Perturbation Bounds forMt/Mt/NQueue with Catastrophes

Zeifman, Alexander; Korotysheva, Anna

doi:10.1080/15326349.2011.614900

Cited by 41 publications

(31 citation statements)

References 22 publications

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“…for any s and t such that 0 ≤ s ≤ t. The next statement follows immediately from Theorem 1 of Zeifman and Korotysheva (2012), see also the general approach of .…”

Section: A(t) Where the Perturbation Matrixâ(t) = A(t) −ā(T)mentioning

confidence: 74%

See 1 more Smart Citation

Ergodicity and Perturbation Bounds for Inhomogeneous Birth and Death Processes with Additional Transitions from and to the Origin

Zeifman

Korotysheva

Satin

et al. 2015

International Journal of Applied Mathematics and Computer Science

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Service life of many real-life systems cannot be considered infinite, and thus the systems will be eventually stopped or will break down. Some of them may be re-launched after possible maintenance under likely new initial conditions. In such systems, which are often modelled by birth and death processes, the assumption of stationarity may be too strong and performance characteristics obtained under this assumption may not make much sense. In such circumstances, timedependent analysis is more meaningful. In this paper, transient analysis of one class of Markov processes defined on non-negative integers, specifically, inhomogeneous birth and death processes allowing special transitions from and to the origin, is carried out. Whenever the process is at the origin, transition can occur to any state, not necessarily a neighbouring one. Being in any other state, besides ordinary transitions to neighbouring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend (except for transition to the origin) on the process state. To the best of our knowledge, first ergodicity and perturbation bounds for this class of processes are obtained. Extensive numerical results are also provided.

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“…for any s and t such that 0 ≤ s ≤ t. The next statement follows immediately from Theorem 1 of Zeifman and Korotysheva (2012), see also the general approach of .…”

Section: A(t) Where the Perturbation Matrixâ(t) = A(t) −ā(T)mentioning

confidence: 74%

“…We can apply the approach of Zeifman and Korotysheva (2012). Namely, rewrite the forward Kolmogorov equation (3) as…”

Section: Proofmentioning

confidence: 99%

Ergodicity and Perturbation Bounds for Inhomogeneous Birth and Death Processes with Additional Transitions from and to the Origin

Zeifman

Korotysheva

Satin

et al. 2015

International Journal of Applied Mathematics and Computer Science

Self Cite

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“…Arbitrary intensity functions instead of periodic ones can be considered in the same manner. Finally, we would like to remark that all of our results can be applied to birth and death process with catastrophes; see perturbation bounds in the work of Zeifman and Korotysheva (2012) and the bounds on the rate of convergence given by Zeifman et al (2013a).…”

Section: Resultsmentioning

confidence: 99%

“…Bounds on the rate of convergence, truncations and stability for this process were obtained by Granovsky and Zeifman (2004), Zeifman (1995b;1995a), Zeifman et al (2006) as well as Zeifman and Korotysheva (2012). Here we improve estimates of the truncation error.…”

Section: T /M T /S Queuementioning

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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“…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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Perturbation Bounds forM_t/M_t/NQueue with Catastrophes

Abstract: This articles focuses on M t /M t /N queue with catastrophes and obtains stability bounds for the main characteristics of the respective queue-length process.

Cited by 41 publications

References 22 publications

Ergodicity and Perturbation Bounds for Inhomogeneous Birth and Death Processes with Additional Transitions from and to the Origin

Ergodicity and Perturbation Bounds for Inhomogeneous Birth and Death Processes with Additional Transitions from and to the Origin

On truncations for weakly ergodic inhomogeneous birth and death processes

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

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