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Insensitive bounds for the stationary distribution of non-reversible Markov chains

Published online by Cambridge University Press:  14 July 2016

Arie Hordijk  and
AD Ridder
Arie Hordijk
Affiliation:
University of Leiden
AD Ridder
Affiliation:
University of Leiden
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Abstract

A general method is developed to compute easy bounds of the weighted stationary probabilities for networks of queues which do not satisfy the standard product form. The bounds are obtained by constructing approximating reversible Markov chains. Thus, the bounds are insensitive with respect to service-time distributions. A special representation, called the tree-form solution, of the stationary distribution is used to derive the bounds. The results are applied to an overflow model.

Keywords

REVERSIBLE MARKOV CHAINSSTATIONARY DISTRIBUTIONTREE-FORM SOLUTIONSTOCHASTIC MONOTONICITYINSENSITIVE BOUNDS
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 1 , March 1988 , pp. 9 - 20
Copyright
Copyright © Applied Probability Trust 1988 

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References

[1] 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
[2] Boxma, O. J. and Konheim, A. G. (1981) Approximate analysis of exponential queueing systems with blocking. Acta Informatica 15, 19–66.Google Scholar
[3] Chandy, K. M. and Sauer, C. H. (1980) Computational algorithms for product form queueing networks. Comm. Assoc. Comput. Mach. 23, 573–583.Google Scholar
[4] Chen, Wai-Kai (1971) Applied Graph Theory. North-Holland, Amsterdam.Google Scholar
[5] Daley, D. J. (1968) Stochastically monotone Markov chains. Z. Wahrscheinlichkeitsth. 10, 305–317.Google Scholar
[6] Gani, J. and Tin, P. (1985) On a class of continuous time Markov processes. J. Appl. Prob. 22, 804–815.Google Scholar
[7] Hordijk, A. (1983) Insensitivity for stochastic networks, Mathematical Computer Performance and Reliability (eds. Iazeolla, G., Courtois, P. J. and Hordijk, A.), North-Holland, Amsterdam, 77–94.Google Scholar
[8] Hordijk, A. and Ridder, A. (1987) Stochastic inequalities for an overflow model. J. Appl. Prob. 24, 696–708.CrossRefGoogle Scholar
[9] Kamae, T., Krengel, U. and O'brien, G. L. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
[10] Keilson, J. and Kester, A. (1977) Monotone matrices and monotone Markov processes. Stoch. Proc. Appl. 5, 231–241.Google Scholar
[11] Kelly, F. P. (1978) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
[12] Kosten, L. (1973) Stochastic Theory of Service Systems. Pergamon Press, Oxford.Google Scholar
[13] Nielsen, F. (1979) Flowgraphs. In Applications of Graph Theory (ed. Wilson, R. J. and Beineke, L. W.), Academic Press, London, 59–80.Google Scholar
[14] Ridder, A. (1986) A note on insensitive bounds for a grading. Report, University of Leiden, TW-86–01.Google Scholar
[15] Schoute, A. L. (1978) Modelling of Virtual Memory Computer Systems with Multiple Classes of Processes. Dissertation, University of Groningen.Google Scholar
[16] Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.Google Scholar
[17] Takahashi, Y., Miyahara, H. and Hasegawa, T. (1980) An approximation method for open restricted queueing networks. Operat. Res. 28, 594–602.Google Scholar
[18] Van Dijk, N. M. (1987) Simple and insensitive bounds for a grading and an overflow model. Submitted for publication.CrossRefGoogle Scholar
[19] Van Dijk, N. and Lamond, B. F. (1987) Bounds for the call congestion of finite single-server exponential tandem queues. Operat. Res. Google Scholar
[20] Whitt, W. (1985) Blocking when service is required from several facilities simultaneously. AT&T Tech. J. 64, 1807–1856.Google Scholar
[21] Whitt, W. (1986) Stochastic comparisons for non-Markov processes. Math. Operat. Res. 11, 608–618.Google Scholar
[22] Whittle, P. (1985) Partial balance and insensitivity. J. Appl. Prob. 22, 168–176.Google Scholar