Quasi-Birth-and-Death Processes with an Infinite Phase Space
by Allan Motyer
Abstract: A quasi-birth-and-death process (QBD) is a two-dimensional Markov chain for which the transition matrix has a block tridiagonal structure, and is a widely studied stochastic model. The first variable of the QBD process is called the level, the second variable the phase. The properties of QBDs with finitely many possible values of the phase variable have been studied extensively. In particular the level process of a positive-recurrent QBD process with finitely many phases has a stationary distribution which decays geometrically with the decay parameter given by the spectral radius of Neuts' R-matrix. The situation is more complicated for a QBD process with countably many possible values of the phase. In this talk we present results for the stationary distribution of QBDs with infinitely many phases and the convergence of phase-truncation schemes.
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