ON ERGODICITY AND STABILITY ESTIMATES FOR SOME NONHOMOGENEOUS MARKOV CHAINS

Abstract

Journal of Mathematical Sciences, Vol. 112, No. 2, 2002ON ERGODICITY AND STABILITY ESTIMATES FOR SOME NONHOMOGENEOUSMARKOV CHAINSD. B. Andreev, M. A. Elesin, E. A. Krylov, A. V. Kuznetsov,andA.I.Zeifman(Vologda, Russia) UDC 519.21. IntroductionEvaluation of the rate of convergence for the ergodic continuous-time Markov chains has been the subject ofinvestigation of the number of authors(see, for example, [1{7]). There isalso a growing interest in time-nonhomogeneousMarkov chains. Such chains model a variety of queuing systems and they are also used in algorithms of simulatedannealing and some other areas. The tool of our study is the method developed in a series of papers by one of theauthors. The method is based on two basic ingredients: the logarithmic norm of a semigroup generated by a linearoperator and a similarity transformation of the reduced matrix of intensities of the considered Markov chain. In thepresent paper, we deal with continuous-time countable Markov chains. More accurate estimates give more preciseconditions for weak ergodicity and stability for the general-type intensity matrix, stability of birth and death processes,and truncations for Markov chains with bounded jumps. We improve the respective bounds of [9, Theorems 1 and 2;10, Theorem 4; 11, Theorem 3].2. General CaseT