Heavy-Traffic Limits for a Fork-Join Network in the Halfin-Whitt Regime

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.

Gaussian Limits for a Fork-Join Network with Nonexchangeable Synchronization in Heavy Traffic

“…That requires tracking the status (in service or in the waiting buffer for synchronization) of each task of jobs initially in the system. This approach can be further developed to study fork-join networks with NES in the many-server regimes, for example, the recent development in the Halfin-Whitt regime in [44], where non-empty initial condition is considered for K = 2. It remains open to investigate if our approach can be used to study scheduling and routing control problems in single class or multi-class models with potentially multiple processing stages.…”

“…We state the proof of the first approach below, and the proof of the second approach in the Appendix B. We think that the techniques developed in the new approach will turn out to be useful to study fork-join networks with NES in the many-server heavy-traffic regimes (see, e.g., [44]). Proof of Lemma 6.2.…”

“…random vectors and properties of multiparameter processes and martingales. This approach is further developed to study (non-Markovian) many-server fork-join networks in the Halfin-Whitt regime in [44].…”

Economies-of-Scale in Many-Server Queueing Systems: Tutorial and Partial Review of the QED Halfin--Whitt Heavy-Traffic Regime

“…These are (i) loss networks and (ii) bandwidth sharing networks. For work on fork-join networks, we refer to [100,101,102] A loss network is an extension of the Erlang B model. Consider a telecommunication network with J links, and suppose that link j, j = 1, .…”

Control of Fork-Join Processing Networks with Multiple Job Types and Parallel Shared Resources

“…1.1. Literature Review Although there are many studies focusing on performance evaluation of the fork-join networks (see Nguyen [25,26], Thomasian [34] and references therein, Lu and Pang [19,20,21]), there are only a few studies focusing on control of fork-join networks (see Atar et al [4], Pedarsani et al [28,29,30], Özkan and Ward [27]). Atar et al [4] consider the control of a specific fork-join network with probabilistic feedback mechanism.…”