Optimization of hybrid stochastic differential systems in communications networks. (August 2015) - Baili, Hana
Abstract: This paper addresses issues from applied stochastic analysis for modeling and solving control problems in communications networks. We consider the problem of optimal scheduling in a wireless system with time varying traffic. The system is handled by a single base station transmitting over time varying channels. This may be the case in practice for a hybrid TDMA–CDMA (Time Division Multiple Access–Code Division Multiple Access) system. Heavy traffic approximation for the physical system yields a problem of optimal control in a constrained hybrid stochastic differential system. Here constrained means bounded or reflected in the K -dimensional positive orthant. In this work we establish a closed form solution for the nascent optimal control problem. The aim is to find a control which minimizes the expected total delay for the users. The control is constrained to satisfy some inequalities. Hence it seems natural that the cost function involve a penalty term for breaking these inequalities. We study this control problem by a dynamic programming approach and we are led to the resolution of a Hamilton–Jacobi–Bellman (HJB) equation in the finite dimensional space R + K . While optimizing it turns out that the optimum falls inside the class of feedback controls. Hence our method consists in finding a smooth solution of Bellman's equation and consequently getting a unique solution for the closed loop. Here a separation of variables enables us to construct an explicit C 2 solution of the HJB equation so that existence of a unique optimal control is proven. Further stochastic analysis of the hybrid stochastic differential system under the optimal control is provided: the Fokker–Planck equation for the distribution density of the state.