In the mathematical theory of probability, the drift-plus-penalty method is used for optimization of queueing networks and other stochastic systems. The technique is for stabilizing a queueing network while also minimizing the time average of a network penalty function. It can be used to optimize performance objectives such as time average power, throughput, and throughput utility.In the special case when there is no penalty to be minimized, and when the goal is to design a stable routing policy in a multi-hop network, the method reduces to backpressure routing.The drift-plus-penalty method can also be used to minimize the time average of a stochastic process subject to time average constraints on a collection of other stochastic processes.This is done by defining an appropriate set of . I