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11 pages, 380 KB

Open AccessArticle

Utilizing Exact Values of Transition Intensities for Better Estimation of the Limiting Characteristics of Inhomogeneous Birth-and-Death Processes

by Yacov Satin, Rostislav Razumchik, Alexander Zeifman and Janos Sztrik

Computation 2026, 14(7), 155; https://doi.org/10.3390/computation14070155 - 10 Jul 2026

Viewed by 374

Abstract

In this paper, consideration is given to the class of birth-and-death processes with possibly state-dependent and time-varying transition intensities and a finite state space. Several techniques are available in the literature for the computation of the long-run (limiting) time-dependent performance characteristics of such [...] Read more.

In this paper, consideration is given to the class of birth-and-death processes with possibly state-dependent and time-varying transition intensities and a finite state space. Several techniques are available in the literature for the computation of the long-run (limiting) time-dependent performance characteristics of such processes. Whenever a solution technique is combined with a limiting regime detection method, its efficiency may be improved. It is intuitively reasonable to expect that, if additional information about the process is available, a limiting regime detection method may allow one to save more computation effort. In this paper, we demonstrate that the logarithmic norm method, which is one of the methods with which to provide ergodicity bounds for continuous-time Markov chains with discrete state space, can be utilized in such a way. When the exact values of the transition intensities of the (ergodic) birth-and-death process are known and are such that it is clear that one group of states is visited less often than the other, the method allows one to detect the limiting regime rapidly. We illustrate numerically the results obtained within the queueing theory context by considering the activity of the total number of customers in a multi-server finite-capacity queue with periodic arrival and service intensities. Full article

(This article belongs to the Section Computational Engineering)

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29 pages, 397 KB

Open AccessFeature PaperArticle

Convergence Guarantees for Time-Inhomogeneous Uniform-Rate Discrete Diffusion Models

by Yuchen Liang, Lifeng Lai, Ness Shroff and Yingbin Liang

Entropy 2026, 28(6), 675; https://doi.org/10.3390/e28060675 - 11 Jun 2026

Viewed by 331

Abstract

Discrete diffusion models have become an important class of generative models for categorical data, yet their theoretical understanding remains largely limited to time-homogeneous noise schedules. In this work, we study uniform-rate discrete diffusion models with time-inhomogeneous continuous-time Markov chain forward processes. We establish [...] Read more.

Discrete diffusion models have become an important class of generative models for categorical data, yet their theoretical understanding remains largely limited to time-homogeneous noise schedules. In this work, we study uniform-rate discrete diffusion models with time-inhomogeneous continuous-time Markov chain forward processes. We establish convergence guarantees for practical reverse-time samplers by directly controlling the total variation distance, avoiding the indirect route of first bounding KL divergence and then applying Pinsker’s inequality. Our analysis decomposes the sampling error into initialization, score-estimation, discretization, and early-stopping errors, and explicitly characterizes how each term depends on the accumulated noise, the local noise rate, and the smoothness of the noise schedule. Under suitable regularity conditions on the noise schedule, we further derive step-complexity guarantees that match the order of existing results for homogeneous samplers. Full article

(This article belongs to the Special Issue Foundations and Frontiers of Information Theory—Dedicated to Professor H. Vincent Poor on the Occasion of His 75th Birthday)

24 pages, 612 KB

Open AccessArticle

Development of a Constructive Method for Verification of Ergodicity and Calculation of the Stationary Distribution of Quasi-Birth-And-Death Processes with Spatially Inhomogeneous Transitions

by Sergei A. Dudin, Alexander N. Dudin and Olga S. Dudina

Mathematics 2026, 14(7), 1148; https://doi.org/10.3390/math14071148 - 29 Mar 2026

Cited by 2 | Viewed by 522

Abstract

The problem of verifying ergodicity and calculating the stationary distribution for continuous-time multidimensional Markov chains with a block tridiagonal generator (Quasi-Birth-and-Death (

Q B D

) processes) is investigated. This work reviews established results for

Q B D

processes with spatially inhomogeneous transitions [...] Read more.

The problem of verifying ergodicity and calculating the stationary distribution for continuous-time multidimensional Markov chains with a block tridiagonal generator (Quasi-Birth-and-Death (

Q B D

) processes) is investigated. This work reviews established results for

Q B D

processes with spatially inhomogeneous transitions (level-independent

Q B D

processes—

L I D Q B D

processes) and level-dependent

Q B D

processes (

L D Q B D

processes) that require specific asymptotic assumptions. We propose an extension of these results to

L D Q B D

processes that do not rely on such assumptions. The practical application of the proposed methodology is demonstrated through the analysis of a multi-server retrial queueing system characterized by dependent arrival and retrial processes. Full article

(This article belongs to the Special Issue Advances in Queueing Theory and Applications, 2nd Edition)

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19 pages, 362 KB

Open AccessArticle

An Approach to Obtain Upper Ergodicity Bounds for Some QBDs with Countable State Space

by Yacov Satin, Rostislav Razumchik and Alexander Zeifman

Mathematics 2025, 13(16), 2604; https://doi.org/10.3390/math13162604 - 14 Aug 2025

Viewed by 813

Abstract

Usually, when the computation of limiting distributions of (in)homogeneous (in)finite continuous-time Markov chains (CTMC) has to be performed numerically, the algorithm has to be told when to stop the computation. Such an instruction can be constructed based on available ergodicity bounds. One of [...] Read more.

Usually, when the computation of limiting distributions of (in)homogeneous (in)finite continuous-time Markov chains (CTMC) has to be performed numerically, the algorithm has to be told when to stop the computation. Such an instruction can be constructed based on available ergodicity bounds. One of the analytical methods to obtain ergodicity bounds for CTMCs is the logarithmic norm method. It can be applied to any CTMC; however, since the method requires a guessing step (search for proper Lyapunov functions), which may not be successful, the obtained bounds are not always meaningful. Moreover, the guessing step in the method cannot be eliminated or automated and has to be performed in each new use-case, i.e., for each new structure of the infinitesimal matrix. However, the simplicity of the method makes attempts to expand its scope tempting. In this paper, such an attempt is made. We present a new technique that allows one to apply, in one unified way, the logarithmic norm method to QBDs with countable state spaces. The technique involves the preprocessing of the infinitesimal matrix of the QBD, finding bounding for its blocks, and then merging them into the single explicit upper bound. The applicability of the technique is demonstrated through a series of examples. Full article

(This article belongs to the Special Issue Advances in Queueing Theory and Applications)

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20 pages, 992 KB

Open AccessFeature PaperReview

Markov-Chain Perturbation and Approximation Bounds in Stochastic Biochemical Kinetics

by Alexander Y. Mitrophanov

Mathematics 2025, 13(13), 2059; https://doi.org/10.3390/math13132059 - 21 Jun 2025

Cited by 5 | Viewed by 4276

Abstract

Markov chain perturbation theory is a rapidly developing subfield of the theory of stochastic processes. This review outlines emerging applications of this theory in the analysis of stochastic models of chemical reactions, with a particular focus on biochemistry and molecular biology. We begin [...] Read more.

Markov chain perturbation theory is a rapidly developing subfield of the theory of stochastic processes. This review outlines emerging applications of this theory in the analysis of stochastic models of chemical reactions, with a particular focus on biochemistry and molecular biology. We begin by discussing the general problem of approximate modeling in stochastic chemical kinetics. We then briefly review some essential mathematical results pertaining to perturbation bounds for continuous-time Markov chains, emphasizing the relationship between robustness under perturbations and the rate of exponential convergence to the stationary distribution. We illustrate the use of these results to analyze stochastic models of biochemical reactions by providing concrete examples. Particular attention is given to fundamental problems related to approximation accuracy in model reduction. These include the partial thermodynamic limit, the irreversible-reaction limit, parametric uncertainty analysis, and model reduction for linear reaction networks. We conclude by discussing generalizations and future developments of these methodologies, such as the need for time-inhomogeneous Markov models. Full article

(This article belongs to the Section D1: Probability and Statistics)

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13 pages, 741 KB

Open AccessArticle

Computation of Transient and Steady-State Characteristics of Queueing Systems with Different Types of Customer

by Alexander Zeifman, Yacov Satin, Ilia Usov and Janos Sztrik

Computation 2025, 13(6), 150; https://doi.org/10.3390/computation13060150 - 14 Jun 2025

Viewed by 1318

Abstract

This paper deals with queueing models, in which the number of customers is described by a (inhomogeneous, in general) birth–death process. Depending on the choice of the type of intensities for the arrival and service of customers, the system can either have impatience [...] Read more.

This paper deals with queueing models, in which the number of customers is described by a (inhomogeneous, in general) birth–death process. Depending on the choice of the type of intensities for the arrival and service of customers, the system can either have impatience (in which, as the queue length increases, the intensities of arrival decrease and the intensities of service increases) or attraction (in which, on the contrary, as the queue length increases, the intensities of the arrival of customers increase and service intensities decrease). In this article, various types of such models are considered, and their transient and limiting characteristics are computed. Furthermore, the rate of convergence and related bounds are also dealt with. Several numerical examples illustrate the proposed procedures. Full article

(This article belongs to the Section Computational Engineering)

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12 pages, 323 KB

Open AccessArticle

On One Approach to Obtaining Estimates of the Rate of Convergence to the Limiting Regime of Markov Chains

by Yacov Satin, Rostislav Razumchik, Alexander Zeifman and Ilya Usov

Mathematics 2024, 12(17), 2763; https://doi.org/10.3390/math12172763 - 6 Sep 2024

Cited by 2 | Viewed by 1601

Abstract

We revisit the problem of the computation of the limiting characteristics of (in)homogeneous continuous-time Markov chains with the finite state space. In general, it can be performed only numerically. The common rule of thumb is to interrupt calculations after quite some time, hoping [...] Read more.

We revisit the problem of the computation of the limiting characteristics of (in)homogeneous continuous-time Markov chains with the finite state space. In general, it can be performed only numerically. The common rule of thumb is to interrupt calculations after quite some time, hoping that the values at some distant time interval will represent the sought-after solution. Convergence or ergodicity bounds, when available, can be used to answer such questions more accurately; i.e., they can indicate how to choose the position and the length of that distant time interval. The logarithmic norm method is a general technique that may allow one to obtain such bounds. Although it can handle continuous-time Markov chains with both finite and countable state spaces, its downside is the need to guess the proper similarity transformations, which may not exist. In this paper, we introduce a new technique, which broadens the scope of the logarithmic norm method. This is achieved by firstly splitting the generator of a Markov chain and then merging the convergence bounds of each block into a single bound. The proof of concept is illustrated by simple examples of the queueing theory. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)

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12 pages, 1109 KB

Open AccessArticle

Numerical Computation of Distributions in Finite-State Inhomogeneous Continuous Time Markov Chains, Based on Ergodicity Bounds and Piecewise Constant Approximation

by Yacov Satin, Rostislav Razumchik, Ilya Usov and Alexander Zeifman

Mathematics 2023, 11(20), 4265; https://doi.org/10.3390/math11204265 - 12 Oct 2023

Cited by 2 | Viewed by 1664

Abstract

In this paper it is shown, that if a possibly inhomogeneous Markov chain with continuous time and finite state space is weakly ergodic and all the entries of its intensity matrix are locally integrable, then, using available results from the perturbation theory, its [...] Read more.

In this paper it is shown, that if a possibly inhomogeneous Markov chain with continuous time and finite state space is weakly ergodic and all the entries of its intensity matrix are locally integrable, then, using available results from the perturbation theory, its time-dependent probability characteristics can be approximately obtained from another Markov chain, having piecewise constant intensities and the same state space. The approximation error (the taxicab distance between the state probability distributions) is provided. It is shown how the Cauchy operator and the state probability distribution for an arbitrary initial condition can be calculated. The findings are illustrated with the numerical examples. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)

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11 pages, 327 KB

Open AccessArticle

Bounds on the Rate of Convergence for

M_{t}^{X}

/

M_{t}^{X}

/1 Queueing Models

by Alexander Zeifman, Yacov Satin and Alexander Sipin

Mathematics 2021, 9(15), 1752; https://doi.org/10.3390/math9151752 - 25 Jul 2021

Cited by 3 | Viewed by 2202

Abstract

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 [...] Read more.

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

l_{1}

. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)

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20 pages, 1658 KB

Open AccessArticle

Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method

by Alexander Zeifman, Yacov Satin, Ivan Kovalev, Rostislav Razumchik and Victor Korolev

Mathematics 2021, 9(1), 42; https://doi.org/10.3390/math9010042 - 27 Dec 2020

Cited by 21 | Viewed by 3488

Abstract

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 [...] Read more.

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 and the length of the “distant time interval” (in the periodic case) on which the solution has to be computed. They can also be helpful whenever the state space truncation is required. In this paper one such analytic method—the logarithmic norm method—is being reviewed. Its applicability is shown within the queueing theory context with three examples: the classical time-varying

M / M / 2

queue; the time-varying single-server Markovian system with bulk arrivals, queue skipping policy and catastrophes; and the time-varying Markovian bulk-arrival and bulk-service system with state-dependent control. In each case it is shown whether and how the bounds on the rate of convergence can be obtained. Numerical examples are provided. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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25 pages, 1526 KB

Open AccessArticle

Queuing System with Two Types of Customers and Dynamic Change of a Priority

by Valentina Klimenok, Alexander Dudin, Olga Dudina and Irina Kochetkova

Mathematics 2020, 8(5), 824; https://doi.org/10.3390/math8050824 - 19 May 2020

Cited by 31 | Viewed by 6115

Abstract

The use of priorities allows us to improve the quality of service of inhomogeneous customers in telecommunication networks, inventory and health-care systems. An important modern direction of research is to analyze systems in which priority of a customer can be changed during his/her [...] Read more.

The use of priorities allows us to improve the quality of service of inhomogeneous customers in telecommunication networks, inventory and health-care systems. An important modern direction of research is to analyze systems in which priority of a customer can be changed during his/her stay in the system. We considered a single-server queuing system with a finite buffer, where two types of customers arrive according to a batch marked Markov arrival process. Type 1 customers have non-preemptive priority over type 2 customers. Low priority customers are able to receive high priority after the random amount of time. For each non-priority customer accepted into the buffer, a timer, which counts a random time having a phase type distribution, is switched-on. When the timer expires, the customer with some probability leaves the system unserved and with the complimentary probability gains the high priority. Such a type of queues is typical in many health-care systems, contact centers, perishable inventory, etc. We describe the behavior of the system by a multi-dimensional continuous-time Markov chain and calculate a number of the stationary performance measures of the system including the various loss probabilities as well as the distribution function of the waiting time of priority customers. The illustrative numerical examples giving insights into the system behavior are presented. Full article

(This article belongs to the Section E: Applied Mathematics)

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12 pages, 850 KB

Open AccessArticle

A Hidden Markov Model to Address Measurement Errors in Ordinal Response Scale and Non-Decreasing Process

by Lizbeth Naranjo, Luz Judith R. Esparza and Carlos J. Pérez

Mathematics 2020, 8(4), 622; https://doi.org/10.3390/math8040622 - 17 Apr 2020

Cited by 5 | Viewed by 4038

Abstract

A Bayesian approach was developed, tested, and applied to model ordinal response data in monotone non-decreasing processes with measurement errors. An inhomogeneous hidden Markov model with continuous state-space was considered to incorporate measurement errors in the categorical response at the same time that [...] Read more.

A Bayesian approach was developed, tested, and applied to model ordinal response data in monotone non-decreasing processes with measurement errors. An inhomogeneous hidden Markov model with continuous state-space was considered to incorporate measurement errors in the categorical response at the same time that the non-decreasing patterns were kept. The computational difficulties were avoided by including latent variables that allowed implementing an efficient Markov chain Monte Carlo method. A simulation-based analysis was carried out to validate the approach, whereas the proposed approach was applied to analyze aortic aneurysm progression data. Full article

(This article belongs to the Special Issue Statistics 2020)

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25 pages, 588 KB

Open AccessFeature PaperReview

Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

by Alexander Zeifman, Victor Korolev and Yacov Satin

Mathematics 2020, 8(2), 253; https://doi.org/10.3390/math8020253 - 14 Feb 2020

Cited by 25 | Viewed by 4752

Abstract

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 [...] Read more.

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 chains associated with queuing models. Several specific models are considered for which the limit characteristics and perturbation bounds for admissible “perturbed” processes are calculated. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

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10 pages, 254 KB

Open AccessArticle

On the Rate of Convergence for a Characteristic of Multidimensional Birth-Death Process

by Alexander Zeifman, Yacov Satin, Ksenia Kiseleva and Victor Korolev

Mathematics 2019, 7(5), 477; https://doi.org/10.3390/math7050477 - 26 May 2019

Cited by 7 | Viewed by 3472

Abstract

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 [...] Read more.

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 case, a non-Markov process is obtained, in which the transitions to neighboring states are possible in small periods of time. For this one-dimensional process, by modifying the method previously developed by the authors of the note, estimates of the rate of convergence in weakly ergodic and null-ergodic cases are obtained. The simplest example of a two-dimensional process of this type is considered. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)

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