Stochastic cyclic flow lines: Non-blocking, Markovian models

In this paper we consider stochastic cyclic flow lines where identical sets of jobs are repeatedly produced in the same loading and processing sequence. Each machine has an input buffer with enough capacity. Processing times are stochastic. We model the shop as a stochastic event graph, a class of Petri nets. We characterise the ergodicity condition and the cycle time. For the case where processing times are exponentially distributed, we present a way of computing queue length distributions. For two-machine cases, by the matrix geometric method, we compute the exact queue length distributions. For general cases, we present two methods for approximately decomposing the line model into two-machine submodels, one based on starvation propagation and the other based on transition enabling probability propagation. We experiment with our approximate methods for various stochastic cyclic flow lines and discuss performance characteristics as well as accuracy of the approximate methods. Finally, we discuss the effects of job processing sequences of stochastic cyclic flow lines.