ESSAYS IN STOCHASTIC MODELING AND OPTIMIZATION

Abstract

Stochastic modeling plays an important role in estimating potential outcomes where randomness or uncertainty is present. This type of modeling forecasts the probability distributions of potential outcomes by allowing for random variation in one or more inputs over time under different conditions. One of the classic topics of stochastic modeling is queueing theory.Hence, the first part of the dissertation is about a stylized queueing model motivated by paid express lanes on highways. There are two parallel, observable queues with finitely many servers: one queue has a faster service rate, but charges a fee to join, and the other is free but slow. Upon arrival, customers see the state of each queue and choose between them by comparing the respective disutility of time spent waiting, subject to random shocks. This framework encompasses both the multinomial logit and exponomial customer choice models. Using a fluid limit analysis, we give a detailed characterization of the equilibrium in this system. We show that social welfare is optimized when the express queue is exactly at (but not over) full capacity; however, in some cases, revenue is maximized by artificially cre- ating congestion in the free queue. The latter behaviour is caused by changes in the price elasticity of demand as the service capacity of the free queue fills up. The second part of the dissertation is about a new optimal experimental design for linear regression models with continuous covariates, where the expected response is interpreted as the value of the covariate vector, and an “error” occurs if a lower- valued vector is falsely identified as being better than a higher-valued one. Our design optimizes the rate at which the probability of error converges to zero using a large deviations theoretic characterization. This is the first large deviations-based optimal design for continuous decision spaces, and it turns out to be considerably simpler and easier to implement than designs that use discretization. We give a practicable sequential implementation and illustrate its empirical potential.