A generalization of little's law to moments of queue lengths and waiting times in closed, product-form queueing networks

Little's theorem states that under very general conditions L = 1 W, where L is the time average number in the system, W is the expected sojourn time in the system, and A is the mean arrival rate to the system. For certain systems it is known that relations of the form E((L)) = 'IE((W)') are also true, where (L) = L(L - 1)... (L - I + 1). It is shown in this paper that closely analogous relations hold in closed, product-form queueing networks. Similar expressions relate Nji and Sfi, where Nji is the total number of classjjobs at center i and Sii is the total sojourn time of a class j job at center i, when center i is a single-server, FCFS center. When center i is a c-server, FCFS center, Qji and Wji are related this way, where Qii is the number of classjjobs queued, but not in service at center i and Wji is the waiting time in queue of a classjjob at center i. More remarkably, generalizations of these results to joint moments of queue lengths and sojourn times along overtake-free paths are shown to hold.