Barabasi Queueing Model and Invasion Percolation on a tree

TL;DR

This paper models queueing dynamics on a tree using invasion percolation, revealing a power-law waiting time distribution and slow convergence to stationarity, with implications for understanding real-world systems.

Contribution

It provides an exact mapping of the Barabási queueing model onto invasion percolation on a Cayley tree, elucidating waiting time distributions and activity bursts.

Findings

Waiting time distribution follows a power law with exponent -3/2.

Approach to stationarity is very slow.

Reveals causal and geometric structure of activity bursts.

Abstract

In this paper we study the properties of the Barab\'asi model of queueing under the hypothesis that the number of tasks is steadily growing in time. We map this model exactly onto an Invasion Percolation dynamics on a Cayley tree. This allows to recover the correct waiting time distribution $P_W(\tau)\sim \tau^{-3/2}$ at the stationary state (as observed in different realistic data) and also to characterize it as a sequence of causally and geometrically connected bursts of activity. We also find that the approach to stationarity is very slow.