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## On convergence rate for homogeneous Markov chains

Veretennikov A., Veretennikova M.

New convergence rate asymptotic bound for a class of homogeneous Markov chains is established.

А. В. Ткаченко, Вестник Московского университета. Серия 1: Математика. Механика 2013 № 2 С. 12-17

This paper is devoted to $M|GI|1|\infty$ queueing system with unreliable server and customer service times depending on the system state. Condition of ergodicity and generating function are found in the stationary state. ...

Added: March 27, 2013

Blank M., Nonlinearity 2014 Vol. 27 No. 5 P. 953-971

We discuss conditions for unique ergodicity of a collective random walk on a continuous circle. Individual particles in this collective motion perform independent (and different in general) random walks conditioned by the assumption that the particles cannot overrun each other. Additionally to sufficient conditions for the unique ergodicity we discover a new and unexpected way ...

Added: November 21, 2014

Kleptsyn V., Alvarez S., Malicet D. et al., Groups with infinitely many ends acting analytically on the circle / Cornell University. Series math "arxiv.org". 2015.

Added: June 22, 2016

Piatnitski A., Pirogov S. A., Zhizhina E., Applicable Analysis 2019 Vol. 98 No. 1-2 P. 217-231

The goal of the paper is to describe the large time behaviour of a symmetric diffusion in a high-contrast periodic environment and to characterize the limit process under the diffusive scaling. We consider separately the C_0 and L^2 settings. ...

Added: December 5, 2020

Shchegolev A., A new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains / Cornell University. Серия math "arxiv.org". 2021.

In the paper, we study a new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains based on the Markov-Dobrushin condition. This result generalizes the convergence estimates for any positive number of transition steps. An example of a class such a process provided indicates that such types of estimates considering several transition steps may ...

Added: October 22, 2021

Tkachenko A., Esaulov D., Applied Mathematical Modelling 2016

This paper is focused on a multichannel queueing system with heterogeneous servers, regenerative input flow, and balking. Service times are random variables but not necessary exponential. If a new customer encounters j other customers in the system, then it joins the queue with probability fj and leaves with probability 1-fj . For this system the ...

Added: October 19, 2016

Veretennikov A., Veretennikova M., Известия РАН. Серия математическая 2022 Т. 86 № 1 С. 98-133

We continue the work of improving the rate of convergence of ergodic homogeneous Markov chains. The setting is more general than in previous papers: we are able to get rid of the assumption about a common dominating measure and consider the case of inhomogeneous Markov chains as well as more general state spaces. We give examples ...

Added: March 14, 2022

Afanasyeva L. G., Tkachenko A., Theory of Probability and Its Applications 2014 Vol. 58 No. 2 P. 174-192

We consider the multichannel queueing system with nonidentical servers and regenerative input flow. The necessary and sufficient condition for ergodicity is established, and functional limit theorems for high and ultra-high load are proved. As a corollary, the ergodicity condition for queues with unreliable servers is obtained. Suggested approaches are used to prove the ergodic theorem ...

Added: August 20, 2014

Blank M., Доклады Академии наук 2013 Т. 448 № 6 С. 629-632

We give conditions for unique ergodicity for a discrete time collective
random walk on a continuous circle. Individual particles in this collective
motion perform independent (and different) random walks conditioned
by the assumption that the particles cannot overrun each other.
Deterministic version of this system is studied as well. ...

Added: November 25, 2014

Tkachenko A., Moscow University Mathematics Bulletin 2014 Vol. 69 No. 1 P. 37-40

This paper is focused on a multichannel queueing system with heterogeneous servers and regenerative input flow operating in a random environment. The environment can destroy the whole system and the system is reconstructed after that. The necessary and sufficient ergodicity condition is obtained for the system. ...

Added: August 20, 2014

Ткаченко А.В., Вестник Московского университета. Серия 1: Математика. Механика 2014 № 1 С. 53-57

This paper is focused on multichannel queueing system with heterogeneous servers and regenerative input flow in a random environment. The environment can destroy all the system and then system is reconstructed. Ergodicity condition of the system is obtained. ...

Added: May 11, 2013

Tkachenko A., Multichannel queuing systems with balking and regenerative input flow / Высшая школа экономики. Series WP BRP "Science, Technology and Innovation". 2013. No. 14.

Motivated by the application to telephone call centers this paper is focused on the multichannel queueing system with heterogeneous servers, regenerative input flow and balking. Servers times are random variables but not necessary exponential. If a new customer encountering j other customers in the system then it stays for service with probability fj and gets ...

Added: August 1, 2013

Veretennikov A., On convergence rate for Erlang--Sevastyanov type models with infinitely many servers / Cornell University. Series cond-mat "arxiv.org". 2014.

Polynomial convergence rate to stationarity is shown for extended Erlang -- Sevastyanov's model. ...

Added: December 16, 2014

Афанасьева Л. Г., Ткаченко А.В., Теория вероятностей и ее применения 2013 Т. 58 № 2 С. 210-234

We consider a multichannel queuing system with heterogeneous servers and regenerative input flow. The necessary and sufficient condition is established and functional limit theorems are proved in overloaded and critically loaded regimes. The ergodicity condition is obtained for multichannel system with unreliable servers. Some approaches for ergodicity of systems with abandonment are discussed. ...

Added: March 31, 2013

Vedenin A., Remizov I., Rapidly Converging Chernoff Approximations to Solution of Parabolic Differential Equation on the Real Line / Cornell University. Series math "arxiv.org". 2020.

Abstract. The method of Chernoff approximation was discovered by Paul Chernoff in 1968 and now is a powerful and flexible tool of contemporary functional analysis. This method is different from grid-based approach and helps to solve numerically the Cauchy problem for evolution equations, e.g., for heat equation and for more general parabolic second-order partial differential equations ...

Added: December 14, 2020

Tumanov M., Piunovskly A., Plakhov A., Optimal Control Applications and Methods 2019 P. 1-21

Using the tools of the Markov Decision Processes, we justify the dynamic programming approach to the optimal impulse control of deterministic dynamical systems. We prove the equivalence of the integral and diﬀerential forms of the optimality equation. The theory is illustrated by an example from mathematical epidemiology. The developed methods can be also useful for ...

Added: February 27, 2020

Galkin O., Remizov I., Upper and lower estimates for rate of convergence in Chernoff's product formula for semigroups of operators / Cornell University. Series math "arxiv.org". 2021. No. 2104.01249.

Эта статья посвящена изучению скорости сходимости черновских приближений к сильно непрерывным однопараметрическим полугруппам. Мы приводим естественные примеры, для которых эта сходимость: произвольно высока; произвольно медленна; выполняется в сильной операторной топологии, но не выполняется в нормальной операторной топологии. Мы также доказываем общую теорему, которая дает оценку сверху для скорости стремления к нулю нормы остатка, появляющегося в черновских приближениях. Мы ...

Added: October 14, 2021

Piterbarg V., Щербакова Ю. А., Теория вероятностей и ее применения 2022 Т. 67 № 1 С. 57-80

We propose a sequence of accompanying laws in the B. V. Gnedenko limit theorem
for maxima of independent random variables with distributions lying in the Gumbel max domain
of attraction. We show that this sequence provides a power-law convergence rate, whereas the
Gumbel distribution provides only the logarithmic rate. As examples, we consider in detail the classes
of Weibull ...

Added: October 27, 2022

Tumanov M., Абдуллин С., Вестник Московского государственного университета леса - Лесной вестник 2015 № 3 С. 173-177

Discusses frequency methods to ensure the accuracy of the control system with model lag and the connection with ergodic theory (the theory of quasi-periodic functions) in the case where the delay model differs from the true. ...

Added: August 27, 2015

Shchegolev A., Управление большими системами: сборник трудов 2021 № 90 С. 36-48

The paper studies an improved estimate for the rate of convergence for nonlinear homogeneous discrete-time Markov chains. These processes are nonlinear in terms of the distribution law. Hence, the transition kernels are dependent on the current probability distributions of the process apart from being dependent on the current state. Such processes often act as limits ...

Added: April 21, 2021

Decrouez G. G., Borovkov K., Gilson M., Journal of Applied Probability 2014 Vol. 51 No. 3 P. 837-857

The paper deals with nonlinear Poisson neuron network models with bounded memory dynamics, which can include both Hebbian learning mechanisms and refractory periods. The state of the network is described by the times elapsed since its neurons fired within the post-synaptic transfer kernel memory span, and the current strengths of synaptic connections, the state spaces ...

Added: September 29, 2014

Vedenin A., Воеводкин В. С., Galkin V. et al., Математические заметки 2020 Т. 108 № 3 С. 463-468

This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations. ...

Added: October 21, 2019

Skripchenko A., Dynnikov I., Journal of Modern Dynamics 2017 Vol. 11 P. 219-248

It is known since a 40-year-old paper by M.Keane that minimality is a generic (i.e., holding with probability one) property of an irreducible interval exchange transformation. If one puts some integral linear restrictions on the parameters of the interval exchange transformation, then minimality may become an "exotic" property. We conjecture in this paper that this ...

Added: April 20, 2018

Borodin A., Olshanski G., Journal of Functional Analysis 2012 Vol. 263 P. 248-303

We construct a four-parameter family of Markov processes on infinite Gelfand–Tsetlin schemes that preserve the class of central (Gibbs) measures. Any process in the family induces a Feller Markov process on the infinite-dimensional boundary of the Gelfand–Tsetlin graph or, equivalently, the space of extreme characters of the infinite-dimensional unitary group U(∞). The process has a ...

Added: February 25, 2013