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

Choi, Bong Dae; Kim, Bara

doi:10.1017/s0021900200018192

Cited by 9 publications

(22 citation statements)

References 4 publications

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“…The case γ 1 = 1 and γ 2 < ∞ has been developed by Postnikov [65], Sect. 25. He established the following two Tauberian theorems.…”

Section: 4)mentioning

confidence: 99%

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Takács’ Asymptotic Theorem and Its Applications: A Survey

Abramov¹

2008

Acta Appl Math

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The book of Lajos Takács Combinatorial Methods in the Theory of StochasticProcesses has been published in 1967. It discusses various problems associated withwhere N n = ν 1 + ν 2 + · · · + ν n is a sum of mutually independent, nonnegative integer and identically distributed random variables, π j = P{ν k = j }, j ≥ 0, π 0 > 0, and ρ(i) is the smallest n such that N n = n − i, i ≥ 1. (If there is no such n, then ρ(i) = ∞.) Equation ( * ) is a discrete generalization of the classic ruin probability, and its value is represented as P k,i = Q k−i /Q k , where the sequence {Q k } k≥0 satisfies the recurrence relation of convolution type: Q 0 = 0 and Q k = k j =0 π j Q k−j +1 . Since 1967 there have been many papers related to applications of the generalized classic ruin probability. The present survey is concerned only with one of the areas of application associated with asymptotic behavior of Q k as k → ∞. The theorem on asymptotic behavior of Q k as k → ∞ and further properties of that limiting sequence are given on pp. 22-23 of the aforementioned book by Takács. In the present survey we discuss applications of Takács' asymptotic theorem and other related results in queueing theory, telecommunication systems and dams. Many of the results presented in this survey have appeared recently, and some of them are new. In addition, further applications of Takács' theorem are discussed.

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“…The case γ 1 = 1 and γ 2 < ∞ has been developed by Postnikov [65], Sect. 25. He established the following two Tauberian theorems.…”

Section: 4)mentioning

confidence: 99%

“…The analytic proofs given in these papers are much more difficult than those by application of Takács' theorem 1.1. We refer to Kim and Choi [48], Choi et al [27] and Simonot [70], where the readers can find the proofs by using the standard analytic techniques.…”

Section: Losses In the Gi/m/m/n Queuementioning

confidence: 99%

Takács’ Asymptotic Theorem and Its Applications: A Survey

Abramov¹

2008

Acta Appl Math

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“…Tijms illustrated the method in many queueing models. Further illustrations are found in [5] and [6], for example. The remainder of this paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Tail Asymptotics for the Queue Size Distribution in an M/G/1 Retrial Queue

Kim¹,

Kim²,

2007

Journal of Applied Probability

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“…It belongs to the open interval (0,1) if λ < µ, and it is equal to 1 otherwise. In the recent papers Choi and Kim [8] and Choi et al [9] study the questions related to the asymptotic behavior of the sequence {π j } as j → ∞. Namely, they study asymptotic behavior of the loss probability p n , n → ∞, as well as obtain the convergence rate of the stationary distributions of the GI/M/1/n queueing system to those of the GI/M/1 queueing system as n → ∞.…”

Section: Introductionmentioning

confidence: 99%

“…Namely, they study asymptotic behavior of the loss probability p n , n → ∞, as well as obtain the convergence rate of the stationary distributions of the GI/M/1/n queueing system to those of the GI/M/1 queueing system as n → ∞. The analysis of [8] and [9] is based on the theory of analytic functions.…”

Section: Introductionmentioning

confidence: 99%

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Abramov

2002

Annals of Operations Research

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This paper provides the asymptotic analysis of the loss probability in the GI/M/1/n queueing system as n increases to infinity. The approach of this paper is alternative to that of the recent papers of Choi and Kim [8] and Choi et al [9] and based on application of modern Tauberian theorems with remainder. This enables us to simplify the proofs of the results on asymptotic behavior of the loss probability of the abovementioned paper of Choi and Kim [9] as well as to obtain some new results.

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

Cited by 9 publications

References 4 publications

Takács’ Asymptotic Theorem and Its Applications: A Survey

Takács’ Asymptotic Theorem and Its Applications: A Survey

Tail Asymptotics for the Queue Size Distribution in an M/G/1 Retrial Queue

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