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/ $$\infty $$ ∞ and BMAP/M/c+M queues

Tetsuya Takine ()
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Tetsuya Takine: Osaka University

Queueing Systems: Theory and Applications, 2016, vol. 84, issue 1, No 3, 49-77

Abstract: 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/ $$\infty $$ ∞ 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/ $$\infty $$ ∞ and BMAP/M/c+M queues are shown.

Keywords: Markov chain of M/G/1-type; Level dependence; Stationary distribution; Computational algorithm; BMAP/M/ $$\infty $$ ∞; BMAP/M/c+M; 60K25; 60J22 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (5)

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DOI: 10.1007/s11134-016-9482-1

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