Application of Method of Differential Inequalities to Bounding the Rate of Convergence for a Class of Markov Chains

Kryukova, Anastasia; Oshushkova, Victoria; Zeifman, Alexander; Satin, Yacov

doi:10.1007/978-3-030-56323-3_8

Cited by 2 publications

(5 citation statements)

References 7 publications

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“…Under essential non-negativity of B * (t) all bounds on the rate of convergence can be obtained via logarithmic norm, see [3]. However, in the general case, this approach may not work, and the method of differential inequalities described in our previous papers, see [3,14], would be more effective.…”

Section: Explicit Forms Of the Reduced Intensity Matricesmentioning

confidence: 99%

“…, x S (t)) T , and let D now be the corresponding finite diagonal matrix. The simplest situation with analytical (in t) coefficients b * ij (t) has been studied in [3,14,15]. The method of estimating under such assumption is based on the fact that, in this case, on any finite interval, each coordinate has a finite number of sign changes, which means that the semiaxis can be divided into intervals, on each of which the signs of the coordinates are constant.…”

Section: Upper Bounds On the Rate Of Convergencementioning

confidence: 99%

“…Note that according to (14) the matrix B * * (t) has nonzero entries only on the main diagonal and at most R diagonals above and below it. Then…”

Section: Upper Bounds On the Rate Of Convergencementioning

confidence: 99%

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Bounds on the Rate of Convergence for MtX/MtX/1 Queueing Models

2021

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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. Such an approach seems very general; the corresponding description and bounds were considered earlier for finite Markov chains with analytical in time intensity functions. Now we generalize this method to locally integrable intensity functions. Special attention is paid to the situation of a countable Markov chain. To obtain these estimates, we investigate the corresponding forward system of Kolmogorov differential equations as a differential equation in the space of sequences l1.

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Section: Explicit Forms Of the Reduced Intensity Matricesmentioning

confidence: 99%

Section: Upper Bounds On the Rate Of Convergencementioning

confidence: 99%

See 1 more Smart Citation

Bounds on the Rate of Convergence for MtX/MtX/1 Queueing Models

2021

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show abstract

“…Under essential non-negativity of B * (t) all bounds on the rate of convergence can be obtained via logarithmic norm, see [16]. However, in the general case, this approach may not work, and the method of differential inequalities described in our previous papers, see [5,16] would be more effective.…”

Section: Explicit Forms Of the Reduced Intensity Matricesmentioning

confidence: 99%

“…, x S (t)) T , and let D now be the corresponding finite diagonal matrix. Firstly we consider a situation where all coefficients b * ij (t) are analytical functions of t, it was considered in [5,14,16]. The proof is based on the fact that in this case, on any finite interval, each coordinate has a finite number of sign changes, which means that the semiaxis can be divided into intervals, on each of which the signs of the coordinates are constant.…”

Section: Upper Bounds On the Rate Of Convergencementioning

confidence: 99%