Size-Aware Dispatching to Fluid Queues

We develop a fluid-flow model for routing problems, where fluid consists of different size particles and the task is to route the incoming fluid to $n$ parallel servers using the size information in order to minimize the mean latency. The problem corresponds to the dispatching problem of (discrete) jobs arriving according to a stochastic process. In the fluid model the problem reduces to finding an optimal path to empty the system in $n$-dimensional space. We use the calculus of variation to characterize the structure of optimal policies. Numerical examples shed further light on the fluid routing problem and the optimal control of large distributed service systems.