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

Abstract

We consider a MAP/G/1 retrial queue where the service time distribution has a finite exponential moment. We derive matrix differential equations for the vector probability generating functions of the stationary queue size distributions. Using these equations, Perron-Frobenius theory, and the Karamata Tauberian theorem, we obtain the tail asymptotics of the queue size distribution. The main result on light-tailed asymptotics is an extension of the result in Kim et al. (J. Appl. Probab. 44:1111-1118, 2007 on the M/G/1 retrial queue.

Queue size distribution in a discrete-time D-BMAP/G/1 retrial queue

“…It is known that det(zI − A(z)) has simple zeros at 1 and and m − 1 zeros (counting multiplicities) in the open unit disk, furthermore, it has no other zeros on {z ∈ C : 1 Յ |z| Յ , z 1, z }. By similar arguments as in Section 2.2 of [13], we have that (zI −Â(z))(zI − A(z)) −1 has a removable singularity at z = 1 and that (zI −Â(z))(zI − A(z)) −1 has a simple pole at z = . Furthermore, by the same procedure as in the proof of Lemma 3 in [12], we can see that the power series ∞ n=0 p n z n has radius of convergence .…”

“…Kim et al [10] studied light-tailed asymptotics of the queue size distribution. Recently, Kim et al [13] generalized the results in [10] to the MAP/G/1 retrial queue. Shang et al [16] and Kim and Kim [11] studied heavy-tailed asymptotics in the M/G/1 retrial queue.…”

Moments of the queue size distribution in the MAP/G/1 retrial queue

“…According to Theorem 1 in Kim et al [17], the vector probability generating functions p * (z) and q * (z) satisfy the following matrix differential equations:…”

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

“…Light-tailed behaviors have been studied by Nobel and Tijms [14] and Kim et al [12,13]. Nobel and Tijms [14] suggested a light-tailed approximation of the waiting time distribution in the M/G/1 retrial queue when the service time distribution has a finite exponential moment.…”

“…Kim et al [12] showed that if the service time distribution has a finite exponential moment then the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function in the M/G/1 retrial queue under an additional condition. The result of [12] was generalized to the MAP/G/1 retial queue by Kim et al [13].…”