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(tau) approximately tau(-delta) , where tau is the interevent time, and delta approximately 1 or 32 . 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(w)(tau) approximately tau(-alpha) with either alpha=1 or 32 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 gamma . For this case, we obtain analytically the exponent alpha , which is not necessarily 1 or 32 and takes nonuniversal values depending on gamma . We develop the generating function formalism to obtain the exponent alpha , which is distinct from the continuous time approximation used in the previous studies.