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Imbedded and non-imbedded stationary characteristics of queueing systems with varying service rate and point processes

Published online by Cambridge University Press:  14 July 2016

D. König  and
V. Schmidt
D. König*
Affiliation:
Bergakademie Freiberg
V. Schmidt*
Affiliation:
Bergakademie Freiberg
*
∗Postal address: Sektion Mathematik, Bergakademie Freiberg, 92 Freiberg (Sachs), DDR.
∗Postal address: Sektion Mathematik, Bergakademie Freiberg, 92 Freiberg (Sachs), DDR.
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Abstract

In this paper a unified approach is used for proving relationships between customer-stationary and time-stationary characteristics of service systems with varying service rate and point processes. This approach is based on an intensity conservation principle for general stationary continuous-time processes with imbedded stationary marked point processes. It enables us to work under weaker independence assumptions than usual in queueing theory.

Keywords

STATIONARY QUEUESVARYING SERVICE RATEPOINT PROCESSESINTENSITY CONSERVATION PRINCIPLECUSTOMER- AND TIME-STATIONARY PROBABILITIESQUEUE LENGTHWAITING TIME
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 3 , September 1980 , pp. 753 - 767
Copyright
Copyright © Applied Probability Trust 1980 

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References

Borovkov, A. A. (1976) Stochastic Processes in Queueing Theory. Springer-Verlag, Berlin. (Translated from the Russian, Izdat. Nauka, Moscow (1972).).Google Scholar
Brill, P. H. and Posner, M. J. M. (1977) Level crossings in point processes applied to queues: single-server case. Operat. Res. 25, 662–674.CrossRefGoogle Scholar
Cohen, J. W. (1969) The Single-Server Queue. North-Holland Publishing Company, Amsterdam.Google Scholar
Cohen, J. W. (1976) On Regenerative Processes in Queueing Theory. Lecture Notes in Economics and Mathematical Systems 121, Springer-Verlag, Berlin.Google Scholar
Cohen, J. W. (1977) On up- and downcrossings. J. Appl. Prob. 14, 405–410.Google 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
Franken, P. and Kalähne, U. (1978) Existence, uniqueness and continuity of stationary distributions for queueing systems without delay. Math. Nachr. 86, 97–115.Google Scholar
Gutjahr, R. (1977a) A remark concerning the existence of infinitely many ‘empty points’ in queueing systems with infinitely many servers (in German). Math. Operationsforsch. Statist., Ser. Optimization 8, 245–251.CrossRefGoogle Scholar
Gutjahr, R. (1977b) Contributions to the Theory of Queueing Systems with General Initial Data (in German). Thesis, University of Jena.Google Scholar
Harris, R. (1974) The expected number of idle servers in a queueing system. Operat. Res. 22, 1258–1259.Google Scholar
Harrison, J. M. and Lemoine, A. J. (1976) On the virtual and actual waiting time distributions of a GI/G/1 queue. J. Appl. Prob. 13, 833–836.Google Scholar
Jankiewicz, M. and Kopocinski, B. (1976) Steady-state distributions of piecewise Markov processes. Zast. Mat. 15, 25–32.Google Scholar
Jankiewicz, M. and Rolski, T. (1977) Piecewise Markov processes on a general state space. Zast. Mat. 15, 421–436.Google Scholar
Kalähne, U. (1976) Existence, uniqueness and some invariance properties of stationary distributions for general single-server queues. Math. Operationsforsch. Statist. 7, 557–575.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., Matthes, K. and Nawrotzki, K. (1967) Generalizations of the Erlang and Engset Formulae (A method in queueing theory) (in German). Akademie-Verlag, Berlin.Google Scholar
König, D., Matthes, K. and Nawrotzki, K. (1971) Insensitivity properties of queueing processes (in German). Supplement to: Gnedenko, B. V. and Kovalenko, I. N. (1971) Einführung in die Bedienungstheorie. Akademie-Verlag, Berlin.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. Operations forsch. Statist., Ser. Optimization 9, 125–141.CrossRefGoogle Scholar
König, D. and Schmidt, V. (1977) Marked point processes with random behaviour between points. Proc. 2nd Vilnius Conf. Prob. Theory Math. Statist. 3, 109–112.Google Scholar
König, D. and Schmidt, V. (1980) Stochastic inequalities between customer-stationary and time-stationary characteristics of service systems with point processes. J. Appl. Prob. 17, 768–777.CrossRefGoogle Scholar
Kopocinska, I. (1977) Repairman problem with arbitrary distributions of repairment and working times. Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 25, 1023–1028.Google Scholar
Kopocinska, I. and Kopocinski, B. (1977) Steady-state distributions of queue length in the GI/G/s queue. Zast. Mat. 16, 39–46.Google Scholar
Kopocinski, B. and Rolski, T. (1977) A note on the virtual waiting time in the GI/G/s queue. Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 25, 1279–1280.Google Scholar
Kuczura, A. (1973) Piecewise Markov processes. SIAM J. Appl. Math. 24, 169–181.Google Scholar
Lemoine, A. J. (1974) On two stationary distributions for the stable GI/G/1 queue. J. Appl. Prob. 11, 849–852.Google Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Wiley, New York.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
Nawrotzki, K. (1978) Some remarks concerning the use of the Palm distribution in queueing theory (in German). Math. Operationsforsch. Statist., Ser. Optimization 9, 241–253.Google Scholar
Rolski, T. (1977) A relation between imbedded Markov chains in piecewise Markov processes. Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 25, 181–189.Google Scholar
Rolski, T. (1978) A rate conservative principle for stationary piecewise Markov processes. Adv. Appl. Prob. 10, 392–410.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
Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
Takács, L. (1963) The limiting distribution of the virtual waiting time and the queue size for a single-server queue with recurrent input and general service times. Sankhya A25, 91–100.Google Scholar