Phase transition in a directed traffic flow network

Abstract

The generic feature of traffic in a network of flowing electronic data packets is a phase transition from a stationary free-flow phase to a continuously growing congested non-stationary phase. In the most simple network of directed oriented square lattice we have been able to observe all crucial features of such flow systems having non-trivial critical behavior near the critical point of transition. The network here is in the shape of a square lattice and data packets are randomly posted with a rate $\rho$ at one side of the lattice. Each packet executes a directed diffusive motion towards the opposite boundary where it is delivered. Packets accumulated at a particular node form a queue and a maximum of $m$ such packets randomly jump out of this node at every time step to its neighbors on a first-in-first-out (FIFO) basis. The phase transition occurs at $\rho_c=m$. The distribution of travel times through the system is found to have a log-normal behavior and the power-spectrum of the load time-series shows $1/f$ like noise similar to the scenario of Internet traffic.