Estimating completion time ratios of a fork-join barrier synchronization

In simulation studies of parallel processors, it is useful to consider the following abstraction of a parallel program. A job is partitioned into n processes, whose running times are proportional to i.i.d. random variables X1,…, Xn. As a measure of performance we consider the normalized job completion time S = max Xi ∑i = 1n Xi. We consider a simple approximation to the expected value of S, valid asymptotically whenever the Xi ʹs are bounded, and assess its accuracy as a function of n both theoretically and experimentally. The approximation is easy to compute and involves only the mean, variance, and maximum of Xi. We also give asymptotic series for the mean and standard deviation of S when the Xi have a uniform distribution. In this case, the standard deviation is asympotically negligible compared to the mean.