Exact results for the Barabási model of human dynamics.

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. Barabasi, Nature (London) 435, 207 (2005)). In this Letter we obtain exact results for the Barabasi 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. selection protocol. In the extremal dynamics limit, when the highest priority task is selected first, numerical simu- lations and heuristic arguments show that most of the tasks are executed in one step, while the waiting time distribu- tion of tasks waiting more than one step exhibits a heavy tail (5). Yet, further research is required to obtain the scaling behavior in the vicinity of this singular point. In this work we obtain exact results for the Barabasi model, allowing us to prove previous conjectures based on heuristic arguments and numerical simulations, and creat- ing a solid background for future research. We calculate the priority and waiting time distribution of those tasks remaining in the list for the case of two active tasks. We corroborate the observation of a singular behavior in the limit when the task with the highest priority is selected first, and derive the corresponding scaling behavior. We also obtain an upper bound for the average waiting time, which is independent of the selection protocol. Based on this result we discuss the asymptotic behaviors of the waiting time distribution. All the results presented here were checked by numerical simulations, providing a per- fect match with the theoretical curves. Barabasi model.—The Barabasi model is defined as follows. A human keeps track of a list with L active tasks that he/she must do. A priority x � 0 is assigned to each active task when it is added to the list, with a probability density function (PDF) � � x� . The list is started at t � 0 by adding L new tasks to it. At each discrete time step t> 0 the task in the list with the highest priority is selected with probability p, and with probability 1 � p a task is selected at random. The selected task is executed, removed from the list, and a new task is added. The control parameter p interpolates between the random selection protocol at p � 0 and the highest priority first selection protocol at p � 1. The numerical simulations indicate that the case L � 2 already exhibits the relevant features of the model (5). Furthermore, if we focus on a single task, such as sending Email, we can model the active tasks list as a list with two tasks, one corresponding to sending Email and the other to