Computable Bounds in Fork-Join Queueing Systems

Abstract

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 multiservice 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. 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.

Collaborative Uploading in Heterogeneous Networks: Optimal and Adaptive Strategies

“…Finally, the data batch reaches the cloud when all of its chunks are received. Such systems are known as FJ queuing systems [1]- [3].…”

“…In the following, we show some illustrative examples with computable k opt before generalizing this allocation scheme to include replication strategies. Mean upload latency (1,1) (2,1) (4,1)…”

“…4, which marks the regimes The lines specify different (3, 2)-allocations. For example, the line joining 1, 5, and 6 corresponds to the allocation (1,5,6). Valid (3, 2)-allocations require the combined size of any 2 chunks to be at least the data size, here, 6.…”

“…Following [1], we define the waiting time of an incoming data batch as the amount of time it waits until the last of its chunks starts getting uploaded. Consider the steady-state waiting time W (precise definition given in [1] and for the sake of completeness, also in Appendix I). It is hard to find out the distribution of the steady-state waiting times in closed form (see [1], [8]).…”

Fork and Join Queueing Networks with Heavy Tails

“…Due to such synchronization and blocking mechanism, exact analyses of FJQN/Bs can be challenging and possess high complexity. Much of the literature focuses on performance properties such as stability, duality, and comparison results, e.g., [6,7,20], approximation or bounding techniques, e.g., [8,71,55,56,64,66], or heavy traffic limits, e.g., [41,42].…”