Static Pricing Guarantees for Queueing Systems
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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.
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