The concert queueing game with a random volume of arrivals

We consider the concert queueing game in the fluid framework, where the service facility opens at a specified time, the customers are particles in a fluid with homogeneous costs that are linear and additive in the waiting time and in the time to service completion, and wish to choose their own arrival times so as to minimize their cost. This problem has recently been analyzed under the assumption that the total volume of arriving customers is deterministic and known beforehand. We consider here the more plausible setting where this volume may be random, and only its probability distribution is known beforehand. In this setting, we identify the unique symmetric Nash equilibrium and show that under it the customer behavior significantly differs from the case where such uncertainties do not exist. While, in the latter case, the equilibrium profile is uniform, in the former case it is uniform up to a point and then it tapers off. We also solve the associated optimization problem to determine the socially optimal solution when the central planner is unaware of the actual amount of arrivals. Interestingly, the Price of Anarchy (ratio of the social cost of the equilibrium solution to that of the optimal one) for this model turns out to be two exactly, as in the deterministic case, despite the different form of the social and equilibrium arrival profiles.