Optimal Load Balancing on Distributed Homogeneous Unreliable Processors

We consider optimal load balancing in a distributed computing environment with several homogeneous unreliable processors that have limited state information. Each processor receives its own arrival process of tasks from outside users, some of which can be redirected to the other processors. Processing times are {\em iid} and arbitrarily distributed. The arrival process of outside tasks to each processor may be arbitrary as long as it is independent of the state of the system. Processors may fail, with arbitrary failure and repair processes that are also independent of the state of the system. The only information available to a processor is the history of its decisions for routing work to other processors, and the arrival times for its own arrival process. We show that the round-robin policy, in which each processor sends the tasks that can be redirected to each of the processors in turn, {\em stochastically} minimizes the $n^{th}$ task completion time for all $n$, and minimizes response times and queue lengths in a separable increasing convex sense, among all policies that balance workload. We also show that if there is a single centralized controller, round-robin is the optimal policy, and a single controller using round-robin routing is better than the optimal distributed system, where «optimal» and «better» are in the sense of stochastically minimizing task completion times and minimizing response times and queue lengths in the separable increasing convex sense.