Ergodic control of resource sharing networks: lower bound on asymptotic costs

Dynamic capacity allocation control for resource sharing networks (RSN) is studied when the networks are in heavy traffic. The goal is to minimize an ergodic cost with a linear holding cost function. Our main result shows that the optimal cost associated with an associated Brownian control problem provides a lower bound for the asymptotic ergodic cost in the RSN for any sequence of control policies. A similar result for an infinite horizon discounted cost has been previously shown for general resource sharing networks in Budhiraja and Conroy (in Modeling, stochastic control, optimization, and applications, 2019) and for general 'unitary networks' in Budhiraja and Ghosh (Ann Appl Probab 16:1962-2006, 2006). The study of an ergodic cost criterion requires different ideas as bounds on ergodic costs only yield an estimate on the controls in a time-averaged sense which makes the time-rescaling ideas of Budhiraja and Conroy (in Modeling, stochastic control, optimization, and applications, 2019) and Budhiraja and Ghosh (Ann Appl Probab 16:1962-2006, 2006) hard to implement. Proofs rely on working with a weaker topology on the space of controlled processes that is more amenable to an analysis for the ergodic cost. As a corollary of the main result, we show that the explicit policies constructed in Budhiraja and Johnson (Math Oper Res 45:797-832, 2020; Ann Appl Probab 34:851-916, 2024) for general RSN are asymptotically optimal for the ergodic cost when the underlying cost per unit time has certain monotonicity properties.