We consider a general queueing model with price-sensitive customers in which the service provider seeks to balance two objectives, maximizing the average revenue rate and minimizing the average queue length. Customers arrive according to a Poisson process, observe an offered price, and decide to join the queue if their valuation exceeds the price. The queue is operated first-in first-out, and the service times are exponential. Our model represents applications in areas like make-to-order manufacturing, cloud computing, and food delivery. The optimal solution for our model is dynamic; the price changes as the state of the system changes. However, such dynamic pricing policies may be undesirable for a variety of reasons. In this work, we provide performance guarantees for a simple and natural class of static pricing policies which charge a fixed price up to a certain occupancy threshold and then allow no more customers into the system. We provide a series of results showing that such static policies can simultaneously guarantee a constant fraction of the optimal revenue with at most a constant factor increase in expected queue length. For instance, our policy for the M/M/1 setting allows bi-criteria approximations of $(0.5, 1), (0.66, 1.16), (0.75, 1.54)$ and $(0.8, 2)$ for the revenue and queue length, respectively. We also provide guarantees for settings with multiple customer classes and multiple servers, as well as the expected sojourn time objective.

翻译：我们考虑一个包含价格敏感型顾客的通用排队模型，服务提供商需平衡两大目标：最大化平均收益率和最小化平均队列长度。顾客按照泊松过程到达，观察报价后，若其估值高于价格则决定加入队列。队列遵循先到先服务原则，服务时间服从指数分布。该模型可应用于按订单生产、云计算和食品配送等领域。本模型的最优解是动态的，即价格随系统状态变化而变化。然而，此类动态定价策略可能因多种原因而不理想。本文针对简单且自然的静态定价策略提供性能保证：该类策略在系统未达设定占用阈值前收取固定价格，超出阈值后则禁止新顾客进入。我们通过系列结果表明，此类静态策略能同时保证最优收益的常数比例，且期望队列长度最多增加常数倍。例如，针对M/M/1场景，我们的策略可实现收益与队列长度的双目标近似值分别为$(0.5, 1)、(0.66, 1.16)、(0.75, 1.54)$和$(0.8, 2)$。我们还针对多顾客类别、多服务器场景以及期望逗留时间目标提供了性能保证。