A Fluid Control Problem in Queueing Networks with General Service Times

We consider a queueing network with several customer classes and several service pools, each consisting of many statistically identical servers. The service requirement of each customer is assumed to be generally distributed and is allowed to depend on both the customer class and the service pool. Customers may also abandon while waiting in queue, with exponentially distributed patience time. For each class, the holding cost per unit time is given as a convex function of the corresponding queue length. It is shown that, under an overload condition and under a certain fluid scaling, the problem of identifying an optimal dynamical scheduling that minimizes the long average holding cost, can be reduced to solving a simple deterministic problem.