Optimal Measurement-based Pricing for an M/M/1 Queue

In this paper, we consider a system modelled as an M/M/1 queue. Several users send jobs to the queue and are characterized by a delay cost per unit of time and a demand function. Our goal is to design an optimal pricing scheme for the queue, where the total charge depends on both the mean delay at the queue and arrival rate of each customer. We also assume that those two values have to be (statistically) measured, introducing errors on the total charge that might avert users from using the system, and then decrease demand. This model can be applied in telecommunication networks, where pricing can be used to control congestion, and the network can be characterized by a single bottleneck queue; the throughput of each user would be determined through passive measurements while the delay would be determined through active measurements.