# Self-Similar Markovian Sources
###### Abstract
We use a number of tools to convert a distribution to a set of exponentially distributed phases--e.g., [3]--and a number of tools to numerically solve the equations for state probabilities--e.g., [4]. However, the representation of general interarrival time distributions is not the only issue; another challenge is the autocorrelation of the input flow intensity. This should also be represented in Markov models if we aim to apply this approach to the evaluation of the Internet; i.e., to predict queueing delays and their variability, as well as the packet drop probability (caused by the congestion of the queues). Such queueing models are also useful for the proper choice of active queueing management mechanisms in routers and for the detection of possible distributed denial of service (DDoS) attacks which influence the self-similarity degree of the traffic; see [5; 6]. In recent years, the problem of the self-similarity and long-range dependence (LRD) of network traffic has become a popular topic of research [5; 6; 7; 8; 9]. Several articles [10; 11] have shown that ignoring these phenomena in computer network analysis causes an underestimation of performance measures. Many studies, both theoretical and empirical, have shown that network traffic exhibits self-similar properties. The self-similarity of the network traffic has a large impact on a network performance; it enlarges the queue occupancy and increases the number of packets dropped in the network nodes [12]. Thus, it is necessary to take this feature into account in the process of creating realistic models of traffic sources [13]. If we want to apply the Markov approach in this case, we require the Markov modelsof the input flow (of the source generating this flow) together with the Markov model of the server. Only then may we construct a Markov chain in which states are defined by the states of the source and the server simultaneously. In statistics, the term \"self-similar\" indicates that a portion of the object can be considered as the rescaled image of the whole. It was first introduced by [PERSON] in 1967 [14]. If we consider a continuous stochastic process, this means that the rescaled process and the original process have the same distribution [15]. Let us denote by \(Y(t)\) a continuous stochastic process. The condition that a process must satisfy to be self-similar can be written as follows [16]:  
\[Y(t)\stackrel{{ d}}{{=}}a^{-H}Y(at), \tag{1}\]  
for \(t\geq 0,a\ \geq 0\) and \(0<H<1\), where \(H\) denotes the Hurst parameter used to estimate the degree of self-similarity. When dealing with network traffic, we usually examine the data in the form of a time series instead of a continuous process [17]. Any measurement of traffic intensity within a certain interval gives us a discrete time stochastic process, determined by the obtained samples; if the sequence \(X_{1},X_{2},\ldots X_{k},\ldots\) is self-similar, then for the aggregated sequence  
\[X_{k}^{(m)}=\frac{1}{m}(X_{kn-m+1}+\ldots+X_{kn}),\ \text{where}\ m>1 \tag{2}\]  
the variance is equal to  
\[Var[X^{(m)}]=\frac{Var[X]}{m^{\beta}},\ \ \ \text{where}\ \ \ 0<\beta<1,\ \ \ H=1-\beta/2 \tag{3}\]  
or  
\[\log\text{Var}\,(X_{k}^{(m)})\approx\log\text{Var}\,(X)-\beta\log m \tag{4}\]  
Thus, the log-log plot of \(\text{Var}\,(X_{k}^{(m)})\) versus \(m\) is a straight line with the slope \(-\beta\). The equations above are used in a well-known method of Hurst parameter estimation known as the aggregated variance method (it is described in [7; 15]). We can use many different methods to estimate this value utilizing the aggregated series presented above or some more complex methods operating on a transformed domain [17]. The notions of self-similarity and long-range dependence (LRD) are often used as equivalents, although they are not [18]. A process that exhibits LRD is in fact an asymptotically second-order self-similar process. LRD means that some temporal similarities occur in the examined data. Let us denote by \(X_{n}\) a second-order stationary process.