The aim of this paper is to introduce a macroscopic fluid dynamic model dealing with the flow of information on a telecommunication network encoded in packets. Taking an intermediate time and space scale, we propose a model similar to that introduced recently for car traffic; see [G. M. Coclite, M. Garavello, and B. Piccoli, SIAM J. Math. Anal., 36 (2005), pp. 1862–1886]. For dynamics at nodes we consider two “routing algorithms” and prove existence of solutions to Cauchy problems. The main difference among the two algorithms is the possibility of redirecting packets of the second algorithm, which in turn implies stability, i.e., Lipschitz continuous dependence on initial data, not granted for solutions using the first algorithm.

This paper proposes a macroscopic fluid dynamic model dealing with the flows of information on a telecommunication network with sources and destinations. The model consists of a conservation law for the packet density and a semilinear equation for traffic distribution functions, i.e., functions describing packet paths. We describe methods to solve Riemann problems at junctions assigning different traffic distribution functions and two “routing algorithms.” Moreover, we prove the existence of solutions to Cauchy problems for small perturbations of network equilibria.

In this paper we deal with a continuous model for supply chains, consisting of a\ud PDE for the density of processed parts and an ODE for the queue buffer occupancy. We discuss\ud the optimal control problem stated as the minimization of the queues and the quadratic difference\ud between the effective outflow and a desired one. Here the input flow is the control and is assumed\ud to have uniformly bounded variation. Introducing generalized tangent vectors to piecewise constant\ud controls, representing shifts of discontinuities, we analyze the dependence of the solution on the\ud control function. Then existence of an optimal control for the original problem is obtained. Finally\ud we study the sensitivity of the cost functional J as function of controlled inflow, providing an estimate\ud of the derivative of J with respect to switching times