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

  • Qi-Ming He (a1) and Marcel F. Neuts (a2)


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.


Corresponding author

Postal address: Department of Industrial Engineering, DalTech, Dalhousie University, Halifax, Nova Scotia, Canada B3J 2X4. Email address:
∗∗ Postal address: Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA.


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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
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