Delay Differential Equation Models for Queueing Theory

Abstract

In many service systems, customers are often presented with information about queue lengths or waiting times. This information has the potential to influence the decisions of customers on whether or not to join a queue to receive service. However, it is often the case that the information that customers receive is not given in real time and has some inherent delay. For example, the information provided could be updated periodically and therefore the customer receives information about the system from some time in the past. In this thesis, we focus our attention on fluid models of queueing systems. Generally, the fluid models in this thesis are the limiting objects of scaled stochastic systems and take the form of delay differential equations. We show that the dynamics of the queueing system can change dramatically depending on how large the delay in information is. Thus, our main goal in this thesis is to explore and understand how the type of delay and the size of the delay impact the underlying dynamics of different queueing models.