We consider a time-slotted job-assignment system consisting of a central server, $N$ task-specific networks of machines, and multiple users. Each network specializes in executing a distinct type of task. Users stochastically generate jobs of various types and forward them to the central server, which routes each job to the appropriate network of machines. Due to resource constraints, the server cannot serve all users' jobs simultaneously, which motivates the design of scheduling policies with possible preemption. To evaluate scheduling performance, we introduce a novel timeliness metric, the age of job, inspired by the well-known metric, the age of information. We study the problem of minimizing the long-term weighted average age of job. We first propose a max-weight policy by minimizing the one-step Lyapunov drift and then derive the Whittle index (WI) policy when the job completion times of the networks of machines follow geometric distributions. For general job completion time distributions, we introduce a Whittle index with max-weight fallback (WIMWF) policy. We also investigate the Net-gain maximization (NGM) policy. Numerically, we show that the proposed WIMWF policy achieves the best performance in the general job completion time setting. We also observe a scaling trend: two different max-weight policies can outperform the NGM policy in small systems, whereas the NGM policy improves as we scale the system size and becomes asymptotically better than max-weight policies. For geometric service times, the WI policy yields the lowest age across all considered system sizes.

翻译：我们考虑一个由中央服务器、$N$个任务特定机器网络以及多个用户组成的时间片作业分配系统。每个网络专门执行不同类型的任务。用户随机生成各种类型的作业并将其转发给中央服务器，服务器将每个作业路由到相应的机器网络。由于资源限制，服务器无法同时服务所有用户的作业，这促使我们设计可能包含抢占的调度策略。为了评估调度性能，我们引入了一种新颖的时效性度量指标——作业年龄，其灵感来源于众所周知的年龄信息指标。我们研究了最小化长期加权平均作业年龄的问题。我们首先通过最小化一步李雅普诺夫漂移提出了一种最大权重策略，然后在机器网络的作业完成时间服从几何分布时推导了惠特尔指数策略。对于一般的作业完成时间分布，我们引入了一种带最大权重回退的惠特尔指数策略。我们还研究了净增益最大化策略。数值实验表明，在一般作业完成时间设置下，所提出的带最大权重回退的惠特尔指数策略实现了最佳性能。我们还观察到一个缩放趋势：在小型系统中，两种不同的最大权重策略可以优于净增益最大化策略；而随着系统规模的扩大，净增益最大化策略的性能得到改善，并渐近地优于最大权重策略。对于几何服务时间，惠特尔指数策略在所有考虑的系统规模中均产生了最低的作业年龄。