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A note on the infinitely deep dam with a Markovian input

Published online by Cambridge University Press:  14 July 2016

R. M. Phatarfod*
Affiliation:
Monash University
*
Postal address: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia.

Abstract

In this paper we consider an infinitely deep dam fed by inputs which form an ergodic Markov chain and whose release M is non-unit. The extension to non-unit release follows on lines similar to the independent inputs case. We show that P(θ) – θ MΙ where P(θ) = (pijθ i) has a maximum of N = M(M + l)/2 non-zero singularities in the unit disc, so that the general solution of the equilibrium equations has N unknown constants. We also show that these constants satisfy N linear constraints, so that the solution is fully determined.

Type
Short Communications
Copyright
Copyright © Applied Probability Trust 1979 

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References

Foster, F. G. (1961) Queues with batch arrivals, I. Acta Math. Acad. Sci. Hungar. 12, 110.Google Scholar
Kingman, J. F. C. (1961) A convexity property of positive matrices. Quart. J. Math. Oxford 12, 283284.Google Scholar
Loynes, R. M. (1963) Stationary waiting-time distributions for single-server queues. Ann. Math. Statist. 34, 13231339.Google Scholar
Pakes, A. G. and Phatarfod, R. M. (1978) The limiting distribution for the infinitely deep dam with a Markovian input. Stoch. Proc. Appl. 8, 199209.Google Scholar
Phatarfod, R. M. (1979) The bottomless dam. J. Hydrology 40, 337363.Google Scholar
Wishart, D. M. G. (1956) A queueing system with ?2 service-time distribution. Ann. Math. Statist. 27, 768779.Google Scholar