Invasion percolation and the time scaling behavior of a queuing model of human dynamics

In this paper we study the properties of the Barabasi model of queuing 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 us to recover the correct waiting time distribution PW(τ)~τ−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.

[1]  Paczuski,et al.  Avalanche dynamics in evolution, growth, and depinning models. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[2]  Sidney Redner,et al.  A guide to first-passage processes , 2001 .

[3]  B. Nickel,et al.  Invasion Percolation on the Cayley Tree: Exact Solution of a Modified Percolation Model , 1983 .

[4]  Caldarelli,et al.  Invasion percolation with temperature and the nature of self-organized criticality in real systems , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  J. G. Oliveira,et al.  Human Dynamics: The Correspondence Patterns of Darwin and Einstein , 2005 .

[6]  R Linsker,et al.  Biased diffusion and universality in model queues. , 2006, Physical review letters.

[7]  S. Redner.  A guide to first-passage processes , 2001 .

[8]  R Linsker,et al.  Power-law and exponential tails in a stochastic priority-based model queue. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Albert-László Barabási,et al.  The origin of bursts and heavy tails in human dynamics , 2005, Nature.

[10]  G Caldarelli,et al.  Invasion percolation and critical transient in the Barabási model of human dynamics. , 2007, Physical review letters.

[11]  M. Marsili.  Run time statistics in models of growth in disrrdered media , 1994 .

[12]  Alexei Vázquez,et al.  Exact results for the Barabási model of human dynamics. , 2005, Physical review letters.

[13]  David Wilkinson,et al.  Invasion percolation: a new form of percolation theory , 1983 .

[14]  L. Pietronero,et al.  Comment on the run time statistics in models of growth in disordered media , 1996 .

[15]  Albert-László Barabási,et al.  Modeling bursts and heavy tails in human dynamics , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Dieter Baum,et al.  An introduction to queueing theory and matrix-analytic methods , 2005 .

[17]  Carl M. Harris,et al.  Fundamentals of queueing theory , 1975 .

[18]  A. Barabasi,et al.  Human dynamics: Darwin and Einstein correspondence patterns , 2005, Nature.