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

TL;DR

This paper introduces a linear transformation technique to approximate and bound the sojourn times in (n, k) fork-join queues, which are common in distributed systems, by relating them to basic fork-join queues.

Contribution

A novel linear transformation method for representing order statistics, enabling improved approximations and bounds for (n, k) fork-join queue sojourn times.

Findings

Effective for moderate n and large k

Bridges approximations from basic to (n, k) queues

Improves upper bounds for purging (n, k) queues

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) fork-join 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.