Simulations

We have simulated a two-machine reentrant system to test and help develop our theory. We began with prime processing times 223 for stage 1, 53 for stage 2, 167 for stage 3, and 71 for stage 4. Under the push policy, as well as two other policies, we found periodic behavior in many cases. In the others, seeming ly aperiodic behavior was found, although simulation cannot distinguish between very long transients and true nonperiodic or quasiperiodic behavior. In all cases we found that transients ceases and periodic behavior begins when one machine's queues contained no batches. In other words, when one machine becomes a bottleneck, forcing the other machine to idle, the dynamics become periodic. The mostly-idle machine processes batches as soon as they enter the queues while the bottleneck machine works constantly and predictably according to the policy. This is as predicted by the theory described above.

An interesting feature of these simulations is the length of the transient. When the bottleneck machine's processing times were decreased or increases, the transient lengthened or shortened, respectively. In fact, at close to balanced machine use, the transient length surpassed 100,000 for system load of 15 or 16 lots. The transient also increases when queue load increases, since it is longer before one machine empties its queues and the other becomes the bottleneck. We also introduced stochastic perturbation of processing times, adding a uniform random variable. Suprisingly, the dynamics was unchanged, suggesting no sensitive dependence on initial conditions. This phenomenon can perhaps be explained in terms of rigidity of periodic orbits for interval exchange maps.