Sample path methods in the control of queues

Sample path methods are now among the most used techniques in the control of queueing systems. However, due to the lack of mathematical formalism, they may appear to be non-rigorous and even sometimes mysterious. The goal of this paper is threefold: to provide a general mathematical setting, to survey the most popular sample path methods including forward induction, backward induction and interchange arguments, and to illustrate the present approach through the study of a number of classical scheduling and routing optimization problems arising in queueing theory.