Dynamic Admission Game into an M/M/1 Queue

Around 50 years ago, P. Naor has derived the optimal social and individual admission rules to an M/M/1 queue. In both cases, the optimal policies were identified to be of a pure threshold type: admit if and only if the number queued upon arrival is below some threshold. The value of the threshold in the individual optimal case was shown to be larger than the one for the social optimal criterion. We make the observation that admitting according to a threshold policy requires only the information of whether the queue is above or below a threshold. We call these "red" and "green" light, respectively, associated with a threshold, say L. The question that we pose in this paper is: what happens if one restricts to the above information pattern but let the threshold level L be chosen by the system which signals to arrivals whether the queue is above or below the threshold. Can one find a choice of a threshold that will induce an equilibrium that performs better than in the case that full information is available? We also examine the question of what is the threshold that maximizes the revenue for the queue. We show that the choice of threshold that maximizes the system's performance at equilibrium is the same as under the full information case if the service in the queue follows the FIFO discipline.