Polling models : from theory to traffic intersections

The subject of the present monograph is the study of polling models, which are queueing models consisting of multiple queues, cyclically attended by one server. Polling models originated in the late 1950s, but did not receive much attention until the 1980s when an abundance of new applications arose in computer networks and in communication protocols. The name polling model originates from this application area, where central bank computers serve ("poll") their branch terminals in a cyclic fashion. Because of this renewed practical relevance, a huge literature on polling models has appeared throughout the years. The contribution of the present monograph is twofold. Firstly, it introduces polling models with features that are well-known in the context of single-queue systems, but have remained unexplored in polling systems. For example, we study polling models with customer priority levels, impatience and abandonments, and varying arrival rates. The second contribution is to provide a new approach to analyse vehicle actuated traffic intersections with exhaustive service, by applying and extending recent results in the polling literature. The thesis is divided into three parts. The first part, which starts after a general introduction and an extensive literature review, studies different types of customer behaviour in polling models. In particular, smart, stupid, and impatient customers are discussed. In the second part we study system behaviour in the sense that we introduce customer priority levels and discuss several service policies based on these priority levels. The third part focusses on traffic intersections and new approaches to apply results from the polling literature to this application area. We discuss the three parts in more detail below. As stated before, Part I introduces new types of customer behaviour in polling models. In Chapter 3 we study a polling model with arrival rates that depend on the server position. This extension destroys the convenient property that an arbitrary arriving customer sees the system in equilibrium, and a novel analysis technique is required to circumvent this complication. The model is practically relevant because it allows customers to choose which queue they want to join, knowing which queue is being served at their arrival epoch. Besides the practical relevance of the model, the analysis and the insights gained from it have great merit for future studies. This is illustrated by the model studied in Chapter 4, which is an extension of a vacation model with customer impatience, studied in existing literature. Using the analysis methodology of Chapter 3, we are able to obtain new results for the vacation model, and extend it to the context of polling models. In polling models, this is the first appearance of customer impatience, in queueing literature often referred to as reneging. Part II studies system behaviour through the introduction of customer priority levels. We study the impact of changing the order in which customers are served based on these priority levels. For example, we illustrate the effect of giving higher priority to customers with less service requirement. In Chapter 5 we study priorities in a standard polling system. Chapter 6 discusses the impact of using different service disciplines for customers with different priority levels, and shows how this can be used as a strategy to give even more advantages to customers with a higher priority level. The third part of the thesis is mainly dedicated to polling models for traffic intersections. Polling models have been applied to road traffic before, but only in the situation that one lane on a single road is blocked and traffic has to share the other lane. In Chapter 7 we develop a novel closed-form waiting time approximation for polling systems. The manner in which this approximation is created lends itself perfectly for adaption to the scenario where groups of multiple queues receive service simultaneously, as is the case in signalised traffic intersections. Chapter 8 presents this adaptation for traffic signals. Another aspect that is new in the analysis of delays at traffic intersections, is the analysis of conflicting traffic flows receiving a green light simultaneously. For this purpose, we use the approximation developed in Chapter 7 in combination with the results on polling models with priorities from Chapter 5.