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Stationary state probabilities at arrival instants for closed queueing networks with multiple types of customers

Published online by Cambridge University Press:  14 July 2016

S. S. Lavenberg  and
M. Reiser
S. S. Lavenberg*
Affiliation:
IBM Thomas J. Watson Research Center, Yorktown Heights
M. Reiser*
Affiliation:
IBM Research Laboratory, Zürich
*
∗Postal address: IBM Thomas J. Watson Research Center, P. O. Box 218, Yorktown Heights, NY 10598, U.S.A.
∗∗Postal address: IBM Research Laboratory, Zürich, Switzerland.
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Abstract

We consider closed networks of interconnected service centers with multiple types of customers and multiple classes, whose stationary state probabilities at arbitrary times have a product form. A customer can change its class but not its type as it traverses the network. We show that the stationary state probabilities at instants at which customers of a particular type arrive at a particular service center and enter a particular class are equal to the stationary state probabilities at arbitrary times for the network with one less customer of that type. Applications of this result are given.

Keywords

QUEUEING NETWORKSMULTIPLE TYPES OF CUSTOMERSSTATIONARY STATE PROBABILITIESSTATIONARY WAITING-TIME DISTRIBUTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 4 , December 1980 , pp. 1048 - 1061
Copyright
Copyright © Applied Probability Trust 

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References

Barbour, A. D. (1976) Networks of queues and the method of stages, Adv. Appl. Prob. 8, 584–591.Google Scholar
Baskett, F., Chandy, K. M., Muntz, R. R. and Palacios, F. G. (1975) Open, closed and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22, 248–260.Google Scholar
Kelly, F. P. (1975) Networks of queues with customers of different types. J. Appl. Prob. 12, 542–554.CrossRefGoogle Scholar
Kelly, F. P. (1976) Networks of queues. Adv. Appl. Prob. 8, 416–432.Google Scholar
Lemoine, A. J. (1977) Networks of queues — A survey of equilibrium analysis. Management Sci. 24, 464–481.Google Scholar
Melamed, B. (1978) On Markov jump processes imbedded at traffic epochs and their queueing-theoretic interpretation. Department of Industrial Engineering and Management Sciences, Northwestern University.Google Scholar
Reiser, M. (1977) Numerical methods in separable queueing networks. Studies in the Management Sciences 7, 113–142.Google Scholar
Reiser, M. and Lavenberg, S. S. (1978) Mean value analysis of closed multichain queueing networks. IBM Research Report RC 7023, Yorktown Heights, New York.Google Scholar
Sevcik, K. C. and Mitrani, I. (1979) The distribution of queuing network states at input and output instants. Proc. 4th Internat. Symp. Modelling and Performance Evaluation of Computer Systems. North-Holland, Amsterdam.Google Scholar