Rigorous exposition of the mathematical foundations of arbitrary input flows, which are flows of information circulating in today's telecommunications networks, with their packet. The theory of recurrent flows, flows with limited aftereffect and so on. For rational functions of the Laplace - Stiltjes transforms distributions of intervals between the moments of input of customers of flow obtained exact analytical solutions for the moments of the number of input customers for the finite fixed time. If the functions are not rational, we obtained the complete asymptotic expansions for these moments. For specialists in the theory of probability and its applications, queuing theory. Reviewer - Professor, doctor of physical and mathematical Sciences G. S. Tsitsiashvili. Viktor Ivnitskiy