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Rate conservation for stationary processes

Published online by Cambridge University Press:  14 July 2016

Josep M. Ferrandiz  and
Aurel A. Lazar
Josep M. Ferrandiz
Affiliation:
Columbia University
Aurel A. Lazar*
Affiliation:
Columbia University
*
∗∗Postal address: Department of Electrical Engineering, 1312 S. W. Mudd Building, Columbia University, New York, NY 10027, USA.
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Abstract

We derive a rate conservation law for distribution densities which extends a result of Brill and Posner. Based on this conservation law, we obtain a generalized Takács equation for the G/G/m/B queueing system that only requires the existence of a stochastic intensity for the arrival process and the residual service time distribution density for the G/GI/1/B queue. Finally, we solve Takács' equation for the N/GI/1/∞ queueing system.

Keywords

PALM PROBABILITYSTOCHASTIC INTENSITYTAKÁCS FORMULAGROUP ARRIVALS
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 1 , March 1991 , pp. 146 - 158
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

∗

Present address: Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS12 6QZ, UK.

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. (1981) Point Processes and Queues: Martingale Dynamics. Springer-Verlag, New York.Google Scholar
[3] 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
[4] 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
[5] Ferrandiz, J. M. (1989) Point Processes in Modeling, Analysis and Control of Integrated Networks. Ph.D. thesis, Columbia University, Department of Electrical Engineering.Google Scholar
[6] Franken, P., König, D., Arndt, U. and Schimdt, V. (1982) Queues and Point Processes. Wiley, Chichester.Google Scholar
[7] Karatzas, I. and Shreve, S. (1988) Brownian Motion and Stochastic Calculus. Springer-Verlag, New York.Google Scholar
[8] Kleinrock, L. (1975) Queueing Systems, Volume 1: Theory. Wiley, New York.Google Scholar
[9] König, D. and Schmidt, V. (1981) Relationships between time and customer stationary characteristics of service systems. In Point Processes and Queueing Problems, eds Bartfai, P. and Tomko, J., North-Holland, Amsterdam, pp. 181–225.Google Scholar
[10] Miyazawa, M. (1985) The intensity conservation law for queues with randomly changed service rate. J. Appl. Prob. 22, 408–418.Google Scholar
[11] Neuts, M. F. (1979) A versatile markovian point process. J. Appl. Prob. 16, 764–769.CrossRefGoogle Scholar
[12] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore.Google Scholar
[13] Ramaswami, V. (1980) The N/G/1 queue and its detailed analysis. Adv. Appl. Prob. 12, 222–261.Google Scholar