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：我们研究一个具有价格敏感顾客的通用排队模型，其中服务提供商寻求平衡两个目标：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，观察提供的价格，并在其估值超过价格时决定加入队列。队列采用先进先出方式运作，服务时间服从指数分布。我们的模型代表了按订单生产制造、云计算和食品配送等领域的应用场景。该模型的最优解是动态的；价格随系统状态变化而调整。然而，由于多种原因，此类动态定价策略可能并不理想。在本研究中，我们为一类简单且自然的静态定价策略提供性能保证，这些策略在达到特定占用阈值前收取固定价格，之后不再允许顾客进入系统。我们通过一系列结果表明，此类静态策略能同时保证获得最优收益的恒定比例，且预期队列长度最多仅以恒定倍数增加。例如，我们在M/M/1场景中提出的策略可分别实现收益与队列长度的双准则近似比为$(0.5, 1)、(0.66, 1.16)、(0.75, 1.54)$和$(0.8, 2)$。我们还为多顾客类别、多服务器场景以及预期逗留时间目标提供了性能保证。