A Mean Field Limit for a Lattice Caricature of Dynamic Routing in Circuit Switched Networks

Simulation studies of circuit switched networks with dynamic alternate routing reveal the existence of hysteresis phenomena, which suggest that such networks can admit more than one regime of operation for the same offered traffic. Such behavior is also suggested by a detailed analytical model due to Marbukh and a simpler model due to Gibbens, Hunt and Kelly. In these models, a limit is taken as the network size becomes large, and one finds a limiting ODE describing the proportions of network links in different states. The possibility of multiple regimes of operation shows up through the fact that the ODE has multiple equilibrium points for certain ranges of parameters. The kinds of limits considered by Marbukh and Gibbens, Hunt and Kelly do not take into account the spatial extent of the network. In an attempt to preserve the spatial characteristics, we consider a lattice model similar to that of Gibbens, Hunt and Kelly. We derive a mean field limit for this lattice model. This is an integrodifferential equation which describes how the spatial distribution of the network evolves in time. The mean field equation also admits multiple spatially homogeneous equilibrium solutions for certain ranges of the parameters, which may be loosely thought of as the different operating regimes. This equation may be particularly useful in understanding the exchange between the operating regimes, that is, questions like "for what parameter values is a hot spot of heavy loading in the system likely to take over the whole network by knock-on effects?"