Instantaneous control for traffic flow

The solution methods for optimal control problems with coupled partial differential equations as constraints are computationally costly and memory intensive; in particular for problems stated on networks, this prevents the methods from being relevant. We present instantaneous control problems for the optimization of traffic flow problems on road networks. We derive the optimality conditions, investigate the relation to the full optimal control problem and prove that certain properties of the optimal control problem carry over to the instantaneous one. We propose a solution algorithm and compare quality of the computed controls and run‐times. Copyright © 2006 John Wiley & Sons, Ltd.     

Car path tracking in traffic flow networks with bounded buffers at junctions

This article deals with the modeling for an individual car path through a road network, where the dynamics is driven by a coupled system of ordinary and partial differential equations. The network is characterized by bounded buffers at junctions that allow for the interpretation of roundabouts or on-ramps while the traffic dynamics is based on first-order macroscopic equations of Lighthill-Whitham-Richards (LWR) type. Trajectories for single drivers are then influenced by the surrounding traffic and can be tracked by appropriate numerical algorithms. The computational experiments show how the modeling framework can be used as navigation device.     

Polling systems in heavy traffic: Higher moments of the delay

We study an asymmetric cyclic polling model with general mixtures of exhaustive and gated service, and with zero switch-over times, in heavy traffic. We derive closed-form expressions for all moments of the steady-state delay at each of the queues, under standard heavy-traffic scalings. The expressions obtained provide new and useful insights into the behavior of polling systems under heavy-load conditions. This revised version was published online in June 2006 with corrections to the Cover Date.     

A dynamical systems approach based on averaging to model the macroscopic flow of freeway traffic

The flow of traffic exhibits distinct characteristics under different conditions, reflecting the congestion during peak hours and relatively free motion during off-peak hours. This requires one to use different mathematical equations to describe the diverse traffic characteristics. Thus, the flow of traffic is best described by a hybrid system, namely different governing equations for the different regimes of response, and it is such a hybrid approach that is investigated in this paper. Existing models for the flow of traffic treat traffic as a continuum or employ techniques similar to those used in the kinetic theory of gases, neither of these approaches gainfully exploit the hybrid nature of the problem. Spurious two-way propagation of disturbances that are physically unacceptable are predicted by continuum models for the flow of traffic. The number of vehicles in a typical section of the highway does not justify its being modeled as a continuum. It is also important to recognize that the basic premises of kinetic theory are not appropriate for the flow of traffic (see [S. Darbha, K.R. Rajagopal, Limit of a collection of dynamical systems: an application to modeling the flow of traffic, Mathematical Models and Methods in Applied Sciences 12 (10) (2002) 1381–1399] for a rationale for the same). A model for the flow of traffic that does not treat traffic as a continuum or use notions from kinetic theory is developed here and corroborated with real-time data collected on US 183 in Austin, Texas. Predictions based on the hybrid system model seem to agree reasonably well with the data collected on US 183.     

Macroscopic models for long-range dependent network traffic

A common way to inject long-range dependence in a stochastic traffic model possessing a weak regenerative structure is to make the variance of the underlying period infinite (while keeping the mean finite). This method is supported both by physical reasoning and by experimental evidence. We exhibit the long-range dependence of such a process and, by studying its second-order properties, we asymptotically match its correlation structure to that of a fractional Brownian motion. By studying a certain distributional limit theorem associated with such a process, we explain the emergence of an extremely skewed stable Lévy motion as a macroscopic model for the aforementioned traffic. Surprisingly, long-range dependence vanishes in the limit, being “replaced” by independent increments and highly varying marginals. The marginal distribution is computed and is shown to match the one empirically obtained in practice. Results on performance of queueing systems with Lévy inputs of the aforementioned type are also reported in this paper: they are shown to be in agreement with pre-limiting models, without violating experimental queueing analysis. This revised version was published online in June 2006 with corrections to the Cover Date.     

An extended multi-anticipative delay model of traffic flow

An extended multi-anticipative delay model is proposed by introducing multiple velocity differences and incorporating the reaction-time delay of drivers. The stability condition of the new model is obtained by applying the linear stability theory, and the modified Korteweg–de Vries (mKdV) equation is derived by the use of the nonlinear analysis method. The analytical and numerical results show that both the reaction-time delay of drivers and the information of multiple velocity differences have an important influence on the stability of the model, and that the stabilization of traffic flow is enhanced by appending the velocity difference information of multiple vehicles ahead or by decreasing the delay time.     

Simulating vehicular traffic in a network using dynamic routing

We present a new numerical code which solves the Lighthill – Whitham model, the classic macroscopic model for vehicular traffic flow, in a network with multi-destinations. We use a high-resolution shock-capturing scheme with approximate Riemann solver to solve the partial differential equations of the Lighthill – Whitham theory. These schemes are very efficient, robust and moreover well adapted to simulations of traffic flows. We develop a theory of dynamic routing including a procedure for traffic flow assignment at junctions which reproduces the correct propagation of irregularities and ensures at the same time conservation of the number of vehicles.     

Analysis of traffic flow by the finite element method

A finite elernent methodology is developed for the numerical solution of traffic flow problems encountered in arterial streets. The simple continuum traffic flow model consisting of the equation of continuity and an equilibrium flow-density relationship is adopted. A Galerkin type finite element method is used to formulate the problem in discrete form and the solution is obtained by a step-by-step time integration in conjunction with the Newton-Raphson method. The proposed finite element methodology, which is of the shock capturing type, is applied to flow traffic problems. Two numerical examples illustrate the method and demonstrate its advantages over other analytical or numerical techniques.     

Dynamic programming—neural network real-time traffic adaptive signal control algorithm

In this paper, an “intelligent” isolated intersection control system was developed. The developed “intelligent” system makes “real time” decisions as to whether to extend (and how much) current green time. The model developed is based on the combination of the dynamic programming and neural networks. Many tests show that the outcome (the extension of the green time) of the proposed neural network is nearly equal to the best solution. Practically negligible CPU times were achieved, and were thus absolutely acceptable for the “real time” application of the developed algorithm.     

A hybrid macroscopic-based model for traffic flow in road networks

Recently, the unquestionable growth of interest to increase the operational efficiency and capability of transportation systems led to the development of a large number of traffic modeling theories. One of the major operational issues when developing a transportation system management model lies in the selection of the appropriate methodological approach with respect to several decisions, such as the selection of the type of input and output data as well as the qualitative representation and the computational power of the model. Despite the considerable effort in the area, there is still not an approach which per se models effectively the various dynamically evolving features of traffic in road networks. The present paper addresses this issue by introducing a new hybrid approach which combines the complementary features and capabilities of both continuum mathematical models e.g. 1, 6, 23 and 26 and knowledge-based models e.g. 7, 22 and 28 in order to describe effectively traffic flow in road networks.     

Polling systems in heavy traffic: Exhaustiveness of service policies

We study the expected delay in cyclic polling models with general ‘branching-type’ service disciplines. For this class of models, which contains models with exhaustive and gated service as special cases, we obtain closed-form expressions for the expected delay under standard heavy-traffic scalings. We identify a single parameter associated with the service discipline at each queue, which we call the ‘exhaustiveness’. We show that the scaled expected delay figures depend on the service policies at the queues only through the exhaustiveness of each of the service disciplines. This implies that the influence of different service disciplines, but with the same exhaustiveness, on the expected delays at the queues becomes the same when the system reaches saturation. This observation leads to a new classification of the service disciplines. In addition, we show monotonicity of the scaled expected delays with respect to the exhaustiveness of the service disciplines. This induces a complete ordering in terms of efficiency of the service disciplines. The results also lead to new rules for optimization of the system performance with respect to the service disciplines at the queues. Further, the exact asymptotic results suggest simple expected waiting-time approximations for polling models in heavy traffic. Numerical experiments show that the accuracy of the approximations is excellent for practical heavy-traffic scenarios. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analytical studies on the instabilities of heterogeneous intelligent traffic flow

It has been widely reported in literature that a small perturbation in traffic flow such as a sudden deceleration of a vehicle could lead to the formation of traffic jams without a clear bottleneck. These traffic jams are usually related to instabilities in traffic flow. The applications of intelligent traffic systems are a potential solution to reduce the amplitude or to eliminate the formation of such traffic instabilities. A lot of research has been conducted to theoretically study the effect of intelligent vehicles, for example adaptive cruise control vehicles, using either computer simulation or analytical method. However, most current analytical research has only applied to single class traffic flow. To this end, the main topic of this paper is to perform a linear stability analysis to find the stability threshold of heterogeneous traffic flow using microscopic models, particularly the effect of intelligent vehicles on heterogeneous (or multi-class) traffic flow instabilities. The analytical results will show how intelligent vehicle percentages affect the stability of multi-class traffic flow.     

Connectivity threshold and recovery time in rank-based models for complex networks

We study a generalized version of the protean graph (a probabilistic model of the World Wide Web) with a power law degree distribution, in which the degree of a vertex depends on its age as well as its rank. The main aim of this paper is to study the behaviour of the protean process near the connectivity threshold. Since even above the connectivity threshold it is still possible that the graph becomes disconnected, it is important to investigate the recovery time for connectivity, that is, how long we have to wait to regain the connectivity.     

Polynomial viscosity methods for multispecies kinematic flow models

Multispecies kinematic flow models are defined by systems of strongly coupled, nonlinear first‐order conservation laws. They arise in various applications including sedimentation of polydisperse suspensions and multiclass vehicular traffic. Their numerical approximation is a challenge since the eigenvalues and eigenvectors of the corresponding flux Jacobian matrix have no closed algebraic form. It is demonstrated that a recently introduced class of fast first‐order finite volume solvers, called polynomial viscosity matrix (PVM) methods [M. J. Castro Díaz and E. Fernández‐Nieto, SIAM J Sci Comput 34 (2012), A2173–A2196], can be adapted to multispecies kinematic flows. PVM methods have the advantage that they only need some information about the eigenvalues of the flux Jacobian, and no spectral decomposition of a Roe matrix is needed. In fact, the so‐called interlacing property (of eigenvalues with known velocity functions), which holds for several important multispecies kinematic flow models, provides sufficient information for the implementation of PVM methods. Several variants of PVM methods (differing in polynomial degree and the underlying quadrature formula to approximate the Roe matrix) are compared by numerical experiments. It turns out that PVM methods are competitive in accuracy and efficiency with several existing methods, including the Harten, Lax, and van Leer method and a spectral weighted essentially non‐oscillatory scheme that is based on the same interlacing property. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1265–1288, 2016     

Generalized M/G/C/C state dependent queueing models and pedestrian traffic flows

The generality and usefulness ofM/G/C/C state dependent queueing models for modelling pedestrian traffic flows is explored in this paper. We demonstrate that the departure process and the reversed process of these generalizedM/G/C/C queues is a Poisson process and that the limiting distribution of the number of customers in the queue depends onG only through its mean. Consequently, the models developed in this paper are useful not only for the analysis of pedestrian traffic flows, but also for the design of the physical systems accommodating these flows. We demonstrate how theM/G/C/C state dependent model is incorporated into the modelling of large scale facilities where the blocking probabilities in the links of the network can be controlled. Finally, extensions of this work to queueing network applications where blocking cannot be controlled are also presented, and we examine an approximation technique based on the expansion method for incorporating theseM/G/C/C queues in series, merge, and splitting topologies of these networks.     

On air traffic flow management with rerouting. Part II: Stochastic case

We present a framework for modeling multistage mixed 0-1 problems for the air traffic flow management problem with rerouting (ATFMRP) under uncertainty in the airport arrival and departure capacity, the air sector capacity and the flight demand. The model allows for flight cancelation, if necessary. It considers several types of objective functions to minimize, namely, total ground and air holding cost, penalization of the alternative routes to the scheduled one for each flight, delay cost for the flights to arrive to the airports and the air sector nodes, and penalization for advancing the arrival of the flights to the airport over the scheduled period. A scenario tree based scheme is used to represent the Deterministic Equivalent Model (DEM) of the stochastic mixed 0-1 program with full recourse. The nonanticipativity constraints that equate the so named common 0-1 and continuous variables from the same group of scenarios in each period are implicitly satisfied in the compact representation of DEM. Some computational experience is reported for medium-scale instances. The model is so tight that none of the instances of the testbed but two of them requires to execute the branch-and-cut phase of the MIP optimization engine of choice.     

A general model of road traffic accidents

A general model that relates road traffic accidents to the number of vehicles involved, and the number of primary causes of such accidents, is presented. The model considers traffic accidents as failures of a road traffic network system to meet social and economic constraints, and therefore as a measure of the unreliability of such a system. The equations apply to accidents in the real time domain as well as to mean values per unit of vehicle exposure time or vehicle distance. They also apply to single vehicles and drivers, groups of drivers and fleets of vehicles, and the entire vehicle and driving population. They can be used for sections of a network or for a whole network.The equations used have a large number of terms, hence bias errors are common in road accident investigations.     

Application of averaging techniques to traffic flow theory

Real traffic data are very versatile and are hard to explain with the so‐called standard fundamental diagram. A simple microscopic model can show that the heterogeneity of traffic results in a reduced mean flow and that the reduction is proportional to the density variance. Standard averaging techniques allow us to evaluate this reduction without having to describe the complex microscopic interactions. Using a second equation for the variance results in a two‐dimensional hyperbolic system that can be put in conservative form. The Riemann problem is completely solved in the case of a parabolic fundamental diagram, and the solutions are compared with the famous second‐order Aw–Rascle–Zhang model in a simulation of lane reduction. Adding a diffusion term results in entropy production, and the diffusive model is studied as well. Finally, a numerical scheme is used and converges to the analytical solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Sharp critical thresholds for a class of nonlocal traffic flow models

We study a class of traffic flow models with nonlocal look-ahead interactions. The global regularity of solutions depend on the initial data. We obtain sharp critical threshold conditions that distinguish the initial data into a trichotomy: subcritical initial conditions lead to global smooth solutions, while two types of supercritical initial conditions lead to two kinds of finite time shock formations. The existence of non-trivial subcritical initial data indicates that the nonlocal look-ahead interactions can help avoid shock formations, and hence prevent the creation of traffic jams.     

A path-based double projection method for solving the asymmetric traffic network equilibrium problem

In this paper we propose a new iterative method for solving the asymmetric traffic equilibrium problem when formulated as a variational inequality whose variables are the path flows. The path formulation leads to a decomposable structure of the constraints set and allows us to obtain highly accurate solutions. The proposed method is a column generation scheme based on a variant of the Khobotov’s extragradient method for solving variational inequalities. Computational experiments have been carried out on several networks of a medium-large scale. The results obtained are promising and show the applicability of the method for solving large-scale equilibrium problems. This work has been supported by the National Research Program FIRB/RBNE01WBBBB on Large Scale Nonlinear Optimization.