We consider a finite-buffer single server queueing system with queue-length dependent vacations where arrivals occur according to a batch Markovian arrival process (BMAP). The service discipline is P-limited service, also called E-limited with limit variation (ELV) where the server serves until either the system is emptied or a randomly chosen limit of

Queueing systems with vacations have found wide applications in the modelling and analysis of computer and communication networks and several other engineering systems in which single server is performing more than one type of jobs. Modelling such systems as single server queues with vacations allows one to analyze each queue in relative isolation since the time the server is attending to other jobs in the system may be modeled as vacation. For more details and versatile implementation of vacation models, one can refer to the comprehensive survey by Doshi [

Traditional teletraffic analysis using Poisson process is not powerful enough to capture the correlated and bursty nature of traffic arising in the present high-speed networks where packets or cells of voice, video, images and data are sent over a common transmission channel on statistical multiplexing basis. The performance analysis of statistical multiplexers whose input consists of superposition of several packetized sources has been done through some analytically tractable arrival process, for example, Markovian arrival process (MAP); see Lucantoni et al. [

In this paper, we analyze BMAP/G/1/

Let us consider a BMAP/G/1/

The input process is BMAP where arrivals are governed by an underlying

Let us define

Let

Consider the system at service completion/vacation termination epochs which are taken as embedded points. Let

Let

Observing the system immediately after each embedded point, we have the transition probability matrix (TPM)

The evaluation of the matrices

If

See Neuts [

The unknown probability vectors

Considering the departure of a customer as an embedded point excluding vacation termination epochs, one may obtain queue-length distributions at departure epoch. Distributions of number of customers in the queue at service completion and departure epochs are proportional. Let

To determine queue-length distribution at arbitrary epoch, we will develop relations between distributions of number of customers in the queue at service completion (vacation termination) and arbitrary epochs. Supplementary variable method has been used, and for that we relate the states of the system at two consecutive time epochs

Consider the following:

Setting

Consider the following:

Differentiating (

We first relate the service completion (vacation termination) epoch probabilities,

Let

One may note here that

Let

Let

In this section, we discuss the various performance measures which are often needed for investigating the behavior of a queueing system. As the state probabilities at which departure, arbitrary and prearrival epochs are known, the corresponding mean queue-lengths can be easily obtained. For example, the average number in the queue at any arbitrary epoch is

Since prearrival epoch probabilities are known, the blocking probability of the first customer of an arriving batch is given by

One may note here that

In this subsection, we develop the total expected cost function per unit time for this queueing system in which

We consider a cost function per unit time for this BMAP/G/1/

Using the definitions of each cost element listed above, the total expected cost function per unit time is given by

To demonstrate the applicability of the results obtained in the previous sections, some numerical results have been presented in the form of graphs showing the nature of some performance measures against the variation of some critical model parameters. We have conducted an experiment on the BMAP/PH/1/

The 2-state BMAP representation is taken as

In Figure

In Figure

In Figure

In Figures

Finally, in Figure _{1}/1/_{2}/1/

In this paper, we have successfully analyzed the BMAP/G/1/

The author received partial financial support from the Department of Science and Technology, New Delhi, India, research Grant SR/FTP/MS-003/2012.