On Sharp Bounds on the Rate of Convergence for Finite Continuous-Time Markovian Queueing Models

Zeifman, Alexander; Сипин, А. С.; Korolev, V. Yu.; Shilova, Galina; Kiseleva, Ksenia; Korotysheva, Anna; Satin, Yacov

doi:10.1007/978-3-319-74727-9_3

Cited by 12 publications

(6 citation statements)

References 18 publications

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“…Now briefly describe the main classes of the chains under consideration. The details concerning the first four classes can be found in [65,66].…”

Section: Basic Notions and Preliminariesmentioning

confidence: 99%

“…This chain describes, for instance, the number of customers in a queueing system in which the customers arrive in groups, but are served one by one (in this case a k (t) is the arrival intensity of a group of k customers, and µ i (t) is the service intensity of the ith customer). The simplest models of this type were considered in [37], also see [65,66].…”

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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“…Now briefly describe the main classes of the chains under consideration. The details concerning the first four classes can be found in [65,66].…”

Section: Basic Notions and Preliminariesmentioning

confidence: 99%

Section: Basic Notions and Preliminariesmentioning

confidence: 99%

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

2020

Self Cite

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“…This means that the method of the logarithmic norm is inconvenient to apply (it gives poor results). That's why we use the method of 'differential inequalities', which was described in [4]- [6].…”

Section: Stationary Markovian Queueing Modelmentioning

confidence: 99%

On the rate of convergence for a class of Markovian queues with group services

Kryukova

2020

Discrete and Continuous Models

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There are many queuing systems that accept single arrivals, accumulate them and service only as a group. Examples of such systems exist in various areas of human life, from traffic of transport to processing requests on a computer network. Therefore, our study is actual. In this paper some class of finite Markovian queueing models with single arrivals and group services are studied. We considered the forward Kolmogorov system for corresponding class of Markov chains. The method of obtaining bounds of convergence on the rate via the notion of the logarithmic norm of a linear operator function is not applicable here. This approach gives sharp bounds for the situation of essentially non-negative matrix of the corresponding system, but in our case it does not hold. Here we use the method of differential inequalities to obtaining bounds on the rate of convergence to the limiting characteristics for the class of finite Markovian queueing models. We obtain bounds on the rate of convergence and compute the limiting characteristics for a specific non-stationary model too. Note the results can be successfully applied for modeling complex biological systems with possible single births and deaths of a group of particles.

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“…Consider the following four classes of finite inhomogeneous continuoustime Markov chains, see (Zeifman et al 2018a), the detailed consideration of the corresponding countable chains is given in (Zeifman et al 2018c).…”

Section: Introductionmentioning

confidence: 99%

On obtaining sharp bounds of the rate of convergence for a class of continuous-time Markov chains

Zeifman

Satin

Kiseleva

2020

Statistics & Probability Letters

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On Sharp Bounds on the Rate of Convergence for Finite Continuous-Time Markovian Queueing Models

Cited by 12 publications

References 18 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

On the rate of convergence for a class of Markovian queues with group services

On obtaining sharp bounds of the rate of convergence for a class of continuous-time Markov chains

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