Approximations and Bounds for (n, k) Fork-Join Queues: A Linear Transformation Approach

Abstract

Compared to basic fork-join queues, a job in (n, k) fork-join queues only needs its k out of all n sub-tasks to be finished. Since (n, k) fork-join queues are prevalent in popular distributed systems, erasure coding based cloud storages, and modern network protocols like multipath routing, estimating the sojourn time of such queues is thus critical for the performance measurement and resource plan of computer clusters. However, the estimating keeps to be a well-known open challenge for years, and only rough bounds for a limited range of load factors have been given. In this paper, we developed a closed-form linear transformation technique for jointly-identical random variables: An order statistic can be represented by a linear combination of maxima. This brand-new technique is then used to transform the sojourn time of non-purging (n, k) forkjoin queues into a linear combination of the sojourn times of basic (k, k), (k + 1, k + 1), ..., (n, n) fork-join queues. Consequently, existing approximations for basic fork-join queues can be bridged to the approximations for non-purging (n, k) fork-join queues. The uncovered approximations are then used to improve the upper bounds for purging (n, k) fork-join queues. Simulation experiments show that this linear transformation approach is practiced well for moderate n and relatively large k.

Performance Evaluation of Blockchain Systems: A Systematic Survey

“…Even though few analytical results exist for fork-join queues, various approximation solutions are known. An example is the linear transformation approach [127], which can be used to approximately compute the sojourn time t(n, k) of a general non-purging (n, k) fork-join queue for the vote-based blockchain system.…”

Latency Bounds of Packet-Based Fronthaul for Cloud-RAN with Functionality Split

“…(n, k) fork-join system. Mean value analysis for (n, k) forkjoin system has been done in [4] and [5]. Authors in [3] used stochastic network calculus to define a stochastic upper bound for distribution of delay in (n, k) fork-join system in a general case.…”

“…On the other hand, 5G traffic classes eMBB (enhanced mobile broadband), URLLC (ultra reliable low latency communications) and Massive machine type communications, each come with varied requirements on reliability that should also be maintained on the FH link; improving reliability often results in increasing latency. To this end, focusing on reliable packet-based FH [2], the aim of this paper is to compute the lower bound and upper bound of latency analytically, using stochastic network calculus [3], [4], [5]. We simulate the reliable packet-based FH, demonstrating where and how delay bounds are achieved for two classes of traffic, which are eMBB and URLLC.…”

Modeling of Request Cloning in Cloud Server Systems using Processor Sharing

“…Cloning implies that the n sub-tasks are identical and k = 1. Approximate analysis and latency bounds have been extensively studied for the general (n, k) fork-join systems [13,21,23], but unfortunately no exact analysis exists when n ≥ 3. This is, however, not the case for cloning.…”