Dynamic control of Brownian networks: State space collapse and equivalent workload formulations
Brownian networks are heavy traffic limits or heavy traffic approximations of queueing network processes or interconnected inventory system processes. The stochastic behaviour of such processes is determined by a multidimensional Brownian motion input process and a control process. These are connected by an input-output matrix, a capacity consumption matrix, and the boundary behaviour (non-negativity) of the processes of interest: Inventory level \(Z\) and cumulative idleness \(U\). Given the initial inventory level, the structure functions, and the Brownian motion, a pair \((Z,U)\) is achievable if there exists a control process whose application results in it. The authors identify a class of control actions called reversible and show that any two state vectors which differ by a reversible displacement only are equivalent with respect to the control problem under consideration. This gives rise to the possibility of reducing the state space of the process by considering only a projection of the process. Control decisions can be based on the projection process only. The main theorem states that any pair \((Z,U)\) is achievable in the original Brownian network iff it is achievable in the reduced network. Several examples are discussed in detail and the authors point out that their results do not use the Brownian motion properties of the driving process.
scientific article; zbMATH DE number 1080212
scientific article