Sufficient conditions for regularity, positive recurrence, and absorption in level‐dependent QBD processes and related block‐structured Markov chains

Abstract

This paper is concerned with level‐dependent quasi‐birth‐death (LD‐QBD) processes, i.e., multi‐variate Markov chains with a block‐tridiagonal ‐matrix, and a more general class of block‐structured Markov chains, which can be seen as LD‐QBD processes with total catastrophes. Arguments from univariate birth‐death processes are combined with existing matrix‐analytic formulations to obtain sufficient conditions for these block‐structured processes to be regular, positive recurrent, and absorbed with certainty in a finite mean time. Specifically, it is our purpose to show that, as is the case for competition processes, these sufficient conditions are inherently linked to a suitably defined birth‐death process. Our results are exemplified with two Markov chain models: a study of target cells and viral dynamics and one of kinetic proof‐reading in T cell receptor signal transduction.