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:10.1239/jap/1014843080

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

8 Citations (Scopus)

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/10.1239/jap/1014843080
Publication status	Published - 2000 Jan 1

Keywords

Convergence rate
Dual sequence
GI/M/1/K queue
Stationary distribution
Stationary measure

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 limK → ∞ σ−K∼∼π(K) − π∼∼ and limK→∞σ−K∼∼q(K) − q∼∼ explicitly.",

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AU - Choi, Bong Dae

AU - Kim, Bara

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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 limK → ∞ 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 limK → ∞ σ−K∼∼π(K) − π∼∼ and limK→∞σ−K∼∼q(K) − q∼∼ explicitly.

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KW - Dual sequence

KW - GI/M/1/K queue

KW - Stationary distribution

KW - Stationary measure

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

Abstract

Keywords

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