Error bounds for augmented truncation approximations of Markov chains via the perturbation method

Liu, Yuanyuan; Li, Wendi

doi:10.1017/apr.2018.28

Cited by 30 publications

(23 citation statements)

References 39 publications

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“…see e.g. [12,25,34,35]. The following result gives perturbation bounds via the Dobrushin's ergodic coefficient, which extends the previous results for finite and countable Markov chains in [25,35].…”

Section: General Results For Perturbation Boundssupporting

confidence: 80%

Perturbation theory and uniform ergodicity for discrete-time Markov chains

Mao,

Song

2020

Preprint

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We study perturbation theory and uniform ergodicity for discrete-time Markov chains on general state spaces in terms of the uniform moments of the first hitting times on some set. The methods we adopt are different from previous ones. For reversible and non-negative definite Markov chains, we first investigate the geometrically ergodic convergence rates. Based on the estimates, together with a first passage formula, we then get the convergence rates in uniform ergodicity. If the transition kernel P is only reversible, we transfer to study the two-skeleton chain with the transition kernel P 2 . At a technical level, the crucial point is to connect the geometric moments of the first return times between P and P 2 .

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Section: General Results For Perturbation Boundssupporting

confidence: 80%

Perturbation theory and uniform ergodicity for discrete-time Markov chains

Mao,

Song

2020

Preprint

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“…Proof. It is implied in [18] and [19] that this proposition holds. However, for completeness, we provide the proof: Using (4.1) and π (N ) = P (N ) π (N ) , we have…”

Section: A Difference Formula Via the Fundamental Deviation Matrixmentioning

confidence: 87%

A subgeometric convergence formula for finite-level M/G/1-type Markov chains: via a block-decomposition-friendly solution for the Poisson equation of deviation matrix

Masuyama¹,

Katsumata²,

Kimura³

2018

Preprint

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This paper considers finite-level M/G/1-type Markov chains. We introduce the fundamental deviation matrix of the infinite-level M/G/1-type Markov chain, which is a solution of the Poisson equation that the deviation matrix satisfies. With the fundamental deviation matrix, we describe a difference formula for the respective stationary distributions of the finite-level chain and its infinite-level limit. From the difference formula, we derive a subgeometric convergence formula for the stationary distribution of the finite-level chain as its maximum level goes to infinity. Using the obtained formula, we show an asymptotic formula for the loss probability in the MAP/G/1/N + 1 queue.

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“…, which implies that F 00 (s)1 < ∞ from (19). Since the chain is transient, it follows that (F 00 (1)1) i < 1 for some i ∈ N m 0 .…”

Section: Theorem 1 Suppose That Bothmentioning

confidence: 93%

“…Mao, Tai, Zhao, and Zou [24] characterized ergodic properties of Markov chains of GI/G/1 type in terms of system parameters. Truncation approximations and computing algorithm of the stationary distributions of Markov chains of M/G/1 type have recently been investigated by Liu et al [19,21] and Masuyama [25] respectively. In addition to the study of ergodicity, some researchers have investigated transient properties for block-structured Markov chains.…”

Section: Introductionmentioning

confidence: 99%

On geometric and algebraic transience for block-structured Markov chains

Liu¹,

2020

J. Appl. Probab.

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Block-structured Markov chains model a large variety of queueing problems and have many important applications in various areas. Stability properties have been well investigated for these Markov chains. In this paper we will present transient properties for two specific types of block-structured Markov chains, including M/G/1 type and GI/M/1 type. Necessary and sufficient conditions in terms of system parameters are obtained for geometric transience and algebraic transience. Possible extensions of the results to continuous-time Markov chains are also included.

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Error bounds for augmented truncation approximations of Markov chains via the perturbation method

Cited by 30 publications

References 39 publications

Perturbation theory and uniform ergodicity for discrete-time Markov chains

Perturbation theory and uniform ergodicity for discrete-time Markov chains

A subgeometric convergence formula for finite-level M/G/1-type Markov chains: via a block-decomposition-friendly solution for the Poisson equation of deviation matrix

On geometric and algebraic transience for block-structured Markov chains

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