Sharp results on convergence rates for the distribution of GI/M/1/K queues as K tends to infinity

Bong Dae Choi; Bara Kim

doi:10.1017/S0021900200018192

Sharp results on convergence rates for the distribution of GI/M/1/K queues as K tends to infinity

Bong Dae Choi, Bara Kim

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, we investigate how fast the stationary distribution π^(K) of an embedded Markov chain (time-stationary distribution q^(K)) of the GI/M/1/K queue converges to the stationary distribution π of the embedded Markov chain (time-stationary distribution q) of the GI/M/1 queue as K tends to infinity. Simonot (1997) proved certain equalities. We obtain sharper results than these by finding limit values lim _{K → ∞} σ^-K ∥π^(K) -π∥ and lim_{K → ∞} σ ^-K ∥q^(K) -q∥ explicitly.

Original language	English
Pages (from-to)	1010-1019
Number of pages	10
Journal	Journal of Applied Probability
Volume	37
Issue number	4
DOIs	https://doi.org/10.1017/S0021900200018192
Publication status	Published - 2000 Jan 1

Keywords

GI/M/1/K queue; dual sequence; stationary measure; stationary distribution; convergence rate

ASJC Scopus subject areas

Statistics and Probability
Mathematics(all)
Statistics, Probability and Uncertainty

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abstract = "In this paper, we investigate how fast the stationary distribution π(K) of an embedded Markov chain (time-stationary distribution q(K)) of the GI/M/1/K queue converges to the stationary distribution π of the embedded Markov chain (time-stationary distribution q) of the GI/M/1 queue as K tends to infinity. Simonot (1997) proved certain equalities. We obtain sharper results than these by finding limit values lim K → ∞ σ-K ∥π(K) -π∥ and limK → ∞ σ -K ∥q(K) -q∥ explicitly.",

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N2 - In this paper, we investigate how fast the stationary distribution π(K) of an embedded Markov chain (time-stationary distribution q(K)) of the GI/M/1/K queue converges to the stationary distribution π of the embedded Markov chain (time-stationary distribution q) of the GI/M/1 queue as K tends to infinity. Simonot (1997) proved certain equalities. We obtain sharper results than these by finding limit values lim K → ∞ σ-K ∥π(K) -π∥ and limK → ∞ σ -K ∥q(K) -q∥ explicitly.

AB - In this paper, we investigate how fast the stationary distribution π(K) of an embedded Markov chain (time-stationary distribution q(K)) of the GI/M/1/K queue converges to the stationary distribution π of the embedded Markov chain (time-stationary distribution q) of the GI/M/1 queue as K tends to infinity. Simonot (1997) proved certain equalities. We obtain sharper results than these by finding limit values lim K → ∞ σ-K ∥π(K) -π∥ and limK → ∞ σ -K ∥q(K) -q∥ explicitly.

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Sharp results on convergence rates for the distribution of GI/M/1/K queues as K tends to infinity

Abstract

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