Redundancy scheduling with scaled Bernoulli service requirements

Redundancy scheduling has emerged as a powerful strategy for improving response times in parallel-server systems. The key feature in redundancy scheduling is replication of a job upon arrival by dispatching replicas to different servers. Redundant copies are abandoned as soon as the first of these replicas finishes service. By creating multiple service opportunities, redundancy scheduling increases the chance of a fast response from a server that is quick to provide service and mitigates the risk of a long delay incurred when a single selected server turns out to be slow. The diversity enabled by redundant requests has been found to strongly improve the response time performance, especially in the case of highly variable service requirements. Analytical results for redundancy scheduling are unfortunately scarce however, and even the stability condition has largely remained elusive so far, except for exponentially distributed service requirements. In order to gain further insight in the role of the service requirement distribution, we explore the behavior of redundancy scheduling for scaled Bernoulli service requirements. We establish a sufficient stability condition for generally distributed service requirements, and we show that, for scaled Bernoulli service requirements, this condition is also asymptotically nearly necessary. This stability condition differs drastically from the exponential case, indicating that the stability condition depends on the service requirements in a sensitive and intricate manner.