When to arrive at a queue with tardiness costs?

A Poisson number of customers seek service during a time interval which starts at time zero and which can be bounded or not. They face the question of when to arrive so as to minimize their waiting plus tardiness costs. We consider two cases; with early birds, namely when seniority is kept among those who arrive prior to opening, and the case without early birds, when it is not. A Nash equilibrium time of arrival (mixed) strategy is derived. This is done through a set of non-linear differential equations which need to be solved numerically. It is then followed by exact solutions for the corresponding fluid models. For these models the socially optimal strategies, as well as the price of anarchy, are determined.