Heavy traffic analysis of controlled multiplexing systems

The paper develops the mathematics of the heavy traffic approach to the control and optimal control problem for multiplexing systems, where there are many mutually independent sources which feed into a single channel via a multiplexer (or of networks composed of such subsystems). Due to the widely varying bit rates over all sources, control over admission, bandwidth, etc., is needed to assure good performance. Optimal control and heavy traffic analysis has been shown to yield systems with greatly improved performance. Indeed, the heavy traffic approach covers many cases of great current interest, and provides a useful and practical approach to problems of analysis and control arising in modern high speed telecommunications. Past works on the heavy traffic approach to the multiplexing problem concentrated on the uncontrolled system or on the use of the heavy traffic limit control problem for applications, and did not provide details of the proofs. This is done in the current paper. The basic control problem for the physical system is hard, and the heavy traffic approach provides much simplification. Owing to the presence of the control as well as to the fact that the cost function of main interest is ‘ergodic’, the problem cannot be fully treated with ‘classical’ methods of heavy traffic analysis for queueing networks. A basic result is that the optimal average costs per unit time for the physical problem converge to the optimal cost per unit time for the limit stationary process as the number of sources and the time interval goes to infinity. This convergence is both in the mean and pathwise senses. Furthermore, a ‘nice’ nearly optimal control for the limit system provides nearly optimal values for the physical system, under heavy traffic, in both a mean and pathwise sense.