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Ergodicity of quasi-birth and death processes. I. (English) Zbl 1113.93103
Summary: Quasi-birth and death processes with block tridiagonal matrices find many applications in various areas. Neuts gave the necessary and sufficient conditions for the ordinary ergodicity and found an expression of the stationary distribution for a class of quasi-birth and death processes. In this paper we obtain the explicit necessary and sufficient conditions for $l$-ergodicity and geometric ergodicity for the class of quasi-birth and death processes, and prove that they are not strongly ergodic.

##### MSC:
 93E15 Stochastic stability 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
##### References:
 [1] Evans, R. V.: Geometric distribution in some two-dimensional queueing systems. Opns. Res., 15, 830–846 (1976) · doi:10.1287/opre.15.5.830 [2] Wallace, V.: The solution of quasi-birth and death processes arising from multiple access computer systems, Ph. D. diss., Systems Engineering Laboratory, University of Michigan, Tech. Rept., no. 07742-6-T, 1969 [3] Neuts, M.: Matrix-Geometric Solutions in StochasticModels: An Algorithmic Approach, The Johns Hopkins University Press, Baltimore, 1981 [4] Hou, Z. T., Liu, Y. Y.: Explicit criteria for several types of ergodicity of the embedded M/G/1 and GI/M/n queues. J. Appl. Probab., 41(3), 778–790 (2004) · Zbl 1065.60134 · doi:10.1239/jap/1091543425 [5] Hou, Z., Li, X.: Explicit criteria for several types of ergodicity of Markov chains in the kind of IPP+M/M/1 queue. submitted for publication [6] Zhang, H. et al.: Strong ergodicity of monotone transition functions. Statistic & Probability Letters, 55, 63–69 (2001) · Zbl 0989.60067 · doi:10.1016/S0167-7152(01)00130-4 [7] Mao, Y. H.: Algebraic convergence for discrete-time ergodic Markov chains. Science in China, 33(2), 152–160 (2003) (Chinese Edition) [8] Hou, Z., Guo, Q.: Homongenerous Denurmerable Markov Processes, Science Press, Beijing, 1978 [9] Tian, N.: Quasi–Birth and Death Processes and Matrix-geometric Solutions, Science Press, Beijing, 2002 (in Chinese) [10] Gantmacher, F. R.: The Theory of Matrices, Chelsea, New York, 1959 [11] Luo, J. H. : Introduction to Matrix Analysis, South China University of Technology Press, Guangzhou, 2002 [12] Lu, W., Fan, X.: Theorem of Implicit Function, Lanzhou University Press, Lanzhou, China, 1986 [13] Chen, M.: From Markov Chains to Non-equilibrium Particle Systems, World Scientific, Singapore, 1992 [14] Anderson, W.: Continuous-time Markov Chains, Springer-Verlag, New York, 1991