Homogenized First-Moment Analysis of Two-Time-Scale Positive Markov Jump Linear Systems

Abstract We address in this paper the mean stability and L ∞ performance of continuous-time Positive Markov Jump Linear Systems (PMJLS). The distinguishing aspect of our approach vis-a-vis the existing literature is that the underlying Markov jump process is a two-time-scale Markov chain, and we consider the singular perturbation setup which arises when a small parameter (which determines the time-scale separation) goes to zero. The interest in this limiting scenario stems from large-scale situations, where complexity reduction is a central issue. To achieve this, we carry out a convergence analysis involving the semigroup that describes the first moment dynamics of the system state. This analysis allows us to subsequently characterize homogenized notions of stability and L ∞ performance, and we show how these can be connected with linear programming methods. A numerical example, regarding a version of the Foschini-Miljanic algorithm for power allocation in a mobile communication system, illustrates the proposed results.