Partially observable point processes and the control of packet radio networks.
William S. Hortos · Deep Blue (University of Michigan) · 1990
An analytical model of the internodal packet flows and link distortions in terms of multivariate point processes is created for the discrete events in packet radio networks. The structure of the point processes is revealed by martingale integral representations based, in part, on the conditional intensities of the arrival, transmission, and routing processes. The innovations method due to Kailath, Jacod, and others is applied to observed and unobserved network processes representing such PRNET phenomena as noise and jamming, collisions, incomplete routing information, and mixed service networks. Examples of the filter of the network state on the observed network history are shown, determined by the martingale calculus. A packet routing control problem over a finite interval is constructed in terms of the network point processes and their integrated intensities. Network performance measures and control constraints are expressed as functions of these processes and their random rates, generalizing metrics such as channel capacity, throughput, and packet delay. The stochastic link arrays define an admissible control class. Control actions are implemented through a mutually absolutely continuous change of probability measure on the network events, wherein the Radon-Nikodym derivative is constructed from the elements of the link arrays and the point-process intensities. A general statement of the optimal routing control with partial state observations is formulated without specific use of point-process structures. A general principle of optimality is shown from which necessary and sufficient conditions for "almost" local optimality of the controls are derived. Optimality conditions are derived using the local description of the network state and observed point process dynamics for partially and completely observed networks. The optimality conditions for the case of complete observations of the network dynamics lead to recursive dynamic programming conditions. These recursive conditions, in turn, form sets of differential-difference equations in the optimal value function.