State Space Collapse in Many‐Server Diffusion Limits of Parallel Server Systems

We consider a class of queueing systems that consist of server pools in parallel and multiple customer classes. Customer service times are assumed to be exponentially distributed. We study the asymptotic behavior of these queueing systems in a heavy traffic regime that is known as the Halfin‐Whitt many‐server asymptotic regime. Our main contribution is a general framework for establishing state space collapse results in this regime for parallel server systems. In our work, state space collapse refers to a decrease in the dimension of the processes tracking the number of customers in each class waiting for service and the number of customers in each class being served by various server pools. We define and introduce a ‘state space collapse’ function, which governs the exact details of the state space collapse. We show that a state space collapse result holds in many‐server heavy traffic if a corresponding deterministic hydrodynamic model satisfies a similar state space collapse condition. Unlike the single‐server heavy traffic setting for multiclass queueing network, our hydrodynamic model is different from the standard fluid model for many‐server queues. Our methodology is similar in spirit to that in Bramson [1998], which focuses on the single‐server heavy traffic regime. We illustrate the applications of our results by establishing state space collapse results in many‐server diffusion limits for V‐model systems under static‐buffer‐priority policy and the threshold policy proposed in the literature.