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We consider the discrete-time

Queueing systems with vacations have been studied by many researchers over the past four decades. A vacation is the period in which a server is not available for some reason: the server may break down, take time to warm up/close down/repair, or be serving other classes of customers and so forth. Such queuing systems are useful in modeling a variety of real-life queueing situations such as those in digital communications, computer networks, and production/inventory systems. Readers are referred to Doshi [

In recent years, there has been a growing interest in the analysis of discrete-time queueing systems due to their applications in a variety of slotted digital communication systems and other related areas. Takagi [

In this paper, we consider a discrete-time queueing system with batch arrivals,

It is well known that discrete-time queueing systems have been extensively applied in computer and digital communication systems. Also, control policies including

This paper is organized as follows. In Section

In discrete-time queueing models, the time axis is segmented into a sequence of equal intervals of unit duration, called slots. It is always assumed that interarrival, service, and vacation times are integer multiples of unit duration. Because nothing is assumed to happen at any time during a slot, the state of the system changes only at a slot boundary

We consider two discrete-time queueing systems: the discrete-time

Let

For this system, we derive PGFs of the stationary queue length, idle period, and busy period.

To obtain the queue-length PGF, we make use of the well-known property of stochastic decomposition [

Let

Now, substituting (

For the discrete-time

The continuous-time counterpart of Theorem

Theorem

In this section, we first consider the idle period and then the busy period. Let

For the discrete-time

Conditioning on the length of the first vacation and the number of arrivals during this vacation (denoted by

Note that

Next, we consider the busy period. Let

For the discrete-time

Along the same lines as presented above, the continuous-time counterparts of Theorems

For this system, we also make use of the well-known property of stochastic decomposition [

Following the procedure presented by Lee et al. [

Now, simply substituting (

For the discrete-time

The continuous-time counterpart of Theorem

Theorem

In this section, we first consider the idle period and then the busy period. Let

For the discrete-time

Conditioning on the length of the first vacation and the number of customers that arrive during this vacation (denoted by

Note that

Next, we consider the busy period. Let

For the discrete-time

Along the same lines as presented above, the continuous-time counterparts of Theorems

In this paper, we consider the discrete-time

Finally, we remark that all the results obtained in this paper for models under AF assumptions also hold for those under DF assumptions. This is because assumptions on the order of simultaneous events at a slot boundary do not affect the system state during a slot (see Kim et al. [

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

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2013S1A5A8022372).