A three-station merge system with unreliable stations and a shared buffer

In this study, a three-station continuous materials flow merge system model with unreliable stations and a shared buffer is presented. Two stations in parallel sharing a finite capacity buffer are placed in series with a third station. The processing rates of the stations are different. Time to failure and time to repair distributions when the buffer is partially full are assumed to be exponential, while no specific assumptions are made for the distributions of the failure time and repair time when the buffer is full or empty. This model will be used as a building block, in a decomposition method to evaluate production systems, that includes merge and split structures. The steady-state performance measures of the model, specifically the production rate and the expected buffer level, are determined. First, a set of differential equations for the interior states is derived, and then these equations are solved subject to some boundary conditions. Second, the boundary processes, i.e., the processes when the buffer is empty and full, are analyzed by using renewal analysis in conjunction with level crossing analysis. Numerical results for a specific boundary process are given. The effect of the buffer capacity on the production rate and also the effects of the station processing rates on the production rate and the expected buffer level are exploited through numerical experiments.