Random Variables with Measurability Constraints with Application to Opportunistic Scheduling

TL;DR

This paper develops a representation theorem for sequences of random elements with complex measurability constraints and applies it to optimize decision-making in opportunistic scheduling for communication networks.

Contribution

It introduces new representation theorems for random variables under measurability constraints and applies them to characterize decision sets in stochastic control and scheduling.

Findings

Refined network capacity theorem without complex state space extensions

Constructible sets theory applied to measurability assumptions

Nonmeasurable scheduling schemes can outperform measurable ones

Abstract

This paper proves a representation theorem regarding sequences of random elements that take values in a Borel space and are measurable with respect to the sigma algebra generated by an arbitrary union of sigma algebras. This, together with a related representation theorem of Kallenberg, is used to characterize the set of multidimensional decision vectors in a discrete time stochastic control problem with measurability and causality constraints, including opportunistic scheduling problems for time-varying communication networks. A network capacity theorem for these systems is refined, without requiring an implicit and arbitrarily complex extension of the state space, by introducing two measurability assumptions and using a theory of constructible sets. An example that makes use of well known pathologies in descriptive set theory is given to show a nonmeasurable scheduling scheme can outperform all measurable scheduling schemes.