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In this paper, we analyze the queueing behaviour of wavelength division multiplexing (WDM) Internet router employing partial buffer sharing (PBS) mechanism with self-similar traffic input. In view of WDM technology in networking, each output port of the router is modelled as multi-server queueing system. To guarantee the quality of service (QoS) in Broadband integrated services digital network (B-ISDN), PBS mechanism is a promising one. As Markov modulated Poisson process (MMPP) emulates self-similar Internet traffic, we can use MMPP as input process of queueing system to investigate queueing behaviour of the router. In general, as network traffic is asynchronous (unslotted) and of variable packet lengths, service times (packet lengths) are assumed to follow Erlang-k distribution. Since, the said distribution is relatively general compared to deterministic and exponential. Hence, specific output port of the router is modelled as MMPP/Ek/s/C queueing system. The long-term performance measures namely high priority and low priority packet loss probabilities and the short-term performance measures namely mean lengths of critical and non-critical periods against the system parameters and traffic parameters are computed by means of matrix-geometric methods and approximate Markovian model. This kind of analysis is useful in dimensioning the router under self-similar traffic input employing PBS mechanism to provide differentiated services (DiffServ) and QoS guarantee.

It is evident from seminal studies that Internet protocol (IP) traffic of both Ethernet traffic and wide area network (WAN) traffic exhibits self-similarity [

Another issue of Internet traffic is to provide Quality of Service (QoS). Internet router with wavelength division multiplexing (WDM) technology is promising one to guarantee QoS. In WDM router, there are N input fiber lines, N output fiber lines, and each fiber line has C wavelength channels and a wavelength converter pool of size s, _{k}/s/C queueing system with PBS mechanism.

The rest of the paper is organized as follows. In section 2, WDM asynchronous router―multi-server queueing model MMPP/E_{k}/s/C employing PBS mechanism with Erlang-k service times is discussed. Computational complexity is briefed in section 3. In section 4, numerical results are presented graphically. Finally, conclusion is given in section 5.

We consider the WDM asynchronous _{k}/s/C queueing system. The operation and multi-server queueing model of the router employing PBS mechanism is shown in

process is in state i, the next departure of class p occurs no later than time t with the process in state j, during the service time there are m packets. We consider the Markov chain

where

In Equations (2) and (3), the elements of row and column outside the matrices

Sojourn time is ignores the matrices of counting functions become independent of time t. The matrices

If the service time distribution

where I is the unit matrix of appropriate dimension. For

For^{th} term of the series on right hand side of Equation (5). Then we have

...……,

Now, multiply on both sides of above equation by

In above equation, equating the coefficients of like powers of z, we obtain,

Equating the coefficients of

The fundamental arrival rate of class p packets is

where

In the paper [

and

This partition of U makes the transition probability matrix P decomposed as follows:

The sub-matrices

Similarly, in the case of critical period, the TPM of the absorbing Markov chain that has transient states

where

and

where

and

The average total number of high priority packets lost during a critical period is

The average total number of low priority packets lost during a critical period is

where

and

The high priority packet loss probability

and

In this section, first we compute the computational complexity of the long-term performance measures, namely, high priority packet loss probability

In order to find the computational complexity of long-term

where 0 and I are the zero and the identity matrices of dimension

The sub-matrices E, F, G, and H are of the dimensions

By the Schur-Banachiewicz inversion formula, where

where

The sub-matrices

Now, we place the algorithmic steps needed for computing the performance measures

Step 1. Compute

Step 2. Compute

Step 3. Compute

Step 4. Compute

Step 5. Compute

Step 6. Compute

The computational complexity of the steps 5-6 in the above algorithm is of the order ^{nd} step is of the order

We next show that the computational complexity of the first step is of the order

The sub-matrices

where

The sub-matrices

where

Using the Equations (15)-(26), the steady state packet loss probabilities and mean length of non-critical and critical periods are computed [

of the plot, which is around

as traffic intensity increases.

In this paper, we have investigated the performance of asynchronous router employing PBS mechanism to provide differentiated services under Markovian modelled self-similar traffic input. To reduce the computational complexity, the original high dimensional MMPP of the low priority packets is approximated by 2-state MAP. The long-term performance measures, namely, the steady state high priority and low priority packet loss probabilities, and the short-term performance measures, namely, average length of non-critical and critical periods, are computed and presented graphically. With this analysis, we can locate the optimal limit (threshold) position of buffer to obtain the greatest performance.

Gudimalla, R.K. and Perati, M.R. (2016) Performance Analysis of Wavelength Division Multiplexing Asynchronous Internet Router Employing Space Priority Mechanism under Self-Similar Traffic Input―Multi-Server Queueing System with Markovian Input and Erlang-k Services. Applied Mathematics, 7, 1707-1725. http://dx.doi.org/10.4236/am.2016.715144