Stochastic Methods and Models in Operations Research and Related Areas

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The nature of the work to be done involves mathematical modeling of real-world systems, such as queues, production-inventories, actuarial risk processes, replacement models that restart when cycles are completed (like a sequence of battery replacements over time), dams (important for railway bridge design, flood control, etc.), pharmacokinetics (the dynamics of the concentration of a drug in the blood stream with multiple dosing over time). These models have important properties in common. They are subject to uncertainty, and their evolution over time can be recorded as a tracing or realization (also called sample path) of key variables, which are random in nature. These key random variables are called state variables. The sample path contains information about the state variable as the process evolves over time. The range of values that the state variable can assume is called the state space. An important, basic property of the state variable is its probability distribution at finite time-points measured from time zero, or as time tends to infinity. A useful analytical tool for obtaining such probability distributions is the level crossing method for stochastic models, which was originated and first applied by the present researcher in 1974. This will be a dominant method of analysis used in the proposed research. We describe the method by considering a typical sample path of the state variable of interest, such as the workload in a queue, or the stock on hand in an inventory, or the monetary surplus of an insurance company in an actuarial risk model. Operating characteristics of such models often depend on the probability distribution of the state variable. In the level crossing method, the procedure for obtaining the probability distribution starts with the sample path, and considers level crossing rates of state-space levels, from time zero until the time point of interest. Level crossing rates turn out to be simple mathematical functions or integral transforms of the probability density function of the state variable. Thus the crossing rates and a simple, intuitive conservation law can be used to construct integral equations (having integrals as terms) by inspection! Explicit formulas for the probability distributions are obtained by solving these equations analytically or numerically. The level crossing method is often faster, easier and more intuitive for deriving the desired probability distributions, than other methods of analysis. The proposed work will also address more complex models and use more general level crossing methods to analyze them. It will also apply other quantitative techniques including: probability theory, applied probability, applied mathematics, differential and integral equations, computer programming, simulation, renewal theory, regenerative processes, Markov processes. It will offer alternative solution methods and points of view, and suggest new directions of research, which will aid researchers, practitioners and students to understand the models better. The proposed research is important because it will help to increase knowledge about a large class of stochastic models, while continuing to develop the level crossing methodology. It will add and apply a set of useful new tools to the scientific literature. It will derive interesting, practical relationships between completely different models, obtain key expected first passage times, and get new results for inventories, risk models, etc. The anticipated outcome is a set of novel, useful analyses of stochastic models in a large scientific category. The work will benefit Canada by training new researchers and producing studies at the forefront of an important field of research.