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Exact Results for the Baraba´si Model of Human Dynamics



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PRL 95 248701 2005 PHYSICAL REVIEW LETTERS week ending 9 DECEMBER 2005 Exact Results for the Baraba si Model of Human Dynamics Alexei Va zquez Department of Physics and Center for Complex Networks Research University of Notre Dame Notre Dame Indiana 46556 USA Received 14 June 2005 published 6 December 2005 Human activity patterns display a bursty dynamics with interevent times following a heavy tailed distribution This behavior has been recently shown to be rooted in the fact that humans assign their active tasks different priorities a process that can be modeled as a priority queueing system A L Baraba si Nature London 435 207 2005 In this Letter we obtain exact results for the Baraba si model with two tasks calculating the priority and waiting time distribution of active tasks We demonstrate that the model has a singular behavior in the extremal dynamics limit when the highest priority task is selected first We find that independently of the selection protocol the average waiting time is smaller or equal to the number of active tasks and discuss the asymptotic behavior of the waiting time distribution These results have important implications for understanding complex systems with extremal dynamics DOI 10 1103 PhysRevLett 95 248701 PACS numbers 89 75 Da 02 50 Le 89 65 Ef Several problems of practical interest require us to understand human activity patterns 1 3 Typical examples are the design of telephone systems or web servers where it is critical to know how many users would use the service simultaneously The traditional approach to characterize the timing of human activities is based in two assumptions the execution of each task is independent from the others and each task is executed at a constant rate 1 4 A specific task such as sending Email or making phone calls is then modeled as a Poisson process 4 characterized by a homogeneous activity pattern More precisely the time interval between two consecutive executions of a task follows an exponential



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