Submodular Returns and Greedy Heuristics for Queueing Scheduling Problems

We consider a range of controlled stochastic systems satisfying conservation laws together with a reducibility property that says that appropriate laws continue to hold when access to the system is restricted to a subset of all possible demand (job, customer) types. We show that for linear objectives, the optimum system-wide performance is a nondecreasing submodular (or supermodular) function of the subset chosen and that these properties are inherited from the geometry of the performance space concerned. These results are of considerable interest in their own right, but they also motivate the use of greedy heuristics for the solution of a range of job selection and scheduling problems which have hitherto seemed intractable. Computational experience suggests that such heuristics perform very well.