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Simple derivations of the invariance relations and their applications

Published online by Cambridge University Press:  14 July 2016

Masakiyo Miyazawa
Masakiyo Miyazawa*
Affiliation:
Science University of Tokyo
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Abstract

In the literature, various methods have been studied for obtaining invariance relations, for example, L = λW (Little's formula), in queueing models. Recently, it has become known that the theory of point processes provides a unified approach to them (cf. Franken (1976), König et al. (1978), Miyazawa (1979)). This paper is also based on that theory, and we derive a general formula from the inversion formula of point processes. It is shown that this leads to a simple proof for invariance relations in G/G/c queues. Using these results, we discuss a condition for the distribution of the waiting time vector of a G/G/c queue to be identical with that of an M/G/c queue.

Keywords

MANY SERVER QUEUESSTATIONARY INPUTSINVARIANCE RELATIONSPOINT PROCESSWAITING TIMEQUEUE LENGTHIDENTIFICATIONM/G/c QUEUE
Type
Research Paper
Information
Journal of Applied Probability , Volume 19 , Issue 1 , March 1982 , pp. 183 - 194
Copyright
Copyright © Applied Probability Trust 1982 

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References

Brumelle, S. L. (1971) On the relation between customer and time averages in queues. J. Appl. Prob. 8, 508–520.CrossRefGoogle Scholar
Chandy, K. M., Howard, J. H. and Towsley, D. F. (1977) Product form and local balance in queueing networks. J. Assoc. Comput. Mach. 24, 205–263.CrossRefGoogle Scholar
Cohen, J. W. (1977) On up- and down-crossings. J. Appl. Prob. 14, 405–410.CrossRefGoogle Scholar
Franken, P. (1976) Einige Anwendungen der Theorie Zufälliger Punktprozesse in der Bedienungstheorie I. Math. Nachr. 70, 309–319.Google Scholar
König, D., Rolski, T., Schmidt, D. and Stoyan, D. (1978) Stochastic processes with imbedded marked point processes (PMP) and their application in queueing. Math. Operationsforsch. Statist., Ser. Optimization 9, 125–141.CrossRefGoogle Scholar
König, D. and Schmidt, V. (1980) Imbedded and non-imbedded stationary characteristics of queueing systems with varying service rate and point processes. J. Appl. Prob. 17, 753–767.CrossRefGoogle Scholar
Little, J. D. C. (1961) A proof for the queueing formula: L =? W. Operat. Res. 9, 383–387.CrossRefGoogle Scholar
Loynes, R. M. (1962) The stability of a queue with nonindependent interarrival and service time. Proc. Camb. Phil. Soc. 58, 497–520.CrossRefGoogle Scholar
Miyazawa, M. (1977) Time and customer processes in queues with stationary inputs. J. Appl. Prob. 14, 349–357.CrossRefGoogle Scholar
Miyazawa, M. (1979) A formal approach to queueing processes in the steady state and their applications. J. Appl. Prob. 16, 332–346.CrossRefGoogle Scholar
Rolski, T. (1978) A rate-conservative principle for stationary piecewise Markov processes. Adv. Appl. Prob. 10, 394–410.CrossRefGoogle Scholar
Ryll-Nardzewski, C. (1961) Remarks on processes of calls. Proc. 4th Berkeley Symp. Math. Statist. Prob. 2, 455–465.Google Scholar
Stoyan, D. (1977) Further stochastic order relations among GI/GI/1 queues with a common traffic intensity. Math. Operationsforsch. Statist., Ser. Optimization 9, 541–548.CrossRefGoogle Scholar