Optimal Control of Fluid Limits of Queuing Networks and Stochasticity Corrections, Lectures in Applied Mathematics

On one hand we present below some examples of uid networks control which illustrate the general method for optimally emptying networks with linear holding costs proposed in Avram, Bertsimas and Ricard(95). On the other hand we attempt to study the changes in the optimal policies for uid models caused by stochasticity e ects. In the case of the usual expected total emptying cost objective, we found out numerically in the particular case of the tandem that the optimal switch curve is very close to being an upward shift of the optimal uid switch line. We give a conjecture for the asymptotical value of this shift, obtained by the perturbation techniques promoted by Matkovski, Knessl, Schuss and Tier. We turn then to another objective which is much easier to work with: the so called "totally risk averse" objective. This is a limiting approach promoted by Whittle and Fleming which replaces stochastic control problems by deterministic differential games. For a tandem with " Gaussian stochasticity", this method yields a quadratic switch curve depending on some "worry" parameters in the objective. The switch curve approaches the line obtained for the uid model when the "worry" parameters tend to 0: We also indicate how the method proposed in [1] may be used to develop an algorithm for nding explicit solutions for the "totally risk averse" optimal scheduling of general Gaussian networks.