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On the use of a fundamental identity in the theory of semi-Markov queues

Published online by Cambridge University Press:  01 July 2016

E. Arjas
E. Arjas*
Affiliation:
The Academy of Finland, Helsinki
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Abstract

In the paper a single server semi-Markov queue is considered. The analysis is based on a fundamental matrix identity due to H. D. Miller. A natural method for the solution of semi-Markov queues is indicated; use is also made of a duality relation, which is discussed in the paper.

Keywords

SEMI-MARKOV QUEUEWAITING TIME PROCESSFUNDAMENTAL IDENTITYDUALITY RELATION
Type
Research Article
Information
Advances in Applied Probability , Volume 4 , Issue 2 , August 1972 , pp. 271 - 284
Copyright
Copyright © Applied Probability Trust 1972 

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References

Ali, H. (1970) Two results in the theory of queues. J. Appl. Prob. 7, 219–226.Google Scholar
Arjas, E. (1972a) On a fundamental identity in the theory of semi-Markov processes. Adv. Appl. Prob. 4, 258–270.Google Scholar
Arjas, E. (1972b) On the asymptotic behaviour of a generalization of Markov renewal processes. Soc. Sci. Fenn. Comment. Phys.-Math. 42, 17–25.Google Scholar
Çinlar, E. (1967a) Time dependence of queues with semi-Markovian services. J. Appl. Prob. 4, 356–364.Google Scholar
Çinlar, E. (1967b) Queues with semi-Markovian arrivals. J. Appl. Prob. 4, 365–379.Google Scholar
Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications. Vol. 1, Third Ed. Wiley, New York.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications. Vol. 2, Second Ed. Wiley, New York.Google Scholar
Greenberg, I. (1969) Some duality results in the theory of queues. J. Appl. Prob. 6, 99–121.CrossRefGoogle Scholar
Henderson, J. and Finch, P. D. (1970) A note on the queuing system E k /G/1. J. Appl. Prob. 7, 473–475.Google Scholar
Kemperman, J. H. B. (1961) The Passage Problem for a Stationary Markov Chain. The University of Chicago Press.Google Scholar
Kingman, J. F. C. (1966) On the algebra of queues. J. Appl. Prob. 3, 285–326.Google Scholar
Loynes, R. M. (1962a) The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Philos. Soc. 58, 497–520.Google Scholar
Loynes, R. M. (1962b) Stationary waiting times for single server queues. Ann. Math. Statist. 33, 1323–1339.Google Scholar
Miller, H. D. (1961) A convexity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32, 1260–1270.Google Scholar
Miller, H. D. (1962a) A matrix factorization problem in the theory of random variables defined on a finite Markov chain. Proc. Camb. Philos. Soc. 58, 268–285.CrossRefGoogle Scholar
Miller, H. D. (1962b) Absorption probabilities for sums of random variables defined on a finite Markov chain. Proc. Camb. Philos. Soc. 58, 286–298.Google Scholar
Neuts, M. F. (1966) The single-server queue with Poisson input and semi-Markov service times. J. Appl. Prob. 3, 202–230.Google Scholar
Prabhu, N. U. (1965) Queues and Inventories. Wiley, New York.Google Scholar
Presman, E. L. (1967) Boundary problems for sums of lattice random variables defined on a finite regular Markov chain. Theor. Probability Appl. 12, 323–328, (English translation).CrossRefGoogle Scholar
Presman, E. L. (1969) Factorization methods and boundary problems for sums of random variables given on Markov chains. Math. USSR-Izv. 3, 815–852, (English translation).Google Scholar
Pyke, R. (1961) Markov renewal processes with finitely many states. Ann. Math. Statist. 32, 1243–1259.Google Scholar