Heavy-tailed asymptotics for a fluid model driven by an M/G/1M/G/1 queue

In this paper, an infinite-buffer fluid queue driven by an M/G/1M/G/1 queue is discussed. The Laplace transform of the distribution of the stationary buffer content is expressed through the minimal positive solution to a crucial equation, similar to the fundamental equation satisfied by the busy period of an M/G/1M/G/1 queue. Furthermore, the distribution of the stationary buffer content is shown to be regularly varying with index −α+1−α+1 if the distribution of the service times is regularly varying with index −α<−1−α<−1. Meanwhile, the first ⌈α⌉−2⌈α⌉−2 moments of the stationary buffer content are given, where ⌈x⌉⌈x⌉ is the ceiling function of the real number xx.