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The derivation of invariance relations in complex queueing systems with stationary inputs

Published online by Cambridge University Press:  01 July 2016

Masakiyo Miyazawa
Masakiyo Miyazawa*
Affiliation:
Science University of Tokyo
*
∗Postal address: Department of Information Sciences. Faculty of Science and Technology, Science University of Tokyo, Noda City, Chiba 278, Japan.
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Abstract

We discuss a method of obtaining invariance relations in complex systems by using the theory of point processes. New formulae are given for obtaining them generally, and in particular in many-stage models such as tandem and network queues. The formulae are shown to be useful by applications to a many-server queue and a tandem queue. Stochastic inequalities in a tandem queue are also discussed using the invariance relations obtained.

Keywords

MANY-SERVER QUEUETANDEM QUEUEMANY-STAGE QUEUESTEADY-STATE PROBABILITYPOINT PROCESSQUEUE LENGTHRESIDUAL SERVICE TIMESTOCHASTIC INEQUALITY
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 4 , December 1983 , pp. 874 - 885
Copyright
Copyright © Applied Probability Trust 1983 

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References

[1] Borokov, A. A. (1978) Ergodicity and stability theorem for a class of stochastic equations and their applications. Theory Prob. Appl. 23, 227–247.CrossRefGoogle Scholar
[2] Bremaud, P. (1981) Point Processes and Queues. Springer-Verlag, New York.CrossRefGoogle Scholar
[3] Finch, P. D. (1959) On the distribution of queue size in queueing problems. Acta Math. Acad. Sci. Hung. 10, 327–336.Google Scholar
[4] Franken, P. (1976) Einige Anwendungen der Theorie zufälliger Punktprozesse in der Bedienungstheorie I. Math. Nachr. 70, 303–319.CrossRefGoogle Scholar
[5] Franken, P., König, D., Arndt, U., and Schmidt, V. (1981) Queues and Point Processes. Akademie-Verlag, Berlin.Google Scholar
[6] König, D. (1980) Method for proving relationships between stationary characteristics of queueing systems with point processes. Elektron. Informationsverarbeit. Kybernetik 16, 521–543.Google Scholar
[7] König, D., Rolski, T., Schmidt, V., and Stoyan, D. (1978) Stochastic processes with imbedded marked point processes (PMP) and their applications in queueing. Math. Operationsforsch. Statist. Ser. Optimization 9, 125–141.CrossRefGoogle Scholar
[8] König, D. and Schmidt, V. (1980a) Imbedded and non-imbedded stationary characteristics of queueing systems with varying service rate and point processes. J. Appl. Prob. 17, 753–767.CrossRefGoogle Scholar
[9] König, D. and Schmidt, V. (1980b) Stochastic inequalities between customer-stationary and time-stationary characteristics of service systems with point processes. J. Appl. Prob. 17, 768–777.Google Scholar
[10] Miyazawa, M. (1976) Stochastic order relations among GI/G/1 queues with a common traffic intensity. J. Operat. Res. Soc. Japan 19, 193–208.Google Scholar
[11] Miyazawa, M. (1977) Time and customer processes in queues with stationary inputs. J. Appl. Prob. 14, 349–357.Google Scholar
[12] Miyazawa, M. (1979) A formal approach to queueing processes in the steady state and their applications. J. Appl. Prob. 16, 332–346.CrossRefGoogle Scholar
[13] Miyazawa, M. (1981) Note on Palm measures in the intensity conservation law and inversion formula in PMP and their applications. Math. Operationsforsch. Statist. Ser. Optimization 12, 281–293.Google Scholar
[14] Nakatsuka, T. (1982) Limiting stability of queues with the unbounded distribution tail of the input. Preprint.Google Scholar
[15] Ryll-Nardzewski, C. (1961) Remarks on processes of calls. Proc 4th Berkeley Symp. Math. Statist. Prob. 2, 455–465.Google Scholar