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

Lu, Hongyuan; Pang, Guodong

doi:10.1287/moor.2015.0740

Cited by 23 publications

(39 citation statements)

References 67 publications

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“…Refined approximations for the mean sojourn time in two-server FJ systems that take the first two moments of the service time distribution are given in [25]; numerical evidence is further provided on the quality of the approximation for different service time distributions. In a recent work, the authors of [30] establish Gaussian limits for the joint distributions of the service and waiting times for synchronization under general arrivals characterized by a limiting Brownian motion.…”

Section: Related Workmentioning

confidence: 99%

“…Given the burstiness condition in (24) and that the function h is monotonically decreasing [9], we can further upper bound the prefactor in (30) as…”

Section: Non-blocking Systemsmentioning

confidence: 99%

“…We can now use (30) and the remark at the end of Section 4.1 to bound the distribution of the steady-state response time r as…”

Section: Non-renewal Arrivalsmentioning

confidence: 99%

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Stochastic bounds in Fork–Join queueing systems under full and partial mapping

2016

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In a Fork-Join (FJ) queueing system an upstream fork station splits incoming jobs into N tasks to be further processed by N parallel servers, each with its own queue; the response time of one job is determined, at a downstream join station, by the maximum of the corresponding tasks' response times. This queueing system is useful to the modelling of multi-service systems subject to synchronization constraints, such as MapReduce clusters or multipath routing. Despite their apparent simplicity, FJ systems are hard to analyze.This paper provides the first computable stochastic bounds on the waiting and response time distributions in FJ systems under full (bijective) and partial (injective) mapping of tasks to servers. We consider four practical scenarios by combining 1a) renewal and 1b) non-renewal arrivals, and 2a) non-blocking and 2b) blocking servers. In the case of non-blocking servers we prove that delays scale as O(log N ), a law which is known for first moments under renewal input only. In the case of blocking servers, we prove that the same factor of log N dictates the stability region of the system. Simulation results indicate that our bounds are tight, especially at high utilizations, in all four scenarios. A remarkable insight gained from our results is that, at moderate to high utilizations, multipath routing "makes sense" from a queueing perspective for two paths only, i.e., response times drop the most when N = 2; the technical explanation is that the resequencing (delay) price starts to quickly dominate the tempting gain due to multipath transmissions.

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Section: Related Workmentioning

confidence: 99%

“…Given the burstiness condition in (24) and that the function h is monotonically decreasing [9], we can further upper bound the prefactor in (30) as…”

Section: Non-blocking Systemsmentioning

confidence: 99%

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Stochastic bounds in Fork–Join queueing systems under full and partial mapping

2016

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“…This buffer is called "unsynchronized queue" or "waiting buffer for synchronization". Tasks are only synchronized if all the parallel tasks of the same job are completed, called "non-exchangeable synchronization" (NES) [3,60,61,33]. After synchronization, jobs will leave the system immediately (the synchronization time is irrelevant in our model).…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

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confidence: 99%

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

Lu¹,

Pang²

2016

Stoch. Syst.

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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.

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Dynamic Control of the Join-Queue Lengths in Saturated Fork-Join Stations

Marin

Rossi

2016

Quantitative Evaluation of Systems

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The analysis of fork-join queueing systems has played an important role for the performance evaluation of distributed systems where parallel computations associated with the same job are carried out and the job is considered served only when all the parallel tasks it consists of are served and then joined. The fork&join nodes that we consider consist of K ≥ 2 parallel servers each of which is equipped with two FCFS queues, namely the service-queue and the join-queue. The latter store the serviced tasks waiting for being joined. This paper addresses the problem that under independent and exponentially distributed service time, the process describing the join-queue lengths becomes instable under heavy load. This is due to the variance of the service time distribution. We propose a simple mechanism that avoids this problem, show that we can analytically study a set of relevant performance indices and study by simulation its robustness.

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

Cited by 23 publications

References 67 publications

Stochastic bounds in Fork–Join queueing systems under full and partial mapping

Stochastic bounds in Fork–Join queueing systems under full and partial mapping

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

Dynamic Control of the Join-Queue Lengths in Saturated Fork-Join Stations

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