The issue addressed is the computation at each arrival time and at each state at any node of the network, of the travel time needed to join any node of the network to this state at this arrival date by a prototypical vehicle and of the regulation law governing such viable evolutions. We assume that is known the time dependent controlled dynamics of a prototypical vehicle, the network on which it is constrained to evolve at each instant, and the nodes from which it starts or through which it passes at prescribed times. The basic question we address is the determination of the arrival sets at each arrival time made of terminal states at which arrive evolutions governed by the control system, viable in the network, starting from the departure set for some travel time or prescribed travel time. A subsidiary problem is to determine the associated nodes, through the Cournot set-valued map we define at the end of this paper. We use viability techniques summarized in an appendix, which, translated in terms of travel time problems, allow us, for instance, to characterize the arrival tubes, define the "homoclinic" pairs at which any two nodes can be connected, prove a Lax-Hopf Formula characterizing these tubes by an easy formula whenever the control system does not depend either on the time or on the state, propose a concept of solution to a "system of Hamilton-Jacobi-Bellman inclusions" of which the arriving tube is the unique solution, adapt to those systems the dual concept of Barron-Jensen/Frankowska extension of viscosity solutions to usual Hamilton-Jacobi-Bellman solutions, optimize intertemporal criteria and minimize travel times. We derive the main properties of the Moskowitz Travel Time model of the Lighthill, Whitham and Richards' theory.