Bayesian Dynamic Pricing in Queueing Systems with Unknown Delay Cost Characteristics

The revenue management literature for queues typically assumes that providers know the distribution of customer demand attributes. We study an observable M/M/1 queue that serves an unknown proportion of patient and impatient customers. The provider has a Bernoulli prior on this proportion, corresponding to an optimistic or pessimistic scenario. For every queue length, she chooses a low or a high price, or turns customers away. Only the high price is informative. The optimal Bayesian price for a queue state is belief-dependent if the optimal policies for the underlying scenarios disagree at that queue state; in this case the policy has a belief-threshold structure. The optimal Bayesian pricing policy as a function of queue length has a zone (or, nested-threshold) structure. Moreover, the price convergence under the optimal Bayesian policy is sensitive to the system size, i.e., the maximum queue length. We identify two cases: prices converge (1) almost surely to the optimal prices in either scenario or (2) with positive probability to suboptimal prices. Only Case 2 is consistent with the typical incomplete learning outcome observed in the literature.