Drift Conditions for the Ergodicity of a Class of Level-Dependent M/G/1-type Processes

Abstract:

Structured Markov chains, such as quasi-birth-and-death (QBD) processes, M/G/1-type processes form the basis of matrix-analytic methods that are often used for the performance evaluation and stability of queueing systems.  While closed-form drift conditions for the ergodicity of level-independent versions of these processes are known, the level-dependent case is not as clear. Recently, Cordeiro, et al. (2019) examined level-dependent QBDs (LDQBDs) by examining the Markov chain embedded at jump epochs in which block terms of the transition probability matrix converge over levels. In this talk, we will describe how a similar Foster-Lyapunov drift approach can be applied to establish analytic ergodicity conditions for the level-dependent version of the $M/G/1$-type process. Lastly, several approaches to weakening the regularity condition for block convergence over levels of the process will be discussed.