Tails of stationary distributions of queued work

It has been observed by many authors that under reasonable regularity conditions, the stationary distribution of queued work in a queueing system has an exponential tail. This has been observed in practise in a great variety of communications systems. The main result of this paper is a general equation for the decay coefficient of this tail. A procedure for estimating the weight of the tail is also described. The proof relies on the assumption that future values of the net input process are independent of values in the past after a sufficiently long delay but the equation does not reflect this assumption and it is conjectured that the formula holds good even when the input process retains some degree of correlation over time intervals of arbitrary length. Several examples are considered including some which do not assume that the distribution of the inputs is Gaussian. The method for determining the decay coefficient for the tail in these examples relies on finding the zero of a sample nonlinear equation