sminaire de Probabilit's, Vol. XXXVIII.Google Scholar
Lee, E. Y., and Kinateder, K. K. J. (2000). The expected wet period of finite dam with exponential inputs. Stoch. Process. Appl. 90, 175–180.Google Scholar
Pistorius, M. R. (2003). On doubly reflected completely asymmetric Lévy processes. Stoch. Process. Appl. 107, 131–143.Google Scholar
Pistorius, M. R. (2004). On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Prob. 17, 183–220.Google Scholar