Abstract We consider a single server queue in continuous time, in which customers must be served before some limit sojourn time of exponential distribution. Customers who are not served before this limit leave the system: they are impatient. The fact of serving customers and the fact of losing them due to impatience induce costs. The fact of holding them in the queue also induces a constant cost per customer and per unit time. The purpose is to decide whether to serve customers or to keep the server idle, so as to minimize costs. We use a Markov Decision Process with infinite horizon and discounted cost. Since the standard uniformization approach is not applicable here, we introduce a family of approximated uniformizable models, for which we establish the structural properties of the stochastic dynamic programming operator, and we deduce that the optimal policy is of threshold type. The threshold is computed explicitly. We then pass to the limit to show that this threshold policy is also optimal in the original model and we characterize the optimal policy. A particular care is given to the completeness of the proof. We also illustrate the difficulties involved in the proof with numerical examples.

中文翻译：

---

在不耐烦和设置成本的队列中对服务中的入场进行优化控制

摘要 我们考虑连续时间中的单个服务器队列，其中必须在指数分布的某个限制逗留时间之前为客户提供服务。在此限制之前未得到服务的客户离开系统：他们不耐烦。为客户服务的事实以及由于不耐烦而失去客户的事实会导致成本。将它们排在队列中的事实也导致每个客户和单位时间的成本恒定。目的是决定是为客户服务还是让服务器保持闲置状态，从而最大限度地降低成本。我们使用具有无限范围和折扣成本的马尔可夫决策过程。由于标准的统一化方法在这里不适用，我们引入了一系列近似的统一化模型，为此我们建立了随机动态规划算子的结构特性，并且我们推导出最优策略是阈值类型的。阈值是明确计算的。然后我们传递到极限以表明该阈值策略在原始模型中也是最优的，并且我们表征了最优策略。特别注意证明的完整性。我们还用数值例子说明了证明中涉及的困难。