Foundations of Stochastic Modeling

This is a core PhD-level course providing an in-depth introduction to probability theory and stochastic processes. The objective is to equip students with the mathematical tools and theoretical understanding necessary to model and analyze complex systems subject to uncertainty. The course begins with a rigorous review of probability theory, followed by an exploration of key stochastic processes including the Poisson process, discrete-time Markov chains, and continuous-time Markov chains. The course also introduces queueing theory as an important application of stochastic modeling, preparing students to tackle real-world problems in service operations. Throughout the course, emphasis is placed on developing rigorous theoretical insights and proof writing skills needed for advanced research in stochastic systems.