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Register a new userA multi-lane TASEP model for crossing pedestrian traffic flows
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A one-way {em street} of width M is modeled as a set of M parallel one-dimensional TASEPs. The intersection of two perpendicular streets is a square lattice of size M times M. We consider hard core particles entering each street with an injection probability alpha. On the intersection square the hard core exclusion creates a many-body problem of strongly interacting TASEPs and we study the collective dynamics that arises. We construct an efficient algorithm that allows for the simulation of streets of infinite length, which have sharply defined critical jamming points. The algorithm employs the `frozen shuffle update, in which the randomly arriving particles have fully deterministic bulk dynamics. High precision simulations for street widths up to M=24 show that when alpha increases, there occur jamming transitions at a sequence of M critical values alphaM,M < alphaM,M-1 < ... < alphaM,1. As M grows, the principal transition point alphaM,M decreases roughly as sim 1/(log M) in the range of M values studied. We show that a suitable order parameter is provided by a reflection coefficient associated with the particle current in each TASEP.
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We study two perpendicular intersecting flows of pedestrians. The latter are represented either by moving hard core particles of two types, eastbound ($symbp$) and northbound ($symbm$), or by two density fields, $rhop_t(brr)$ and $rhom_t(brr)$. Each flow takes place on a lattice strip of width $M$ so that the intersection is an $Mtimes M$ square. We investigate the spontaneous formation, observed experimentally and in simulations, of a diagonal pattern of stripes in which alternatingly one of the two particle types dominates. By a linear stability analysis of the field equations we show how this pattern formation comes about. We focus on the observation, reported recently, that the striped pattern actually consists of chevrons rather than straight lines. We demonstrate that this `chevron effect occurs both in particle simulations with various different update schemes and in field simulations. We quantify the effect in terms of the chevron angle $Deltatheta_0$ and determine its dependency on the parameters governing the boundary conditions.
A two-lane extension of a recently proposed cellular automaton model for traffic flow is discussed. The analysis focuses on the reproduction of the lane usage inversion and the density dependence of the number of lane changes. It is shown that the single-lane dynamics can be extended to the two-lane case without changing the basic properties of the model which are known to be in good agreement with empirical single-vehicle data. Therefore it is possible to reproduce various empirically observed two-lane phenomena, like the synchronization of the lanes, without fine-tuning of the model parameters.
First we consider a unidirectional flux omega_bar of vehicles each of which is characterized by its `natural velocity v drawn from a distribution P(v). The traffic flow is modeled as a collection of straight `world lines in the time-space plane, with overtaking events represented by a fixed queuing time tau imposed on the overtaking vehicle. This geometrical model exhibits platoon formation and allows, among many other things, for the calculation of the effective average velocity w=phi(v) of a vehicle of natural velocity v. Secondly, we extend the model to two opposite lanes, A and B. We argue that the queuing time tau in one lane is determined by the traffic density in the opposite lane. On the basis of reasonable additional assumptions we establish a set of equations that couple the two lanes and can be solved numerically. It appears that above a critical value omega_bar_c of the control parameter omega_bar the symmetry between the lanes is spontaneously broken: there is a slow lane where long platoons form behind the slowest vehicles, and a fast lane where overtaking is easy due to the wide spacing between the platoons in the opposite direction. A variant of the model is studied in which the spatial vehicle density rho_bar rather than the flux omega_bar is the control parameter. Unequal fluxes omega_bar_A and omega_bar_B in the two lanes are also considered. The symmetry breaking phenomenon exhibited by this model, even though no doubt hard to observe in pure form in real-life traffic, nevertheless indicates a tendency of such traffic.
Tolls are collected on many highways as a means of traffic control and revenue generation. However, the presence of tollbooths on highway surely slows down traffic flow. Here, we investigate how the presence of tollbooths affect the average car speed using a simple-minded single lane deterministic discrete traffic model. More importantly, the model is exactly solvable.
Motivated by interest in pedestrian traffic we study two lanes (one-dimensional lattices) of length $L$ that intersect at a single site. Each lane is modeled by a TASEP (Totally Asymmetric Exclusion Process). The particles enter and leave lane $sigma$ (where $sigma=1,2$) with probabilities $alpha_sigma$ and $beta_sigma$, respectively. We employ the `frozen shuffle update introduced in earlier work [C. Appert-Rolland et al, J. Stat. Mech. (2011) P07009], in which the particle positions are updated in a fixed random order. We find analytically that each lane may be in a `free flow or in a `jammed state. Hence the phase diagram in the domain $0leqalpha_1,alpha_2leq 1$ consists of four regions with boundaries depending on $beta_1$ and $beta_2$. The regions meet in a single point on the diagonal of the domain. Our analytical predictions for the phase boundaries as well as for the currents and densities in each phase are confirmed by Monte Carlo simulations.
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