Optimality of routing and servicing in dependent parallel processing systems

In a system of dependent, parallel processing service stations, when is it optimal to route customers to the shortest queue and to devote auxiliary capacity to serve the longest queue? We show that this RSQ/SLQ policy is optimal for a wide class of Markovian systems, where the arrival and service rates at the stations, which may depend on the numbers of customers at all the stations, satisfy certain symmetry and monotonicity conditions. Under this policy, the queue lengths will be stochastically smaller in the weak submajorization ordering than the queue lengths under any other policy. Furthermore, this policy minimizes standard discounted and average cost functionals over finite and infinite horizons.