Tail asymptotics for head of line priority queues handling a large number of independent stationary sources

In this paper we study the asymptotics of the tail of the buffer occupancy distribution in buffers accessed by a large number of stationary independent sources and which are served according to a strict HOL priority rule. As in the case of single buffers, the results are valid for a very general class of sources which include long-range dependent sources with bounded instantaneous rates. We first consider the case of two buffers with one of them having strict priority over the other and we obtain asymptotic upper bound for the buffer tail probability for lower priority buffer. We discuss the conditions to have asymptotic equivalents. The asymptotics are studied in terms of a scaling parameter which reflects the server speed, buffer level and the number of sources in such a way that the ratios remain constant. The results are then generalized to the case of M buffers which leads to the source pooling idea. We conclude with numerical validation of our formulae against simulations which show that the asymptotic bounds are tight. We also show that the commonly suggested reduced service rate approximation can give extremely low estimates.