Section: Application Domains
Traffic modeling is a domain where maxplus algebra appears naturally : – at microscopic level where we follow the vehicles in a network of streets, – at macroscopic level where assignment are based on computing smallest length paths in a graph, – in the algebraic duality between stochastic and deterministic assignments.
We develop free computing tools and models of traffic implementing our experience on optimization and discrete event system modeling based on maxplus algebra.
Microscopic Traffic Modeling
Let us consider a circular road with places occupied or not by a car symbolized by a 1. The dynamic is defined by the rule 1001 that we apply simultaneously to all the parts of the word m representing the system. For example, starting with m1 = 1010100101 we obtain the sequence of works (mi) :
For such a system we can call density d the number of cars divided by the number of places called p that is d = n/p . We call flow f(t) at time t the number of cars at this time period divided by the place number. The fundamental traffic law gives the relation between f(t) and d .
If the density is smaller than 1/2 , after a transient period of time all the cars are separated and can go without interaction with the other cars. Then f(t) = n/p that can be written as function of the density as f(t) = d
On the other hand if the density is larger than 1/2 , all the free places are separated after a finite amount of time and go backward freely. Then we have p-n car which can go forward. Then the relation between flow and density becomes
This can be stated formally: it exists a time T such that for all tT , f(t) stays equal to a constant that we call f with
The fundamental traffic law linking the density of vehicles and the flow of vehicles can be also derived easily from maxplus modeling : – in the deterministic case by computing the eigenvalue of a maxplus matrix, – in the stochastic case by computing a Lyapounov exponent of stochastic maxplus matrices.
The main research consists in developing extensions to systems of roads with crossings. In this case, we leave maxplus linear modeling and have to study more general dynamical systems. Nevertheless these systems can still be defined in matrix form using standard and maxplus linear algebra simultaneously.
With this point of view efficient microscopic traffic simulator can be developed in Scilab.
Given a transportation network and a set of transportation demands from an origin to a destination , the traffic assignment problem consists in determining the flows fa on the arcs of the network when the times ta spent on the arcs a are given functions of the flows fa .
We can distinguish the deterministic case — when all the travel times are known by the users — from the stochastic cases — when the users perceive travel times different from the actual ones.
When the travel times are deterministic and do not depend on the link flows, the assignment can be reduced to compute the routes with shortest travel times for each origin-destination pair.
When the travel times are deterministic and depend on the link flows, Wardrop equilibriums are defined and computed by iterative methods based on the previous case.
When the perceived travel times do not depend on the link flows but are stochastic with error distribution — between the perceived time and the actual time — satisfying a Gumbel distribution, the probability that a user choose a particular route can be computed explicitly. This probability has a Gibbs distribution called logit in transportation literature. From this distribution the arc flows — supposed to be deterministic — can be computed using a matrix calculus which can be seen as the counterpart of the shortest path computation (of the case 1) up to the substitution of the minplus semiring by the Gibbs-Maslov semiring, where we call Gibbs-Maslov semiring the set of real numbers endowed with the following two operations :
When the perceived travel times are stochastic and depend on the link flows — supposed to be deterministic quantities — stochastic equilibriums are defined and can be computed using iterative methods based on the logit assignments discussed in the case 3.
Based on this classification, a toolbox dedicated to traffic assignment is available and maintained in Scilab.