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Abstract
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 multi-server 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 as “non-exchangeable synchronization”. Once synchronized, jobs will leave the system immediately. Service times of the 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. We study the joint dynamics of the queueing and service processes at all stations and the associated unsynchronized queueing processes. The main mathematical challenge lies in the “resequencing” of arrival orders after service completion at each station. As in Lu and Pang (2015) for the infinite-server fork-join network model, the dynamics of all the aforementioned processes can be represented via a multiparameter sequential empirical process driven by the service vectors for the parallel tasks of each job. We consider the system in the Halfin-Whitt regime, and prove a functional law of large number and a functional central limit theorem for queueing and synchronization processes. In this regime, although the delay for service at each station is asymptotically negligible, the delay for synchronization is of the same order as the service times.
Highlights
We consider a fundamental fork-join network with a single class of jobs that will fork into a fixed number of parallel tasks upon their arrival, and join after service completion
In this paper we have developed a methodology to study the multi-server fork-join networks with the NES constraints in the Halfin–Whitt regime
The fluid limits are proved for the networks with an empty initial condition, in which each job is split into K ≥ 2 parallel tasks, and the arrival rate can be time-varying
Summary
Publication details, including instructions for authors and subscription information: http://pubsonline.informs.org. To cite this article: Hongyuan Lu, Guodong Pang (2016) Heavy-Traffic Limits for a Fork-Join Network in the Halfin-Whitt Regime. Full terms and conditions of use: https://pubsonline.informs.org/Publications/Librarians-Portal/PubsOnLine-Terms-andConditions. Descriptions of, or references to, products or publications, or inclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, or support of claims made of that product, publication, or service. With 12,500 members from nearly 90 countries, INFORMS is the largest international association of operations research (O.R.) and analytics professionals and students. INFORMS provides unique networking and learning opportunities for individual professionals, and organizations of all types and sizes, to better understand and use O.R. and analytics tools and methods to transform strategic visions and achieve better outcomes. For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org
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