Spectral analysis for performance evaluation in a bus network

Abstract

This paper deals with the performance evaluation of a public transportation system in terms of waiting times at various connection points. The behaviour of a bus network is studied in the framework of Discrete Event Systems (DES). Two possible operating modes of buses can be observed at each connection stop: periodic and non-periodic mode. Two complementary tools, Petri nets and (max, +) algebra, are used to describe the network by a non-stationary linear state model. This one can be solved after solving the structural conflicts associated to the graphical representation. From the characteristic matrix of the mathematical model, we determine eigenvalues and eigenvectors that we use to evaluate the connection times of passengers. This work is finally illustrated with a numerical example.

Introduction

Transportation is an essential component of contemporary economics (Commission, 2001). It has to face with two contradictions: a society that expects always more mobility and a public opinion that cannot bear any more chronic delays and the poor quality of the performance of some services. Indeed, the flexibility of individual transportation modes grew for some years whereas the public transport offer is not sometimes up to the demand. This partially explains the rise of urban traffic involving more pollution and risks of accidents.

To improve urban quality of life, one solution consists in making more attractive the collective transportation modes. It may also be promising to provide more security or more information to users or to ensure a better synchronization between public transport vehicles (bus–bus, train–train, bus–train) so as to reduce passengers waiting times and make the displacements as fast as possible. Nevertheless, before trying to improve any performance of a collective transportation mode, it is necessary to evaluate and analyse the strength and weakness of the existing offer, so as to identify critical points of the network (connection stops on a bus network for example). Then a comparison between supply and demand will allow operators to concentrate their efforts on those critical points, and will lead to specific and more efficient actions. This second phase may be performed in a predictive planning of the system (timetabling or resources assignment), or in a real time control of the network (re-scheduling in case of perturbations).

This paper deals with this first evaluation and analysis step with the purpose to improve service quality of public transport in urban centres. More precisely we consider the connection management of a bus network for which we study two possible operating modes of vehicles on the lines: in one hand, all buses perform their rounds according to a periodic timetable. On the other hand, an extension to the non-periodic working mode is worked out. Moreover, we consider a general non-synchronized behaviour of the buses which do not wait for each other at common interchange points.

Modelling, performance analysis and control of collective transportation networks are issues arousing an ever-increasing interest in many researches (Olsder et al., 1998, Nait et al., 2003, Houssin et al., 2006). There exist many research activities in the same field based on various modelling and analysis approaches. But most of existing works about planning and/or performance evaluation of transportation systems mainly concern railway networks in a periodic working case (Braker, 1991, Bussieck et al., 1997, De Vries et al., 1998, Olsder et al., 1998, Böcker et al., 2001). Such systems are synchronized ones as trains have to wait for each other so as to prevent passengers from missing their connection. Moreover, all these studies consider various criteria like punctuality and real time control rather than connection time minimization. For example De Vries et al. (1998) search for waiting times and propose robust solutions in case of weak perturbations with the aim to evaluate consequences of delays on future connections. Olsder et al. (1998) study the improvement of initial periodic timetable for an existing network with a fixed number of trains. Some studies about bus networks have also been carried out. For example, Karlaftis et al. (2004) propose a decision support system for planning bus operations using mathematical programming. Houssin et al. (2006) develop a timetable synthesis using (max, +) algebra.

The work reported in this paper is a part of a more general study that aims at evaluating and controlling public transportation systems by transposing, via dioid algebra, some classical analysis and control methods from systems theory using (max, +) algebra. (Max, +) algebra is well known to be rather adapted to problems which can be modelled with event graphs, and then which suppose the absence of conflicts. Nevertheless, as it can be seen for manufacturing systems, problems related with transportation networks often come from the occurrence of phenomena like synchronization between resources and conflicts that occur when sharing of resources is necessary. Then modelling and solving the studied problem in particular involves to apply an a priori arbitrating of those ones (Nait et al., 2002), or an a posteriori approach (Spacek et al., 1999).

To model the studied system, we first use a subclass of Petri nets which is known as the Dynamic Timed Event Graph with Withdrawal of Tokens (DTEGWT) (David and Alla, 1992, Lahaye, 2000). It represents an adequate tool which enables one to describe the concrete working of the network. Indeed, this class allows us to model conflicts and synchronization, for the two considered operating modes. Then we describe the obtained model by a state representation in (max, +) algebra, in spite of structural conflicts associated with the Petri net model. In spite of the complexity of this system, the use of dioid algebra allows us, on one hand, to obtain a linear state model and, on the other hand, to derive some properties quite easily. Besides, structural conflicts involve that the state model is expressed with a non-stationary system in (max, +) algebra.

In order to analyse the system and to evaluate the connection times of each passenger, we can use two approaches from the (max, +) state representation. The first one classically consists in solving this state model (Nait et al., 2005). In this paper we propose an alternative approach which avoids solving this system, and is based on spectral theory of the characteristic matrix of the (max, +) model. It consists in determining eigenvalues and eigenvectors that enable us to evaluate the passengers waiting times.

This paper is organised as follows: in Section 2, we describe the studied transportation network. In Section 3, we propose a modelling of the considered bus network. In Section 4, we propose a routing policy adapted to the periodic operating mode; this policy aims at arbitrating a priori the identified structural conflicts. Section 5 deals with the system performance analysis using spectral theory in dioid algebra. In Section 6, we extend our study to the non-periodic operating mode. In this case, another routing policy is provided to solve conflicts and spectral theory is also applied. To illustrate obtained results, a numerical example is worked out in Section 7. Last section gives some conclusions and suggestions for further researches.