Equilibrium strategies in queues based on time or index of arrival

In most decision models dealing with unobservable stochastic congested environments, one looks for a (Nash) equilibrium behavior among customers. This is a strategy that, if adopted by all, then under the resulting steady-state conditions; the best response for an individual is to adopt this strategy too. The purpose of this article is to look for a simple decision problem but where the assumption of steady-state conditions is removed. Specifically, we consider an M/M/N/N loss model in which one pays for trying to get service but is rewarded only if one finds an available server. The initial conditions at time 0 are common knowledge and each customer possesses his arrival time as his private information. The equilibrium profile tells each arrival whether to try (randomization allowed) given his time of arrival. We show that all join up to some point of time. At this point, there is a quantum drop in the joining probability from one to some fraction. From then on, their joining probability continuously converges to the equilibrium joining probability under the model that assumes steady state