Abstract
This article introduces "counting processes with multiple randomness", which appear naturally in applications and differ essentially from known stochastic processes in the literature. Unlike times between consecutive "events" of a usual counting process, inter-event times of a counting process with multiple randomness are defined on proper subsets of the sample space, and not eligible to have marginal distributions. With examples in queuing theory, the existence of this new type of counting processes is demonstrated, and their properties are illustrated.