### 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 |
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Pages (from-to) | 1010-1019 |

Number of pages | 10 |

Journal | Journal of Applied Probability |

Volume | 37 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2000 Jan 1 |

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### 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

### Cite this

**Sharp results on convergence rates for the distribution of GI/M/1/K queues as K tends to infinity.** / Choi, Bong Dae; Kim, Bara.

Research output: Contribution to journal › Article

*Journal of Applied Probability*, vol. 37, no. 4, pp. 1010-1019. https://doi.org/10.1239/jap/1014843080

}

TY - JOUR

T1 - Sharp results on convergence rates for the distribution of GI/M/1/K queues as K tends to infinity

AU - Choi, Bong Dae

AU - Kim, Bara

PY - 2000/1/1

Y1 - 2000/1/1

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.

KW - Convergence rate

KW - Dual sequence

KW - GI/M/1/K queue

KW - Stationary distribution

KW - Stationary measure

UR - http://www.scopus.com/inward/record.url?scp=0034354521&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034354521&partnerID=8YFLogxK

U2 - 10.1239/jap/1014843080

DO - 10.1239/jap/1014843080

M3 - Article

VL - 37

SP - 1010

EP - 1019

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 4

ER -