On the stochastic and asymptotic improvement of First-Come First-Served and Nudge scheduling

Recently it was shown that, contrary to expectations, the First-Come-First-Served (FCFS) scheduling algorithm can be stochastically improved upon by a scheduling algorithm called {\it Nudge} for light-tailed job size distributions. Nudge partitions jobs into 4 types based on their size, say small, medium, large and huge jobs. Nudge operates identical to FCFS, except that whenever a {\it small} job arrives that finds a {\it large} job waiting at the back of the queue, Nudge swaps the small job with the large one unless the large job was already involved in an earlier swap. In this paper, we show that FCFS can be stochastically improved upon under far weaker conditions. We consider a system with $2$ job types and limited swapping between type-$1$ and type-$2$ jobs, but where a type-$1$ job is not necessarily smaller than a type-$2$ job. More specifically, we introduce and study the Nudge-$K$ scheduling algorithm which allows type-$1$ jobs to be swapped with up to $K$ type-$2$ jobs waiting at the back of the queue, while type-$2$ jobs can be involved in at most one swap. We present an explicit expression for the response time distribution under Nudge-$K$ when both job types follow a phase-type distribution. Regarding the asymptotic tail improvement ratio (ATIR) , we derive a simple expression for the ATIR, as well as for the $K$ that maximizes the ATIR. We show that the ATIR is positive and the optimal $K$ tends to infinity in heavy traffic as long as the type-$2$ jobs are on average longer than the type-$1$ jobs.