Optimal moments on redundancies in job cloning

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 \textit{balanced} assignment schemes have the same expectation on the number of distinct tasks received and then study the variance. We show constructions using a generalization of ``Balanced Incomplete Block Design''\cite{doi:10.1111/j.1469-1809.1939.tb02219.x,sprott1955} minimizes the variance, and constructions based on repetition coding schemes attain the largest variance.