Iterative solution methods for obtaining the steady-state probability distributions of Markovian multi-echelon repairable item inventory systems

Abstract

The Jacobi, Gauss-Seidel and bi-conjugate gradient methods are used to compute the steady-state probability distributions for finite state-space continuous time Markov processes (closed queueing networks) that arise in the modeling of two- and three-echelon repairable item inventory systems. Alternative systems of linear equations that express the steady-state conditions are examined, generation and storage of the transition rate matrix are discussed briefly and various initializations and stopping criteria are tested. Numerical results are given for problems with up to one million states. Good results are obtained with a two-phase algorithm that uses the Gauss-Seidel method first and the bi-conjugate gradient method subsequently.