Abstract

We study a general k dimensional infinite server queues process with Markov switching, Poisson arrivals and where the service times are fat tailed with index $\alpha \in (0,1).$ When the arrival rate is sped up by a factor $n^{\gamma },$ the transition probabilities of the underlying Markov chain are divided by $n^{\gamma }$ and the service times are divided by n, we identify two regimes (”fast arrivals”, when $\gamma >\alpha ,$ and” equilibrium”, when $\gamma =\alpha$) in which we prove that a properly rescaled process converges pointwise in distribution to some limiting process. In a third” slow arrivals” regime, $\gamma <\alpha ,$ we show the convergence of the two first joint moments of the rescaled process.

摘要

我们研究了具有马尔可夫切换、泊松到达以及服务时间与索引肥尾的一般k维无限服务器队列过程$\alpha \in (0,1).$ 当到达率提高一个倍数时 $n^{\gamma },$ 底层马尔可夫链的转移概率除以 $n^{\gamma }$并且服务时间除以n，我们确定了两种制度（“快速到达”，当$\gamma >\alpha ,$ 和“平衡”，当 $\gamma =\alpha$)，其中我们证明了适当重新缩放的过程在分布中逐点收敛到某个限制过程。在第三种“缓慢到达”制度中，$\gamma <\alpha ,$ 我们展示了重新缩放过程的两个第一个关节矩的收敛性。