### 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

- 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