Analysis of exponential reliable production lines using Kronecker descriptors

Abstract

This paper presents a solution procedure for reliable production lines with service times distributed according to an exponential distribution, based on a Markovian formulation with a Kronecker structured representation (sum of tensor products). Specifically, structured Markovian formalisms are used to reduce the impact of the well-known state explosion problem associated with other methods of solution. Such formalisms combined with the Kronecker representation deliver memory efficiency in storing very large models, i.e. models with more than states. The exact steady-state solutions of these models may be obtained using efficient existing software packages. The proposed solution procedure is illustrated with two detailed examples, and generalised with a model construction algorithm. The computed throughput for several examples of production lines with perfectly reliable machines, as well as the computational costs in terms of CPU time to solve them with PEPS2007 and GTAexpress software packages, are also presented. In effect the paper demonstrates the power of the use of the Kronecker descriptor analysis applied to the derivation of the exact solution of the particular class of production lines considered. The Kronecker descriptor methodology is well-known to analysts concerned with computer and communication systems.