Jean-Sébastien Tancrez, Philippe Chevalier and Pierre Semal. Modelling Queueing Networks with Blocking using Probability Masses Fitting    (abstract only)   information

Abstract. In this paper, we are interested in the modelling of queueing networks with split and merge operations, and without loops. Such a network maybe used to model assembly/disassembly manufacturing systems, for example. The buffers are finite, and can thus lead to blocking in the network. No assumption is taken on service time distributions.
Such queueing networks, with general service time distributions, cannot be modeled exactly. In order to use analytical models, the distributions have to be transformed to “tractable” distributions, i.e. phase-type distributions most of the time. Our originality mainly lies in this stage of the modelling process. We propose “probability masses fitting”, to build tractable discrete phase-type distributions. Then, the queueing network can be modeled by a discrete Markov chain, from which the performance measures can be deduced.
Probability masses fitting (PMF) is quite intuitive: the probability masses on regular intervals are computed and aggregated on a single value in the corresponding interval, leading to a discrete distribution. PMF has some interesting characteristics: it conserves the shape of the distribution (but not the moments, as do moments fitting), it can be refined, and it is particularly natural when the data about the processing time distributions is collected in the form of histograms.
Probability masses fitting leads to various results. First, it allows to compute lower and upper refinable bounds on the throughput of the network. Second, the distribution of the cycle time can be estimated and shows to be realistic. Finally, we show by computational experiments that good accuracy levels are reached in the evaluation of various performances (cycle time, work-in-progress, flow time).