On the variance of the production output of transfer lines

The variability factor, (i.e., the variance) of the cumulative production, Σ(t), delivered by a line composed of failure prone machines is studied in the fluid modeling approach. In this context, the evolution of Σ(t) is described by a stochastic differential equation in which the noise source describes the random failures of the machines. We calculate the fluctuations of the production for three different situations, namely for a single nonMarkovian machine, for unbuffered networks of Markovian machines and for production dipoles composed of two machines separated by one buffer. The dynamics of the production dipole is approached via the introduction of a random environment model. Thanks to this new model, we can explicitly take into account the buffer induced correlations phenomena which directly influence the variability of Σ(t). The probabilistic properties of the random time needed to complete a batch of fixed size are also explicitly derived