Economies of scale in long and short buffers of large multiplexers

We examine phenomenological and theoretical aspects of scaling laws of the queue-length distributions in large multiplexers. Anick, Mitra and Sondhi (1982) have shown that the queue length Q^L in a multiplexer of L sources served at constant load (i.e. independent of L) has asymptotics for large L of the form p[Q^L>Lb]≈e^-LI(b), where I is some function (we call it the shape function) depending on the source processes and the load. If I is known, then we can predict the queue length distribution at large L. With a model for the sources, I can be calculated. The procedure for this is outlined. Alternatively we can estimate I. One way to do this is by using the above equation as a definition of I based upon measurements at some L which is both small enough for measurements to be made, but also large enough for the approximation above to be accurate. We illustrate the application of this idea with some empirical queue length distributions obtained by simulation at various L. We find that the most accurate predictions are obtained by taking finite-size effects into account in a simple manner in the approximation