We consider a new scheduling problem on parallel identical machines in which the number of machines is initially not known, but it follows a given probability distribution. Only after all jobs are assigned to a given number of bags, the actual number of machines is revealed. Subsequently, the jobs need to be assigned to the machines without splitting the bags. This is the stochastic version of a related problem introduced by Stein and Zhong [SODA 2018, TALG 2020] and it is, for example, motivated by bundling jobs that need to be scheduled by data centers. We present two PTASs for the stochastic setting, computing job-to-bag assignments that (i) minimize the expected maximum machine load and (ii) maximize the expected minimum machine load (like in the Santa Claus problem), respectively. The former result follows by careful enumeration combined with known PTASs. For the latter result, we introduce an intricate dynamic program that we apply to a suitably rounded instance.

翻译：我们研究一种新的并行同构机器调度问题，其中机器数量初始未知，而是服从给定的概率分布。仅当所有作业被分配到指定数量的任务包后，实际机器数量才被揭示。随后，需要在不拆分任务包的前提下将作业分配到各机器上。这是Stein和Zhong[SODA 2018, TALG 2020]所提出相关问题的随机化版本，其应用场景包括数据中心需要调度的批量作业打包。针对该随机化场景，我们提出了两种多项式时间近似方案（PTAS），分别用于计算满足以下目标的作业到任务包分配方案：（i）最小化期望最大机器负载；（ii）最大化期望最小机器负载（类似圣诞老人问题）。前者结果通过精细枚举结合已知PTAS技术实现。对于后者结果，我们引入了一种精密的动态规划方法，并将其应用于经过适当舍入处理的问题实例。