Journal of Applied Mathematics and Stochastic Analysis
Volume 5 (1992), Issue 1, Pages 83-97
A first passage problem and its applications to the analysis of a class of stochastic models
1Department of Mathematics, Loyola Marymount University, Los Angeles 90045, CA, USA
2Department of Applied Mathematics, Florida Institute of Technology, Melbourne 32901, FL, USA
Received 1 July 1991; Revised 1 December 1991
Copyright © 1992 Lev Abolnikov and Jewgeni H. Dshalalow. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.