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

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 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.

Preface to the Special Issue on “Control, Optimization, and Mathematical Modeling of Complex Systems”

“…Zeifman et al [1] consider 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.…”

“…Numerical simulation in physical, social, and life sciences [1][2][3][4]; • Modeling and analysis of complex systems based on mathematical methods and AI/ML approaches [5,6]; • Control problems in robotics [3,[7][8][9][10][11][12]]; • Design optimization of complex systems [13]; • Modeling in economics and social sciences [4,14]; • Stochastic models in physics and engineering [1,[15][16][17][18]; • Mathematical models in material science [19]; • High-performance computing for mathematical modeling [20].…”

Bounds on the Rate of Convergence for MtX/MtX/1 Queueing Models

“…The problem of estimating the rate of convergence, like the very fact of convergence, is very important for studying the long-run (limiting) behavior of continuous time Markov chains with time varying intensities, see detailed discussion, examples and references in [7]. The simplest and most convenient for studying the rate of convergence to the limiting regime is the method of the logarithmic norm, see, for example [1,3,8].…”

“…Let us now return to a countable system (7) and consider the corresponding truncated system d dt u(n, t) = B * (n, t)u(n, t),…”

Ergodicity and perturbation bounds forMt/Mt/1 queue with balking, catastrophes, server failures and repairs

“…In [5] DiCrescenzo et al construed the a time-non-homogeneous for double-ended queue subject to catastrophes and repairs, as this is an extension of their previous work in [4]. Some other nonstationary models were studied by a number of authors, see for instance [6,7,8,10,11,23].…”