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An elementary proof of Sengupta's invariance relation and a remark on Miyazawa's conservation principle

Published online by Cambridge University Press:  14 July 2016

P. Brémaud
P. Brémaud*
Affiliation:
Laboratoire des Signaux et Systèmes, CNRS
*
∗ Postal address: Laboratoire des Signaux et Systèmes, CNRS-ESE, Plateau de Moulon, 91190 Gif-sur-Yvette, France.
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Abstract

In this short note we derive Sengupta's (1989) invariance relation using elementary arguments and we show that Miyazawa's (1983), (1985) conservation principle, on which Sengupta's proof is based, admits the Palm inversion formula as a consequence. This contrasts with Miyazawa's first proof based on the inversion formula. We also show that Neveu's (1976) cycle formula is a direct consequence of Miyazawa's principle.

Keywords

POINT PROCESSQUEUEING PROCESSPALM PROBABILITY
Type
Short Communications
Information
Journal of Applied Probability , Volume 28 , Issue 4 , December 1991 , pp. 950 - 954
Copyright
Copyright © Applied Probability Trust 1991 

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References

Baccelli, F. and Bremaud, P. (1987) Palm Probabilities and Stationary Queues. Lecture Notes in Statistics 41, Springer, New York.10.1007/978-1-4615-7561-0Google Scholar
Bremaud, P. (1989) Characteristics of queueing systems observed at events and the connection between stochastic intensity and Palm probability. QUESTA 5, 99–112.Google Scholar
Loynes, R. M. (1962) The stability of queues with non independent interarrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
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
Miyazawa, M. (1985) The intensity conservation law for queues with randomly changed service rate. J. Appl. Prob. 22, 408–418.Google Scholar
Neveu, J. (1976) Sur les mesures de Palm de deux processus ponctuels stationnaires. Z. Wahrscheinlichkeitsth. 34, 199–203.10.1007/BF00532703Google Scholar
Sakasegawa, H. and Wolff, R. (1990) The equality of the virtual delay and attained time distributions. Adv. Appl. Prob. 22, 257–259.10.2307/1427611Google Scholar
Sengupta, B. (1989) An invariance relationship for the G/G/1/8 queue. Adv. Appl. Prob. 21, 956–957.Google Scholar