The average number of joint hops in a shortest-path multicast tree from a root to m arbitrary chosen group member nodes is studied. A general theory for all graphs, hence including the graph representation of the Internet, is presented which quantifies the multicast reduction in network links compared to m times unicast. For two special types of graphs, the random graph G/sub p/(N) and the k-ary tree, exact and asymptotic results are derived. Comparing these explicit results with previously published Internet measurements indicates that the number of routers in the Internet that can be reached from a root grows exponentially in the number of hops with an effective degree of approximately 3.2.

We present a recursive formula for the moments of phase noise in communication systems. The phase noise is modeled using continuous Brownian motion. The recursion is simple and valid for an arbitrary initial phase value. The moments obtained bp the recursion are used to calculate approximations to the probability density function of the phase noise, using orthogonal polynomial series expansions and a maximum entropy criterion.