Social and monopoly optimization in observable queues

Naor's celebrated paper studies customer decisions in an observable M/M/1 queue in which joining-customers utility is linearly decreasing with the joining position. Naor derives the optimal threshold strategies for the individuals, social planner, and monopolist and proves that the monopoly optimal threshold is (weakly) smaller than the socially optimal threshold, which is (weakly) smaller than the individually optimal one. Studies show, based on numerical observations and/or ad hoc proof techniques, that this triangular relation holds within various specific setups, in which the queuing process is not M/M/1 and/or when the utility is not linear. We point out properties that imply the aforementioned result in Naor's model and its extensions and suggest model applications for our findings. Our formulation gives strictly stronger results than those currently appearing in the literature. We further provide simple examples in which the inequality does not hold.