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This paper analyzes a finite buffer size discrete-time

Discrete-time queues with classical vacation policies have been explored more in depth during the last few decades due to their widespread application in telecommunication system, electronic information network, production system, and so on (see Takagi [

These mentioned research works all concentrated on working-vacation queueing models with infinite buffer size; however, the finite buffer size counterparts received little attention. In real situations, queues with finite buffer size are more suitable than queues with infinite buffer space as it is used to store arrived customers if server is busy. Among the existing references, very few papers considered the working-vacation queue with finite buffer size; see Goswami and Samanta [

In addition, as far as the queueing systems with working vacations are concerned, the assumption assumes in general that the customers arrive in system at a fixed rate. However, the customer’s choice of entering into system or not usually depends on the system’s status what they see at the arrival epoch. For example, in a make-to-order production system where the system information such as server’s status and queue length is fully observable to an arriving customer, the arriving customer with rate of

On account of the introduction mentioned above, we study the finite buffer

The rest of this paper is organized as follows. Section

We consider a discrete-time single server queue with vacations, in which a potential customer arrives in time interval

Various time epochs in a late arrival system with delayed access (LAS-DA).

To describe the system state, the following random variables are introduced:

In this section, by combining embedded Markov chain and supplementary variable methods, the queue-length distribution at arbitrary epoch is obtained.

Firstly, we develop the usual Chapman-Kolmogorov (C-K) difference equations by regarding the remaining service time as the supplementary variable. Generally, using one-step transition probability, the system can get to the state of

Substituting (

Let

For

In order to find the solution of

One may note that if we could obtain the queue-length distribution at departure epoch,

Gain the probabilities of

Obtain the arbitrary epoch probabilities of

Get the value of

Achieve the values of

So, in the following subsection, using the embedded Markov chain technique, we investigate the queue-length distribution at a departure epoch.

Let