Mathematics in Transport Planning and Control

At roundabouts where entering traffic is required to give way to traffic circulating in the roundabout, the traffic capacity of each entry is a function of the flow of traffic circulating past it. This relationship has previously been analysed in two main ways: using a linear relationship based on regression and using a non-linear relationship based on a model of entering drivers' acceptance of gaps in the circulating traffic. The linear analysis has previously been extended to estimation of the reserve capacity or degree of overload of the roundabout as a whole in relation to a given pattern of approaching traffic. The non-linear analysis is extended similarly in this paper.

The relationships between entry capacity and circulating flow imply in turn that the capacity of each entry is a function of the entering flows and the proportions of traffic making various movements through the junction from some or all of the entries. Equations are established for determining derivatives of capacity or delay on each entry with respect to the demand flow for each movement. In particular, it is shown that when the roundabout is overloaded the capacity of an entry can depend upon the demand flow on that same entry, giving rise to a corresponding term in the derivative of the delay-flow relationship for the approach concerned.

The need to reduce traffic congestion is becoming increasingly important. The means of achieving this aim involves optimising parameters such as traffic signal green-times, road prices and public transport fares. This paper will describe a new bilevel method of optimising traffic signals and prices. The method uses the steepest descent direction together with projections in order to define a descent direction which will reduce the objective function subject to the overriding necessity to be in equilibrium.

The paper will provide a description of the bilevel method together with results on two simple problems. Optimisation is performed on two functions simultaneously; the equilibrium function E (which must have value zero for equilibrium) and the objective function Z which is minimised subject to the constraint that E is zero. For most traffic problems equilibrium is not mathematically well behaved and therefore the method approaches equilibrium in stages, at each stage it minimises Z whilst avoiding the difficult equilibrium region.

The increasing awareness of the importance of the wider objectives of traffic management and control has led to the work described in this paper. The aim of the study is to develop a flexible signal controller which may be configured so that it embodies the objectives appropriate for the situation in which it is to be used. This paper describes the optimisation of a prototype fuzzy logic signal controller with respect to several criteria simultaneously. Having demonstrated the controller's sensitivity to changes in its parameters, a multiobjective genetic algorithm (MOGA) optimisation technique is used to derive a family of solutions, each of which is optimal with respect to at least one of the criteria, whilst minimising the trade-off with respect to the other criteria.

Traffic signal control is one of the oldest application areas of fuzzy sets in transportation. In general, fuzzy control is found to be superior in complex problems with multi-objective decisions. In traffic signal control, several traffic flows compete for the same time and space, and different priorities are often set to different traffic flows or vehicle groups

The public transport priorities are a very important part of the effective traffic signal control. Normally, the public transport priorities are programmed by using special algorithms, which are tailor-made for each intersection. The experiences have proved that this kind of algorithms can be very effective if some compensation algorithms and the traffic-actuated control mode are used. We believe that using the fuzzified public transport priority algorithms, the measures of effectiveness of traffic signal control can be even better. In this paper, our fuzzy control algorithm of the public transport priorities will be presented.

A bi-level programming approach has been used to tackle an area traffic control optimisation problem subject to user equilibrium traffic flows. In the upper level problem, the signal timing plan for coordinated fixed time control has been defined. In the lower level problem, user equilibrium traffic assignment obeying Wardrop's first principle has been formulated as a variational inequality problem. Mathematical expressions for various components of the performance index in the upper level problem and the average delay in the lower level problem have been derived and reported (Chiou 1997a). A mixed search procedure has been proposed as the solution method to the bi-level problem and a range of numerical tests have been carried out (Chiou 1997b, 1998a,b). In this paper, further numerical tests are performed on Allsop and Charlesworth's (1977) road network in which various traffic loads are taken into account. Effectiveness in terms of the robustness and reliability of the mixed search procedure in congested and uncongested road networks is thus investigated further. Comparisons of the performance index resulting from the mixed search procedure and that of mutually consistent TRANSYT-optimal signal settings and traffic flows are made for the congested road network.

This paper addresses the problem of bunching of buses and how it may be counter-acted. An algorithm is presented for providing selective priority to buses at traffic signals according to their headways, the highest levels of priority being given to those buses with the highest headways, i.e. those buses which are running late or falling behind the bus in front.

Alternative selective priority strategies are evaluated in terms of their effects on bus journey time regularity, bus delay and general traffic delay. A simulation model SPLIT (Selective Priority for Late buses Implemented at Traffic signals) has been developed to investigate the performance of different priority strategies. The paper describes the details of the model, including bus stop dwell times and overlapping bus services, and compares results obtained from the model for a number of different priority strategies.

This work was driven by the keen interest in bus priority applications in London within the EC DGVII project INCOME.

In urban areas where roadworks are required, single lane shuttle operation is applied, especially where there is limited road space. There are operational problems relating to the site such as site geometry, visibility, length of roadworks zone, position of signs with other traffic control devices and signal timing. Other problems are mainly related to drivers’ behaviour and their compliance with traffic controls on site.

The reduced road width caused by the works will interrupt the free flow of traffic and it can also add to the risks to road users. In addition, shuttle operation may introduce long queues and increase delays especially during peak periods.

There is a need to identify those parameters and behaviours which might influence traffic performance in terms of safety and capacity. An investigation of four roadworks sites in urban roadworks within the Greater Manchester area was undertaken for this purpose. Parameters included in the examination were position of the STOP sign, signal timing, weather conditions, time headway, vehicle speed and percentages of heavy goods vehicles (HGV) in the traffic stream. Statistical analysis and comparisons between sites were conducted. Other factors related to the operation of the shuttle-lane were provided based on site observations.

The need to provide efficient public transport services in urban areas has led to the implementation of bus priority measures in many congested cities. Much interest has recently centred on priority at signal controlled junctions, including the concept of pre-signals, where traffic signals are installed at or near the end of a with-flow bus lane to provide buses with priority access to the downstream junction. Although a number of pre-signals have now been installed in the UK, particularly in London, there has been very little published research into the analysis of benefits and disbenefits to both buses and non-priority vehicles at pre-signalised intersections. This paper addresses these points through the development of analytical procedures which allow pre-implementation evaluation of specific categories of pre-signals.

The paper considers a discrete-time, Markov, stochastic process model of drivers' day-to-day evolving route choice, the evolving ‘state’ of such a system being governed by the traffic interactions between vehicles, and the adaptive behaviour of drivers in response to previous travel experiences. An approximating deterministic process is proposed, by approximating both the probability distribution of previous experiences—the “memory filter”—and the conditional distribution of future choices. This approximating process includes both flow means and variances as state variables. Existence and uniqueness of fixed points of this process are examined, and an example used to contrast these with conventional stochastic equilibrium models. The elaboration of this approach to networks of an arbitrary size is discussed.

This paper explores the use of some stochastic models for traffic assignment in the case of homogeneous traffic and simple networks. For non-dynamic routing we obtain asymptotic results in the form of paths representing time dependent evolution of traffic over routes. A functional limit theorem gives integral equations for the limiting fluid path which converges to an assignment satisfying Wardrop's first principle as time goes to infinity. For linear cost functions we are able to use the theory of large deviations to examine the way in which rare network overload events occur. In the case of dynamic assignment, we discuss the use of heavy traffic limits and Brownian models to examine the efficiency of network capacity usage when drivers choose routes according to conditions obtaining on entrance to the network. In particular we discuss the phenomenon of resource pooling.

Town centre redevelopment provides an opportunity to redesign the topology of the road network. Environmental and safety considerations will usually entail some streets being reserved for pedestrians only and others for access to car parks and service bays close to the pedestrian precinct. The effect of the new topology on traffic flow can and should be an input to the planning process. One way to appraise alternative schemes is to find the extent to which traffic could be routed to reduce the amount of conflict there will be between the different streams of traffic at the junctions; conflict at junctions is a simple measure indicating the potential for both accidents and congestion with its attendant pollution.

The problem of finding an assignment of flow such that conflict is minimised is a constrained optimisation problem. The main constraints are provision of routes for a demand specified by an origin-destination (OD) matrix. The objective function which measures the amount of conflict between the different streams is simple to formulate; it depends only on the topology of the network and the OD matrix of demand. But it is a quadratic function of a type which is not amenable to existing programming methods. However, the author has developed an iterative method for obtaining good, if not necessarily optimal, solutions.

The purpose of the paper is to demonstrate the appraisal of hypothetical networks involving four external zones. The OD matrix has been chosen to model some through as well as mostly inbound traffic flow. Various alternative schemes are appraised ranging from those which merely allow one access point to a car park within a ring of streets surrounding the pedestrian precinct, to those which retain one link across the ring and up to two car parks. This is a pilot study for a larger one involving the systematic analysis of hypothetical town centre topologies. So far it has shown the advantage of two access points over one, and the greater advantage of two separate car parks each with a single access point which allows each car park to be in a separate cell of the road network. The relative positions of the access points and the link across the ring also affect the amount of conflict.

In this paper we consider two traffic control strategies relying on user response to information and/or flow restriction. Ultimately, the control strategies are designed to function in real time, hence provide command values based on actual conditions and requiring little computational effort. The proposed control strategies are based on the idea that the network load, as measured by instantaneous travel times for instance, should be shared as equally as possible between paths. In order to achieve such an aim, the commands are designed to make the system state converge towards a state in which instantaneous travel times of paths relative to any given OD tend to be equal.

This paper investigates the temporal inflow profile that minimises the total cost of travel for a single route. The problem is formulated to consider the case in which the total demand to be serviced is fixed. The approach used here is a direct calculation of the first order variation of total system cost with respect to variations in the inflow profile. Two traffic models are considered; the bottleneck with deterministic queue and the kinematic wave model. For the bottleneck model a known solution is recovered. The wave model proves more difficult and after eliminating the possibility of a smooth inflow profile the restricted case of constant inflow is solved. As the space of possible profiles is finite dimensional in this case, the standard techniques of calculus apply. We establish a pair of equations that are satisfied simultaneously by the optimal inflow and time of first departure.

This paper develops a new algorithmic approach to equilibrium road traffic assignment which, by directly estimating differences, can more accurately estimate the impact of (relatively) small traffic schemes or changes in the demand pattern. Comparing the outputs of two independent traffic assignments to “with” and “without” scheme networks very often masks the effect of the scheme due to the “noise” in the resulting solutions. By contrast an incremental approach attempts to directly estimate the changes in link flows - and hence costs - resulting from (relatively) small perturbations to the network and/or trip matrix. The algorithms are based firstly on “route flows” as opposed to “link flows“, and secondly, they use a variant of the standard Frank-Wolfe algorithm known as “Social Pressure” which gives a greater weight to those O-D path flows whose costs are well above the minimum costs as opposed to those which are already at or near minimum. Tests on a set of five “real” networks demonstrate that the Social Pressure Algorithm is marginally better than Frank-Wolfe for single assignments but is very much faster and more accurate in predicting the impact of small network changes.

The traffic assignment problem aims to predict driver route choice, and is typically applied in the assessment of road schemes. The authors have previously published an SUE (Stochastic User Equilibrium) assignment algorithm, i.e. one which models variation in driver perception, and cost variation due to congestion. The algorithm works by minimising a function given by Sheffi and Powell (1982); in this paper the three terms of the function are investigated separately, and the possibility explored of constructing more sophisticated versions of the SUE algorithm.

This paper deals with two problems in transport network planning and control: trip matrix estimation and traffic signal optimisation. These two problems have both been formulated as bi-level programming problems with the User Equilibrium assignment as the second-level programming problem. One currently used method for solving the two problems consists of alternate optimisation of the two sub-problems until mutually consistent solutions are found. However, this alternate procedure does not converge to the solution of the bi-level programming problem. In this paper, a new algorithm will be developed and will be applied to two road networks.

The paper describes the conceptual and mathematical background of a tool for modelling and optimising transportation networks. The tool has been created to model resources and cargo flows in port terminal networks. The concept behind the tool is characterised by three features. Firstly the nodes are non-stationary and can be adapted to accommodate changed requirements in the transportation system. Secondly a high degree of complexity is built into the operational modes of the nodes. Thirdly, the concept incorporates an information network interacting closely with nodes and links. All networks can be optimised employing powerful algorithms and tools from system theory, neural networks and combinatorial graph theory.

This paper outlines a steady state multi-modal equilibrium transportation model which contains elastic demands and deterministic route-choices. The model may readily be extended to include some stochastic route-choice or mode choice. Capacity constraints and queueing delays are permitted; and signal green-times and prices are explicitly included. The paper shows that, under natural linearity and monotonicity conditions, for fixed control parameters the set of equilibria is the intersection of convex sets. Using this result the paper outlines a method of designing appropriate values for these control parameters; taking account of travellers' choices by supposing that the network is in equilibrium. The method may be applied to non-linear monotone problems by linearising about a current point. An outline justification of the method is given; a rigorous proof of convergence is as yet missing. Thus the method must now be regarded as a heuristic.

A method is derived for estimating a discrete choice model incorporating heteroscedasticities to reflect repeated measurement problems. Heterogeneity of each observation is characterised by a specific scale function and individual heterogeneity is introduced in the random utility choice model. This research proves that the unobserved influences affecting a specific individuals' mode choice are correlated from one of his or her selections to the next repeated questions. This research also suggest a strong evidence of learning effect, implying variances would be decrease as the responses faces repeated questions.

The appropriateness of traditional travel models has recently been questioned on the grounds that behaviour change represents an ongoing process subject potentially to the effect of new policy initiatives in the long term. The travel market is a dynamic function of change in lifecycle stage as well as underlying demographic, economic and level of service effects. Belfast has recently experienced the opening of two new rail projects. The decision to proceed with these projects was informed by two modelling exercises. The paper considers not only the models, forecasts and outturn travel behaviour change but the appropriateness of the models in the context of a dynamic market.

It is a feature of suburbanised towns and villages in the proximity of larger towns or cities, that the former patterns of rural travel behaviour change significantly. The nature of travel behaviour is mutable as a consequence of rural planning policy where small towns and villages are influenced by government area plans. Northern Ireland offers numerous examples of this phenomenon; and, as part of a wider investigation on sustainable development, this paper reports on a prototype study that assesses the pertinent views of residents within one suburbanised village under the remit of a defined area plan. Primarily, it describes the overall modelling strategy with an emphasis on:

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  differences, in respect of travel behaviour, between newcomers and established residents, and

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  pull factor weightings that illustrate why a suburbanised village is chosen by its residents.

differences, in respect of travel behaviour, between newcomers and established residents, and

pull factor weightings that illustrate why a suburbanised village is chosen by its residents.

More particularly, the paper discusses the merits and demerits of using a simplified hierarchical cross impact technique in modelling the perception of residents in their choice of dormitory settlement. In supporting this technique, the application of the Gaussian neighbourhood consensus function (Dodd, 1993) is introduced as an opinion profiling device.

Some cycle routes have proved unpopular because at several points a cyclist has to stop or slow very appreciably. Cyclists are discouraged by the resulting extra journey time. Three alternative hypotheses are made for a cyclist's performance in response to an imposed stop. In all he brakes uniformly from a cruising speed. In one, this speed is a personal standard, and he accelerates uniformly to regain it. In the second, because his energy is used in accelerating, he has less to use in cruising. The third hypothesis is that a cyclist's power output in accelerating is the same as in cruising, except at very low speeds, so that he attains his normal cruising speed asymptotically.

Trials were conducted on a lightly trafficked 2.5-km circuit with seven roundabouts. Cyclists made at least two circuits each, one non-stop and one stopping at roundabouts. Pairs of times for each cyclist were compared with derivations from each hypothesis. It was found that the estimates derived from the second hypothesis were much closer to the trial times than those given by the others. The conclusion is that making an adult cyclist stop is equivalent to extending the journey time by as much as an extra 50 m, approximately, would take. In reckoning the lengths of routes in order to select one for development, an equivalent distance of about 50 m should be added for each stop or near stop, unless they are close together.

This study contains an assessment of air pollution levels in Trafalgar Road, Greenwich. This is a congested road on a main route into central London, which has achieved notoriety since its residents took legal action in an attempt to restrict traffic. The four types of dispersion model used in the assessment are briefly described. All four models show that there is a likelihood that air quality standards for a number of pollutants, particularly PM₁₀, will be exceeded in the vicinity of busy congested roads in London. Zones where standards are exceeded are restricted to regions within 10m or so from the road. Any assessment has to take account of concentrations on a very fine scale (at distances of 10m from a road).

Geographic information for the home address of the accident casualty is obtained from the home-address post-code for each casualty. This allows the STATS 19 data base, the UK police system for reporting accidents, for the former Lothian Region in Scotland, 1990 to 1992, to be linked to social and economic indicators in the 1991 UK census and to the corresponding digitised boundaries at the smallest census geographical level (Output Areas, OAs) and post-code sector level in Scotland. For each post-code sector Standardised Casualty Ratio (SCR) which is commonly used in epidemiology to study rare diseases is calculated from the ratio of the number of casualties observed to that expected in the area. Adjusted SCRs are calculated, they are the ratios of the numbers of casualties predicted by social and economic factors that are measured at the census using Poisson regression to the expected numbers. Empirical Bayes Estimates (EMEs) are applied to prevent the results from areas with small populations being shown as too extreme. Results from the analysis indicate that accident risk to residents from deprived areas is high compared with those from affluent areas. Finally maps that can be used to identify areas in Lothian where there is relatively high SCRs are presented.

The aim of this paper is to review the structure of road accident data, the database framework within which it is stored, and the potential for exploiting a hierarchical structure using multilevel statistical models. Most national accident databases regard accidents as the primary units of observation, with other characteristics stored as attributes. But it is more natural to picture the network, accidents, and other variables as a collection of related objects within a hierarchical system, which can be achieved using object-oriented database technology within a Geographical Information System (GIS) framework. This would permit more efficient data capture and storage, facilitate analysis of accident frequencies as a function of road layout, and facilitate the development of multilevel statistical models.

The technique of graphical modelling (Whittaker, 1990) can be used to identify the dependence relationships between variables representing characteristics of recorded road accidents. It allows large multi-dimensional tables to be analysed by looking for conditional independence relationships among the variables. The variables under study can often be divided into groups that are ordered in time or by a hypothesised causal assumption. For these situations graphical chain models (Whittaker, 1990) are used to explore causal relationships between the variables. Some examples are given for a six-dimensional and a ten-dimensional contingency table.

Increasing growth in car ownership and private car usage is leading to high levels of congestion and pollution in many busy towns and cities. A reliable and efficient public transport service is one of the solutions for future transport provision. The development of a novel cell-based Automatic Vehicle Locationing system for public transport application is presented in this paper. The technique uses the existing AVL system and combines it with cellular technologies. Little modification is required to turn the existing AVL sequential polling system to the simultaneous cell-based AVL polling system. The fundamental advantage of the cell-based approach to AVL is that direct communication between the vehicles and the cell-based station enables the vehicle fleet to be polled simultaneously. Cell-based AVL System offers possibility for improved public transport service and public transport priority through frequent update of vehicle position and vehicle information. Modelling of cell-based AVL systems using a simulation method is given in this paper. The results show that the polling time for a bus fleet using the cell-based technique is considerably lower than the conventional sequential polling AVL system.

The modelling and prediction of traffic flow is one of the future challenges for science. We present a simulation tool for an urban road network based on real-time traffic data and a cellular automaton model for traffic flow. This tool has been applied to the inner city of Duisburg. The quality of the reproduced traffic states is investigated with regard to vehicle densities and typical features of urban traffic.

The paper considers the application of neural networks to model driver decisions to change lane on a dual carriageway road. The lane changing process is treated as consisting of two decisions, namely motivation and opportunity. Separate backpropagation neural networks are applied to represent each of the two decisions. The trained motivation and opportunity neural network models are linked to produce a layered network which represents the complete lane changing process. Separate models are developed to represent the nearside to offside lane changing decision, and the offside to nearside lane changing decision. This paper describes the development of the model of the nearside to offside lane changing decision.

For model development, data were collected from several subject vehicle drivers. The results are presented and the implications considered. Selected data were applied to train the neural networks and then an independent subset of data were used to assess performance. When the complete nearside lane changing neural network model was presented with the unseen test examples, 93.3% of the examples were correctly predicted as a lane change or no lane change. These results are shown to be a considerable improvement on those obtained previously.

This contribution discusses a continuum model of large discrete networks in planar domains. For this model, the Kirchhoff law, boundary conditions and capacity constraints lead in a system optimisation approach to a infinite dimensional constrained optimisation problem and to “mixed” variational inequalities. Mixed finite element methods can be formulated for these variational inequalities such that computable discretizations of the continuum problem are obtained.

The mathematical models used to describe the dynamical behaviour of a group of closely-spaced road vehicles travelling in a single lane without overtaking are known as car-following models. This paper presents a novel car-following model, which differs from the traditional models by having an equilibrium solution that corresponds to consecutive vehicles having not only zero relative velocity, but also travelling at a certain desired distance apart. This new model is investigated using both numerical and analytical techniques. For many parameter values the equilibrium solution is stable to a periodic perturbation but, for certain parameter values, chaotic motion results. This shows that in congested traffic, even drivers attempting to follow a safe driving strategy, may find themselves driving in an unpredictable fashion.

We use a combination of continuum and car-following models to explore the potential impact of speed-controls on (i) decreasing travel times at times of congested flow; and (ii) increasing the safety of motorway flow approaching the site of an accident.

This research was prompted by work undertaken by the author on the efficiency of shipping operations in the Suez Canal. The physical limitations of the Canal allow only one-way movement of ships for the greater part of its length, and thus ships are organised in convoys. These convoys have fixed starting times, with normally just one convoy per day operating in each direction. When traffic is heavy in the southbound direction, a second (smaller) relief convoy is organised to reduce waiting times which can otherwise exceed 24 hours. The process can be analysed by means of a bulk-service queueing model, where convoys of ships correspond to service batches of customers.

The model has application in the many other fields of transport where relief services are supplied. For example, a coach or train operator will often provide a relief service when customer demand is high. The process may be extended to cover cases where relief is provided for the relief service, resulting in a “cascade” of relief service queues.

The estimation of queue length and delays in queues that are oversaturated for some part of a study period is of substantial importance in a range of traffic engineering applications. Whiting’s co-ordinate transformation has provided the basis for several approaches to this. We analyse this approach and present an explicit form for the derivative of queue length with respect to time, which we then use to establish various properties. We also report the results of numerical comparisons with exact formulae for certain special cases and show that these offer little or no advantage over the co-ordinate transformation approximations and can be computationally impractical in study periods of moderate duration.

Real-time simulation trials involving operational air traffic controllers are an essential stage in the validation of new ideas and computer assistance tools for air traffic control. This paper describes the business and technical background to such trials and then highlights two statistical issues which continue to complicate the design and reporting of trials:

Reconciling objective, subjective, quantitative and qualitative data: striking the right balance between controlled measurement and expert opinion;

Correlation and independence in sequences of data: designing cost-effective trials without over-sampling.

Currently, the European air transport system is experiencing an annual growth of 7%. With an increasing number of flights, airports are reaching their capacity limits and are becoming a bottleneck in the system. Mantea is a European Commission funded project dealing with this issue. This paper focuses on planning decision support tools for airport traffic controllers.

The objective of our planning tools is to achieve a better use of the available airport infrastructure (taxiways and runways). To generate a safe plan, many rules must be taken into account that restrict the usage of airport tarmac: international regulations, airport operational procedures, aircraft performance, weather conditions and sometimes even controller “usual practices”. To generate a realistic plan, extensive monitoring of the traffic situation as well as suitable timing must be achieved. In the life cycle of a flight, 11 out of 15 possible causes of delay occur in an interval of 10-20 minutes, between aircraft start-up request and push-back. This means that precise planning before the end of this period is highly improbable. On the other hand, planning after this period implies the need for fast responses from the system.

In the Mantea project, an architecture is proposed in which a co-operative approach is taken towards planning aircraft movements at the airport. Controllers will be supported by planning tools that help assigning routes and departure times to controlled vehicles, in planning runway allocation (departure sequence) and occupancies, and in monitoring plan progress during flight phases. The planning horizon relates to medium term operations, i.e. 2-20 minutes ahead. The Mantea planning tools implement the following functions: runway departure planning, routing, and plan conformance monitoring. The tools will reduce the controller's workload, increase the level of safety for airport surface movements, and reduce the number of delays and operating costs for the airliners.

In this paper, we will focus on the constraint satisfaction programming techniques used in Mantea for (1) runway departure planning, (2) itinerary search and taxi planning functions. The airport tarmac and runway vicinity air routes have been modelled as a graph. Real time constraints have brought us to develop an algorithm linear in complexity for the itinerary search problem. Operational pressure has led us to develop fast search strategies for scheduling (i.e. use of heuristics, hill climbing…).

This paper explores the use of the theory of games as a tool for modelling road usage decisions. By explicitly considering the interdependence among road users and their ‘payoffs’ (utility) from road use, an adapted symmetric form of the ‘game of chicken’ is developed. The paper explores the various possible equilibria in this game. The mixed strategy symmetric Nash equilibrium is derived and a probability of road usage by a single user is calculated from this. The latter depends explicitly upon the generalised costs of road usage that include both the monetary and time costs of travel. From the single road user probability of usage, measures of the expected number of users are derived. An examination is made of how these vary with variations in the level of the generalised cost of usage. This provides an insight into how the tools of game theory may aid the understanding of the generation and regulation of road congestion.

Road user charging has been proposed as a solution to the uncontrolled growth of traffic and congestion in urban areas. In the absence of evidence from real world applications, modelling techniques provide the best information about the potential impacts and benefits of different charging approaches. This research has employed an elastic demand network equilibrium model, as part of the well established SATURN suite of computer programs, to represent a series of alternative road user charging systems which have been proposed for practical application.

Results have been obtained for both the impacts of charges on the volume and spatial distribution of road travel demand and for aggregate measures of network performance, such as travel distances, times and costs. Some interesting issues have emerged regarding the overall performance of charging systems in comparison with prior expectations and the specific impacts of charges related to travel conditions, which attempt to approximate the economic theories of marginal cost pricing. In addition, doubts are raised regarding the ability of steady-state equilibrium models to provide plausible representations of behavioural responses to charges which may vary significantly in time and space. It is suggested that alternative modelling techniques may provide superior user response predictions.