具有重试速率控制策略的M/G/1重试排队的尾渐近性

Tail Asymptotics for the M/G/1 Retrial Queue with Retrial Rate Control Policy

本文研究了一种具有重试速率控制策略的重试排队系统,在该重试策略下,单个顾客的重试速率与重试轨道中的顾客数目成反比。虽然已有文献研究了该排队系统的轨道队长的平稳分布的概率生成函数,但其结果是隐式的,难以直接得到轨道队长的概率分布,所以本文在此基础上,研究轨道队长的平稳分布的尾渐近性。在服务时间的平衡分布属于次指数分布族的情形下,我们基于轨道队长的条件概率生成函数,使用穷举随机分解方法得到轨道队长的平稳分布的尾渐近性,结果表明,轨道队长的平稳分布具有次指数的尾部。此外,在服务时间的平稳分布具有正规变化的尾部这一特殊情况下,我们证明了轨道队长的平稳分布具有正规变化的尾部。最后,我们通过数值例子验证推导结果的正确性。本文的研究结果刻画了轨道队长的衰减效果,弥补了现有研究的不足,为现实情况中的重试排队系统提供参考。In this paper, we study an M/G/1 retrial queue with retrial rate control policy, where the retrial rate is inversely proportional to the number of customers in the orbit. Although there have been studies on the probability generation function of the stationary distribution of the orbital length, the expressions are implicit and it is difficult to obtain the corresponding probability distribution. Therefore, we study the tail asymptotics for the stationary distribution of the orbital length. Assuming that the equilibrium distribution of the service time is subexponential, we aim to characterize the tail asymptotics of the orbit queue length. Based on the conditional probability generating functions, we adopt an exhaustive version of the stochastic decomposition method and prove that the corresponding distributions have subexponential tails. As a special case, we consider the service time has a distribution with regularly varying tail, and show that the corresponding distributions also have regularly varying tails, while they are heavier than that of the service time. Our results depict the decay effect of orbit length, which supplements existing research and provides reference for retrial queuing systems in reality.