Some general models of traffic flow in an isolated network

Starting form basic principles, we obtain mathematical models that describe the traffic of material objects in a network represented by a graph. We analyze existence, uniqueness, and positivity of solutions for some implicit models. Also, some linear models and their equilibria are analyzed. Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

A mixed ODE–PDE model for vehicular traffic

We present a traffic flow model consisting of a gluing between the Lighthill–Whitham and Richards macroscopic model with a first‐order microscopic following the leader model. The basic analytical properties of this model are investigated. Existence and uniqueness are proved, as well as the basic estimates on the dependence of solutions from the initial data. Moreover, numerical integrations show some qualitative features of the model, in particular the transfer of information among regions where the different models are used. Copyright © 2014 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.     

Window flow control in FIFO networks with cross traffic

We focus on window flow control as used in packet-switched communication networks. The approach consists in studying the stability of a system where each node on the path followed by the packets of the controlled connection is modeled by a FIFO (First-In-First-Out) queue of infinite capacity which receives in addition some cross traffic represented by an exogenous flow. Under general stochastic assumptions, namely for stationary and ergodic input processes, we show the existence of a maximum throughput allowed by the flow control. Then we establish bounds on the value of this maximum throughput. These bounds, which do not coincide in general, are reached by time-space scalings of the exogenous flows. Therefore, the performance of the window flow control depends not only on the traffic intensity of the cross flows, but also on fine statistical characteristics such as the burstiness of these flows. These results are illustrated by several examples, including the case of a nonmonotone, nonconvex and fractal stability region. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

Estimation and monitoring of traffic intensities with application to control of stochastic systems

The development of optimal control strategies for many stochastic models relies on the observed traffic intensity. However, implementation of such control strategies is often infeasible because of high operating costs induced by the fluctuations of traffic flows. In this study, we propose a framework for estimating and monitoring the traffic intensities of stochastic systems. The framework does not require knowledge of any input traffic statistics, and it allows us to adaptively estimate the intensity function over time and simultaneously detect its significant changes so that the control strategy can be adjusted accordingly without requiring high operating costs. Finally, a canonical queueing system with various types of input traffic is used to evaluate the effectiveness of the proposed framework. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

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.     

Monotonicity of optimal flow control for failure-prone production systems

In this paper, we consider the problem of the optimal flow control for a production system with one machine which is subject to failures and produces one part type. In most previous work, it has been assumed that the machine has exponential up and down times, i.e., its state process is a Markov process. The system considered in our study has general machine up and down times. Our main result is establishing monotone properties for the optimal control policy.This work was partially supported by the National Science Foundation under Grants DDM-9215368 and EDI-9212122. The authors thank two anonymous reviewers for helpful comments and suggestions.     

Optimal control of generalized multiobjective games with application to traffic networks modeling

The purpose of this paper is to study some new results on the existence and convergence of the solutions to controlled systems of generalized multiobjective games, controlled systems of traffic networks, and optimal control problems (OCPs). First, we introduce the controlled systems of generalized multiobjective games and establish the existence of the solutions for these systems using Browder-type fixed point theorem in the noncompact case and the $C_i$-quasi-concavity. Results on the convergence of controlled systems of the solutions for such problems using the auxiliary solution sets and the extended $C_i$-convexity of the objective functions are studied. Second, we investigate OCPs governed by generalized multiobjective games. The existence and convergence of the solutions to these problems are also obtained. Finally, as a real-world application, we consider the special case of controlled systems of traffic networks. Many examples are given for the illustration of our results.     

Erratum: flow control in gas networks: exact controllability to a given demand

We correct a technical error in the paper of Gugat, Herty, Schleper, Math. Methods Appl. Sci. 34 (2011), where a framework for controllability of quasi–linear hyperbolic systems has been studied. The application to the case of gas networks is specified in more detail in the current work. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Light traffic for workload in queues

General exact light traffic limit theorems are given for the distribution of steadystate workloadV, in open queueing networks having as input a general stationary ergodic marked point process {(t _n ,K _n )n0 (where t_n denotes the arrival time and K_n the routing and service times of the nth customer). No independence assumptions of any kind are required of the input. As the light traffic regime, it is only required that the Palm distribution for the exogenous interarrival time converges weakly to infinity (while the service mechanism is not allowed to change much). As is already known in the context of a single-server queue, work is much easier to deal with mathematically in light traffic than is customer delayD, and consequently, our results are far more general than existing results forD. We obtain analogous results for multi-channel and infinite-channel queues. In the context of open queueing networks, we handle both the total workload in the network as well as the workload at isolated nodes.Research supported in part by the Japan Society for the Promotion of Science during the author's fellowship in Tokyo, and by NSF Grant DDM 895 7825.     

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.     

Optimal Control for Traffic Flow Networks

We consider traffic flow models for road networks where the flow is controlled at the nodes of the network. For the analytical and numerical optimization of the control, the knowledge of the gradient of the objective functional is useful. The adjoint calculus introduced below determines the gradient in two ways. We derive the adjoint equations for the continuous traffic flow network model and derive also the adjoint equations for a discretized model. Numerical examples for the solution of problems of optimal control for traffic flow networks are presented.This author was supported by Deutsche Forschungsgemeinschaft (DFG), Grant KL 1105/5.     

A new cellular automaton model for traffic flow

Establishment of effective traffic models to reveal fundamental traffic characteristics is an essential requirement in the design, planning and operation of transportation systems. In 1992 Nagel and Schreckenberg presented a cellular automaton model describing traffic flow of N cars on a single lane and applied it in the famous project TRANSIMS on transportation simulation. In this paper, the author proposes a new model for the same problem and gives a comparison of simulation results with the former ones. The comparison shows that the new model works better under the condition of high traffic density.     

Optimal control laws for traffic flow

Optimal “on–off” laws for the traffic signals are developed based on the bilinear control problem with the binary constraints. A Lyapunov function based feedback law for regulating traffic congestions is developed. Also, a real-time optimal signal law is developed using a novel binary optimization method. Both methods are tested and compared, and our tests demonstrate that the both methods provide very effective and efficient traffic control laws.     

Instantaneous Shrinking of Supports for Nonlinear Reaction-convection Equations

Support analysis is performed for solutions of nonlinear reaction-convection equations in which both singular flux and source term are included. The positivity versus instantaneous shrinking for the solutions is determined by the relative strength of the flux and the source, as well as the decay rate at infinity of initial value of solutions. As an application of the analysis, the case of power-type nonlinearities is checked in details.     

Pricing and distributed QoS control for elastic network traffic

We study a processor-sharing model in which users choose between a high- and a low-priority service, based on their utility functions and prices charged by the service provider. The latter aims at revenue maximization. The model is motivated by file transmissions in data networks with distributed congestion control.