Large Deviations for the Single-Server Queue and the Reneging Paradox
研究了M/M/1+M排队模型中长期放弃率的大偏差行为，发现放弃概率的衰减率不依赖于个体放弃率，并给出了该衰减率的显式表达式。
For the M/M/1+M model at the law-of-large-numbers scale, the long-run reneging count per unit time does not depend on the individual (i.e., per customer) reneging rate. This paradoxical statement has a simple proof. Less obvious is a large deviations analogue of this fact, stated as follows: the decay rate of the probability that the long-run reneging count per unit time is atypically large or atypically small does not depend on the individual reneging rate. In this paper, the sample path large deviations principle for the model is proved and the rate function is computed. Next, large time asymptotics for the reneging rate are studied for the case when the arrival rate exceeds the service rate. The key ingredient is a calculus of variations analysis of the variational problem associated with atypical reneging. A characterization of the aforementioned decay rate, given explicitly in terms of the arrival and service rate parameters of the model, is provided yielding a precise mathematical description of this paradoxical behavior.