Virtues of Patience in Strategic Queuing Systems

ABSTRACTWe 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. In classical work for learning in repeated games, the learners evaluate the outcome of their strategy in each step---in our context, this means that queues estimate their success probability at each server. Earlier work by the authors [in 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 myopic way of evaluating outcomes is suboptimal: if more patient queues choose strategies that selfishly maximize their long-run success rate, stability can be ensured with just e/e-1 ~1.58 times extra capacity, strictly better than what is possible assuming the no-regret property. As these systems induce highly dependent random processes, our analysis draws heavily on techniques from the theory of stochastic processes to establish various game-theoretic properties of these systems. Though these systems are random even under fixed stationary policies by the queues, we show using careful probabilistic arguments that surprisingly, under such fixed policies, these systems have essentially deterministic and explicit asymptotic behavior. We show that the growth rate of a set can be written as the ratio of a submodular and modular function, and use the resulting explicit description to show that the subsets of queues with largest growth rate are closed under union and non-disjoint intersections, which we use in turn to prove the claimed sharp bicriteria result for the equilibria of the resulting system. Our equilibrium analysis relies on a novel deformation argument towards a more analyzable solution that is quite different from classical price of anarchy bounds. While the intermediate points in this deformation will not be Nash, the structure will ensure the relevant constraints and incentives similarly hold to establish monotonicity along this continuous path.