Invasion Percolation and Critical Transient in the Barabási Model of Human Dynamics

We introduce an exact probabilistic description for $L=2$ of the Barabási model for the dynamics of a list of $L$ tasks. This permits us to study the problem out of the stationary state and to solve explicitly the extremal limit case where a critical behavior for the waiting time distribution is observed. This behavior deviates at any finite time from that of the stationary state. We study also the characteristic relaxation time for finite time deviations from stationarity in all cases showing that it diverges in the extremal limit, confirming that these deviations are important at all time.