<lb>We consider the non-cooperative choice of arrival times by individual users, who<lb>seek service at a first-come first-served queueing system that opens up at a given time.<lb>Each user wishes to obtain service as early as possible, while minimizing the expected<lb>wait in the queue. This problem was recently studied within a simplified fluid-scale<lb>model. Here we address the unscaled stochastic system, assuming a finite (possibly<lb>random) number of homogeneous users, exponential service times, and linear cost<lb>functions. In this setting we establish that there exists a unique Nash equilibrium,<lb>which is symmetric across users, and characterize the equilibrium arrival-time dis-<lb>tribution of each user in terms of a corresponding set of differential equations. We<lb>further establish convergence of the Nash equilibrium solution to that of the associ-<lb>ated fluid model as the number of users is increased. We finally consider the price of<lb>anarchy in our system and show that it exceeds 2, but converges to this value for a<lb>large population size.