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

Abstract

We consider an M/G/1 retrial queue, where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. The result is obtained by investigating analytic properties of probability generating functions for the queue size and the server state.

Analyzing retrial queues by censoring

“…2.7 of [8]). Tail asymptotic analysis for the stationary distribution of a retrial queue has been reported recently on an M/G/1 retrial queue [15] and its discrete time counterpart model Geo/G/1 retrial queue [13] and also on a discrete-time D-BMAP/G/1 retrial queue [14].…”

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

“…By applying Theorem 2 to the M/G/1 retrial queue, we obtain the following corollary which is the same as Theorem 1 in [10]. Corollary 1 (Kim et al [10]) We consider the M/G/1 retrial queue.…”

“…Corollary 1 (Kim et al [10]) We consider the M/G/1 retrial queue. If there is a real number σ satisfying β(λσ − λ) = σ , 1 < σ < 1 + δ λ , then P(N = n, S = 0) ∼ c 0 n a−1 σ −n as n → ∞,…”

“…Tail behaviors of the queue size and the waiting time distributions in retrial queues began to be investigated recently. Nobel and Tijms [14] and Kim et al [10] studied light-tailed asymptotic behaviors in the M/G/1 retrial queue where the service time distribution has a finite exponential moment. Nobel and Tijms [14] suggested a lighttailed approximation of the waiting time distribution.…”

“…Nobel and Tijms [14] suggested a lighttailed approximation of the waiting time distribution. 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.…”