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On the convergence and limits of certain matrix sequences arising in quasi-birth-and-death Markov chains

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Qi-Ming He  and
Marcel F. Neuts
Qi-Ming He*
Affiliation:
Dalhousie University
Marcel F. Neuts*
Affiliation:
University of Arizona
*
∗ Postal address: Department of Industrial Engineering, DalTech, Dalhousie University, Halifax, Nova Scotia, Canada B3J 2X4. Email address: [email protected]
∗∗ Postal address: Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA.
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Abstract

We study the convergence of certain matrix sequences that arise in quasi-birth-and-death (QBD) Markov chains and we identify their limits. In particular, we focus on a sequence of matrices whose elements are absorption probabilities into some boundary states of the QBD. We prove that, under certain technical conditions, that sequence converges. Its limit is either the minimal nonnegative solution G of the standard nonlinear matrix equation, or it is a stochastic solution that can be explicitly expressed in terms of G. Similar results are obtained relative to the standard matrix R that arises in the matrix-geometric solution of the QBD. We present numerical examples that clarify some of the technical issues of interest.

Keywords

matrix analytic methodsquasi-birth-and-death Markov chainnonnegative matrix

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 2 , June 2001 , pp. 519 - 541
Copyright
Copyright © Applied Probability Trust 2001 

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References

Alfa, A. S., and Chakravarthy, S. R. (eds) (1998). Advances in Matrix Analytic Methods for Stochastic Models. Notable Publications Inc., Neshanic Station, NJ.Google Scholar
Alfa, A. S., Sengupta, B., and Takine, T. (1998). The use of non-linear programming in matrix analytic methods. Stoch. Models 14, 351–367.Google Scholar
Asmussen, S., and Ramaswami, V. (1990). Probabilistic interpretation of some duality results of the matrix paradigms in queueing theory. Stoch. Models 6, 715–734.Google Scholar
Bini, D. A., and Meini, B. (1996). On the solution of a nonlinear matrix equation arising in queueing problems. SIAM J. Matrix Anal. Appl. 17, 906–926.Google Scholar
Favati, P., and Bini, D. A. (1998). On factional iteration methods for solving M/G/1 type Markov chains. In Advances in Matrix Analytic Methods for Stochastic Models. Notable Publications Inc., Neshanic Station, NJ, pp. 39–54.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. Stoch. Models 10, 1–43.Google Scholar
Gail, H. R., Hantler, S. L., and Taylor, B. A. (1997). Non-skip-free M/G/1 and G/M/1 type Markov chains. Adv. Appl. Prob. 29, 733–758.Google Scholar
Gantmacher, F. R. (1959). The Theory of Matrices. Chelsea, New York.Google Scholar
He, Q.-M. (1995). Differentiability of the matrices R and G in the matrix analytic methods. Stoch. Models 11, 123–132.Google Scholar
Latouche, G. (1987). A note on two matrices occurring in the solution of quasi-birth-and-death processes. Stoch. Models 3, 251–257.Google Scholar
Latouche, G. (1994). Newton's iteration for nonlinear equations in Markov chains. IMA J. Num. 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. SIAM, Philadelphia.Google Scholar
Latouche, G., and Taylor, P. G. (2000). Truncation and augmentation of level-independent QBD processes. Unpublished manuscript.Google Scholar
Meini, B. (1997). New convergence results on functional iteration techniques for the numerical solution of M/G/1 type Markov chains. Num. Math. 78, 39–58.CrossRefGoogle Scholar
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models: an Algorithmic Approach. The Johns Hopkins University Press, Baltimore.Google Scholar
Neuts, M. F. (1986). The caudal characteristic curve of queues. Adv. Appl. Prob. 18, 221–254.Google Scholar
Neuts, M. F. (1989). Structured Stochastic Matrices of M/G/1 Type and their Applications. Marcel Dekker, New York.Google Scholar
Purdue, P. (1973). Non-linear matrix integral equation of Volterra type in queueing theory. J. Appl. Prob. 10, 644–651.CrossRefGoogle Scholar
Ramaswami, V. (1988). Nonlinear matrix equations in applied probability: solution techniques and open problems. SIAM Rev. 30, 256–263.Google Scholar
Ramaswami, V. (1990). A duality theorem for the matrix paradigms in queueing theory. Stoch. Models 6, 151–161.Google Scholar
Ramaswami, V., and Taylor, P. G. (1995). Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases. Stoch. Models 12, 143–164.Google Scholar
Seneta, E. (1973). Non-Negative Matrices: an Introduction to Theory and Applications. John Wiley, New York.Google Scholar
Wallace, V. (1969). The solution of quasi birth and death processes arising from multiple access computer systems. Doctoral Thesis, Systems Engineering Laboratory, University of Michigan.Google Scholar