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

Yacov Satin, Rostislav Razumchik, Ilya Usov and Alexander Zeifman ()
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Yacov Satin: Department of Applied Mathematics, Vologda State University, 160000 Vologda, Russia
Rostislav Razumchik: Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 119133 Moscow, Russia
Ilya Usov: Department of Applied Mathematics, Vologda State University, 160000 Vologda, Russia
Alexander Zeifman: Department of Applied Mathematics, Vologda State University, 160000 Vologda, Russia

Mathematics, 2023, vol. 11, issue 20, 1-12

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 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.

Keywords: Markov models; birth–death process; ergodicity; bounds; limiting characteristics (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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