Optimal Moments on Redundancies in Noisy Parallel Computing Setup

We consider the problem of job assignment where a master server aims to compute some tasks and is provided a few child servers to compute under a uniform straggling pattern where each server is equally likely to straggle. We distribute tasks to the servers so that the master is able to receive most of the tasks even if a significant number of child servers fail to communicate. We first show that all balanced assignment schemes have the same expectation on the number of distinct tasks received and then study the variance. The variance or the second moment is a useful metric to study as there could be a high variation in the number of distinct tasks received. We show constructions using a generalization of ``Balanced Incomplete Block Design'' 11, 40 minimizes the variance, and constructions based on repetition coding schemes attain the largest variance. Both minimum variance and maximum variance attaining designs have their own use cases depending on whether the master aims for a heavy-tailed or light-tailed distribution on the number of distinct jobs. We further show the equivalence between job and server-based assignment schemes when the number of jobs and child servers are equal.