Construction of positive recurrent semimartingale reflecting Brownian motions with divergent fluid paths using predator prey models

Abstract

Semimartingale reflecting Brownian motions (SRBMs) are an integral part of queueing theory. Fluid paths are a standard tool for demonstrating positive recurrence of SRBMs, and their convergence provides both necessary and sufficient conditions for 3 and fewer dimensional SRBMs. Previous examples have been given in the literature for 6 and greater dimensional positive recurrent SRBMs with divergent linear fluid paths. This thesis first examines certain predator prey models for dependent and independent Brownian motions, where the expected capture time of the prey is finite. Using these predator prey models, this thesis provides a family of 4 dimensional positive recurrent SRBMs with divergent linear fluid paths, and a family of 5 dimensional positive recurrent SRBMs with divergent linear fluid paths and identity covariance matrix. This closes the gap on the number of dimensions under which the positive recurrence of an SRBM does and does not imply convergence of the associated fluid paths.