Time-Varying Priority Queuing Models for Human Dynamics

Queuing models provide insight into the temporal inhomogeneity of human dynamics, characterized by the broad distribution of waiting times of individuals performing tasks. We study the queuing model of an agent trying to execute a task of interest, the priority of which may vary with time due to the agent’s “state of mind.” However, its execution is disrupted by other tasks of random priorities. By considering the priority of the task of interest either decreasing or increasing algebraically in time, we analytically obtain and numerically confirm the bimodal and unimodal waiting time distributions with power-law decaying tails, respectively. These results are also compared to the updating time distribution of papers in the arXiv.org and the processing time distribution of papers in Physical Review journals. Our analysis helps to understand human task execution in a more realistic scenario.

💡 Research Summary

The paper introduces a minimal two‑task queuing model that captures the temporal variability of human task execution by allowing the priority of a focal task (type‑A) to evolve over time, while a competing random task (type‑B) arrives anew at each discrete time step. Building on Barabási’s priority‑queue framework, the authors relax two key assumptions: (i) all tasks are of the same type, and (ii) priorities are static. In the model, the queue length is fixed at two; the first slot holds the type‑A task introduced at t = 0 with an initial priority x₀ drawn uniformly from (0, 1). The second slot is filled at every step by a fresh type‑B task whose priority rₜ is also uniform on (0, 1). At each step the agent compares x(t) with rₜ. If x(t) ≥ rₜ the type‑A task is executed; otherwise the type‑B task is executed. A parameter p governs deterministic versus stochastic execution: with probability p the highest‑priority task is chosen, while with probability 1 − p a task is selected at random (the “trembling‑hand” effect).

Two families of time‑dependent priorities are examined.

1. Algebraically decreasing priority:
   x(t) = x₀ (t + 1)^{‑γ}.
   When γ = 0 the model reduces to the classic fixed‑priority case, yielding a waiting‑time distribution P(τ) ∝ τ^{‑2} (α = 2). For 0 < γ < 1 the priority decays slowly; analytic treatment (Eqs. 5‑6) shows that for large τ the distribution follows a power law with exponent α = 2 − γ, multiplied by an exponential cutoff τ_c ≈ (2ε)^{‑1} where ε = 1 − p. As γ exceeds 1 the priority collapses rapidly toward zero, and the dominant contribution to P(τ) becomes proportional to