Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T12:36:18.927Z Has data issue: false hasContentIssue false
  • English
  • Français

Markov Chains with Hybrid Repeating Rows - Upper-Hessenberg, Quasi-Toeplitz Structure of the Block Transition Probability Matrix

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Alexander Dudin ,
Chesoong Kim  and
Valentina Klimenok
Alexander Dudin*
Affiliation:
Belarusian State University
Chesoong Kim*
Affiliation:
Sangji University
Valentina Klimenok*
Affiliation:
Belarusian State University
*
∗Postal address: Laboratory of Applied Probabilistic Analysis, Department of Applied Mathematics and Computer Sciences, Belarusian State University, 4 Independence Avenue, 220030 Minsk-30, Belarus.
∗∗∗Postal address: Department of Industrial Engineering, Sangji University, Wonju, Kangwon, 220-702, Korea. Email address: dowoo@sangji.ac.kr
∗Postal address: Laboratory of Applied Probabilistic Analysis, Department of Applied Mathematics and Computer Sciences, Belarusian State University, 4 Independence Avenue, 220030 Minsk-30, Belarus.
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 this paper we consider discrete-time multidimensional Markov chains having a block transition probability matrix which is the sum of a matrix with repeating block rows and a matrix of upper-Hessenberg, quasi-Toeplitz structure. We derive sufficient conditions for the existence of the stationary distribution, and outline two algorithms for calculating the stationary distribution.

Keywords

Multidimensional Markov chainqueueing systemergodicityalgorithmcensoring

MSC classification

Secondary: 60K25: Queueing theory 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 211 - 225
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Bini, D. A., Latouche, G. and Meini, B. (2005). Numerical Methods for Structured Markov Chains. Oxford University Press.Google Scholar
[2] Chakravarthy, S. (1993). Analysis of a finite MAP/G/1 queue with group services. Queueing Systems 13, 385–407.Google Scholar
[3] Dudin, A. N. and Klimenok, V. I. (1999). Multi-dimensional quasi-Toeplitz Markov chains. J. Appl. Math. Stoch. Anal. 12, 393–415.Google Scholar
[4] Dudin, A. N. and Nishimura, S. (1999). A BMAP/SM/1 queueing system with Markovian arrival of disasters. J. Appl. Prob. 36, 868–881.Google Scholar
[5] Dudin, A. N. and Semenova, O. V. (2004). Stable algorithm for stationary distribution calculation for a BMAP/SM/1 queueing system with Markovian input of disasters. J. Appl. Prob. 42, 547–556.Google Scholar
[6] 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
[7] Gantmakher, F. R. (1967). The Matrix Theory. Science, Moscow.Google Scholar
[8] Grassmann, W. K. and Heyman, D. P. (1990). Equilibrium distribution of block-structured Markov chains with repeating rows. J. Appl. Prob. 27, 557–576.Google Scholar
[9] Kemeni, J. G., Snell, J. L. and Knapp, A. W. (1966). Denumerable Markov Chains. Van Nostrand, New York.Google Scholar
[10] Klimenok, V. I. (2001). On the modification of Rouche's theorem for queueing theory problems. Queueing Systems 38, 431–434.Google Scholar
[11] Klimenok, V. I. and Dudin, A. N. (2006). Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory. Queueing Systems 54, 245–259.Google Scholar
[12] Meini, B. (1997). An improved FFT-based version of Ramaswami's formula. Commun. Statist. Stoch. Models 13, 223–238.Google Scholar
[13] Moustafa, M. D. (1957). Input–output Markov process. In Proc. Koninkl. Net. Akad. Wetensch. A60, pp. 112–118.Google Scholar
[14] Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach. Johns Hopkins University Press, Baltimore, MD.Google Scholar
[15] Neuts, M. F. (1989). Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Dekker, New York.Google Scholar
[16] Neuts, M. F. (1994). An interesting random walk on the non-negative integers. J. Appl. Prob. 31, 48–58.Google Scholar
[17] Ramaswami, V. (1988). A stable recursion for the steady-state vector in Markov chains of M/G/1 type. Commun. Statist. Stoch. Models 4, 183–188.Google Scholar
[18] Ramaswami, V. (1997). Matrix analytic methods: a tutorial overview with some extensions and new results. In Matrix-Analytic Methods in Stochastic Models (Lecture Notes Pure Appl. Math. 183), Marcel Dekker, New York, pp. 261–296.Google Scholar
[19] Riska, A. (2002). Aggregate matrix-analytic techniques and their applications. , The College of William and Mary, Williamsburg.Google Scholar
[20] Zhao, Y. Q. (2000). Censoring technique in studying block-structured Markov chains. In Advances in Algorithmic Methods for Stochastic Models, Notable Publications, Neshanic Station, NJ, pp. 417–433.Google Scholar