# Meeting Details

Title: Heavy-traffic limits for a fork-join network in the Halfin-Whitt regime Probability and Financial Mathematics Seminar Hongyuan Lu, PSU, Industr. & Manufact. Engineering We study a fork-join network with a single class of jobs, which are forked into a fixed number of parallel tasks upon arrival to be processed at the corresponding parallel service stations. After service completion, each task will join a buffer associated with the service station waiting for synchronization, called unsynchronized queue". The synchronization rule requires that all tasks from the same job must be completed, referred to non-exchangeable synchronization". Once synchronized, jobs will leave the system immediately. Service times of the associated parallel tasks of each job can be correlated and form a sequence of i.i.d. random vectors with a general continuous joint distribution function. Each service station has multiple statistically identical parallel servers. We consider the system in the Halfin-Whitt (Quality-and-Efficiency-Driven, QED) regime, in which the arrival rate of jobs and the number of servers in each station get large appropriately so that all service stations become critically loaded asymptotically. We develop a new method to study the joint dynamics of the service processes and the unsynchronized queueing processes at all stations and the synchronized process. The waiting processes for synchronization after service depend on the service dynamics at all service stations, and thus are extremely difficult to analyze exactly. The main mathematical challenge lies in the resequencing of arrival orders after service completion at each station. We represent the dynamics of all the aforementioned processes via a multiparameter sequential empirical process driven by the service vectors of the parallel tasks. We show a functional law large number (FLLN) and a functional central limit theorem (FCLT) for these processes. Both the service and unsynchronized queueing processes in the limit can be characterized as unique solutions to the associated integral convolution equations driven by the arrival limit process and a generalized multiparameter Kiefer process driven by the service vectors.