Stability and Learning in Strategic Queuing Systems

Abstract

Bounding the price of anarchy, which quantifies the damage to social welfare due to selfish behavior of the participants, has been an important area of research in algorithmic game theory. In this paper, we study this phenomenon in the context of a game modeling queuing systems: routers compete for servers, where packets that do not get service will be resent at future rounds, resulting in a system where the number of packets at each round depends on the success of the routers in the previous rounds. We model this as an (infinitely) repeated game, where the system holds a state (number of packets held by each queue) that arises from the results of the previous rounds. We assume that routers satisfy the no-regret condition, e.g. they use learning strategies to identify the server where their packets get the best service.Classical work on repeated games makes the strong assumption that the subsequent rounds of the repeated games are independent (beyond the influence on learning from past history). The carryover effect caused by packets remaining in this system makes learning in our context result in a highly dependent random process. We analyze this random process and find that if the capacity of the servers is high enough to allow a centralized and knowledgeable scheduler to get all packets served even when service rates are halved, and queues use no-regret learning algorithms, then the expected number of packets in the queues will remain bounded throughout time, assuming older packets have priority. This paper is the first to study the effect of selfish learning in a queuing system, where the learners compete for resources, but rounds are not all independent: the number of packets to be routed at each round depends on the success of the routers in the previous rounds. CCS Concepts: • Mathematics of computing → Queueing theory; • Theory of computation → Online learning theory; Multi-agent learning; Regret bounds; Quality of equilibria; Convergence and learning in games.

Virtues of Patience in Strategic Queuing Systems

“…The main result of this paper is to show that with patience by the queues, a factor of −1 ≈ 1.58 extra server capacity over what is needed in the centralized setting suffices to guarantee the stability of the system despite selfishness. This is in contrast to the main result of [5], where it is shown that if queues use no-regret learning, the system needs a factor of 2 extra server capacity. In this setting, the no-regret property implies that queues send to queues with the highest empirical success rates, without accounting for how alternate choices affect these rates.…”

“…Recall that our goal is to ensure stability in any Nash equilibrium, assuming some relationship on the service rates to the arrival rates. Our main result shows that the correct constant of system slack is −1 ≈ 1.58, beating the best achievable constant of 2 in the no-regret setting of [5]: Theorem 1.3 (Main, Corollary 5.2, informal). If the service capacity is large enough so that the system would remain feasible when centrally managed even if capacities are scaled down by −1 , then in every equilibrium of this game, all queues are stable.…”

“…We further assume that servers attempt to serve the oldest packet it receives in each round 1 , thus giving priority to queues that have not been able 1 An alternate choice is to assume that servers select packets uniformly at random among packets it receives in a given round to attempt to serve. However, it is shown in [5] that adding even poly( ) slack to the central feasibility conditions (given by (1)) cannot guarantee stability in general, and patience by the queues does not help in this case either.…”

“…Recent work by the authors [5] initiates the study of price-of-anarchy-style bounds when the repeated games carry state in a discrete-time queuing system. In this model, queues must compete for servers to clear their own packets which arrive at heterogeneous rates, while servers give priority to older packets.…”

“…This priority scheme induces strong dependencies between rounds because a queue that fails to clear packets will have priority in future rounds. The main result of [5] is that if the queues use no-regret algorithms, then just a factor of 2 higher service rates than what is necessary for centralized stability is sufficient to ensure that this selfish system also remains stable.…”