Applications of differential inequalities to bounding the rate of convergence for continuous-time Markov chains

Zeifman, Alexander; Satin, Yacov; Kiseleva, Ksenia; Kryukova, Anastasia

doi:10.1063/1.5114074

Cited by 10 publications

(9 citation statements)

References 5 publications

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“…The papers [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] present various methods of studying the non-stationary mode of queuing systems. All of them are reduced either to a numerical solution of the corresponding Kolmogorov system, or to the use of Laplace transforms.…”

Section: The Matrix Methods For Solving the Kolmogorov Equationsmentioning

confidence: 99%

“…In recent years, queuing systems with periodically varying parameters of input flows and service time for customers have been actively studied [8,9]. One paper [8] considered the problem of convergence for a nonstationary two-processor system with catastrophes, server failures, and repairs, when all parameters are harmonic functions of time.…”

Section: Introductionmentioning

confidence: 99%

“…One paper [8] considered the problem of convergence for a nonstationary two-processor system with catastrophes, server failures, and repairs, when all parameters are harmonic functions of time. Another paper [9] proposed three different analytical methods for calculating upper bounds for the rate of convergence to a stationary mode of a process given by a continuous inhomogeneous Markov chain. The first method is based on the logarithmic norm of a linear operator function, the second uses the simplest Lyapunov functions, and the third uses differential inequalities.…”

Section: Introductionmentioning

confidence: 99%

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Transient Behavior of the MAP/M/1/N Queuing System

2021

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This paper investigates the characteristics of the MAP/M/1/N queuing system in the transient mode. The matrix method for solving the Kolmogorov equations is proposed. This method makes it possible, in general, to obtain the main characteristics of the considered queuing system in a non-stationary mode: the probability of losses, the time of the transient mode, the throughput, and the number of customers in the system at time t. The developed method is illustrated by numerical calculations of the characteristics of the MAP/M/1/3 system in the transient mode.

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Section: The Matrix Methods For Solving the Kolmogorov Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Transient Behavior of the MAP/M/1/N Queuing System

2021

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“…Below we describe one method, called the "logarithmic norm" method, which is applicable in the situations, when the discrete state space of the Markov chain cannot be replaced by the continuous one and the transition intensities are such that the chain is either null or weakly ergodic. The method is based on the notion of the logarithmic norm (see e.g., [27,28]) and utilizes the properties of linear systems of differential equations.…”

Section: Logarithmic Norm Methodsmentioning

confidence: 99%

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

et al. 2020

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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.

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“…In the next section, one outlines another approach, which is based on the direct applications of the differential inequalities. It was firstly considered for a finite Markovian queueing model in [19].…”

Section: Model Description and Basic Transformationsmentioning

confidence: 99%