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Multichannel queueing systems with infinite waiting room and stochastic control

Published online by Cambridge University Press:  14 July 2016

Jewgeni Dshalalow
Jewgeni Dshalalow*
Affiliation:
Florida Institute of Technology
*
∗ Postal address: Florida Institute of Technology, Department of Applied Mathematics, Melbourne, FL 32901-6988, USA.
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Abstract

A wide class of multichannel queueing models appears to be useful in practice where the input stream of customers can be controlled at the moments preceding the customers' departures from the source (e.g. airports, transportation systems, inventories, tandem queues). In addition, the servicing facility can govern the intensity of the servicing process that further improves flexibility of the system. In such a multichannel queue with infinite waiting room the queueing process {Zt; t ≧ 0} is under investigation. The author obtains explicit formulas for the limiting distribution of (Zt) partly using an approach developed in previous work and based on the theory of semi-regenerative processes. Among other results the limiting distributions of the actual and virtual waiting time are derived. The input stream (which is not recurrent) is investigated, and distribution of the residual time from t to the next arrival is obtained. The author also treats a Markov chain embedded in (Zt) and gives a necessary and sufficient condition for its existence. Under this condition the invariant probability measure is derived.

Keywords

CONTROLLED INPUTCONTROLLED SERVICEQUEUEING PROCESSLIMITING DISTRIBUTIONWAITING TIMESEMI-REGENERATIVE PROCESSMARKOV CHAININPUT PROCESSWEAK MARKOV PROCESSSEMI-MARKOV PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 2 , June 1989 , pp. 345 - 362
Copyright
Copyright © Applied Probability Trust 1989 

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Footnotes

This paper is partly based upon work supported by the National Science Foundation under Grant No. DMS-8706186 and by Allegheny Research and Development Center, University of Pittsburgh at Bradford.

References

Abolnikov, L. and Dukhovny, A. (1987) Necessary and sufficient conditions for the ergodicity of Markov chains with transition J. Appl. Math. Simul. 1, 13–24.Google Scholar
Cinlar, E. (1975) Introduction to Stochastic Processes. Prentice Hall, Englewood Cliffs, N.J. Google Scholar
Cohen, J. (1982) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
Dshalalow, J. (1978) Many-server feedback queueing systems. Engineering Cybernetics, Proc. Acad. Sci. USSR. 16, 78–88.Google Scholar
Dshalalow, J. (1983) Queues with Feedback. Doctoral Dissertation, Technische Universität Berlin, West Germany.Google Scholar
Dshalalow, J. (1985) On the multiserver queue with finite waiting room and controlled input. Adv. Appl. Prob. 17, 408–423.CrossRefGoogle Scholar
Dshalalow, J. (1987a) On a multi-channel transportation loss system with controlled input and controlled service. J. Appl. Math. Simul. 1, 41–55.CrossRefGoogle Scholar
Dshalalow, J. (1987b) Infinite channel queueing system with controlled input. Math. Oper. Res. 12, 665–677.CrossRefGoogle Scholar
Dshalalow, J. (1988) Multi-channel queueing systems with infinite waiting room and stochastic control. Techn. Report No. MA-0188, Florida Inst. Technology, 1–32.Google Scholar
Gihman, I. and Skorohod, A. (1974) The Theory of Stochastic Processes, 1. Springer-Verlag, Berlin.Google Scholar
Takács, L. (1958) Some probability questions in the theory of telephone traffic. Mag. Tud. Akad. Mat. Fiz. Oszl. Közl. 8, 151–210.Google Scholar
Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar