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Yacov Satin

Publications and source records attributed to Yacov Satin.

8 recordsLinked to original sources
arXiv2021

Upper bound on the rate of convergence and truncation bound for nonhomogeneous birth and death processes on $\mathbb{Z}$

We consider the well-known problem of the computation of the (limiting) time-dependent performance characteristics of one-dimensional continuous-time birth and death processes on $\mathbb{Z}$ with time varying and possible state-dependent intensities. First in the literature upper bounds on the rate of convergence along with one new concentration inequality are provided. Upper bounds for the error of truncation are also given. Condition under which a limiting (time-dependent) distribution exists is formulated but relies on the quantities that need to be guessed in each use-case. The developed theory is illustrated by two numerical examples within the queueing theory context.

math.PR↗
arXiv2021

Bounds on the Rate of Convergence for $M^X_t/ M^X_t/1$ Queueing Models

We apply the method of differential inequalities for the computation of upper bounds for the rate of convergence to the limiting regime for one specific class of (in)homogeneous continuous-time Markov chains. To obtain these estimates, we investigate the corresponding forward system of Kolmogorov differential equations.

math.PR↗
arXiv2020

Ergodicity and Perturbation Bounds for $M_t/M_t/1$ Queue with Balking, Catastrophes, Server Failures and Repairs

In this paper, we display methods for the computation of convergence and perturbation bounds for $M_t/M_t/1$ system with balking, catastrophes, server failures and repairs. Based on the logarithmic norm of linear operators, the bounds on the rate of convergence, perturbation bounds, and the main limiting characteristics of the queue-length process are obtained. Finally, we consider the application of all obtained estimates to a specific model.

math.PR↗
arXiv2020

On the Three Methods for Bounding the Rate of Convergence for some Continuous-time Markov Chains

Consideration is given to the three different analytical methods for the computation of upper bounds for the rate of convergence to the limiting regime of one specific class of (in)homogeneous continuous-time Markov chains. This class is particularly suited to describe evolutions of the total number of customers in (in)homogeneous $M/M/S$ queueing systems with possibly state-dependent arrival and service intensities, batch arrivals and services. One of the methods is based on the logarithmic norm of a linear operator function; the other two rely on Lyapunov functions and differential inequalities, respectively. Less restrictive conditions (compared to those known from the literature) under which the methods are applicable, are being formulated. Two numerical examples are given. It is also shown that for homogeneous birth-death Markov processes defined on a finite state space with all transition rates being positive, all methods yield the same sharp upper bound.

math.PR↗
arXiv2019

Two approaches to the construction of perturbation bounds for continuous-time Markov chains

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.

math.PR↗
arXiv2016

Bounds on the rate of convergence for inhomogeneous $M/M/S$ systems with either state-dependent transitions, or batch arrivals and service, or both

In this paper one presents method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically one considers inhomogeneous $M/M/S$ queueing system with possibly state-dependent arrival and service intensities and additionally possible batch arrivals and batch service. The unified approach based on logarithmic norm of linear operators for obtaining sharp upper and lower bounds on the rate of convergence and corresponding sharp perturbation bounds is described. As a side result, one shows by virtue of numerical examples that the approach based on logarithmic norm can also be used for approximation of limiting characteristics (idle probability and mean number of customers in the system) of the considered systems with given approximation error. Extensive numerical examples are provided.

math.PR↗
arXiv2016

Ergodicity and uniform in time truncation bounds for inhomogeneous birth and death processes with additional transitions from and to origin

In this paper one presents the extension of the transient analysis of the class of continuous-time birth and death processes defined on non-negative integers with special transitions from and to the origin. From the origin transitions can occur to any state. But 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 on the state of the process. We improve previously known ergodicity and truncation bounds for this class of processes which were known only for the case when transitions from the origin decay exponentially (other intensities must have unique uniform upper bound). We show how the bounds can be obtained the decay rate is slower than exponential. Numerical results are also provided.

math.PR↗