PhD Defense: Lu Wang

Lu Wang

Stochastic Models for Service and Taxi Systems

(Under the direction of Vidyadhar Kulkarni)

This dissertation consists of two topics: inventory systems and service systems.

The first topic involves a single-item inventory system with two demand classes with backorders. We introduce a four-parameter (A, B, C₁, C₂) policy to manage such a system. Under this policy, we place an order of size A if the on-hand inventory level is less than or equal to B, and reject demands of class k if the inventory level is below or at C_k (k = 1, 2). We develop methods of computing the long-run average cost that can be used to numerically obtain the four parameters that minimize this average cost. When the demands arrive according to Poisson processes and the production lead times are exponential and the order size A is fixed, we formulate the problem as a Markov Decision Process. We prove structural properties when A = 1, and numerically show that the four-parameter policies are optimal when A > 1. We also study the cases where demand interarrival times or production lead times are generally distributed. The numerical optimality is done using the Genetic Algorithm.

The second topic considers a system of customers and taxis with Poisson arrivals and exponential patience times and delayed matching. We formulate the system as a Continuous Time Markov Chain and study the fluid and diffusion approximations. We consider Kurtz’s Approximation and Gaussian Approximation, and compare their performance numerically with simulation. We next formulate an optimal control problem to maximize the total net revenue over a fixed time horizon by controlling the arrival rate of taxis. We solve the optimal control problem numerically and compare its performance to simulation. We propose a heuristic control policy. We show that the expected regret of the heuristic policy is a bounded function of the horizon.