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Stochastic inequalities between customer-stationary and time-stationary characteristics of queueing systems with point processes

Published online by Cambridge University Press:  14 July 2016

D. König  and
V. Schmidt
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Abstract

By means of a general intensity conservation principle for stationary processes with imbedded marked point processes (PMP) stochastic inequalities are proved between customer-stationary and time-stationary characteristics of queueing systems G/G/s/r.

Keywords

QUEUESCUSTOMER- AND TIME-STATIONARY PROBABILITIESINEQUALITIESPOINT PROCESSESINTENSITY CONSERVATION PRINCIPLENBUE (NWUE)
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 3 , September 1980 , pp. 768 - 777
Copyright
Copyright © Applied Probability Trust 1980 

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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
Franken, P. (1976) Some applications of the theory of stochastic point processes in queueing theory (in German). Math. Nachr. 70, 309–316.Google Scholar
Grandell, J. (1976) Doubly Stochastic Poisson Processes. Lecture Notes in Mathematics 529, Springer-Verlag, Berlin.CrossRefGoogle Scholar
König, D. (1976) Stochastic processes with basic stationary marked point processes. Buffon Bicentenary Symp. Stochastic Geometry and Directional Statistics, Erevan. (Abstract: Adv. Appl. Prob. 9 (1978), 440–442.).Google Scholar
König, D., Rolski, T., Schmidt, V. 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.Google 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.Google Scholar
König, D., Schmidt, V. and Stoyan, D. (1976) On some relations between stationary distributions of queue lengths and imbedded queue lengths in G/G/s queueing systems. Math. Operationsforsch. Statist. 7, 577–586.Google Scholar
Marshall, A. W. and Proschan, F. (1970) Classes of distributions applicable in replacement, with renewal theory implications. Proc. 6th Berkeley Symp. Math. Statist. Prob. 1, 495–515.Google Scholar
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
Miyazawa, M. (1979) A formal approach to queueing processes in the steady state and their applications. J. Appl. Prob. 16, 332–346.Google Scholar
Mori, M. (1975) Some bounds for queues. J. Operat. Res. Soc. Japan 18, 152–181.Google Scholar
Neuts, M. F. (1977) Some explicit formulas for the steady-state behaviour of the queue with semi-Markovian service times. Adv. Appl. Prob. 9, 141–157.Google Scholar
Rodhe, H. and Grandell, J. (1972) On the removal time of aerosol particles from the atmosphere by precipitation scavenging. Tellus 24, 443–454.Google Scholar
Ross, S. M. (1978) Average delay in queues with non-stationary Poisson arrivals. J. Appl. Prob. 15, 602–609.Google Scholar
Schmidt, V. (1978) On some relations between stationary time and customer state probabilities for queueing systems G/GI/s/r. Math. Operationsforsch. Statist., Ser. Optimization 9, 261–272.Google Scholar
Stoyan, D. (1977a) Qualitative Properties and Bounds of Stochastic Models (in German). Akademie-Verlag, Berlin.Google Scholar
Stoyan, D. (1977b) Further stochastic order relations among GI/GI/1 queues with a common traffic intensity. Math. Operationsforsch. Statist., Ser. Optimization 8, 541–548.Google Scholar