Papers by Galina Zverkina
On polynomial convergence rate of the availability factor to its stationary value
arXiv (Cornell University), Nov 17, 2021
arXiv (Cornell University), Dec 30, 2021
The study of the behaviour of stochastic processes in queuing theory and related fields is often ... more The study of the behaviour of stochastic processes in queuing theory and related fields is often based on the study of the behaviour of regenerative Markov processes. Often these regenerative processes are a combination of piecewise linear stochastic processes. We discuss the definition of a piecewise linear process which is comfortable to study the complex stochastic models currently being studied. In many cases, a piecewise linear Markov process has an embedded renewal process, and hence the study of the behaviour of a piecewise linear Markov process can be based on an analysis of the behaviour of this embedded renewal process. But the behaviour of a complex queuing system or a complex reliability model can be described by a non-regenerative stochastic process, but this non-regenerative process can be in some sense "close" to some regenerative process. In this case, embedded processes that would correspond to embedded renewal processes turn out to be in some sense "close" to some "classical" renewal processes. Therefore, in this paper, we introduce the concept of quasi-renewal and quasi-regenerative processes. And we describe an example of such a stochastic model whose behaviour is described by a quasi-regenerative process, and an example of a stochastic model having embedded quasi-renewal processes. We also propose a method for obtaining the upper bounds for the convergence rate of the distribution of a regenerative and quasi-regenerative process to a stationary distribution, if this process is ergodic.
Communications in computer and information science, 2019
The mathematical model of the restorable system with warm reserve considered, in the case when al... more The mathematical model of the restorable system with warm reserve considered, in the case when all working and repair times are bounded by exponential random variable (upper and lower), and working and repair times can be dependent. The exponential upper bounds for the convergence rate of the distribution of this system. The bounds for the convergence rate of the availability factor are estimated.
Communications in computer and information science, 2016
We give strong bounds for the rate of convergence of the regenerative process distribution to the... more We give strong bounds for the rate of convergence of the regenerative process distribution to the stationary distribution in the total variation metric. These bounds are obtained by using coupling method. We propose this method for obtaining such bounds for the queueing regenerative processes.
arXiv (Cornell University), Jun 5, 2013
The well-known Leibniz theorem (Leibniz Criterion or alternating series test) of convergence of a... more The well-known Leibniz theorem (Leibniz Criterion or alternating series test) of convergence of alternating series is generalized for the case when the absolute value of terms of series are "not absolutely monotonous ly" convergent to zero. Questions of accuracy of the estimation for the series remainder are considered. Leibniz theorem (Leibniz Criterion or alternating series test) give the possibility to demonstrate the convergence of an alternating series with decreasing to zero components. However in some cases values of series components decrease to zero fluctuating. In some this case it can use the facts proved below.
arXiv (Cornell University), Jun 3, 2017
We give a scheme of using the coupling method to obtain strong bounds for the convergence rate of... more We give a scheme of using the coupling method to obtain strong bounds for the convergence rate of the distribution of the backward renewal process in the total variation distance. This scheme can be applied to a wide class of regenerative processes in queuing theory.
arXiv (Cornell University), Jan 12, 2023
Communications in computer and information science, 2020
Generalization of the Lorden's inequality is an excellent tool for obtaining strong upper bounds ... more Generalization of the Lorden's inequality is an excellent tool for obtaining strong upper bounds for the convergence rate for various complicated stochastic models. This paper demonstrates a method for obtaining such bounds for some generalization of the Markov modulated Poisson process (MMPP). The proposed method can be applied in the reliability and queuing theory.
arXiv (Cornell University), Oct 16, 2021
arXiv (Cornell University), Oct 12, 2020
It is more important to estimate the rate of convergence to a stationary distribution rather than... more It is more important to estimate the rate of convergence to a stationary distribution rather than only to prove the existence one in many applied problems of reliability and queuing theory. This can be done via standard methods, but only under assumptions about an exponential distribution of service time, independent intervals between recovery times, etc. Results for such simplest cases are well-known. Rejection of these assumptions results to rather complex stochastic processes that cannot be studied using standard algorithms. A more sophisticated approach is needed for such processes. That requires generalizations and proofs of some classical results for a more general case. One of them is the generalized Lorden's inequality proved in this paper. We propose the generalized version of this inequality for the case of dependent and arbitrarily distributed intervals between recovery times. This generalization allows to find upper bounds for the rate of convergence for a wide class of complicated processes arising in the theory of reliability. The rate of convergence for a two-component process has been obtained via the generalized Lorden's inequality in this paper.
arXiv (Cornell University), Oct 8, 2019
arXiv (Cornell University), Dec 3, 2015
Аннотация A computable estimate of the readiness coefficient for a standard binarystate system is... more Аннотация A computable estimate of the readiness coefficient for a standard binarystate system is established in the case where both working and repair time distributions possess heavy tails.
arXiv (Cornell University), May 13, 2018
The upper bound for the convergence rate of the distribution of a queuing system state with infin... more The upper bound for the convergence rate of the distribution of a queuing system state with infinitely many servers is obtained for the case when the intensities of the incoming and service flows depend on the state of the system.
arXiv (Cornell University), Sep 21, 2015
We prove an ergodic theorem for a linearwise Markov process, and we give estimation for the avera... more We prove an ergodic theorem for a linearwise Markov process, and we give estimation for the average of renewal process amount of overjump; these results are based on Smith's Key Renewal Theorem. Also we discuss the possibility of using these results for studying some type of queueing systems.
Journal of Mathematical Sciences, Mar 22, 2021
An upper bound for the convergence rate of the distribution of a queuing system state with infini... more An upper bound for the convergence rate of the distribution of a queuing system state with infinitely many servers is obtained for the case where the intensities of the incoming and service flows depend on the state of the system.
Journal of Mathematical Sciences, May 16, 2022
Exponential upper bounds for the convergence rate of the distribution of restorable element with ... more Exponential upper bounds for the convergence rate of the distribution of restorable element with partially energized standby redundancy are found, in the case where all working and repair times are bounded by an exponential random variable (upper and lower), and working and repair times can be dependent. The convergence rate of the availability factor is estimated.
arXiv (Cornell University), Jun 10, 2013
An elementary rigorous justification of Dynkin's identity with an extended generator based on the... more An elementary rigorous justification of Dynkin's identity with an extended generator based on the idea of a complete probability formula is given for queueing systems with a single server and discontinuous intensities of arrivals and serving. This formula is applied to the analysis of ergodicity and, in particular, to polynomial bounds of convergence rate to stationary distribution.
arXiv (Cornell University), Apr 16, 2017
We propose a new modification of the coupling method for renewal process in continuous time. We c... more We propose a new modification of the coupling method for renewal process in continuous time. We call this modification "the stationary coupling method", and construct it primarily to obtain the bounds for convergence rate of the distribution of the regenerative processes in the total variation metrics. At the same time this modification of the coupling method demonstrates an improvement of the classical result of polynomial convergence rate of the distribution of the regenerative process-in the case of a heavy tail. keywords Renewal process, Regenerative process, Rate of convergence, Coupling method subclass MSC 60B10 MSC 60J25 MSC 60K15 * The author is supported by the RFBR, project No 17-01-00633 A.
Some Extensions of Alternating Series Test and Its Applications
Lecture Notes in Computer Science, 2017
The well-known Leibniz Criterion or alternating series test of convergence of alternating series ... more The well-known Leibniz Criterion or alternating series test of convergence of alternating series is generalized for the case when the absolute value of terms of series are “not absolutely monotonously” convergent to zero. Questions of accuracy of the estimation for the series remainder are considered.
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Papers by Galina Zverkina