We consider non-preemptive scheduling problems on parallel identical machines where machines change their status from being available to being unavailable and vice versa along the time horizon. The particular form of unavailability we consider is when the starting time of each downtime depends upon the cardinality of the job subset processed on that machine since the previous downtime. We consider the problem of minimizing the makespan in such scenarios as well as its dual problem where we have a fixed common deadline of $1$ and the goal is to minimize the number of machines for which there is a feasible schedule. We develop an EPTAS for the first variant and an AFPTAS for the second variant.

翻译：我们考虑并行同构机器上的非抢占式调度问题，其中机器状态随时间在可用与不可用之间切换。本文研究的不可用时段具有特殊形式：每次停机时间取决于该机器自上次停机以来所处理作业子集的基数。我们考虑两类问题：一类是最小化此类场景中的最大完工时间，另一类是其对偶问题——在固定共同截止期为$1$的条件下，最小化存在可行调度方案的机器数量。针对第一类问题，我们提出了一个高效多项式时间近似方案(EPTAS)；针对第二类问题，我们提出了一个渐近完全多项式时间近似方案(AFPTAS)。