Journal of Applied Mathematics and Stochastic Analysis
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.
Lev Abolnikov and Jewgeni H. Dshalalow, “A first passage problem and its applications to the analysis of a class of stochastic models,” Journal of Applied Mathematics and Stochastic Analysis, vol. 5, no. 1, pp. 83-97, 1992. doi:10.1155/S1048953392000066