Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-06-03T07:04:11.004Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

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.)

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