Tail asymptotics for the queue length in an M/G/1 retrial queue

Abstract

In this paper, we study the tail behavior of the stationary queue length of an M/G/1 retrial queue. We show that the subexponential tail of the stationary queue length of an M/G/1 retrial queue is determined by that of the corresponding M/G/1 queue, and hence the stationary queue length in an M/G/1 retrial queue is subexponential if the stationary queue length in the corresponding M/G/1 queue is subexponential. Our results for subexponential tails also apply to regularly varying tails, and we provide the regularly varying tail asymptotics for the stationary queue length of the M/G/1 retrial queue

Tail Asymptotics for the Queue Size Distribution in an M/G/1 Retrial Queue

“…On the other hand, there are recent works on heavy-tailed asymptotics in the M/G/1 retrial queue: Shang et al [14] showed that the stationary distribution of the queue length in the M/G/1 retrial queue is subexponential if the stationary distribution of the queue length in the corresponding ordinary M/G/1 queue is subexponential. As a corollary of this property, Shang et al proved that the stationary distribution of the queue length in the M/G/1 retrial queue has a regularly varying tail if the service time distribution has a regularly varying tail.…”

Tail asymptotics for the queue size distribution in the MAP/G/1 retrial queue

“…Kim et al [10] showed that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. On the other hand, Shang et al [16] and Kim and Kim [9] studied heavy-tailed asymptotics in the M/G/1 retrial queue. Shang et al [16] showed that the stationary distribution of the queue size in the M/G/1 retrial queue is subexponential if the stationary distribution of the queue size in the corresponding ordinary M/G/1 queue is subexponential.…”

“…On the other hand, Shang et al [16] and Kim and Kim [9] studied heavy-tailed asymptotics in the M/G/1 retrial queue. Shang et al [16] showed that the stationary distribution of the queue size in the M/G/1 retrial queue is subexponential if the stationary distribution of the queue size in the corresponding ordinary M/G/1 queue is subexponential. As a corollary of this property, they proved that the stationary distribution of the queue size has a regularly varying tail if the service time distribution has a regularly varying tail.…”

Regularly varying tail of the waiting time distribution in M/G/1 retrial queue

“…See, for example, [5,6,11,16] and references therein. However, for the heavy-tailed asymptotics in retrial queues, it seems that Shang et al [15] is the only known result in the open literature. Shang et al [15] showed that the stationary distribution of the queue length in the M/G/1 retrial queue is subexponential if the stationary distribution of the queue length in the corresponding ordinary M/G/1 queue is subexponential.…”

“…However, for the heavy-tailed asymptotics in retrial queues, it seems that Shang et al [15] is the only known result in the open literature. Shang et al [15] showed that the stationary distribution of the queue length in the M/G/1 retrial queue is subexponential if the stationary distribution of the queue length in the corresponding ordinary M/G/1 queue is subexponential. As a corollary of this property, they proved that the stationary distribution of the queue length has a regularly varying tail if the service time distribution has a regularly varying tail.…”