Performance implications of very large service-time variances

Abstract

Measurements of file sizes transported on the World-Wide-Web have led some researchers to propose describing them by probability distributions with infinite variance. The M/G/1 queue often arises as a performance model for components of the WWW, and the service times correspond to file sizes; the infinite variance of the file sizes becomes the variance of the service times. In this paper the effects of very large service-time variances on some performance measures for the M/G/1 queue are explored via numerical examples and analytic arguments. The first main conclusion is that it is the form of the service-time distribution over a wide finite range that controls the steady-state queueing performance, so distributions with very large finite variances can yield the same behavior as distributions with infinite variances. The second main conclusion is that very large service-time variances cause the rate of approach to steady-state performance to be so slow that steady-state performance measures are not likely to be of engineering interest. A third conclusion is that a common device of using the probability that the work in an infinite queue exceeds the level b to approximate the probability that a finite buffer of size b overflows may be very inaccurate. The approximation works better for the fat-tailed distributions studied than for the others. The most important engineering implication of these results is that when service times have a very large variance (such as file transfers on the WWW), performance criteria other than steady-state measures have to be used.