Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-14T12:39:03.213Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on level-crossing analysis for the excess, age, and spread distributions

Part of:Special processesStochastic processes

Published online by Cambridge University Press:  14 July 2016

Tsuyoshi Katayama
Tsuyoshi Katayama*
Affiliation:
Toyama Prefectural University
*
∗ Postal address: Department of Electronics and Informatics, Faculty of Engineering, Toyama Prefectural University, Kosugi-Machi, Toyama 939-0398, Japan. Email address: katayama@pu-toyama.ac.jp
Get accessRights & Permissions [Opens in a new window]

Abstract

In this paper, we show that the time-average distributions of excess, age, and spread are given by the solution of first-order differential equations. These differential equations can be directly derived in a simple, unified way using a general level-crossing formula based on the balance of up and down crossings on sample paths, which may be helpful for the intuitive interpretation.

Keywords

Renewal processlevel-crossing methoddifferential equationtime average

MSC classification

Primary:60G17: Sample path properties60K05: Renewal theory60K25: Queueing theory

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 39 , Issue 4 , December 2002, pp. 896 - 900
Copyright
Copyright © Applied Probability Trust 2002 

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.)

Article purchase

Temporarily unavailable

References

Brill, P. H. (2001). Level crossing methods. In Encyclopedia of Operations Research and Management Science, 2nd edn, eds Gass, S. I. and Harris, C. M., Kluwer, Dordrecht, pp. 448–450.CrossRefGoogle Scholar
Doshi, B. (1992). Level-crossing analysis of queues. In Queueing and Related Models, eds Bhat, U. N. and Basawa, I. V., Oxford University Press, pp. 3–33.Google Scholar
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York.Google Scholar
Miyazawa, M. (1985). The intensity conservation law for queues with randomly changed service rate. J. Appl. Prob. 22, 408–418.CrossRefGoogle Scholar
Miyazawa, M. (1994). Rate conservation laws: a survey. Queueing Systems 15, 1–58.CrossRefGoogle Scholar
Sigman, K. (1994). Stationary Marked Point Processes: An Intuitive Approach. Chapman and Hall, New York.Google Scholar
Wolff, R. W. (1988). Sample-path derivations of the excess, age, and spread distributions. J. Appl. Prob. 25, 432–436.CrossRefGoogle Scholar
Zazanis, M. A. (1992). Sample path analysis of level crossings for the workload process. Queueing Systems 11, 419–428.CrossRefGoogle Scholar