Optimality of weighted least squares load balancing

The authors consider a model comprising several servers, with possibly different services speeds, each equipped with its own queue. Each server receives a dedicated arrival stream of jobs; there is also a stream of generic jobs that arrive at a job scheduler and can be individually allocated to any of the servers. It is shown that if the arrival streams are all Poisson, and all jobs have the same exponentially distributed service requirements, then the probabilistic splitting of the generic stream that minimizes the average job response time is one that balances the server idle times in a weighted least squares sense, where the weighting coefficients are related to the service speeds of the servers. The corresponding result holds for nonexponentially distributed service times, if the service speeds are all equal