Analysis and computation of the stationary distribution in a special class of Markov chains of level-dependent M/G/1-type and its application to BMAP/M/∞ and BMAP/M/c+M queues

Analysis and computation of the stationary distribution in a special class of Markov chains of level-dependent M/G/1-type and its application to BMAP/M/∞ and BMAP/M/c+M queues

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Article ID: iaor20163713
Volume: 84
Issue: 1
Start Page Number: 49
End Page Number: 77
Publication Date: Oct 2016
Journal: Queueing Systems
Authors:
Keywords: queues: applications, markov processes, simulation
Abstract:

This paper considers a special class of continuous‐time Markov chains of level‐dependent M/G/1‐type, where block matrices representing downward jumps in the infinitesimal generator are nonsingular. This special class naturally arises in the analysis of BMAP/M/ equ1 queues and BMAP/M/c queues with exponential impatience times (BMAP/M/c+M). We first formulate the boundary probability vector in terms of a solution of a system of infinitely many linear inequalities. We then reveal that in the above special class, this infinite system is regarded as a nested sequence of simplices, and we identify their vertices. Based on these results, we develop a simple yet efficient computational algorithm for the stationary distribution conditioned that the level is not greater than a predefined N. Note that for a large N, the conditional distribution will provide a good approximation to the stationary distribution. Some numerical examples for BMAP/M/ equ2 and BMAP/M/c+M queues are shown.

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