Stochastic Systems

Heavy-traffic limits for a fork-join networkin the Halfin-Whitt regime

Hongyuan Lu and Guodong Pang

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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.

Article information

Stoch. Syst., Volume 6, Number 2 (2016), 519-600.

Received: October 2015
First available in Project Euclid: 22 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60H20: Stochastic integral equations 60K25: Queueing theory [See also 68M20, 90B22] 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 90B15: Network models, stochastic 90B22: Queues and service [See also 60K25, 68M20]

Fork-join networks non-exchangeable synchronization resequencing Halfin-Whitt (QED) regime functional law of large numbers (FLLN) functional central limit theorem (FCLT) multiparameter sequential empirical process generalized multiparameter Kiefer process

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Lu, Hongyuan; Pang, Guodong. Heavy-traffic limits for a fork-join networkin the Halfin-Whitt regime. Stoch. Syst. 6 (2016), no. 2, 519--600. doi:10.1214/15-SSY206.

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