On the equilibrium probabilities of deterministic flow lines with random arrivals

Flow lines serve as fundamental models for many manufacturing systems. However, while they have been studied for decades, for flow lines with randomness, exact performance measures are only available when there are three servers or less. In this paper, we study the steady state behavior of flow lines with deterministic service and a renewal arrival process. Using a recursion based on an exact channel decomposition of the system, we demonstrate that the delays in each server possess the Markovian property. Restricting our scope to discrete-time flow lines under a renewal arrival process, we can exploit this property to obtain a multidimensional discrete-time time-homogeneous Markov chain for the server delays. For this Markov chain, it is possible to obtain a finite collection of balance equations that can be solved numerically for the equilibrium probabilities. As an example, we demonstrate how to derive the equilibrium probabilities for single channel flow lines with geometric interarrival times. To our knowledge, these are the first exact results that can be applied to flow lines consisting of more than three servers.