Before revision:  When the estimation error distribution is known, the Gittins' Index policy is known to be optimal in minimizing the mean sojourn time in an M/G/1 queue. For a multiplicative error model, we first characterize the Gittins' Index policy for any estimation error distribution. We then use insights from the Gittins' Index policy to construct a simple heuristic, Size Estimate Hedging (SEH), that only uses jobs' estimated processing times for scheduling while exhibiting near-optimal performance .
After revision:  We propose a simple heuristic, Size Estimate Hedging (SEH), that only uses jobs' estimated processing times for scheduling while exhibiting near-optimal performance .