^{1}

^{*}

^{2}

^{*}

^{3}

^{*}

^{3}

^{*}

Most of the classical self-similar traffic models are asymptotic in nature. Therefore, it is crucial for an appropriate buffer design of a switch and queuing based performance evaluation. In this paper, we investigate delay and loss behavior of the switch under self-similar fixed length packet traffic by modeling it as CMMPP/D/1 and CMMPP/D/1/K, respectively, where Circulant Markov Modulated Poisson Process (CMMPP) is fitted by equating the variance of CMMPP and that of self-similar traffic. CMMPP model is already the validated one to emulate the self-similar characteristics. We compare the analytical results with the simulation ones.

An effective traffic model has, at least, to reproduce the first and second order statistics of the original traffic trace. The second order statistics play an important role in traffic modeling, because traffic correlation is an important factor in packet losses due to buffer and bandwidth limitations. However, the first two order statistics may not be sufficient to characterize real data traces that are known to be bursty and spiky in nature. It has been shown through experimental evidence that network traffic may exhibit properties of self-similarity and/or long range dependence (LRD) [1-3]. Self-similar traffic shows identical statistics’ characteristics over a wide range of time scales, which have significant impact on network performance. Therefore, it is important to make frequent measurement of packet flows and to decide them through appropriate traffic models. Characterizing the statistical behavior of traffic is crucial for proper buffer design of switch in the network traffic to provide the quality of service (QoS). Various stochastic models have been proposed to emulate the statistical nature of self-similar network traffic over certain range of time scales. Traffic models such as Fractional Brownian Motion (FBM), Fractional Auto Regressive Integrated Moving Average (FARIMA) and Chaotic maps are proposed to characterize the self-similarity. These models describe the self-similar behavior in a relatively simple manner. Although these processes have less number of parameters, they are less effective in the context of queuing based performance evaluation. Traditional traffic models, such as Markovian models, can still be used to model traffic exhibiting LRD. In [4-7], Markovian arrival process (MAP) is employed to model the self-similar behavior over the desired time scales. These fitting models equate the second order statistics of self-similar traffic and superposition of several 2-state Interrupted Poisson Processes (IPPs). The said models hold well for voice traffic as IPP consists of two states talkspurt and silence. On the other hand, the Circulant Markov Modulated Poisson Process (CMMPP) is a Poisson process, the rate of which is changed according to circulant Markov chain [

In addition to traditional data services, multimedia and real-time applications are becoming indispensable services offered by the best-effort Internet. The future Internet is expected to offer a certain QoS guarantee to some important applications, which are the best effort today. As is well understood, packets may suffer some delay and loss at the network nodes during their traversal across a packet-switched network. Therefore, packet loss and end-to-end delay are two crucial performance metrics for Internet QoS. In the present paper, we investigate delay and loss behavior of the resultant CMMPP/D/1/K queueing system and compare with that of simulation results.

The paper is organized as follows. In Section 2, we first overview the fundamentals of self-similar process and Circulant Markov modulated Poisson process. In Section 3, the generalized fitting procedure is given. In Section 4, Queuing systems and numerical results are presented. Finally, some conclusions are made in Section 5.

In this section, we first overview the definition of the exact second order self-similar process and summarize some characteristics of CMMPP and then, we make some remarks.

The definition of exact second-order self-similar processes is given as follows. If we consider

CMMPP is a doubly stochastic process in which arrival rate is given by

The mean and variance of

The variance of

Since the index of dispersion for counts (IDC) is defined as

From (3) and (4), we can obtain

We then could obtain the following remarks:

1)

2)

3)

4) Steady state distribution of 2-state CMMPP is

The first and second order statistics of N_{t} in the case of MMPP and CMMPP are listed in the following table.

Generalized variance-based fitting method is a procedure to find out the traffic model parameters, that match the variance of self-similar and that of model traffic [4-6,11]. The fitted model emulating self-similar traffic consists of a superposition of

Superposition of above

In (7),

Let

Put

Using (4), we obtain the variance of the ith CMMPP as

Also

From (9)-(11) and using the fact that superposition of independent sub-processes preserves the variance, we obtain

where

Using (1) and (12), we can match the variance at _{i},

where

Now, we assume the following relations between

That is,

There are due to the fact that a Self-similar process takes the same in any time scale. Because of this assumption, we can reduce the number of parameters to be determined. That is, if we determine

In this section, synchronous input traffic of fixed length h (in time units) is modeled as CMMPP/D/1queuing system. In CMMPP/D/1 system, the packets of fixed length arrive according to CMMPP. The performance metrics in this case involve

where ^{2} = 0.6 over the time scales [10^{2}, 10^{6}] [10^{2}, 10^{7}] [10^{2}, 10^{8}]. In all the above cases, the number of two state CMMPPs, d, is equal to 4. Numerical calculations are performed using the MATLAB and the results are shown in the Figures 1-6. ^{2}, 10^{6}], [10^{2}, 10^{7}] and [10^{2}, 10^{8}]. In this case, analytical results are validated with that of simulation. From this figure, it is clear that, the mean waiting time increases as the traffic intensity increases. ^{2}, 10^{6}], [10^{2}, 10^{7}], and [10^{2}, 10^{8}]. From the figure, it is clear that mean waiting time decreases as the time increases. ^{2}, 10^{8}]. From this we infer that the mean waiting time increases as H increases. ^{2}, 10^{6}], [10^{2}, 10^{7}], and [10^{2}, 10^{8}]. From this figure, we conclude that packet loss probability increases as the traffic intensity increases. Figures 5-6 depict the packet loss probability against the traffic intensitiesfor the cases of H =

0.7, H = 0.8, H = 0.9, over the time scales [10^{2}, 10^{6}], and [10^{2}, 10^{7}], respectively. From these figures, we conclude that packet loss probability decreases as the time scale increases, and packet loss probability decreases as the Hurst parameter decreases.

Most of the parsimonious self-similar traffic models proposed earlier are asymptotic in nature, therefore, they are less effective in the context of queuing based performance evaluation. Markovian models emulating selfsimilar traffic are proposed, as they hold well for queueing theory. These models are based on second order statistics. In this paper, we investigated queuing delay and loss behavior over different time scales and for different Hurst parameters. It is found from the numerical results that self-similar can be well represented by the proposed model. Our numerical results reveal that time-scale does have impact on packet loss probability. Packet loss probability increases as H and ρ increase. Based on the analysis presented in this paper, one could select the appropriate time-scale in the generalized variance based fitting method to meet the QoS requirement. This kind of analysis is useful in dimensioning the switch under self-similar input traffic.