Optimal routing policies in deterministic queues in tandem

In this paper, we address the problem of routing a stream of customers in two parallel networks of queues in tandem with deterministic service times in order to minimize the average response time. We show that the optimal routing is a Sturmian word whose density depends on the maximum service time on each route. In order to do this, we particularly study the output process of deterministic queues when the input process is Sturmian.