# Self-Similar Markovian Sources
## 2 Markov Chains and the Markov Traffic Sources
### Markov Modulated Poisson Process (MMPP)  
The Markov modulated Poisson process is a Poisson process in which the intensity of arrivals is defined by the state of a modulator. In the simplest case, the modulator has two states, and if there is no traffic in one of these states, which means that \(\lambda=0\), we can refer to an ON-OFF process. One of the properties of the Markov modulated Poisson process is that a superposition of MMPPs is also an MMPP. A sum of \(d\) ON-OFF processes, as proposed in [40], may be used to represent a self-similar process. The obtained MMPP is a process with \(2^{d}\) states. This model consists of \(d\) ON-OFF processes, and each of them may be parametrized by two square matrices:  
\[\mathbf{D}_{\mathbf{0}}^{i}=\left[\begin{array}{cc}-(c_{1i}+\lambda_{1i})&c _{1i}\\ c_{2i}&-(c_{2i}+\lambda_{2i})\end{array}\right]\,\qquad\mathbf{D}_{\mathbf{1}}^{i}=\left[ \begin{array}{cc}\lambda_{1i}&0\\ 0&\lambda_{2i}\end{array}\right],\quad i=1,\dots d.\]  
The element \(c_{1i}\) is the transition rate (if multiplied by \(dt\), it becomes the transition probability) from the ON state to OFF of the \(i\)-th source, and \(c_{2i}\) is the rate from the OFF state to ON. \(\lambda_{1i}\) is the traffic rate when the \(i\)-th source is ON. In the OFF state, a source is inactive. The sum of \(\mathbf{D}_{\mathbf{0}}^{i}\) and \(\mathbf{D}_{\mathbf{1}}^{i}\) is an irreducible infinitesimal generator \(\mathbf{Q}^{i}\) with the stationary probability vector \(\pi_{i}\), which can be defined as follows:  
\[\boldsymbol{\pi}_{i}=\left(\frac{c_{2i}}{c_{1i}+c_{2i}},\frac{c_{1i}}{c_{1i}+c_{ 2i}}\right)\]  
To determine the parameter matrices \(\mathbf{D_{0}}\) and \(\mathbf{D_{1}}\) of an MMPP obtained as a superposition of \(d\) MMPPs with two states, it is possible to use the Kronecker sum of those \(d\) two-state MMPPs [41]:  
\[(\mathbf{D_{0}},\mathbf{D_{1}})=\left(\oplus_{i=1}^{d}\mathbf{D_{0}}^{i}, \oplus_{i=1}^{d}\mathbf{D_{1}}^{i}\right)\]  
To fit the examined model to the specific data, two approaches can be used: moment-based approach and likelihood-based approach [42]. It has been presented in [40] that it is possible for the fitting procedure for a superposition of two-state MMPPs to be based not only on moments, but also on a self-similar measure--the Hurst parameter.  
The input parameters that are required in the fitting method can be described as follows [32]:  
* \(\lambda^{*}\)--the mean rate of the process to be modeled,
* \(n\)--number of time scales,
* \(d\)--number of active MMPPs,
* \(H=1-\frac{\beta}{2}\)--the Hurst parameter,
* \(\rho\)--lag 1 correlation.  
The MMPP model may be integrated with any CTMC queueing model. We have successfully used this approach, for example, to model an IP router in which a system of four MMPPs reflected data published by CAIDA and which came from the link Equinix Chicago, collected during one hour on 18 February 2016, with 22,644,654 packets belonging to 1,174,515 IPv4 flows. The service time distribution at the server corresponded to the histogram of IP packet sizes and was represented by a hyper-Erlang distribution, with over 1400 phases in total [43].