We consider a time-slotted job-assignment system with a central server, N users and a machine which changes its state according to a Markov chain (hence called a Markov machine). The users submit their jobs to the central server according to a stochastic job arrival process. For each user, the server has a dedicated job queue. Upon receiving a job from a user, the server stores that job in the corresponding queue. When the machine is not working on a job assigned by the server, the machine can be either in internally busy or in free state, and the dynamics of these states follow a binary symmetric Markov chain. Upon sampling the state information of the machine, if the server identifies that the machine is in the free state, it schedules a user and submits a job to the machine from the job queue of the scheduled user. To maximize the number of jobs completed per unit time, we introduce a new metric, referred to as the age of job completion. To minimize the age of job completion and the sampling cost, we propose two policies and numerically evaluate their performance. For both of these policies, we find sufficient conditions under which the job queues will remain stable.

翻译：本文研究一个具有中央服务器、N个用户及状态遵循马尔可夫链变化的机器（故称为马尔可夫机器）的时隙化作业分配系统。用户根据随机作业到达过程向中央服务器提交作业。针对每个用户，服务器设有专用作业队列。当服务器接收到来自用户的作业时，将其存储于对应队列中。当机器未处理服务器分配的作业时，其可能处于内部繁忙或空闲状态，这些状态的动态变化遵循二元对称马尔可夫链。服务器通过采样机器状态信息，若识别到机器处于空闲状态，则调度某一用户并从该用户的作业队列中向机器提交作业。为最大化单位时间内完成的作业数量，我们引入称为"作业完成时效"的新度量指标。为最小化作业完成时效与采样成本，提出两种策略并通过数值模拟评估其性能。针对这两种策略，我们找到了确保作业队列保持稳定的充分条件。