The Single Server Queue with Periodic Arrival Process and Deterministic Service Times

We derive an algorithm for computing the exact delay distribution for the following first-come-first-served single server queue. The arrival process is the superposition of
N + 1
independent equivalent deterministic arrival streams, and the service times are deterministic. Numerical results are used to compare this system with an approximating system which has been used in the absence of exact results: the
M/D/1
queue (Poisson input, deterministic service time). A comparison with a finite source model is also given. This study was motivated by the following issue germane to packet switching data networks. There are
N + 1
packet processes, arriving over identical trunks, which must share the transmission capacity of a single high-speed trunk in a store-and-forward manner. The idealized queueing model above analyzes the worst-delay situation for this problem when the incoming trunks are fully utilized and the packets are maximum sized. The results indicate that an
M/D/1
approximation can be quite pessimistic in predicting the performance of such a system.