Inhomogeneous CTMC Birth-and-Death Models Solved by Uniformization with Steady-State Detection

Abstract

Time-inhomogeneous queueing models play an important role in service systems modeling. Although the transient solutions of corresponding continuous-time Markov chains (CTMCs) are more precise than methods using stationary approximations, most authors consider their computational costs prohibitive for practical application. This article presents a new variant of the uniformization algorithm that utilizes a modified steady-state detection technique. The presented algorithm is applicable for CTMCs when their stationary solution can be efficiently calculated in advance, particularly for many practically applicable birth-and-death models with limited size. It significantly improves computational efficiency due to an early prediction of an occurrence of a steady state, using the properties of the convergence function of the embedded discrete-time Markov chain. Moreover, in the case of an inhomogeneous CTMC solved in consecutive timesteps, the modification guarantees that the error of the computed probability distribution vector is strictly bounded at each point of the considered time interval.