A problem of the first passage of a cumulative random process with generally distributed discrete or continuous increments over a fixed level is considered in the article as an essential part of the analysis of a class of stochastic models (bulk queueing systems, inventory control and dam models).Using direct probability methods the authors find various characteristics of this problem: the magnitude of the first excess of the process over a fixed level, the shortage before the first excess, the

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... levels of the first and pre-first excesses, the index of the first excess and others. The results obtained are illustrated by a number of numerical examples and then are applied to a bulk queueing system with a service delay discipline.