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 4 Â 10 7 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.
Introduction and literature reviewModelling formalisms are usually applied to describe real (and large) systems, capturing their behaviour and computing performance indices. A well-known modelling formalism used for this purpose is Markov chains (Stewart 1994(Stewart , 2009). Markov chain (MC) models are present in many domains such as chemistry, bioinformatics, economics and social sciences, to cite a few. MC models use simple primitives (states and transitions) to exemplify the system's evolution and operational semantics. Associated with large systems is the well-known state explosion problem which effectively limits the use of the MC model mapping formalism for such systems using traditional methods of solution.Over recent years a number of structured modelling formalisms have been proposed for the analysis of continuoustime MCs (CTMCs), taking advantage of the particular structure of the associated transition matrices.Queueing networks (Kaufman 1983) is certainly the most well-known structured formalism in the performance evaluation area. The popularity of this formalism is based on a very intuitive idea of customers passing by queues. Many extensions to this formalism have been proposed, providing solution approximations and even product-form solutions for some queueing networks (QN) models (Dallery, Liu, and Towsley 1994;Levantesi, Matta, and Tolio 2003;Bariş and Gershwin 2011). However, most real-life systems do not satisfy the necessary conditions for product-type solutions, such as finite capacity queues.The exact solution of small production lines was initiated by Hunt (1956) followed by Buzacott (1972), Gershwin and Berman (1981) and Gershwin and Schick (1981), among others. Sol...