Jamming of directed traffic on a square lattice

Phase transition from a free-flow phase to a jammed phase is an important feature of traffic networks. We study this transition in the case of a simple square lattice network for different values of data posting rate $(\rho)$ by introducing a parameter $p$ which selects a neighbour for onward data transfer depending on queued traffic. For every $\rho$ there is a critical value of $p$ above which the system become jammed. The $\rho-p$ phase diagram shows some interesting features. We also show that the average load diverges logarithmically as $p$ approaches $p_c$ and the queue length distribution exhibits exponential and algebraic nature in different regions of the phase diagram.