Dynamics and Stochasticity in Transportation Systems

Tools for Transportation Network Modelling

Giulio E Cantarella, David Watling, Stefano de Luca, Roberta Di Pace

• Publisher: Elsevier
• Year: 2019
• ISBN:
  9780128143544
• Topic: Law
• Subtopic: Law Theory & Practice
• Index: Law

Dynamics and Stochasticity in Transportation Systems: Solutions for Transportation Network Modeling breaks new ground on the topics, providing consistent and comprehensive coverage of steady state equilibrium and dynamic assignment within a common strategy. The book details the most recent advances in network assignment, including day-to-day and within-day dynamics, providing a solid foundation to help transportation planners solve transient overload and other problems. Users will find a book that fills the gap in knowledge with its description on how to use and employ the latest dynamic network models for evaluation of traffic and transport demand interventions.

This book demystifies the many different dynamic traffic assignment approaches and requires no previous knowledge on the part of the reader. All results are fully described and proven, thus eliminating the need to seek out other references. The skills described will appeal to transportation professionals, researchers and graduate students alike.

• Presents a consistent and comprehensive theory on steady state equilibrium assignment and day-to-day dynamic assignment models within a common framework
• Describes and solves modeling calculations in detail, with no need to reference other sources
• Includes numerical and graphical examples, text boxes and summaries at the end of each chapter to help readers better understand theoretical components
• Includes primary mathematical tools necessary for each dynamic model, easing comprehension

Chapter 1

Introduction

Giulio Erberto Cantarella University of Salerno, Salerno, Italy

Abstract

In this chapter the reader may find the basic definitions and assumptions used to develop the models presented in the next chapters. This is consistent with the methodology common to applied sciences whose first steps are a preliminary analysis aimed at providing a simplified description of the system under study and a statement of assumptions about space and time modelling possibly including stochastic approaches.

Keywords

Transportation networks; Dynamics in transportation systems; Stochasticity in transportation systems; Travel demand assignment

The knowledge of first principles, as space, time, motion, number,

is as sure as any of those which we get from reasoning.

And reason must trust these intuitions of the heart,

and must base on them every argument.

Blaise Pascal, Pensées (translated by W. F. Trotter).

Outline. In this chapter the reader may find the basic definitions and assumptions used to develop the models presented in the next chapters. This is consistent with the methodology common to applied sciences whose first steps are a preliminary analysis aimed at providing a simplified description of the system under study and a statement of assumptions about space and time modelling possibly including stochastic approaches.

A model for transportation system analysis tries to describe the state of a transportation system as resulting from the interaction between travellers willing to travel and transportation supply systems providing opportunities to them. In the this section and the following ones we outline the methodology, common to all modern applied sciences such as engineering and economics, applied in this book and the main assumption supporting it. Reader of this book is assumed familiar with fundamentals of Calculus, of Theory of Probability, and of Inferential Statistics.

Through all this book we keep clearly separated features of the real world, say (data from) observations, and those of the modelling tools trying to use different words as far as we can; for instance location over space and evolution over time come from observations of the real word, while spatial and dynamic are adjectives only used to denote models trying to describe these phenomena. Similar considerations hold with regards to observed variations or fluctuations over space or time in real world and to adjectives such as random or stochastic only used to denote models trying to describe these variations as well as other sources of uncertainty. The following examples may help understanding this point and to introduce some basic elements of modelling theory.

Suppose that you are walking a path in a wood looking for seeds of horse-chestnut trees to play conkers with a friend you expect will pay you a visit in the near future (or for any other reason). You may observe the number of seeds at the foot of each tree along the path in a given day, location over space, and/or at the foot of a tree in consecutive days, evolution over time. The sample of observations can be collected in an array of variables, such that the index describes the tree (space) or the day (time).

If the number of observations is too large to be easily remembered you may compute the observed mean to have a roughly idea of the number seeds you may find together with the observed variance to describe how dispersed the observations are around the observed mean. Both these values, as well as others indicators from descriptive statistics, are to be considered as observations since they can be computed from them through (simple algebraic) equations. [This aggregation procedure introduces a source of uncertainty somehow different from the one discussed below.]

A more sophisticated approach is based on probabilistic models based on random variables, such that the model mean and the model variance try to reproduce the observed mean or variance, and more generally the probability (density) function tries to mimic the observed frequency distribution; an example are models based on the Poisson random variable, popular models for both location over space and evolution over time. These models are not considered proper spatial or dynamic models since they lack an explicit description of location over space or of evolution over time.

Spatial or dynamic models are those used to explicitly forecast the location over space (in a future day) or of the evolution over time (in a given location). Models including random variables are used when forecasting is affected by uncertainty due to the lack of enough information and possibly other sources of uncertainty, see below so that effective deterministic models cannot be specified; according to these considerations dynamic models are named deterministic or stochastic processes. [This source of uncertainty due to lack of enough information is somehow different from the one due to the aggregation procedure discussed above.]

After its specification, any kind of model should be calibrated against observed data, say its parameters should be estimated through inference statistics methods, before a practical application is possible. This issue is out of the scope of this book.

The modelling approach discussed above can be extended multi-class models which also take into account the distribution over other quantitative features, the size of seeds besides their number, or qualitative features, say different types of items to collect such as walnut seeds or mushrooms. Multi-type models occur when the distribution of these features is duly modelled, usually through a probabilistic model.

All the above considerations hold in other fields of application as well: if we change the path with a long urban street, the seeds with cars, and the trees with links we get the main elements of Traffic Theory, briefly presented in Appendix B to this book.

Before any (mathematical) model can be developed a preliminary analysis should be carried out aiming at highlighting the most relevant features and providing a simplified (verbal qualitative) description of the system under study, as briefly reviewed in the beginning of Section 2.2 in the Preface for transportation systems. Main elements are repeated below for reader's convenience together with new considerations.

Users can be travellers, or persons, and freight, or goods. This book main emphasis is on travellers, but most described models can relatively easily adapted to freight transportation. In the following a user may mean a walking person, a person riding a bicycle, a vehicle, a ton of freight, a (20 ft long) container, …, thus pronoun “it” is utilised.

Several types of user choice behaviour occur in real life; this book mostly focuses on two of them only [HYP ①]:

• • driving, concerning interactions between users moving on the same facility (called congestion) and their effects on travel time, …;
• • routing, concerning connections between origin and destination of the journey, parking location and type, possibly departing time, … .

User driving behaviour is usually modelled within the transportation supply models, which describe (if and) how routing user choices affect level of service, say travel time, delay at junctions, monetary cost, …. On the other hand, user routing behaviour is usually modelled within the travel demand models, which describe how provided level of service affects routing user choices.

Transportation supply systems can be distinguished between those providing continuous over space and time services (walk, bicycles, cars, vans, trucks, …) or discrete services (buses, trains, airplanes, ferries, …), often requiring quite different modelling approaches; the geographical scale, urban/metropolitan areas vs. extra-urban connections, is also a relevant features. Discrete service transportation systems operating in urban areas are often called transit systems.

Making a sharp distinction, a discrete service system may be called:

• • frequency-based: most users arrive at random at stops without any pre-trip planning since they do not precisely know the timetable or the frequency is so high that it does not matter, thus users perceive the system as a set of l...