Analysis of Fork-Join Scheduling on Heterogeneous Parallel Servers

This paper investigates the $(k,k)$ fork-join scheduling scheme on a system of n parallel servers comprising both slow and fast servers. Tasks arriving in the system are divided into k sub-tasks and assigned to a random set of k servers, where each task can be assigned independently to a distinct slow or fast server with selection probability $p_{s}$ or $1-p_{s}$ , respectively. Our analysis demonstrates that the joint distribution of the stationary workload across any set of k queues becomes asymptotically independent as the number of servers n grows, with k scaling as $o\left ({{n^{\frac {1}{4}}}}\right)$ . Under asymptotic independence, the limiting mean task completion time can be expressed as an integral. However, it is analytically challenging to compute the optimal selection probability $p_{s}^{\ast } $ that minimizes this integral. To address this, we provide an upper bound on the limiting mean task completion time and identify the selection probability $\hat {p}_{s}$ that minimizes this bound. We validate that this selection probability $\hat {p}_{s}$ yields a near-optimal performance through numerical experiments.