Histogram based bounds and approximations for production lines

We present a modelling method for the analysis of production lines with generally distributed processing times and finite buffers. We consider the complete modelling process, from the data collection to the performance evaluation. First, the data about the processing times is supposed to be collected in the form of histograms. Second, tractable discrete phase-type distributions are built. Third, the evolution of the production line is described by a Markov chain, using a state model. Our originality mostly comes from the way the phase-type distributions are built: the “grouping at the end” discretization aggregates the probability mass in a time step at its end. The method allows to compute refinable upper and lower bounds on the throughput. Furthermore, we propose some approximations and show how the method performs on simple examples. We argue that the way the distributions are discretized, called “probability masses fitting”, can be thought as a valuable alternative in order to build tractable distributions.