Optimal stochastic scheduling in multiclass parallel queues

In this paper we consider the problem of scheduling different classes of customers on multiple distributed servers to minimize an objective function based on per-class mean response times. This problem arises in a wide range of distributed systems, networks and applications. Within the context of our model, we observe that the optimal sequencing strategy at each of the servers is a simple static priority policy. Using this observation, we argue that the globally optimal scheduling problem reduces to finding an optimal routing matrix under this sequencing policy. We formulate the latter problem as a nonlinear programming problem and show that any interior local minimum is a global minimum, which significantly simplifies the solution of the optimization problem. In the case of Poisson arrivals, we provide an optimal scheduling strategy that also tends to minimize a function of the per-class response time variances. Applying our analysis to various static instances of the general problem leads us to rederive many results, yielding simple approximation algorithms whose guarantees match the best known results.