Markovian modeling of multiclass deterministic flow lines with random arrivals: The case of a single-channel

Although there has been some success in the exact analysis of tandem queueing networks with finite intermediate buffers, equilibrium probabilities for the waiting time of customers remain elusive. Recently, for deterministic flow lines with random arrivals and a single customer class, exact channel decomposition has enabled Markovian modeling of the waiting time probabilities. Although exact channel decomposition results have been obtained for certain types of multi-class deterministic flow lines, stochastic analysis of customer delays remains unresolved. Here we demonstrate that certain types of single channel multi-class flow lines also possess a Markovian property for their customer delays. The explicit recursive relationship between the delays from one customer to the next is developed. Due to the complexity of the recursive relationship, we provide some guidance for constructing the state space and transition probabilities of a Markov chain modeling the delays. A computational example is provided. As flow lines can serve as good models for certain types of semiconductor manufacturing equipment, the results may ultimately lead to useful analytic models for such systems.