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Homogeneous row-continuous bivariate markov chains with boundaries

Published online by Cambridge University Press:  14 July 2016

J. Keilson  and
M. Zachmann
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Abstract

The matrix-geometric results of M. Neuts are extended to ergodic row-continuous bivariate Markov processes [J(t), N(t)] on state space B = {(j, n)} for which: (a) there is a boundary level N for N(t) associated with finite buffer capacity; (b) transition rates to adjacent rows and columns are independent of row level n in the interior of B. Such processes are of interest in the modelling of queue-length for voice-data transmission in communication systems.

One finds that the ergodic distribution consists of two decaying components of matrix-geometric form, the second induced by the finite buffer capacity. The results are obtained via Green's function methods and compensation. Passage-time distributions for the two boundary problems are also made available algorithmically.

Keywords

MATRIX-GEOMETRICGREEN'S FUNCTIONCOMPENSATION
Type
Part 6 - The Analysis of Stochastic Phenomena
Information
Journal of Applied Probability , Volume 25 , Issue A , 1988 , pp. 237 - 256
Copyright
Copyright © Applied Probability Trust 1988 

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References

[1] Debreu, G. and Herstein, I. N. (1953) Non-negative square matrices. Econometrica 21, 597–607.Google Scholar
[2] Friedman, D. (1981) Queueing analysis of a shared voice-data link. MIT LIDS Report TH 1161.Google Scholar
[3] Graves, S. C. and Keilson, J. (1981) The compensation method applied to a one-product production/inventory problem. Math. Operat. Res. 6, 246–262.Google Scholar
[4] Keilson, J. (1965) Green's Function Methods in Probability Theory. Griffin, London.Google Scholar
[5] Keilson, J. (1965) The role of Green's functions in congestion theory. In Symposium on Congestion Theory, University of North Carolina Press, Chapel Hill.Google Scholar
[6] Keilson, J. (1979) Markov Chain Models—Rarity and Exponentiality. Springer-Verlag, New York.Google Scholar
[7] Keilson, J. and Kester, A. (1977) A circulatory model for human metabolism. Graduate School of Management, University of Rochester, Working Paper Series No. 7724.Google Scholar
[8] Keilson, J. and Wishart, D. M. G. (1964) A central limit theorem for processes defined on a finite Markov chain. Proc. Camb. Phil. Soc. 60, 547–567.Google Scholar
[9] Keilson, J. and Wishart, D. M. G. (1965) Boundary problems for additive processes defined on a finite Markov chain. Proc. Camb. Phil. Soc. 61, 173–190.Google Scholar
[10] Keilson, J., Sumita, U. and Zachmann, M. (1987) Row-continuous finite Markov chains—structure and algorithms. J. Operat. Res. Soc. Japan. Google Scholar
[11] Kleinrock, L. (1976) Queueing Systems, Vol. 2 Computer Applications. Wiley, New York.Google Scholar
[12] Neuts, M. F. (1978) Markov chains with applications in queueing theory which have a matrix-geometric invariant vector. Adv. Appl. Prob. 10, 185–212.Google Scholar
[13] Neuts, M. F. (1981) Matrix Geometric Solutions in Stochastic Models—An Algorithmic Approach. Johns Hopkins University Press, Baltimore, MD.Google Scholar