Multinode broadcast in hypercubes and rings with randomly distributed length of packets

Multinode broadcast (MNB) in a hypercube and in a ring network of processors is considered. It is assumed that the lengths of the packets that are broadcast are not fixed, but are distributed according to some probabilistic rule, and the optimal times required to execute the MNB are compared for variable and for fixed packet lengths. For large hypercubes, it is shown, under very general probabilistic assumptions on the packet lengths, that the MNB is completed in essentially the same time as when the packet lengths are fixed. In particular, the MNB is completed by time (1+δ)T_s with probability at least 1-ε, for any positive ε and δ, where T _s is the optimal time required to execute the MNB when the packet lengths are fixed at their mean, provided that the size of the hypercube is large enough. In the case of the ring, it is proved that the average time required to execute a MNB when the packet lengths are exponentially distributed exceeds by a factor of ln n the corresponding time for the case there the packet lengths are fixed at their mean, where n is the number of nodes of the ring