The paper studies the behavior of an (1 + 3)th-dimensional, delayed renewal process with dependent components, the first three (called active) of which are to cross one of their respective thresholds. More specifically, the crossing takes place when at least one of the active components reaches or exceeds its assigned level. The values of the other two active components, as well as the rest of the components (passive), are to be registered. The analysis yields the joint functional of the "crossing level" and other characteristics (some of which can be interpreted as the first passage time) in a closed form, refining earlier results of the author. A brief, informal discussion of various applications to stochastic models is presented.
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 drect 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 prefirst excesses, the index of the first excess and others. The results obtained are llustrated by a number of numerical examples and then are applied to a bulk queueing system with a service delay discipline.Key words: first passage problem, fluctuation theory, delayed renewal process, first excess level, pre-first excess level, shortage before the first excess, index of the first excess, bulk queues, inventory control, dam.
This article has as an objective to analyze the behavior of multivariate, delayed stationary marked Cox processes with mutually dependent components about some critical levels. The original problems arise in biology, computer engineering, computer networks, software reliability testing, and stock market. The process under investigation can describe the evolution of stocks, indexes, cancer cells, proliferation of bacteria, inventories, military conflicts, in which the process is being observed only restrictively, i.e., at some specified random epochs. Given this (sometimes limited) information, it is possible to "predict" the "first passage time" when the process crosses the critical level (or levels) and see the main probability characteristics (such as distribution) of the components of the process upon the first passage time that occurs at one of the observation times. Among various questions to arise, one is how to choose the frequency of observations to provide more accurate information but not to "exceed the budget" (a quint essence of reliability analysis). On the other hand, there are ways to scrutinize the available information, as to making it analytically more "time sensitive," without any additional efforts, which is one of the primary goals of this investigation. We formalize and provide preliminary results for the work to be continued in [J. Math. Anal. Appl. 293 (2004) 14-27] (about time sensitive functionals) and give closed-form expressions. Many examples from science and technology are presented.
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