R. Eddie Wilson

Traffic jams: dynamics and control

This introductory paper reviews the current state-of-the-art scientific methods used for modelling, analysing and controlling the dynamics of vehicular traffic. Possible mechanisms underlying traffic jam formation and propagation are presented from a dynamical viewpoint. Stable and unstable motions are described that may give the skeleton of traffic dynamics, and the effects of driver behaviour are emphasized in determining the emergent state in a vehicular system. At appropriate points, references are provided to the papers published in the corresponding Theme Issue.

Mechanisms for spatio-temporal pattern formation in highway traffic models

A key qualitative requirement for highway traffic models is the ability to replicate a type of traffic jam popularly referred to as a phantom jam, shock wave or stop-and-go wave. Despite over 50 years of modelling, the precise mechanisms for the generation and propagation of stop-and-go waves and the associated spatio-temporal patterns are in dispute. However, the increasing availability of empirical datasets, such as those collected from motorway incident detection and automatic signalling system (MIDAS) inductance loops in the UK or the next-generation simulation trajectory data (NGSIM) project in the USA, means that we can expect to resolve these questions definitively in the next few years. This paper will survey the essence of the competing explanations of highway traffic pattern formation and introduce and analyse a new mechanism, based on dynamical systems theory and bistability, which can help resolve the conflict.

Reconstructing the Traffic State by Fusion of Heterogeneous Data

This paper presents an advanced interpolation method for estimating smooth spatiotemporal profiles for local highway traffic variables such as flow, speed and density. The method is based on the “adaptive smoothing method” which takes as input stationary detector data as typically collected by traffic control centers. The authors generalize this method to allow for fusion with floating car data or other traffic information. The resulting profiles display transitions between free and congested traffic in great detail, as well as fine structures such as stop-and-go waves. The authors establish the accuracy and robustness of the method and demonstrate three potential applications: (1) compensation for gaps in data caused by detector failure; (2) separation of noise from dynamic traffic information; and (3) the fusion of floating car data with stationary detector data.

Estimates of the bistable region in metal cutting

The classical model of regenerative vibration is investigated with new kinds of nonlinear cutting force characteristics. The standard nonlinear characteristics are subjected to a critical review from the nonlinear dynamics viewpoint based on the experimental results available in the literature. The proposed nonlinear model includes finite derivatives at zero chip thickness and has an essential inflexion point. In the case of the one degree-of-freedom model of orthogonal cutting, the existence of unstable self-excited vibrations is proven along the stability limits, which is strongly related to the force characteristic at its inflexion point. An analytical estimate is given for a certain area below the stability limit where stable stationary cutting and a chaotic attractor coexist. It is shown how this domain of bistability depends on the theoretical chip thickness. The comparison of these results with the experimental observations and also with the subcritical Hopf bifurcation results obtained for standard nonlinear cutting force characteristics provides relevant information on the nature of the cutting force nonlinearity.

Theoretical Maximum Capacity as Benchmark for Empty Vehicle Redistribution in Personal Rapid Transit

A personal rapid transit system uses compact, computer-guided vehicles running on dedicated guideways to carry individuals or small groups directly between pairs of stations. Vehicles move on demand when a passenger requests service at his or her origin station. Because the number of trips requested from a station need not equal the number of trips ending there, some vehicles must run empty to balance the flows. The empty vehicle redistribution (EVR) problem is to decide which empty vehicles to move and when and where to move them; an EVR algorithm makes these decisions in real time, as passengers arrive and request service. A method was developed for finding the theoretical maximum demand (with a given spatial distribution) that a given system could serve with any EVR algorithm, which provides a benchmark against which particular EVR algorithms can be compared. The maximum passenger demand that a particular EVR algorithm can serve can be determined by simulation and then compared with the benchmark. The method is applied to two simple EVR heuristics on two example systems. The results suggest that this is a useful method for determining the strengths and weaknesses of a variety of EVR heuristics across a range of networks, passenger demands, and fleet sizes.

Criteria for convective versus absolute string instability in car-following models

The linear stability properties of car-following models of highway traffic are analysed. A general family of models is introduced and the subsequent analysis developed in terms of its partial derivatives. Two measures of wave propagation, namely (i) the group velocity and (ii) the signal velocity, are introduced and computed. These measures are used to classify how instability propagates disturbances, measured relative to the frame of the road along which the vehicles drive. Detector data suggest that disturbances should propagate only in an upstream direction (convective upstream instability), and it is shown how to parametrize models to agree with data and avoid unrealistic downstream propagation (absolute and convective downstream instability).

Basins of attraction in nonsmooth models of gear rattle

This paper is concerned with the computation of the basins of attraction of a simple one degree-of-freedom backlash oscillator using cell-to-cell mapping techniques. This analysis is motivated by the modeling of order vibration in geared systems. We consider both a piecewise-linear stiffness model and a simpler infinite stiffness impacting limit. The basins reveal rich and delicate dynamics, and we analyze some of the transitions in the systems behavior in terms of smooth and discontinuity-induced bifurcations. The stretching and folding of phase space are illustrated via computations of the grazing curve, and its preimages, and manifold computations of basin boundaries using DsTool (Dynamical Systems Toolkit).

Dynamical pair approximation for cellular automata with shuffle update

The random shuffle update method for the asymmetric exclusion process (ASEP) is introduced, and the dynamical pair technique is extended in order to analyse its dynamics. A sequence of approximate models is introduced, the first element of which corresponds to the classical parallel update rule whose pair dynamics is reviewed. It is then shown how the argument may be extended inductively to solve for the two-cell configuration probabilities for each element of the sequence of approximate models. A formal limit is then taken, and macroscopic velocities and flow rates are derived.

Explicit periodic solutions in a model of a relay controller with delay and forcing

In this paper we use a combination of numerical and analytical methods to find and construct solutions of a cameo model of relay control, formulated as a piecewise-constant delay differential equation (DDE). Numerical solutions of a related equation, where the discontinuities of the original DDE are smoothed out, are used to guide the construction of explicit solutions of the original DDE. On the other hand, the construction of explicit solutions provides starting data for numerical continuation of the smoothed equation. The stability of the explicit solutions can also be inferred from the numerical approach.

Sampling of Redistribution of Empty Vehicles for Personal Rapid Transit

A personal rapid transit (PRT) system uses compact, computer-guided vehicles running on dedicated guideways to carry individuals or small groups directly between pairs of stations. PRT vehicles operate on demand when a passenger requests service at his or her origin station. Because the number of trips requested from a station need not equal the number of trips ending there, some empty vehicles must move to balance the flows. The empty vehicle redistribution (EVR) problem prompts the decision to move empty vehicles reactively in response to known requests or proactively in anticipation of requests. This paper develops a new algorithm for the EVR problem called “sampling and voting” (SV). SV results in the choice of reactive movements with the use of a simple nearest-neighbor rule; the selection of proactive movements is derived by the generation of an ensemble of possible sequences of future passenger requests, the solution of a deterministic optimization problem for each sequence individually, and the discovery of the empty vehicle movements that are common among the sequences. Moving vehicles proactively is essential for providing short passenger waiting times. The new SV algorithm was tested in simulation with several case study systems, and it produced significantly shorter passenger waiting times than did existing EVR algorithms. Variants of the SV method developed here for PRT are also applicable to conventional taxi systems and emergency response systems.