We consider the problem of selfish agents in discrete-time queuing systems, where competitive queues try to get their packets served. In this model, a queue gets to send a packet each step to one of the servers, which will attempt to serve the oldest arriving packet, and unprocessed packets are returned to each queue. We model this as a repeated game where queues compete for the capacity of the servers, but where the state of the game evolves as the length of each queue varies, resulting in a highly dependent random process. Earlier work by the authors [EC'20] shows that with no-regret learners, the system needs twice the capacity as would be required in the coordinated setting to ensure queue lengths remain stable despite the selfish behavior of the queues. In this paper, we demonstrate that this way of evaluating outcomes is myopic: if more patient queues choose strategies that selfishly maximize their long-run success rate, stability can be ensured with just $\frac{e}{e-1}\approx 1.58$ times extra capacity, better than what is possible assuming the no-regret property. As these systems induce highly dependent processes, our analysis draws heavily on techniques from probability theory. Though these systems are random under any fixed policies by the queues, we show that, surprisingly, these systems have deterministic and explicit asymptotic behavior. We show that the asymptotic growth rates of queues can be written as a ratio of a submodular and modular function, which provides significant game-theoretic properties. Our equilibrium analysis then relies on a novel deformation argument towards a more analyzable solution that differs significantly from previous price of anarchy results. While the intermediate points will not be equilibria, this analytic structure will ensure that this deformation is monotonic along this continuous path.