Exact tail asymptotics for the M/M/m retrial queue with nonpersistent customers

Abstract

We consider the M/M/m retrial queue with nonpersistent customers. Liu et al. (2011) [12] provided the asymptotic lower and upper bounds for the stationary distribution of the number of customers in the orbit. In this paper we strengthen Liu, Wang and Zhao’s result by finding the exact tail asymptotic formula.

Introduction

Retrial queues are queueing systems in which arriving customers who find all servers occupied may retry for service again after a random amount of time. Retrial queues have been widely used to model many problems in telephone systems, call centers, telecommunication networks, computer networks and computer systems, and in daily life. Detailed overviews for retrial queues can be found in the bibliographies [1], [2], [3], the surveys [6], [11], [15], and the books [4], [7].

Multi-server retrial queues are characterized by the following feature: if there is a free server when a customer arrives from outside the system, this customer begins to be served immediately and leaves the system after the service is completed. On the other hand, any customer who finds all servers busy upon arrival joins a retrial group, called an orbit, and then attempts service after a random amount of time. If there is a free server when a customer from the orbit attempts service, this customer receives service immediately and leaves the system after the service completion. Otherwise the customer comes back to the orbit immediately and repeats the retrial process.

We consider the M/M/m retrial queue with nonpersistent customers. For this queueing system, when $m>1$, there is no explicit closed form expression for the stationary distribution of the number of customers in the orbit and the number of busy servers. On the other hand, the standard M/M/m retrial queue does not have explicit closed form expression for the stationary distribution, when $m>2$. Therefore, tail asymptotic analysis for the stationary distribution of the queue length (i.e., the number of customers in the orbit) has attracted a lot of interest and has become more important. Tail asymptotic analysis for the stationary distribution of the queue length in single server retrial queues can be found in [10], [8], [14] and references therein.

For the standard M/M/m retrial queue, tail asymptotics for the stationary distribution of the queue length have been studied by Liu and Zhao [13] and Kim et al. [9]. Liu and Zhao [13] provided the asymptotic lower and upper bounds for the stationary distribution of the queue length. More precisely, Liu and Zhao [13] found the decay function ${\tilde{h}}_i\left(n\right)$ satisfying

$$0<\underset{n\to \infty }{lim\; inf}\frac{{\pi }_{ni}}{{\tilde{h}}_i\left(n\right)}\le \underset{n\to \infty }{lim\; sup}\frac{{\pi }_{ni}}{{\tilde{h}}_i\left(n\right)}<\infty \text{,}$$

where ${\pi }_{ni},n\ge 0,i=0,1,\dots ,m$, is the joint probability that there are $n$ customers in the orbit and $i$ busy servers, in the steady state. Kim et al. [9] obtained the exact tail asymptotic formula

$$\underset{n\to \infty }{lim}\frac{{\pi }_{ni}}{{\tilde{h}}_i\left(n\right)}=\frac{\tilde{c}}{i!}{\left(\frac{\nu }{\mu }\right)}^i\text{,}i=0,1,\dots ,m\text{,}$$

with a constant $\tilde{c}>0$, service rate $\mu$ and retrial rate $\nu$.

This paper focuses on the tail asymptotics for the stationary distribution of the queue length in the M/M/m retrial queue with nonpersistent customers. Liu et al. [12] obtained the same formula as in (1), but with a different decay function $h_i\left(n\right)$ from ${\tilde{h}}_i\left(n\right)$:

$$0<\underset{n\to \infty }{lim\; inf}\frac{{\pi }_{ni}}{h_i\left(n\right)}\le \underset{n\to \infty }{lim\; sup}\frac{{\pi }_{ni}}{h_i\left(n\right)}<\infty \text{.}$$

The main contribution of this paper is that we find the following exact tail asymptotic formula for the stationary distribution ${\pi }_{ni}$ of the queue length:

$$\underset{n\to \infty }{lim}\frac{{\pi }_{ni}}{h_i\left(n\right)}=\frac{c}{i!}{\left(\frac{\nu }{\mu }\right)}^i\text{,}i=0,1,\dots ,m\text{,}$$

with a constant $c>0$. Obviously our result strengthens Liu et al.’s result.