Optimal Routing 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 of the whole system. We show that the optimal routing is a Sturmian word which density depends on the decomposition in continuous fraction of the maximum service time on each route. In order to do this we study the output process of deterministic queues when the input process is Sturmian