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(n(1/4)) . 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 p(s)<^> that minimizes this bound. We validate that this selection probability p(s)<^> yields a near-optimal performance through numerical experiments.