Priority queues with bursty arrivals of incoming tasks

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) ∼ τ-δ, where τ is the interevent time, and δ 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 Pw (τ) ∼ 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 power-law 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.