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Multiple randomness is a new kind of random phenomena appeared in experiments concerning practical applications of sciences and engineering disciples. But phenomena with multiple randomness have been ignored in the existing literature, because correctly derived results in mathematics are used to interpret, incorrectly, outcomes obtained by experiments designed to solve practical problems in the real world, leading to questionable theorems in practice. Multiple randomness is characterized by unique properties of the sample space consisting of all possible outcomes obtained by the corresponding random experiment and differs essentially from any known phenomenon observed. This study aims to demonstrate the existence of such phenomena. To this end, the general definition of multiple randomness in a random experiment performed to count the number of ``events'' is given, and specific examples in queuing theory are provided to illustrate the properties of multiple randomness in counting processes. More phenomena with multiple randomness in other stochastic processes and in other application fields of sciences and engineering disciples may also be identified, which may help scientists and engineers to explain weired phenomena and solve puzzling problems in the real world.