Hostname: page-component-7dd5485656-j7khd Total loading time: 0 Render date: 2025-10-30T11:57:11.169Z Has data issue: false hasContentIssue false
  • English
  • Français

On rate conservation for non-stationary processes

Published online by Cambridge University Press:  14 July 2016

Ravi Mazumdar ,
Raghavan Kannurpatti  and
Catherine Rosenberg
Ravi Mazumdar*
Affiliation:
INRS, Université du Québec
Raghavan Kannurpatti*
Affiliation:
INRS, Université du Québec
Catherine Rosenberg*
Affiliation:
Ecole Polytechnique, Montréal
*
∗ Postal address: INRS Télécommunications, Université du Québec, Ile des Soeurs, PQ, Canada H3E 1H6.
∗ Postal address: INRS Télécommunications, Université du Québec, Ile des Soeurs, PQ, Canada H3E 1H6.
∗∗ Postal address: Départment de Génie Electrique, Ecole Polytechnique, Montréal, PQ, Canada H3C 3A7.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper extends the rate conservation principle to cadlag processes whose jumps form a non-stationary point process with a time-dependent intensity. It is shown that this is a direct consequence of path integration and the strong law of large numbers for local martingales. When specialized to mean rates a non-stationary version of Miyazawa's result is obtained which is recovered in the stationary case. Some applications of the result are also given.

Keywords

POINT PROCESSESNON-STATIONARITYMARTINGALES

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 4 , December 1991 , pp. 762 - 770
Copyright
Copyright © Applied Probability Trust 1991 

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

[1] Baccelli, F. and Bremaud, P. (1987) Palm Probabilities and Stationary Queues. Lecture Notes in Statistics 41, Springer-Verlag, Heidelberg.Google Scholar
[2] Bremaud, P. (1989) Characteristics of queueing systems observed at events and the connections between stochastic intensity and Palm probability. QUESTA 5, 99–112.Google Scholar
[3] Bremaud, P. (1991) An elementary proof of Sengupta's invariance relation and a remark on Miyazawa's conservation principle. J. Appl. Prob. 28, 950–954.Google Scholar
[4] Bremaud, P. (1981) Point Processes and Queues: Martingale Dynamics. Springer-Verlag, New York.10.1007/978-1-4684-9477-8Google Scholar
[5] Bremaud, P., Kannurpatti, R. and Mazumdar, R. (1992) Event and time averages: a review. Adv. Appl. Prob. 24(2).Google Scholar
[6] Brill, P. and Posner, M. (1977) The system point method in exponential queues: a level crossing approach. Math. Operat. Res. 6, 31–39.Google Scholar
[7] Ferrandiz, J. and Lazar, A. (1991) Rate conservation for stationary point processes. J. Appl. Prob. 28, 146–158.Google Scholar
[8] Jacod, J. (1975) Multivariate point processes; predictable projection, Radon-Nikodym derivatives, representation of martingales. Z. Wahrschlichkeitsth. 31, 235–253.Google Scholar
[9] König, D. and Schmidt, V. (1981) Relationships between time and customer stationary characteristics of service systems. In Point Processes and Queueing Systems, ed. Bartfai, P. and Tomko, J., North-Holland, Amsterdam, pp. 181–225.Google Scholar
[10] Liptser, R. S. and Shiryayev, A. N. (1978) Statistics of Random Processes, Vol. 2. Springer-Verlag, New York.Google Scholar
[11] Liptser, R. S. and Shiryayev, A. N. (1989) Theory of Martingales. Kluwer, Dordrecht.Google Scholar
[12] Miyazawa, M. (1983) The derivation of invariance relations in complex queueing systems with stationary inputs. Adv. Appl. Prob. 15, 874–885.10.2307/1427329Google Scholar
[13] Miyazawa, M. (1985) The intensity conservation law for queues with randomly changed service rate. J. Appl. Prob. 22, 408–418.Google Scholar
[14] Rolski, T. (1981) Stationary Random Processes Associated with Point Processes. Lecture Notes in Statistics, 5, Springer-Verlag, New York.10.1007/978-1-4684-6268-5Google Scholar