Performance of a pipeline array of automata with periodic cycle-time

We consider a pipeline array of processors where local communications are based on a producer-consumer mechanism. It is assumed that any processor contains a buffer of size one, and that the cycle-time of any processor is a periodic sequence of period s(j),s(j + 1),...,s(n),...,s(j - 1), where s is a fixed permutation of 1,2,...,n, and j may vary from one processor to another. We give upper and lower bounds for the performance of the system.