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The Beňes equations for the distribution of excursions for general storage processes

Part of: Stochastic processes Special processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Fabrice Guillemin  and
Ravi Mazumdar
Fabrice Guillemin*
Affiliation:
CNET, Lannion
Ravi Mazumdar*
Affiliation:
Université de Québec
*
∗ Postal address: Centre National d'Etudes des Télécommunications Lannion-A, Route de Trégastel, 22300 Lannion, France.
∗∗ Postal address: INRS-Télécommunications, Université du Québec, 16 Place du Commerce, Ile-des-Soeurs, P.Q. H3E 1H6, Canada.
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Abstract

In this paper we obtain the Beňes equation for the evolution of the probability distribution of the excursion process associated with the level crossings of a general storage process. We then show that under stationarity and ergodicity assumptions on the process we can recover the well-known rate conservation law (RCL). Using the stationary solution we then show that the existence of an invariant solution can be studied in terms of an operator equation and we show how this characterization leads to a very simple explicit computation of the stationary distribution.

Keywords

SEMIMARTINGALEPOINT PROCESSESRATE CONSERVATION

MSC classification

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 60G17: Sample path properties 90B22: Queues and service
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 2 , June 1994 , pp. 418 - 429
Copyright
Copyright © Applied Probability Trust 1994 

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References

[1] Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
[2] Benes, V. E. (1963) General Stochastic Processes in the Theory of Queues. Addison-Wesley, Reading, MA.Google Scholar
[3] Borovkov, A. A. (1976) Stochastic Processes in Queueing Theory. Springer-Verlag, New York.CrossRefGoogle Scholar
[4] Ferrandiz, J. and Lazar, A. (1991) Rate conservation for stationary point processes. J. Appl. Prob. 28, 146–158.Google Scholar
[5] Mazumdar, R., Kannurpatti, R. and Rosenberg, C. (1991) On rate conservation for nonstationary processes. J. Appl. Prob. 28, 762–770.CrossRefGoogle Scholar
[6] Mazumdar, R., Badrinath, V., Guillemin, F. and Rosenberg, C. (1993) Pathwise rate conservation for a class of semi-martingales. Stoch. Proc. Appl. 47, 119–130.Google Scholar
[7] Miyazawa, ?. (1983) The derivation of invariance relations in complex queueing systems with stationary inputs. Adv. Appl. Prob. 15, 875–885.Google Scholar
[8] Miyazawa, M. (1985) The intensity conservation law for queues with randomly changed service rate. J. Appl. Prob. 22, 408–418.CrossRefGoogle Scholar
[9] Norros, I., Roberts, J., Simonian, A. and Virtamo, J. (1991) The superposition of variable bit rate sources in an ATM multiplexer. IEEE Sel. Areas in Commun. 9, 378–387.Google Scholar
[10] Rosenberg, C., Guillemin, F. and Mazumdar, R. (1993) On quantile measures for traffic characterization. Proc. ITC Seminar on Teletraffic Methods, Bangalore, India.Google Scholar