Abstract
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 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.