Large-scale computations for high dimensional control systems

We investigate a multiplexing communications system, with many independent users competing for space in the transmitter buffer. The control consists in the deletion of selected low priority cells, or various formally equivalent forms. Under reasonable conditions, such systems can be approximated by diffusion type processes. This approximation is the basis of numerical methods for the associated control problems, which are generally much simpler than what one would have by working directly with the original system. Markov chain approximations are used. They have the structure of the original problem, but are generally much simpler. The numerical data shows that performance can be greatly improved over standard operating procedures by the use of optimal controls or reasonable approximations to them. Details algorithms for 4D control problems are discussed. For high dimensional models, the numerical approximation might have millions of states. One needs to compute quantities which are the equivalent of probabilities of the order of 10^-6 or smaller. Once the basic form of the mathematical algorithm is fixed, efficiency in coding is essential. General software codes are discussed. The effectiveness of multigrid-type solution methods is demonstrated as well its limitations. We also investigate the performance of dynamic memory allocation