Optimal pricing and information sharing strategies in a single-server queue

We study optimal pricing in a single-server queueing system that can be observable or unobservable, depending on how customers receive information to estimate sojourn time. Our primary objective is to determine whether the service provider is better off making the system observable or unobservable under optimal pricing. We formulate the optimal pricing problem using Markov decision process (MDP) models for both observable and unobservable systems. For unobservable systems, the problem is studied using an MDP with a fixed-point equation as equilibrium constraints. We show that the MDPs for both observable and unobservable queues are special cases of a generalized arrival-based MDP model, in which the optimal arrival rate (rather than price) is set in each state. Then, we show that the optimal policy that solves the generalized MDP exhibits a monotone structure in that the optimal arrival rate is non-increasing in the queue length, which allows for developing efficient algorithms to determine optimal pricing policies. Next, we show that if no customers overestimate sojourn time in the observable system, it is in the interest of the service provider to make the system observable. We also show that if all customers overestimate sojourn time, the service provider is better off making the system unobservable. Lastly, we learn from numerical results that when customers are heterogeneous in estimating their sojourn time, the service provider is expected to receive a higher gain by making the system observable if on average customers do not significantly overestimate sojourn time.