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

Rizk, Amr; Poloczek, Felix; Ciucu, Florin

doi:10.1007/s11134-016-9486-x

Cited by 35 publications

(57 citation statements)

References 42 publications

(88 reference statements)

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“…Corollary 1.1 demonstrates that, with fixed µ(θ) and ψ(θ), the upper bound for the violation probability P (W (t) ≥ w) grows with increasing m. The observation coincides with the result mentioned in [16] and [17] that the delay roughly scales up linearly with the number of independent paths, especially when the end-to-end delay on each path is small.…”

Section: A Delay Bound For Traffic Dispersionsupporting

confidence: 77%

“…Clearly, both the traffic dispersion and network densification schemes are promising and competitive candidates for lowlatency mm-wave communications. Though there are many research contributions in low-latency communications based on above two principles, the existing literature focus on either the dispersion scheme (e.g., [12]- [17]) or multi-hop relaying scheme (e.g., [18]- [21]). It is not clear yet which scheme can provide better delay performance.…”

Section: B Motivationmentioning

confidence: 99%

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Low-Latency Millimeter-Wave Communications: Traffic Dispersion or Network Densification?

Yang

Xiao

Poor

2018

IEEE Trans. Commun.

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Low latency is critical for many applications in wireless communications, e.g., vehicle-to-vehicle (V2V), multimedia, and industrial control networks. Meanwhile, for the capability of providing multi-gigabits per second (Gbps) rates, millimeter-wave (mm-wave) communication has attracted substantial research interest recently. This paper investigates two strategies to reduce the communication delay in future wireless networks: traffic dispersion and network densification. A hybrid scheme that combines these two strategies is also considered. The probabilistic delay and effective capacity are used to evaluate performance. For probabilistic delay, the violation probability of delay, i.e., the probability that the delay exceeds a given tolerance level, is characterized in terms of upper bounds, which are derived by applying stochastic network calculus theory. In addition, to characterize the maximum affordable arrival traffic for mmwave systems, the effective capacity, i.e., the service capability with a given quality-of-service (QoS) requirement, is studied. The derived bounds on the probabilistic delay and effective capacity are validated through simulations. These numerical results show that, for a given sum power budget, traffic dispersion, network densification, and the hybrid scheme exhibit different potentials to reduce the end-to-end communication delay. For instance, traffic dispersion outperforms network densification when high sum power budget and arrival rate are given, while it could be the worst option, otherwise. Furthermore, it is revealed that, increasing the number of independent paths and/or relay density is always beneficial, while the performance gain is related to the arrival rate and sum power, jointly. Therefore, a proper transmission scheme should be selected to optimize the delay performance, according to the given conditions on arrival traffic and system service capability.• It is demonstrated that, traffic dispersion, network densification and the hybrid scheme have respective advantages, and resulting end-to-end delay performance depends on the sum power budget and the density of data traffic (e.g., average arrival rate). For instance, when the given sum power is large, traffic dispersion, the hybrid scheme, and network densification are suitable for the scenarios with heavy, medium, and light arrival traffic, respectively. However, when the given sum power is small, the corresponding strengths of above three schemes significantly change with respect to arrival traffic. These observations provide interesting insights for mm-wave network designs and implementations. That is, the transmission scheme for low-latency performance should be properly selected according to the density of arrival traffic and/or the feasible system gain. The remainder of this paper is outlined as follows. In Sec. II, preliminaries for MGF-based stochastic network calculus are provided. In Sec. III, we give system models for traffic dispersion, network densification, and the hybrid scheme, respectively, and present ...

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Section: A Delay Bound For Traffic Dispersionsupporting

confidence: 77%

Section: B Motivationmentioning

confidence: 99%

Low-Latency Millimeter-Wave Communications: Traffic Dispersion or Network Densification?

Yang

Xiao

Poor

2018

IEEE Trans. Commun.

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“…In this section we discuss prior work on the limited fork-join model and some other related models. Prior work on the limited forkjoin model [19,28] has focused on the non-asymptotic regime and derived bounds on job delay. However, the bounds in [19,28] do not have tightness guarantees.…”

Section: Related Workmentioning

confidence: 99%

Delay Asymptotics and Bounds for Multi-Task Parallel Jobs

Wang

Jiang

Scheller‐Wolf

et al. 2019

SIGMETRICS Perform. Eval. Rev.

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We study delay of jobs that consist of multiple parallel tasks, which is a critical performance metric in a wide range of applications such as data file retrieval in coded storage systems and parallel computing. In this problem, each job is completed only when all of its tasks are completed, so the delay of a job is the maximum of the delays of its tasks. Despite the wide attention this problem has received, tight analysis is still largely unknown since analyzing job delay requires characterizing the complicated correlation among task delays, which is hard to do.We first consider an asymptotic regime where the number of servers, n, goes to infinity, and the number of tasks in a job, k (n) , is allowed to increase with n. We establish the asymptotic independence of any k (n) queues under the condition k (n) = o(n 1/4 ). This greatly generalizes the asymptotic-independence type of results in the literature where asymptotic independence is shown only for a fixed constant number of queues. As a consequence of our independence result, the job delay converges to the maximum of independent task delays.We next consider the non-asymptotic regime. Here we prove that independence yields a stochastic upper bound on job delay for any n and any k (n) with k (n) ≤ n. The key component of our proof is a new technique we develop, called "Poisson oversampling". Our approach converts the job delay problem into a corresponding balls-and-bins problem. However, in contrast with typical balls-andbins problems where there is a negative correlation among bins, we prove that our variant exhibits positive correlation.

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“…However, exact results are known only for few specific systems, such as two parallel M|M|1 queues [7], [8]. For more complex systems, approximation techniques, e.g., [5], [8]- [14], and bounds, using stochastic orderings [4], martingales [15], or stochastic burstiness constraints [16], have been explored. Given the difficulties posed by single-stage fork-join systems, few works consider This work was supported in part by the European Research Council (ERC) under Starting Grant UnIQue (StG 306644).…”

Section: Introductionmentioning

confidence: 99%

“…A notable exception is [13] where an approximation for closed fork-join networks is developed. Related synchronization problems also occur in the case of load balancing using parallel servers and in the case of multi-path routing of packet data streams [17] using multipath protocols [15]. The tail behavior of delays in multi-path routing is investigated in [17] as well as in [18], [19] where large deviation results of resequencing delays for parallel M|M|1 queues are derived.…”

Section: Introductionmentioning

confidence: 99%

Non-Asymptotic Delay Bounds for Multi-Server Systems with Synchronization Constraints

Fidler

Walker

Jiang

2018

IEEE Trans. Parallel Distrib. Syst.

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Multi-server systems have received increasing attention with important implementations such as Google MapReduce, Hadoop, and Spark. Common to these systems are a fork operation, where jobs are first divided into tasks that are processed in parallel, and a later join operation, where completed tasks wait until the results of all tasks of a job can be combined and the job leaves the system. The synchronization constraint of the join operation makes the analysis of fork-join systems challenging and few explicit results are known. In this work, we model fork-join systems using a max-plus server model that enables us to derive statistical bounds on waiting and sojourn times for general arrival and service time processes. We contribute end-to-end delay bounds for multi-stage fork-join networks that grow in O(h ln k) for h fork-join stages, each with k parallel servers. We perform a detailed comparison of different multiserver configurations and highlight their pros and cons. We also include an analysis of single-queue fork-join systems that are nonidling and achieve a fundamental performance gain, and compare these results to both simulation and a live Spark system.

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

Cited by 35 publications

References 42 publications

Low-Latency Millimeter-Wave Communications: Traffic Dispersion or Network Densification?

Low-Latency Millimeter-Wave Communications: Traffic Dispersion or Network Densification?

Delay Asymptotics and Bounds for Multi-Task Parallel Jobs

Non-Asymptotic Delay Bounds for Multi-Server Systems with Synchronization Constraints

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