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

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 modeling 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 $$\mathcal {O}(\log N)$$ 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 $$\log N$$ 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 $$N=2$$ N = 2 ; the technical explanation is that the resequencing (delay) price starts to quickly dominate the tempting gain due to multipath transmissions.