Universal heavy-tailed behavior in waiting time distribution in a queue

We study the waiting time distribution of queuing models operating in first-in-first-out (FIFO) and priority-based protocols by mapping the dynamics onto random-walk problems. The number of tasks in the queue in the long time limit gives the initial condition of a random walker, and the waiting time of a task is related with its first passage time (FPT). The formalism for the FIFO protocol is established first, successfully reproducing the exponential waiting time distribution, and then the priority-based case by Grinstein and Linsker [1] is reviewed with minor corrections, yielding the power-law waiting time distributions. We also discuss the universality in random systems, comparing the queuing model with other systems from the viewpoint of the ubiquitous exponent 1.5 in the FPT distributions.