Randomized routing schemes for large processor sharing systems with multiple service rates

We consider randomized job routing techniques for a system consisting of a large number of parallel processor sharing servers with heterogeneous server speeds. In particular, a scheme, that routes an incoming job request to the server providing the highest instantaneous processing rate per job among two servers, chosen uniformly at random, is proposed. We show that, unlike the homogeneous case, in the heterogeneous case, such randomized dynamic schemes need not always perform better than the optimal static scheme (in which jobs are assigned to servers with fixed probabilities independent of server states) in terms of reducing the mean response time of jobs. Specifically, we show that the stability region under the proposed scheme is a subset of that under the optimal static routing scheme. We also obtain the stationary tail distribution of server occupancies for the proposed scheme in the limit as the system size grows to infinity. This distribution has been shown to be insensitive to job length distribution and decay super-exponentially