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Significant Research Contribution

Published on: Fri, 12/09/2016
Last Modified: Wed, 05/02/2018 - 12:23pm


During January-August1974, Percy Brill pondered, originated, developed and first applied the (then) new methodology System Point Level Crossing Theory (abbreviated SPLC or LC) for obtaining stationary probability density functions (pdfs) of state variables in stochastic models. (Working papers followed soon after.) LC is based on sample-path structure and System Point (leading point of the sample path thought of dynamically over time--denoted by SP) motion in the state space. Theorems show that the rates at which the SP moves across state-space levels, boundaries, thresholds, and/or enters and exits state-space sets, are equal to simple algebraic expressions of the pdf. Such SP rates are also in one-to-one correspondence with the algebraic terms in a system of Volterra integral equations of the second kind for the pdfs, which contain mathematical expressions of the pdf. LC applies the principle of rate balance to construct the integral equations for the pdfs of the state variable. Thus, the equations can be written term by term by inspection of the sample path! The method is intuitive; it gives a “physical” geometrical meaning to every term in the integral equation(s). Experience by many researchers has since shown that LC can save a great deal of time when developing the integral equations for the pdf(s), compared with classical methods such as Lindley recursions and embedded Markov chains. (Nevertheless, these classical methods of analysis are extremely useful in Applied Probability.) LC works completely within the time domain, thus bypassing Laplace transforms and their subsequent inversion to get formulas back in the time domain. (Nevertheless Laplace transforms may be useful for solving the obtained integral equations if needed, because some of the integrals in the equations often appear as convolutions.) To date, LC has been applied in the analyses of many types of stochastic models, by hundreds of researchers world-wide.

The original work on LC in 1974 dealt with stationary pdfs. Later work by Percy Brill led to LC techniques to obtain time-dependent pdfs in queues, dams, inventories and other models with piecewise continuous sample paths (some of these transient techniques are given in the book “Level Crossing Methods in Stochastic Models, by Percy H. Brill--first ed. 2008, second ed. 2017", references therein, and related working papers in the list of Publications in this web page.)  

 




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