Perturbation bounds and truncations for a class of Markovian queues

Zeifman, Alexander; Korolev, V. Yu.; Satin, Yacov; Korotysheva, Anna; Bening, V. E.

doi:10.1007/s11134-013-9388-0

Cited by 35 publications

(44 citation statements)

References 16 publications

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“…Let a i+k,i (t) = a k (t), a i,i+k (t) = b k (t) for k ≥ 1. This process appears in the description of a system with group arrival and group service, for earlier studies see [39,40,61].…”

Section: Basic Notions and Preliminariesmentioning

confidence: 99%

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Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

2020

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The paper is largely of a review nature. It considers two main methods used to study stability and obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Several specific models are considered for which the limit characteristics and perturbation bounds for admissible "perturbed"processes are calculated.

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“…Let a i+k,i (t) = a k (t), a i,i+k (t) = b k (t) for k ≥ 1. This process appears in the description of a system with group arrival and group service, for earlier studies see [39,40,61].…”

Section: Basic Notions and Preliminariesmentioning

confidence: 99%

“…It is worth noting that for estimates (61) and (62) to hold, only the condition of the smallness of perturbations is required and no additional information concerning the structure of the intensity matrix is required.…”

Section: Examplesmentioning

confidence: 99%

Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

2020

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“…Zeifman et al [67,69] studied the truncation of a weakly ergodic non-time-homogeneous birthand-death process with bounded transition rates (see also Zeifman and Korolev [66], Zeifman et al [68]). Hart and Tweedie [24] discussed the convergence of the stationary distribution vectors of the augmented northwest-corner truncations of continuous-time Markov chains with monotonicity or exponential ergodicity.…”

Section: Q(n −mentioning

confidence: 99%

Error Bounds for Last-Column-Block-Augmented Truncations of Block-Structured Markov Chains

Masuyama

2017

JORSJ

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This paper discusses the error estimation of the last-column-block-augmented northwest-corner truncation (LC-block-augmented truncation, for short) of block-structured Markov chains (BSMCs) in continuous time. We first derive upper bounds for the absolute difference between the time-averaged functionals of a BSMC and its LC-block-augmented truncation, under the assumption that the BSMC satisfies the general f -modulated drift condition. We then establish computable bounds for a special case where the BSMC is exponentially ergodic. To derive such computable bounds for the general case, we propose a method that reduces BSMCs to be exponentially ergodic. We also apply the obtained bounds to level-dependent quasi-birth-and-death processes (LD-QBDs), and discuss the properties of the bounds through the numerical results on an M/M/s retrial queue, which is a representative example of LD-QBDs. Finally, we present computable perturbation bounds for the stationary distribution vectors of BSMCs.

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“…A general approach and related bounds for the study on the convergence rate was considered by Zeifman (1995a), who also first mentioned computation of the limiting characteristics for the process via truncations (Zeifman, 1988), and later considered it in detail (Zeifman et al, 2006). The first results for more general Markovian queueing models have been obtained recently by Zeifman et al (2014).…”

Section: Introductionmentioning

confidence: 99%

On truncations for weakly ergodic inhomogeneous birth and death processes

Zeifman

Satin

Korolev

et al. 2014

International Journal of Applied Mathematics and Computer Science

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We investigate a class of exponentially weakly ergodic inhomogeneous birth and death processes. We consider special transformations of the reduced intensity matrix of the process and obtain uniform (in time) error bounds of truncations. Our approach also guarantees that we can find limiting characteristics approximately with an arbitrarily fixed error. As an example, we obtain the respective bounds of the truncation error for an Mt/Mt/S queue for any number of servers S. Arbitrary intensity functions instead of periodic ones can be considered in the same manner.

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Perturbation bounds and truncations for a class of Markovian queues

Cited by 35 publications

References 16 publications

Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

Error Bounds for Last-Column-Block-Augmented Truncations of Block-Structured Markov Chains

On truncations for weakly ergodic inhomogeneous birth and death processes

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