In queueing theory one seeks to predict in quantitative terms the congestion delays that occur when jobs or customers complete for processing resources. At present no satisfactory methods exist for the analysis of systems that allow simultaneous performance of tasks associated with a single job or customer. We present a heavy traffic analysis for the class of homogeneous fork-join networks in which jobs are routed in a feedforward deterministic fashion. We show that under certain regularity conditions the vector of total job count processes converges weakly to a multidimensional reflected Brownian motion (RBM) whose state space is a polyhedral cone in the nonnegative orthant. Furthermore, the weak limits of workload levels and throughput times are shown to be simple transformations of the RBM. As will be explained, the "steady-state throughput time" (a random variable) is expressed in terms of workload levels via the "longest path functional" of classical PERT/CPM analysis.
"Processing Networks with Parallel and Sequential Tasks: Heavy Traffic Analysis and Brownian Limits." Ann. Appl. Probab. 3 (1) 28 - 55, February, 1993. https://doi.org/10.1214/aoap/1177005506