Zero range model of traffic flow

A multi--cluster model of traffic flow is studied, in which the motion of cars is described by a stochastic master equation. Assuming that the escape rate from a cluster depends only on the cluster size, the dynamics of the model is directly mapped to the mathematically well-studied zero-range process. Knowledge of the asymptotic behaviour of the transition rates for large clusters allows us to apply an established criterion for phase separation in one-dimensional driven systems. The distribution over cluster sizes in our zero-range model is given by a one--step master equation in one dimension. It provides an approximate mean--field dynamics, which, however, leads to the exact stationary state. Based on this equation, we have calculated the critical density at which phase separation takes place. We have shown that within a certain range of densities above the critical value a metastable homogeneous state exists before coarsening sets in. Within this approach we have estimated the critical cluster size and the mean nucleation time for a condensate in a large system. The metastablity in the zero-range process is reflected in a metastable branch of the fundamental flux--density diagram of traffic flow. Our work thus provides a possible analytical description of traffic jam formation as well as important insight into condensation in the zero-range process.Comment: 10 pages, 13 figures, small changes are made according to finally accepted version for publication in Phys. Rev.

Hybrid stochastic kinetic description of two-dimensional traffic dynamics

In this work we present a two-dimensional kinetic traffic model which takes into account speed changes both when vehicles interact along the road lanes and when they change lane. Assuming that lane changes are less frequent than interactions along the same lane and considering that their mathematical description can be done up to some uncertainty in the model parameters, we derive a hybrid stochastic Fokker-Planck-Boltzmann equation in the quasi-invariant interaction limit. By means of suitable numerical methods, precisely structure preserving and direct Monte Carlo schemes, we use this equation to compute theoretical speed-density diagrams of traffic both along and across the lanes, including estimates of the data dispersion, and validate them against real data

The Traffic Phases of Road Networks

We study the relation between the average traffic flow and the vehicle density on road networks that we call 2D-traffic fundamental diagram. We show that this diagram presents mainly four phases. We analyze different cases. First, the case of a junction managed with a priority rule is presented, four traffic phases are identified and described, and a good analytic approximation of the fundamental diagram is obtained by computing a generalized eigenvalue of the dynamics of the system. Then, the model is extended to the case of two junctions, and finally to a regular city. The system still presents mainly four phases. The role of a critical circuit of non-priority roads appears clearly in the two junctions case. In Section 4, we use traffic light controls to improve the traffic diagram. We present the improvements obtained by open-loop, local feedback, and global feedback strategies. A comparison based on the response times to reach the stationary regime is also given. Finally, we show the importance of the design of the junction. It appears that if the junction is enough large, the traffic is almost not slowed down by the junction.Comment: 37 page

A unified approach to the performance analysis of caching systems

We propose a unified methodology to analyse the performance of caches (both isolated and interconnected), by extending and generalizing a decoupling technique originally known as Che's approximation, which provides very accurate results at low computational cost. We consider several caching policies, taking into account the effects of temporal locality. In the case of interconnected caches, our approach allows us to do better than the Poisson approximation commonly adopted in prior work. Our results, validated against simulations and trace-driven experiments, provide interesting insights into the performance of caching systems.Comment: in ACM TOMPECS 20016. Preliminary version published at IEEE Infocom 201

Boltzmann-type models with uncertain binary interactions

In this paper we study binary interaction schemes with uncertain parameters for a general class of Boltzmann-type equations with applications in classical gas and aggregation dynamics. We consider deterministic (i.e., a priori averaged) and stochastic kinetic models, corresponding to different ways of understanding the role of uncertainty in the system dynamics, and compare some thermodynamic quantities of interest, such as the mean and the energy, which characterise the asymptotic trends. Furthermore, via suitable scaling techniques we derive the corresponding deterministic and stochastic Fokker-Planck equations in order to gain more detailed insights into the respective asymptotic distributions. We also provide numerical evidences of the trends estimated theoretically by resorting to recently introduced structure preserving uncertainty quantification methods

Hamiltonian traffic dynamics in microfluidic-loop networks

Recent microfluidic experiments revealed that large particles advected in a fluidic loop display long-range hydrodynamic interactions. However, the consequences of such couplings on the traffic dynamics in more complex networks remain poorly understood. In this letter, we focus on the transport of a finite number of particles in one-dimensional loop networks. By combining numerical, theoretical, and experimental efforts, we evidence that this collective process offers a unique example of Hamiltonian dynamics for hydrodynamically interacting particles. In addition, we show that the asymptotic trajectories are necessarily reciprocal despite the microscopic traffic rules explicitly break the time reversal symmetry. We exploit these two remarkable properties to account for the salient features of the effective three-particle interaction induced by the exploration of fluidic loops

Uncertainty damping in kinetic traffic models by driver-assist controls

In this paper, we propose a kinetic model of traffic flow with uncertain binary interactions, which explains the scattering of the fundamental diagram in terms of the macroscopic variability of aggregate quantities, such as the mean speed and the flux of the vehicles, produced by the microscopic uncertainty. Moreover, we design control strategies at the level of the microscopic interactions among the vehicles, by which we prove that it is possible to dampen the propagation of such an uncertainty across the scales. Our analytical and numerical results suggest that the aggregate traffic flow may be made more ordered, hence predictable, by implementing such control protocols in driver-assist vehicles. Remarkably, they also provide a precise relationship between a measure of the macroscopic damping of the uncertainty and the penetration rate of the driver-assist technology in the traffic stream