This article is devoted to traffic flow networks including traffic lights at intersections.
Mathematically, we consider a nonlinear dynamical traffic model where traffic lights are modeled
as piecewise constant functions for red and green signals. The involved control problem is to
find stop and go configurations depending on the current traffic volume.
We propose a numerical solution strategy and present computational results.
In this paper, we focus on production network models based on ordinary and partial differential equations that are coupled to semi-Markovian failure rates for the processor capacities. This modeling approach allows for intermediate capacity states in the range of total breakdown to full capacity, where operating and down times might be arbitrarily distributed. The mathematical challenge is to combine the theory of semi-Markovian processes within the framework of conservation laws. We show the existence and uniqueness of such stochastic network solutions, present a suitable simulation method and explain the link to the common queueing theory. A variety of numerical examples emphasizes the characteristics of the proposed approach.
In this work we introduce a novel model for the tracking of a thief moving through a road network. The modeling equations are given by a strongly coupled system of scalar conservation laws for the road traffic and ordinary differential equations for the thief evolution. A crucial point is the characterization at intersections, where the thief has to take a routing decision depending on the available local information. We develop a numerical approach to solve the thief tracking problem by combining a time-dependent shortest path algorithm with the numerical solution of the traffic flow equations. Various computational experiments are presented to describe different behavior patterns.
We introduce the optimal inflow control problem for buffer restricted production systems involving a conservation law with discontinuous flux.
Based on an appropriate numerical method inspired by the wave front tracking algorithm, we present two techniques to solve the optimal control problem efficiently. A numerical study compares the different optimization procedures and comments on their benefits and drawbacks.
This article is devoted to the discussion of the boundary layer which arises from the one-dimensional parabolic elliptic type Keller-Segel system to the corresponding aggregation system on the half space case. The characteristic boundary layer is shown to be stable for small diffusion coefficients by using asymptotic analysis and detailed error estimates between the Keller-Segel solution and the approximate solution. In the end, numerical simulations for this boundary layer problem are provided.
We consider a supply network where the flow of parts can be
controlled at the vertices of the network. Based on a coarse grid
discretization provided in
 we derive discrete
adjoint equations which are subsequently validated by the continuous
adjoint calculus. Moreover, we present numerical results concerning
the quality of approximations and computing times of the presented
In this article, an evacuation model describing the egress in case of danger is considered. The underlying evacuation model is based on continuous network flows, while the spread of some gaseous hazardous material relies on an advection-diffusion equation.
The contribution of this work is twofold.
First, we introduce a continuous model coupled to the propagation of hazardous material where special cost functions allow for incorporating the predicted spread into an optimal planning of the egress. Optimality can thereby be understood with respect to two different measures: fastest egress and safest evacuation.
Since this modeling approach leads to a pde/ode-restricted optimization problem,
the continuous model is transferred into a discrete network flow model under some linearity assumptions.
Second, it is demonstrated that this reformulation results in an efficient algorithm always leading to the global optimum. A computational case study shows benefits and drawbacks of the models for different evacuation scenarios.
We discuss a numerical discretization of Hamilton--Jacobi equations on networks. The latter arise for example as reformulation
of the Lighthill--Whitham--Richards traffic flow model. We present coupling conditions for the Hamilton--Jacobi equations
and derive a suitable numerical algorithm. Numerical computations of travel times in a round-about are given.
In this paper, we illustrate how second order traffic flow models, in our case the Aw-Rascle equations, can be used to reproduce empirical observations such as the capacity drop at merges and solve related optimal control problems. To this aim, we propose a model for on-ramp junctions and derive suitable coupling conditions. These are associated to the first order Godunov scheme to numerically study the well-known capacity drop effect, where the outflow of the system is significantly below the expected maximum. Control issues such as speed and ramp meter control are also addressed in a first-discretize-then-optimize framework.
Our main objective is the modelling and simulation of complex production networks originally
introduced in [15, 16] with random breakdowns of individual processors. Similar to
, the breakdowns of processors are exponentially distributed. The resulting network
model consists of coupled system of partial and ordinary differential equations with Markovian switching
and its solution is a stochastic process. We show our model to fit into the framework of piecewise
deterministic processes, which allows for a deterministic interpretation of dynamics between a
multivariate two-state process. We develop an efficient algorithm with an emphasis on accurately
tracing stochastic events. Numerical results are presented for three exemplary networks, including
a comparison with the long-chain model proposed in .