Computational methods for CTMCs

Abstract

This article concerns the computation of stationary and transient distributions of continuous‐time Markov chains (CTMCs). Once the problem has been formulated, it is shown how computational methods for computing stationary distributions of discrete‐time Markov chains can be applied in the continuous‐time case. This is not so for the case of transient distributions, which turns out to be a much more difficult problem in general. Different approaches to computing transient distributions of CTMCs are explored, from the simple and efficient uniformization method, through matrix decomposition and powering techniques, to ordinary differential equation (ODE) solvers. This latter approach is the only one currently available for nonhomogeneous CTMCs. The basic concept is explained using simple Euler methods, but formulae for more advanced and efficient single‐step Runge–Kutta and implicit multistep BDF methods are provided.