The moments of the extended compound point process with applications

The Concept of Compound point(or counting)process is generalized to account for situations in which the compound events are described by a stochastic process rather than a random variable. Quasi–closed form expressions for the mean and the variance of this extended compounded point process are obtained in terms of the first two moments of the underlying point process and of the process describing the compound events. An expression for the higher moments is also obtained. The results are applied to obtain the mean and variance of the number of busy channels in a pp^x/G/∞ queue. (PP=point process) which generalises results of the M(t)^x/G/∞ and GI^x/G/∞ queue. Other applications include the derivation of the first two moments of the total backlog and of the revenues from telephone traffic in a PP/G/∞ queue. Finally, a special empirical model is considered