We analyze a discrete-time

During the past few decades, retrial queues have been widely investigated due to their important applications in modeling many practical problems in computer systems, telecommunication networks, and telephone switching systems. In a typical retrial queueing model, an arriving customer who finds the server unavailable may leave the service area and joins a retrial group (called orbit) in order to retry to get the service later after some random time. For more detailed review of the main results and the literature on this topic, the readers are referred to [

In recent years, there has been a growing interest in studying retrial queueing systems with vacations. According to the vacation policy, the retrial queues with vacation can be divided into two categories: retrial queues with Bernoulli vacation and retrial queues with exhaustive server vacations. Particularly, in retrial queues with Bernoulli vacation, the server will take a single vacation with fixed probability once the service of a customer is finished. Li and Yang [

Most of literatures about retrial queues focus on the continuous-time models. In contrast to the continuous case, the discrete-time retrial queues received much less attention in literatures. However, the discrete-time retrial models are suitable for the design and analysis of slotted time communication systems such as asynchronous transfer mode (ATM) based systems in broadband integrated services digital network (B-ISDN) and circuit-switched time-division multiple access (TDMA) systems. Yang and Li [

Most recently, Wu and Yin [

A possible application of our model is in mobile cellular networks. For an accurate analysis of a mobile cellular network, it can not be ignored that blocked calls are able to redial after some random time [

The rest of the paper is organized as follows. In Section

In this paper, we consider a discrete-time

Customers arrive according to a geometrical arrival process with parameter

The service time is assumed to follow a general probability distribution variable

It is assumed that the server can take two different types of vacations. The first type of vacation is called nonexhaustive vacation; that is, the server may take an urgent vacation with probability

The urgent vacation time (no-exhaustive vacation) is assumed to follow a general probability distribution

Finally, we suppose that various stochastic processes involved in the system are assumed to be independent of each other.

In this section, we will show the steady-state analysis for the considered queueing system. Firstly, the Markov chain underlying the considered queueing system and Kolmogorov equations of the steady-state probabilities are obtained. Then, we derive the generating functions of the numbers of customers of the system. Finally, some performance measures are given.

At time

Firstly, we define the stationary probabilities of the Markov chain

To solve (

Now, we can solve (

The following inequalities hold:

If

If

By using Lemmas

If

Multiplying (

In order to find

Setting

Using Lemmas

In this subsection, we give some performance measures based on Theorem

(1) The marginal generating function of the number of customers in the orbit when the server is idle or on vacation is given by

(2) The marginal generating function of the number of customers in the orbit when the server is busy is given by

(3) The marginal generating function of the number of customers in the orbit when the server is on urgent vacation is given by

(4) The marginal generating function of the number of customers in the orbit when the server is on vacation is given by

(5) The generating function of the number of customers in the orbit is given by

(6) The probability generating function of the number of customers in the system is given by

(1) The probability that system is empty is

(2) The probability that the server is idle is

(3) The probability that the server is busy is

(4) The probability that the server is on urgent vacation is

(5) The probability that the server is on normal vacation is

(6) The mean number of customers in the orbit is

(7) The mean number of customers in the system is

Consider some special cases.

(i) When

(ii) When

The total number of customers

After some algebra operation,

In order to prove that

Solving the above equations, we can get that the generating function of the system size is

In this section, we prove that our model can be used to approximate the corresponding continuous-time

Suppose that the time is divided into intervals of equal length

Let

In this section the results of some numerical examples are given to illustrate the effect of some parameters on the characteristics of the system. Specifically we consider two performance measures: the probability that the system is empty

We assume that the retrial times, the service time, the urgent vacation, and the normal vacation time are all geometric distributions with parameters

For convenience, we choose the arrival rate

For different values of the mean urgent vacation time

In Figures

It is observed in Figure

In this work, we study discrete-time retrial queues with two different types of vacations in which the server can take exhaustive single vacation and nonexhaustive urgent vacation. We firstly analyze the Markov chain underlying the considered queueing system and present some performance measures of the system such as the generating functions of system state distribution, the mean orbit size, and system size. Secondly, a stochastic decomposition result and the relationship between our model and the corresponding continuous-time model are given. Finally, we show the effects of different parameters on some of the main performance measures through some numerical examples. The waiting time and busy period in our model are quite difficult to obtain due to the possible nonexhaustive vacations of the server. In case of server’s nonexhaustive vacation, the service process of a customer may be interrupted and the customer enters the orbit. This complicates the analysis of waiting time distribution of a tagged customer in the orbit. Thus, the waiting time distribution and the busy period deserve further investigation in the future.

Define the functions

Let

The authors declare that there is no conflict of interests regarding the publication of this paper.

^{X}/G/

_{1}, MAP

_{2})/(PH

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_{2})/N