Stability of Queueing Systems with Impatience, Balking and Non-Persistence of Customers

Abstract

The operation of many queueing systems is adequately described by the structured multidimensional continuous-time Markov chains. The most well-studied classes of such chains are level-independent Quasi-Birth-and-Death processes, GI/M/1 type and M/G/1 type Markov chains, generators of which have the block tri-diagonal, lower- and upper-Hessenberg structure, respectively. All these classes assume that the matrices of transition rates are quasi-Toeplitz. This property greatly simplifies their analysis but makes them inappropriate for the study of many important systems, e.g., retrial queues with a retrial rate depending on the number of customers in orbit, queues with impatient customers, etc. The importance of such systems attracts significant interest to their analysis. However, in the literature, there is a methodological gap relating to the ergodicity condition of the corresponding Markov chains. To fulfill this gap and facilitate the analysis of a wide range of such systems, we show that under non-restrictive assumptions, the following hold true: (i) if the customers can balk or are impatient or non-persistent, then the Markov chain describing the behavior of the system belongs to the class of asymptotically quasi-Toeplitz Markov chains; (ii) this chain is ergodic; (iii) known algorithms can be applied for the calculation of the stationary distribution of the corresponding queueing system.

Keywords

• ergodicity
• multidimensional continuous-time asymptotically quasi-Toeplitz Markov chain
• impatience
• retrials