Upper bounds for the travel time on traffic systems
  A key measure of performance and comfort in a road traffic network is the travel time that the users of the network experience to complete their journeys. Travel times on road traffic networks are stochastic, highly variable, and dependent on several parameters. It is, therefore, necessary to have good indicators and measures of their variations. In this article, we extend a recent approach for the derivation of deterministic bounds on the travel time in a road traffic network (Farhi, Haj-Salem and Lebacque 2013). The approach consists in using an algebraic formulation of the cell-transmission traffic model on a ring road, where the car-dynamics is seen as a linear min-plus system. The impulse response of the system is derived analytically, and is interpreted as what is called a service curve in the network calculus theory (where the road is seen as a server). The basic results of the latter theory are then used to derive an upper bound for the travel time through the ring road.   We consider in this article open systems rather than closed ones. We define a set of elementary traffic systems and an operator for the concatenation of such systems. We show that the traffic system of any road itinerary can be built by concatenating a number of elementary traffic systems. The concatenation of systems consists in giving a service guarantee of the resulting system in function of service guarantees of the composed systems. We illustrate this approach with a numerical example, where we compute an upper bound for the travel time on a given route in a urban network. 
Nadir Farhi, Habib Haj-Salem and Jean-Patrick Lebacque
math.DS math.OC