Numerical methods for controlled and uncontrolled multiplexing and queueing systems

Abstract

We deal with a very useful numerical method for both controlled and uncontrolled queuing and multiplexing type systems. The basic idea starts with a heavy traffic approximation, but it is shown that the results are very good even when working far from the heavy traffic regime. The underlying numerical method is a version of what is known as the Markov chain approximation method. It is a powerful methodology for controlled and uncontrolled stochastic systems, which can be approximated by diffusion or reflected diffusion type systems, and has been used with success on many other problems in stochastic control. We give a complete development of the relevant details, with an emphasis on multiplexing and particular queueing systems. The approximating process is a controlled or uncontrolled Markov chain which retains certain essential features of the original problem. This problem is generally substantially simpler than the original physical problem, and there are associated convergence theorems. The non-classical associated ergodic cost problem is derived, and put into a form such that reliable and good numerical algorithms, based on multigrid type ideas, can be used. Data for both controlled and uncontrolled problems shows the value of the method.