Numerical Methods for Analyzing Queues with Heavy-Tailed Distributions

In many Internet-type queues, arrival and service distributions are heavy-tailed. A difficulty with analyzing these queues is that heavy-tailed distributions do not generally have closed-form Laplace transforms. A recently proposed method, the Transform Approximation Method (TAM), overcomes this by numerically approximating the transform. This paper investigates numerical issues of implementing the method for simple queueing systems. In particular, we argue that TAM can be used in conjunction with the Fourier-series method for inverting Laplace transforms, even though TAM is a discrete approximation and the Fourier method requires a continuous distribution. We apply concepts to compute the waiting time distribution of an M/G/1 priority queue. Introduction Many probability distributions associated with telecommunications or Internet traffic are heavy-tailed. Roughly speaking, a heavy-tailed random variable has a non-trivial probability of being extremely large. Traffic data have strongly suggested that many random variables associated with the Internet are heavy-tailed. For example, file sizes, requested file transmission times, connection durations, session sizes, dial-up Internet durations, etc., all have heavy tails (Crovella et al., 1998, Leland et al., 1994, Naldi, 1999, Park et al., 1997). Technically, a distribution function G(x) is heavy-tailed if and only if 0 , 1 ) ( ) ( lim ≥ = +