This paper applies a matrix-analytical approach to analyze the temporal behavior of Markovian-modulated batch-service queue with discrete-time batch Markovian arrival process (DBMAP). The service process is correlated and its structure is presented through discrete-time batch Markovian service process (DBMSP). We examine the temporal behavior of packet loss by means of conditional statistics with respect to congested and noncongested periods that occur in an alternating manner. The congested period corresponds to having more than a certain number of packets in the buffer; noncongested period corresponds to the opposite. All of the four related performance measures are derived, including probability distributions of a congested and noncongested periods, the probability that the system stays in a congested period, the packet loss probability during congested period, and the long term packet loss probability. Queueing systems of this type arise in the domain of wireless communications.

The queueing systems under various types of arrival and service processes have been investigated due to their applicability in wireless multimedia networks. The bursty nature of multimedia traffic incurs bursty packet losses, making it impossible for long term loss behavior to quantify precisely the QoS of multimedia services over the internet. However, the quality of real-time multimedia applications perceived by the end user is highly sensitive to the bursty loss behavior, necessitating the supplementing of traditional long term QoS metrics with short term based ones. Therefore, short term performance metrics such as distribution of the length of congested periods and noncongested periods, that is, the likelihood that the system remains in the congested period and loss probability during the congested period, will seriously degrade the quality of service perceived by the end users, which is especially relevant for systems supporting real-time multimedia services. Consequently, short term performance metrics profoundly impact the evaluation of wireless based multimedia applications. Exploring the short term loss behavior of multimedia traffic in wireless multimedia networks is thus of priority concern.

Because the traditional Markov model can not adequately capture the property of complex multimedia traffic, such as web browsing, VoIP, and teleconferencing, it is necessary to propose a suitable traffic model to describe multimedia applications over wireless networks. Traffic with certain bursty characteristics can be qualitatively modeled by a DBMAP, as confirmed in [

Like the DBMAP, discrete-time batch Markovian service process (DBMSP) is a versatile service process and can capture the correlation among the service times. Therefore, DBMSP is suitable to model wireless systems employing multiple transmission modes in the physical layer, each of which corresponds to a particular modulation and coding scheme. Based on this observation, service process for wireless multimedia networks is modeled as a DBMSP in this paper.

So far, much research has focused on the queue with Markovian service process (MSP). In [

This paper is organized as follows. In Section

In this section, we introduce the arrival process, service process, and queueing model.

The arrival processes discussed in this paper are assumed to be DBMAPs since time is assumed to be slotted. A DBMAP can be described by a special type of discrete-time Markov chain. Let

A DBMSP can be described by a special type of discrete-time Markov chain. Let

The queueing model must be specified as the late arrival model, in which a packet arrives to the queue before a slot boundary. Consider the embedded Markov chain

where

To characterize the real packet loss behavior of wireless communication, it is not adequate to examine only the long term packet loss probabilities. For example, a packet stream may experience the loss of a string of consecutive packets followed by bursty arrivals, though the long term packet loss probability is small. This phenomenon makes the traffic source suffer from a significant QoS degradation in that time period. Therefore, in light of the high correlation among consecutive packet arrivals in the wireless multimedia networks, it is necessary to study the packet loss behavior during a short term interval, that is, the conditional packet loss behavior, as well as during long term intervals, in order to characterize the real packet loss behavior of a wireless communication queueing system.

The level of buffer occupancy of a queueing system passes through alternating congested and noncongested periods. To study the short term loss behavior of a queue during a congested period, we decompose the state space

Next, noncongested and congested periods are characterized by deriving the steady state probabilities for the initial state of each transient Markov chain, as denoted by vector

To investigate the packet loss behavior during a congested period, the submatrix

Note that the behavior of the queueing system during a congested period can be described by the transient Markov chain. For a state

Let

Let

We consider an end-to-end wireless transmission system with adaptive modulation and coding (AMC) scheme in the physical layer to provide streaming media service. We experiment an arrival process described by a DBMAP which can capture time correlations commonly observed in VBR traffic such as MPEG coded streaming video. On the transmitter side, streaming media packets are buffered in a queue with finite length

In AMC scheme, the modulation mode and coding rate are chosen depending on the time-varying channel conditions. AMC is employed at the physical layer with

For flat channels, we adopt the general Nakagami-

Finally, consider a Nakagami-

In this section, we present some numerical results obtained on the basis of an FSMC of packet transmission mode induced by the Rayleigh fading [

The time is slotted such that the unit time is equal to the packet transmission time, which is equal to

In [

Figure

Short term packet loss probability with buffer capacity

Figure

Average length of noncongested period and congested period with buffer capacity

According to Figure

Probability that queuing system stays in congested period with buffer capacity

Long term packet loss probability with buffer capacity

This paper applies matrix-analytic approach to investigate the loss behavior of Markovian-modulated batch-service queueing model with DBMAP. We have examined the congestion nature of packet loss by means of conditional statistics with respect to alternating congestion and non-congestion periods and evaluated the long term packet loss probabilities. By the conditional statistics, all of the four related performance measures are derived, including

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

This work was supported by the National Science Council, Taiwan, under Grant NSC92-2213-E-027-047.