Bounding the Flow Time in Online Scheduling with Structured Processing Sets

Replication in distributed key-value stores makes scheduling more challenging, as it introduces processing set restrictions, which limits the number of machines that can process a given task. We focus on the online minimization of the maximum response time in such systems, that is, we aim at bounding the latency of each task. When processing sets have no structure, Anand et al. (Algorithmica, 2017) derive a strong lower bound on the competitiveness of the problem: no online scheduling algorithm can have a competitive ratio smaller than $\Omega(m)$, where $m$ is the number of machines. In practice, data replication schemes are regular, and structured processing sets may make the problem easier to solve. We derive new lower bounds for various common structures, including inclusive, nested or interval structures. In particular, we consider fixed sized intervals of machines, which mimic the standard replication strategy of key-value stores. We prove that EFT (Earliest Finish Time) scheduling is ($3-2/k$)-competitive when optimizing max-flow on disjoint intervals of size $k$. However, we show that the competitive ratio of EFT is at least $m-k+1$ when these intervals overlap, even when unit tasks are considered. We compare these two replication strategies in simulations and assess their efficiency when popularity biases are introduced, i.e., when some machines are accessed more frequently than others because they hold popular data. Even though overlapping intervals suffer from a bad worst-case in theory, they enable clusters to reach a maximum load that is up to 50% higher than with disjoint sets.