  Recently it was shown that the response time of First-Come-First-Served
(FCFS) scheduling can be stochastically and asymptotically improved upon by the
{\it Nudge} scheduling algorithm in case of light-tailed job size
distributions. Such improvements are feasible even when the jobs are
partitioned into two types and the scheduler only has information about the
type of incoming jobs (but not their size).
  In this paper we introduce Nudge-$M$ scheduling, where basically any incoming
type-1 job is allowed to pass any type-2 job that is still waiting in the queue
given that it arrived as one of the last $M$ jobs. We prove that Nudge-$M$ has
an asymptotically optimal response time within a large family of Nudge
scheduling algorithms when job sizes are light-tailed. Simple explicit results
for the asymptotic tail improvement ratio (ATIR) of Nudge-$M$ over FCFS are
derived as well as explicit results for the optimal parameter $M$. An
expression for the ATIR that only depends on the type-1 and type-2 mean job
sizes and the fraction of type-1 jobs is presented in the heavy traffic
setting.
  The paper further presents a numerical method to compute the response time
distribution and mean response time of Nudge-$M$ scheduling provided that the
job size distribution of both job types follows a phase-type distribution (by
making use of the framework of Markov modulated fluid queues with jumps).
