The geometry of big queues

We use Hamilton equations to find optimal paths to big queues in Jackson networks. They are shown to be given by fluid trajectories of the dual network. The fluid equations are shown to be dual to the Hamilton equations. Thus, a version of the extended Onzager-Machlup principle is proved: a fluctuation trajectory is the time reversal of a relaxation trajectory of the dual dynamic. The case of a two-node network is given a geometric treatment.