An Urn Model Arising From All-Optical Networks

An occupancy model that arose in the investigation of randomized distributed schedules in all-optical networks is considered. The model consists of B initially empty urns, and at state j of the process d(j) less than or equal to B balls are placed in distinct urns with uniform probability. Let M(i,j) denote the number of urns containing i balls at the end of stage j. An explicit expression for the joint factorial moments of M (0,j) and M(1,j) is obtained. A multivariate generating function for the joint factorial moments of M(i,j), for i less than or equal to I, is derived. Finally, the case in which the d(j) are independent identically distributed random variables is investigated.