Static Pricing Guarantees for Queueing Systems

We consider a general queueing system 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, can have multiple servers, 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 non-asymptotic 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. Despite the mixed-sign objective, we are able to show our policy can guarantee a constant fraction of the optimal dynamic pricing policy in the worst-case. We also show that our policy yields a family of bi-criteria approximations that 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 can be set so that its worst-case guarantees is at least 50, 66, 75, or 80% of the optimal revenue and at most a 0, 16, 54, or 100% increase in the optimal queue length, respectively. We also provide guarantees for settings with multiple servers as well as the expected sojourn time objective. In a large simulation, we show that our class of policies is at most 4% sub-optimal on average.