# Alexander ZeifmanVologda State University, Vologda, Russia

Alexander Zeifman

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195

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## Publications

Publications (195)

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 Z with the time–varying and possibly state-dependent intensities. First in the literature upper bounds on the rate of convergence are provided. Upper bounds for the trunca...

The data transmission in wireless networks is usually analyzed under the assumption of non-stationary rates. Nevertheless, they strictly depend on the time of day, that is, the intensity of arrival and daily workload profiles confirm this fact. In this article, we consider the process of downloading a file within a single network segment and unstea...

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. Firth in the literature upper bounds on the rate of convergence along with one new concentration i...

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 analytica...

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...

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.

In this note, a general approach to the study of non-stationary Markov chains with catastrophes and the corresponding queuing models is considered, as well as to obtain estimates of the limiting regime itself. As an illustration, an example of a queuing model is studied.

In this paper we revisit the Markovian queueing system with a single server, infinite capacity queue and one special queue skipping policy. Customers arrive in batches but are served one by one in any order. The size of the arriving batch becomes known upon its arrival and at any time instant the total number of customers in the system is also know...

In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev ζ-metric is used. The obtained results extend the known convergence rate estimates for geometric random sums (in the famous Rényi theorem) to...

The problem considered is the computation of the (limiting) time-dependent performance characteristics of one-dimensional continuous-time Markov chains with discrete state space and time varying intensities. Numerical solution techniques can benefit from methods providing ergodicity bounds because the latter can indicate how to choose the position...

We consider a general nonstationary Markovian queueing model under additional assumption of possibility of catastrophes of the system. As a rule this assumption is sufficient for ergodicity of the corresponding queue-length process.

In applied probability, the normal approximation is often used for the distribution of data with assumed additive structure. This tradition is based on the central limit theorem for sums of (independent) random variables. However, it is practically impossible to check the conditions providing the validity of the central limit theorem when the obser...

An inhomogeneous continuous-time Markov chain X(t) with finite or countable state space under some natural additional assumptions is considered. As a consequence, we study a number of problems for the corresponding forward Kolmogorov system, which is the linear system of differential equations with special structure of the matrix A(t). In the count...

We consider the linear system of differential equations , which is the forward Kolmogorov system, for a class of Markov chains with ‘batch’ births and single deaths. We apply the method of differential inequalities for obtaining bounds on the rate of convergence for the system. A specific queueing model is considered and the corresponding limiting...

The forward Kolmogorov system for a general nonstationary Markovian queueing model with possible batch arrivals, possible catastrophes and state-dependent control at idle time is considered. We obtain upper bounds on the rate of convergence for corresponding models (nonstationary queue without catastrophes with the special resurrection intensities...

A version of the central limit theorem is proved for sums with a random number of independent and not necessarily identically distributed random variables in the double array limit scheme. It is demonstrated that arbitrary normal mixtures appear as the limit distribution. This result is used to substantiate the log-normal finite mixture approximati...

In this paper we revisit the Markovian queueing system with a single server, infinite capacity queue and the special queue skipping policy. Customers arrive in batches, but are served one by one according to any conservative discipline. The size of the arriving batch becomes known upon its arrival and at any time instant the total number of custome...

Given the limited frequency band resources and increasing volume of data traffic in modern multiservice networks, finding new and more efficient radio resource management (RRM) mechanisms is becoming indispensable. One of the implemented technologies to solve this problem is the licensed shared access (LSA) technology. LSA allows the spectrum that...

In this paper, we display a method for the computation of convergence bounds for a non-stationary two-processor heterogeneous system with catastrophes, server failures and repairs when all parameters varying with time. Based on the logarithmic norm of linear operators, the bounds on the rate of convergence and the main limiting characteristics of t...

In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions. Multivariate analogs of the Mittag–Leffler distribution are introduced. Some properties of these distributions are discussed. The main focus is on the representations of the corresponding random vectors...

Consideration is given to the two finite capacity time varying Markov queues: the analogue of the well-known time varying M/M/S/0 queue with S servers each working at rate \(\mu (t)\), no waiting line, but with the arrivals happening at rate \(\lambda (t)\) only in batches of size 2; the analogue of another well-known time varying \(M/M/1/(S-1)\) q...

We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace, and generalized Mittag–Leffler laws. In particular, we prove that the generalized Linnik distribution is a normal scale mixture with the generalized Mittag–Leffler mixing distribution. Based on these representations, we prove some limit theor...

Inhomogeneous continuous-time Markov chain with a special structure of infinitesimal matrix is considered as the queue-length process for the corresponding queueing model with possible batch arrivals, possible catastrophes and state-dependent control at idle time. For two wide classes of such processes we suppose an approach is proposed for obtaini...

In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev $\zeta$-metric is used. The obtained results extend the known convergence rate estimates for geometric random sums (in the famous R{\'e}nyi th...

This paper is largely a review. It considers two main methods used to study stability and to 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...

We study inhomogeneous continuous-time weakly ergodic Markov chains with a finite state space. We introduce the notion of a Markov chain with the regular structure of an infinitesimal matrix and study the sharp upper bounds on the rate of convergence for such class of Markov chains.

In this paper, we display a method for the computation of convergence bounds for a non-stationary two-processor heterogeneous system with catastrophes, server failures and repairs when all parameters varying with time. Based on the logarithmic norm of linear operators, the bounds on the rate of convergence and the main limiting characteristics of t...

The authors consider nonstationary queuing models, the number of customers in which is described by finite Markov chains with periodic intensities. For many classes of such models, the methods of obtaining upper bounds on the rate of convergence to the limiting regime were developed in previous papers of the authors. Using these methods, one can fi...

A cyclic random motion at finite velocity with orthogonal directions is considered in the plane and in $\mathbb{R}^3$. We obtain in both cases the explicit conditional distributions of the position of the moving particle when the number of switches of directions is fixed. The explicit unconditional distributions are also obtained and are expressed...

In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate elliptically contoured stable distributions. It is demonstrated that these distributions form a special subclass of scale mixtures of multivariate elliptically contoured normal distributions. Some properties of these distribu...

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...

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...

In this paper, we study a wide and flexible family of discrete distributions, the so-called generalized negative binomial (GNB) distributions, which are mixed Poisson distributions with the mixing laws belonging to the class of generalized gamma (GG) distributions. This family was introduced by E.W. Stacy as a particular family of lifetime distribu...

Consideration is given to the nonstationary analogue of M / M / 1 queueing model in which the service happens only in batches of size 2, with the arrival rate λ ( t ) and the service rate μ ( t ) . One proposes a new and simple method for the study of the queue-length process. The main probability characteristics of the queue-length process are com...

In this paper we study the applications of differential inequalities to bounding the rate of convergence for finite (inhomogeneous) continuous-time Markov chains.

We study a non-stationary Markovian queueing model of a two-processor heterogeneous system and obtain basic limiting characteristics for this model. Some specific examples are considered illustrated by the corresponding plots.

We consider a multidimensional inhomogeneous birth-death process. In this paper, a general situation is studied in which the intensity of birth and death for each coordinate (“each type of particle”) depends on the state vector of the whole process. A one-dimensional projection of this process on one of the coordinate axes is considered. In this ca...

We study inhomogeneous continuous-time weakly ergodic Markov chains on a finite state space. We introduce a notion of Markov chain with the regular structure of an infinitesimal matrix and study the sharp upper bounds on the rate of convergence for this class.

The main purpose of this work is the analysis of some stochastic algorithms to determine values of harmonic functions at points of a bounded domain of Euclidean space. To solve the Dirichlet problem we use a Random Walk on Spheres algorithm. The Neumann problem is solved by means of integral equations of potential theory.

The article provides new mixture represenations for the generalized Mittag-Leffler distribution. In particular, it is shown that for values of the “generalizing” parameter not exceeding one, the generalized Mittag-Leffler distribution is a scale mixture of the half-normal distribution laws, classic Mittag-Leffler distributions, or generalized Mitta...

The paper deals with a Markovian retrial queueing system with a constant retrial rate and two servers. We present the detailed description of the model as well as establish the sufficient conditions for null ergodicity and strong ergodicity of the corresponding process and obtain the upper bounds on the rate of convergence for both situations.

We consider ergodicity and truncation bounds for a retrial queueing model.

We consider a multidimensional inhomogeneous birth-death process (BDP) and obtain bounds on the rate of convergence for the corresponding one-dimensional processes.

We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace and generalized Mittag-Leffler laws. In particular, we prove that the generalized Linnik distribution is a normal scale mixture with the generalized Mittag-Leffler mixing distribution. Based on these representations, we prove some limit theore...

In this paper we study a non-stationary Markovian queueing model of a two-processor heterogeneous system with time-varying arrival and service rates. We obtain the bounds on the rate of convergence and find the main limiting characteristics of the queue-length process.

An approach is proposed to the construction of general lower bounds for the rate of convergence of probability characteristics of continuous-time inhomogeneous Markov chains with a finite state space in terms of special “weighted” norms related to total variation. We study the sharpness of these bounds for finite birth–death–catastrophes process an...

The model of a two-dimensional birth-death process with possible catastrophes is studied. The upper bounds on the rate of convergence in some weighted norms and the corresponding perturbation bounds are obtained. In addition, we consider the detailed description of two examples with 1-periodic intensities and various types of death (service) rates....

In this paper we present a method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically, we consider an inhomogeneous M/M/S queueing system with possible state-dependent arrival and service intensities, and additionally possible batch arrivals and batch ser...

Nonstationary countable Markov chains with continuous time and absorption at zero are considered. We study the convergence rate to the limit mode. As examples, we consider simple nonstationary random walks.

Finite inhomogeneous continuous-time Markov chains are studied. For a wide class of such processes an approach is proposed for obtaining sharp bounds on the rate of convergence to the limiting characteristics. Queueing examples are considered.

The expected growth in the mobile video (including streaming video, video downloading, conferencing, etc.) now is the key driver for rapid development of 5G wireless network technologies. This paradigm forces wireless networks to manage their resources as effectively as possible. One of the most appropriate solutions for video traffic that may prov...

We consider nonstationary Markovian queueing models with batch arrivals and group services. We study the mathematical expectation of the respective queue-length process and obtain the bounds on the rate of convergence and error of truncation of the process.

In the paper general theorems concerning the asymptotic deficiencies of sample median based on the sample of random size are presented. These results make it possible to compare the quality of the sample median constructed from samples with both random and non-random sizes in terms of additional observations. The cases of the binomial distribution...

Some new stochastic algorithms are suggested to determine the minimal eigenvalue for the first boundary value problem for Laplace operator.

In the paper, the concepts of π-mixed geometric and π-mixed binomial distributions are introduced within the setting of Bernoulli trials with a random probability of success. A generalization of the Rényi theorem concerning the asymptotic behavior of rarefied renewal processes is proved for doubly stochastic rarefaction resulting in that the limit...

An inhomogeneous retrial queueing model is studied. Bounds for the rate of convergence in null ergocic case are obtained.

A new class of discrete GG-mixed Poisson distributions is considered as the family of mixed Poisson distributions in which the mixing laws belong to the class of generalized gamma (GG) distributions. The latter was introduced by E. W. Stacy as a special family of lifetime distributions containing gamma, exponential power and Weibull distributions....

Homogeneous birth and death processes with a finite number of states are studied. We analyze the slowest and fastest rates of convergence to the limit mode. Estimates of these bounds for some classes of mean-field models are obtained. The asymptotics of the convergence rate for some models of chemical kinetics is studied in the case where the numbe...

Inhomogeneous birth and death processes with intensities close to periodic are studied. Limit mean and double mean of such processes are analyzed, their estimates are obtained, and the method of their approximate calculation is developed. Also some examples from queuing systems theory are considered.

Weakly ergodic continuous-time countable Markov chains are studied. We obtain uniform in time bounds for approximations via truncations by analogous smaller chains under some natural assumptions.

Linnik distributions (symmetric geometrically stable distributions) are widely applied in financial mathematics, telecommunication systems modeling, astrophysics, and genetics. These distributions are limiting for geometric sums of independent identically distributed random variables whose distribution belongs to the domain of normal attraction of...

A problem related to the Bernoulli trials with a random probability of success is considered. First, as a result of the preliminary experiment, the value of the random variable π ∈ (0, 1) is determined that is taken as the probability of success in the Bernoulli trials. Then, the random variable N is determined as the number of successes in k ∈ N B...

We prove some new product representations for random variables with the
Linnik, Mittag-Leffler and Weibull distributions. The main result is the
representation of the Linnik distribution as a normal scale mixture with the
Mittag-Leffler mixing distribution. As a corollary, we obtain the known
representation of the Linnik distribution as a scale mix...

One of most popular experimental techniques for investigation of brain
activity is the so-called method of evoked potentials: the subject repeatedly
makes some movements (by his/her finger) whereas brain activity and some
auxiliary signals are recorded for further analysis. The key problem is the
detection of points in the myogram which correspond...

We consider a multidimensional inhomogeneous birth-death process and obtain bounds for the probabilities of the corresponding one-dimensional processes.

An improved and corrected version of the functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes (compound Cox processes) to Lévy processes in the Skorokhod space under more realistic moment conditions. As corollaries, theorems are proved on convergence of random wal...