Controlled stochastic networks in heavy traffic: Convergence of value functions

TL;DR

This paper proves that the value functions of scaled controlled stochastic networks in heavy traffic converge to those of diffusion control problems, providing a rigorous foundation for using diffusion models to approximate complex network control.

Contribution

It establishes a general convergence result for the value functions of controlled stochastic networks to diffusion control problems under broad conditions.

Findings

Value functions of scaled networks converge to diffusion control problem solutions.

Provides a mathematical basis for using diffusion models as approximations.

Suggests a strategy for near-optimal control construction using diffusion models.

Abstract

Scheduling control problems for a family of unitary networks under heavy traffic with general interarrival and service times, probabilistic routing and an infinite horizon discounted linear holding cost are studied. Diffusion control problems, that have been proposed as approximate models for the study of these critically loaded controlled stochastic networks, can be regarded as formal scaling limits of such stochastic systems. However, to date, a rigorous limit theory that justifies the use of such approximations for a general family of controlled networks has been lacking. It is shown that, under broad conditions, the value function of the suitably scaled network control problem converges to that of the associated diffusion control problem. This scaling limit result, in addition to giving a precise mathematical basis for the above approximation approach, suggests a general strategy for constructing near optimal controls for the physical stochastic networks by solving the associated diffusion control problem.