Continuous versus discrete flow of parts in a production dipole. Exact transient analysis

In the dynamical description of production lines, the hydrodynamical framework to describe production flows of parts has been recognized to yield accurate results in numerous actual applications. The "hydrodynamic approximation" consists in describing the population level in the buffers by a continuous variable. The aim of this paper is to compare analytically the results obtained in a simple production line composed of two identical failure prone machines separated by a buffer. In one case, a continuous flux of parts is produced by both machines while in the other case, we assume that each machine produces parts one by one with a processing time which is exponentially distributed. A direct comparison of the exact analytical results obtained in both situations enables us to characterize the operation regime for which the hydrodynamical model is a consistent analytical tool to model discrete flows, even for short time regimes. In this paper, we deal directly with calculation of probabilities and the resulting moments. We supplement our analysis with a comparison of the sojourn time of the population of parts inside the boundary of the buffer.