Heavy-Traffic Limits of Queueing Networks with Polling Stations: Brownian Motion in a Wedge

We consider two serial single–server stations in heavy traffic. There are two job types: All jobs visit station 1 and then station 2. Station 1 processes jobs in an exhaustive service or gated service fashion; station 2 uses an arbitrary nonidling service discipline. Neither station incurs switchover delays. We prove two heavy–traffic limit theorems for the diffusion–scaled, two–dimensional total workload process: one for when the first station implements exhaustive service and the other for when gated service is employed. Our limiting processes are two–dimensional Brownian motions in a wedge, a type of reflected Brownian motion. The limiting process under exhaustive service is equal in distribution to the limiting process that one obtains when the first station performs one of two buffer priority policies.