Functional approximation theorems for controlled queueing networks

Presents a functional law of large numbers and a functional central limit theorem for a class of closed, non-Markovian queueing networks. The authors first consider a closed network of K queues and N customers. The service rates at each node are scaled by a function of the relative distribution of the customers in the various queues. The functional law of large numbers, often referred to as a fluid approximation theorem, expresses the convergence of the state process to the solution of an ordinary differential equation (ODE). The functional central limit theorem, or diffusion approximation theorem, establishes that the error in the law of large numbers converges weakly to the solution of a stochastic differential equation with time-varying coefficients. The authors then present similar theorems for open networks in which both the service and the arrival rates at each node are scaled by a function of the current customer distribution in the network and the previous arrivals