  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 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 round. 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 with double the packet arrival rate,
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.
