Priority queues with bursty arrivals of incoming tasks

Abstract

Recently increased accessibility of large-scale digital records enables one to monitor human activities such as the interevent time distributions between two consecutive visits to a web portal by a single user, two consecutive emails sent out by a user, two consecutive library loans made by a single individual, etc. Interestingly, those distributions exhibit a universal behavior, $D(τ)\sim τ^{-δ}$, where $τ$ is the interevent time, and $δ\simeq 1$ or 3/2. The universal behaviors have been modeled via the waiting-time distribution of a task in the queue operating based on priority; the waiting time follows a power law distribution $P_{\rm w}(τ)\sim τ^{-α}$ with either $α=1$ or 3/2 depending on the detail of queuing dynamics. In these models, the number of incoming tasks in a unit time interval has been assumed to follow a Poisson-type distribution. For an email system, however, the number of emails delivered to a mail box in a unit time we measured follows a powerlaw distribution with general exponent $γ$. For this case, we obtain analytically the exponent $α$, which is not necessarily 1 or 3/2 and takes nonuniversal values depending on $γ$. We develop the generating function formalism to obtain the exponent $α$, which is distinct from the continuous time approximation used in the previous studies.