Distributed control of loss network systems: Independent subnetwork behaviour in infinite networks

Abstract

Call admission and routing control of loss (circuit- switched) networks can be formulated as optimal stochastic control (OSC) problems in case of a class of integral cost functions. The resulting Hamilton-Jacobi-Bellman (HJB) equation for such OSC problems consists of a collection of coupled first order PDEs linked by sets of integral coefficients. Unfortunately, the implementation of optimal control laws even for medium size systems is not feasible since the computational complexity of the HJB equations increases exponentially with network size. In this paper, we study admission control problems for a specific class of radial network systems composed of a group of weakly coupled subnetwork systems. We delineate the so-called (asymptotic) subnetwork (stochastic state) independence property as the network size goes to infinity. In particular this implies that the acceptance of incoming and outgoing call requests are asymptotically independent of all other state processes of the mass (i.e. overall) system. Under the general class of Markovian feedback control laws, we show the basic property of asymptotic sustainability of independent subnetwork behaviour holds as the network size goes to infinity. Based upon this class of network system models, distributed OSC problems may be formulated whereby each subnetwork system implements a local optimal control. This methodology leads to an application of the Nash certainty equivalence (NCE) principle itself proven quite useful within the LQG framework in M. Huang, et al., (Dec. 2006).