Hostname: page-component-784d4fb959-4lvlm Total loading time: 0 Render date: 2025-07-15T07:11:07.778Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Dual Relationship Between Markov Chains of GI/M/1 and M/G/1 Type

Part of: Operations research and management science Markov processes

Published online by Cambridge University Press:  01 July 2016

P. G. Taylor  and
B. Van Houdt
P. G. Taylor*
Affiliation:
University of Melbourne
B. Van Houdt*
Affiliation:
University of Antwerp
*
∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia. Email address: p.taylor@ms.unimelb.edu.au
∗∗ Postal address: Performance Analysis of Telecommunications Systems Research Group, Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, Antwerp, B 2020, Belgium. Email address: benny.vanhoudt@ua.ac.be
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 1990, Ramaswami proved that, given a Markov renewal process of M/G/1 type, it is possible to construct a Markov renewal process of GI/M/1 type such that the matrix transforms G(z, s) for the M/G/1-type process and R(z, s) for the GI/M/1-type process satisfy a duality relationship. In his 1996 PhD thesis, Bright used time reversal arguments to show that it is possible to define a different dual for positive-recurrent and transient processes of M/G/1 type and GI/M/1 type. Here we compare the properties of the Ramaswami and Bright dual processes and show that the Bright dual has desirable properties that can be exploited in the design of algorithms for the analysis of Markov chains of GI/M/1 type and M/G/1 type.

Keywords

Matrix-analytic modelsquasi-birth-and-death processprocesses of GI/M/1 typeprocesses of M/G/1 type

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 90B22: Queues and service

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 1 , March 2010 , pp. 210 - 225
Copyright
Copyright © Applied Probability Trust 2010 

References

Akar, N. and Sohraby, K. (1997). An invariant subspace approach in M/G/1 and G/M/1 type Markov chains. Commun. Statist. Stoch. Models 13, 381–416.Google Scholar
Asmussen, S. (1989). Aspects of matrix Wiener–Hopf factorisation in applied probability. Math. Scientist 14, 101–116.Google Scholar
Asmussen, S. and Ramaswami, V. (1990). Probabilistic interpretations of some duality results for the matrix paradigms in queueing theory. Commun. Statist. Stoch. Models 6, 715–733.Google Scholar
Bini, D. A. and Meini, B. (1995). On cyclic reduction applied to a class of Toeplitz-like matrices arising in queueing problems. In Computations with Markov Chains, ed. Stewart, W. J., Kluwer, Boston. pp. 21–28.Google Scholar
Bini, D. A, Latouche, G. and Meini, B. (2005). Numerical Methods for Structured Markov Chains. Oxford University Press.Google Scholar
Bini, D.A., Meini, B. and Ramaswami, V. (2008). A probabilistic interpretation of cyclic reduction and its relationships with logarithmic reduction. Calcolo 45, 207–216.Google Scholar
Bini, D. A., Meini, B., Steffé, S. and Van Houdt, B. (2006). Structured Markov chains solver: software tools. In Proc. SMCTools Workshop (Pisa, 2006; ACM Internat. Conf. Proc. Ser. 201), ACM Press, New York.Google Scholar
Bright, L. (1996). Matrix-analytic methods in applied probability. , University of Adelaide.Google Scholar
Falkenberg, E. (1994). On the asymptotic behaviour of the stationary distribution of Markov chains of M/G/1-type. Commun. Statist. Stoch. Models 10, 75–97.Google Scholar
Gail, H. R., Hantler, S. L. and Taylor, B. A. (1994). Solutions of the basic matrix equation for M/G/1 and G/M/1 type Markov chains. Commun. Statist. Stoch. Models 10, 1–44.Google Scholar
Gail, H. R., Hantler, S. L. and Taylor, B. A. (1996). Spectral analysis of M/G/1 and G/M/1 type Markov chains. Adv. Appl. Prob. 28, 114–165.Google Scholar
Gail, H. R., Hantler, S. L. and Taylor, B. A. (1998). Matrix-geometric invariant measures for G/M/1 type Markov chains. Commun. Statist. Stoch. Models 14, 537–569.CrossRefGoogle Scholar
Kelly, F. P. (1983). Invariant measures and the q-matrix. In Probability, Statistics and Analysis (London Math. Soc. Lecture Notes 79), eds Kingman, J. F. C. and Reuter, G. E. H., Cambridge University Press, pp. 143–160.Google Scholar
Latouche, G. (1993). Algorithms for infinite Markov chains with repeating columns. In Linear Algebra, Markov Chains, and Queueing Models (IMA Vol. Math. Appl. 48), Springer, New York, pp. 231–265.CrossRefGoogle Scholar
Latouche, G. (1994). Newton's iteration for non-linear equations in Markov chains. IMA J. Numer. Anal. 14, 583–598.Google Scholar
Latouche, G. and Ramaswami, V. (1993). A logarithmic reduction algorithm for quasi-birth-and-death processes. J. Appl. Prob. 30, 650–674.Google Scholar
Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. Society for Industrial and Applied Mathematics, Philadelphia, PA.CrossRefGoogle Scholar
Neuts, M. F. (1978). Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector. Adv. Appl. Prob. 10, 185–212.CrossRefGoogle Scholar
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. John Hopkins University Press, Baltimore, MD.Google Scholar
Neuts, M. F. (1989). Structured Stochastic Matrices of M/G/1 and Their Applications. Marcel Dekker, New York.Google Scholar
Ramaswami, V. (1990). A duality theorem for the matrix paradigm in queueing theory. Commun. Statist. Stoch. Models 6, 151–161.Google Scholar