Abstract In an observable queue, customers joining decisions may be influenced by wait-aversion and crowd-attraction. These opposing phenomena and the diversity of arriving customers lead to an arrival process that depends on the number of present customers. For the system manager, having more customers may be beneficial as it can increase future arrivals due to the attraction generated. It may also saturate the system, resulting in long waits. Rejection at arrival may then be employed as a way to obtain a trade-off between these conflicting objectives. With this in mind, we developed a Markov decision process approach to determine how to optimally reject customers in a queueing system with state-dependent arrivals. When the arrival rate is bounded, we compute the optimal policy from a value iteration approach. When the arrival rate is decreasing and convex, we prove that it has a threshold form. When the arrival rate is increasing and potentially unbounded, uniformization may not apply. In dealing with this case, we restrict the analysis to stationary policies and prove the optimality of threshold policies from a computational approach. In addition, we show how to compute the optimal threshold within a finite number of iterations and prove that the long-run expected cost is decreasing and convex in the number of servers. We finally illustrate the applicability of our results through the analysis of a linearly increasing arrival rate, determining the main drivers of control decisions.

中文翻译：

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使用与状态相关的到达率来标注队列

摘要 在可观察队列中，顾客加入决策可能会受到等待厌恶和人群吸引的影响。这些相反的现象和到达客户的多样性导致到达过程取决于现有客户的数量。对于系统管理员来说，拥有更多的客户可能是有益的，因为它可以增加未来的吸引力，因为它会产生吸引力。它还可能使系统饱和，导致长时间等待。然后可以采用到达时拒绝作为在这些相互冲突的目标之间进行权衡的一种方式。考虑到这一点，我们开发了一种马尔可夫决策过程方法来确定如何在具有状态相关到达的排队系统中以最佳方式拒绝客户。当到达率有界时，我们通过值迭代方法计算最优策略。当到达率递减且凸出时，我们证明它具有阈值形式。当到达率增加并且可能无限时，统一化可能不适用。在处理这种情况时，我们将分析限制在静态策略上，并通过计算方法证明阈值策略的最优性。此外，我们展示了如何在有限的迭代次数内计算最佳阈值，并证明长期预期成本在服务器数量上是递减的和凸的。我们最终通过分析线性增加的到达率来说明我们结果的适用性，确定控制决策的主要驱动因素。我们将分析限制在静态策略上，并通过计算方法证明阈值策略的最优性。此外，我们展示了如何在有限的迭代次数内计算最佳阈值，并证明长期预期成本在服务器数量上是递减的和凸的。我们最终通过分析线性增加的到达率来说明我们结果的适用性，确定控制决策的主要驱动因素。我们将分析限制在静态策略上，并通过计算方法证明阈值策略的最优性。此外，我们展示了如何在有限的迭代次数内计算最佳阈值，并证明长期预期成本在服务器数量上是递减的和凸的。我们最终通过分析线性增加的到达率来说明我们结果的适用性，确定控制决策的主要驱动因素。