Modeling Exponential Reliable Production Lines using Kronecker Descriptors

This paper presents a solution procedure for production lines, based on a Markovian formulation with a Kronecker structured representation (sum of tensor products). Production lines with phase-type distributions for the processing and machine repair times and Erlang distribution times to failures of multiple machines in parallel at each workstation are traditionally modeled as Markovian continuous time discrete state processes. In this paper structured Markovian formalisms are used to manage the well-known state explosion problem associated with other methods of solution. Such formalisms combined with the Kronecker representation deliver impressive memory efficiency in storing very large models, i.e., models with more than 109 states. The exact transient and stationary solutions of these models, stored in this fashion, may be obtained using very efficient existing software packages. In this paper, a three-station exponential production line with perfectly reliable machines is considered in order to demonstrate how the equivalent Stochastic Automata Network (SAN) model is derived from the corresponding Finite Capacity Queueing Network model. The throughput of this production line and of other single perfectly reliable machine production lines, with many more states, are derived using Kronecker based software packages (PEPS2007 and GTAexpress) and comparisons with known results are made.