We study a mean‐field approximation of the M/M/
queueing system. The problem we deal is quite different from standard games of congestion as we consider the case in which higher congestion results in smaller costs per user. This is motivated by a situation in which some TV show is broadcast so that the same cost is needed no matter how many users follow the show. Using a mean‐field approximation, we show that this results in multiple equilibria of threshold type which we explicitly compute. We further derive the social optimal policy and compute the price of anarchy. We then study the game with partial information and show that by appropriate limitation of the queue‐state information obtained by the players, we can obtain the same performance as when all the information is available to the players. We show that the mean‐field approximation becomes tight as the workload increases, thus the results obtained for the mean‐field model well approximate the discrete one.
