# On Edge Pruning of Communication Networks under an Age-of-Information Framework
## 3 Proposed Model
### Phase-Type Distribution
Another more special distribution is the Erlang distribution which represents the mixture of adding up \(k\) identical and independent random distributions with parameter \(\lambda\). It is denoted as \(Erl(k,\lambda)\). We can write the Erlang distribution in the form of a phase-type distribution by making \(\mathbf{S}\) a \(k\times k\) matrix with diagonal elements \(-\lambda\) and super-diagonal elements \(\lambda\), with the probability of starting in state 1 equal to 1. For example, \(Erl(6,\lambda)\) is  
\[\boldsymbol{\alpha}=(1,0,0,0,0,0) \tag{6}\]  
and  
\[\mathbf{S}= \begin{bmatrix}-\lambda&\lambda&0&0&0&0\\ 0&-\lambda&\lambda&0&0&0\\ 0&0&-\lambda&\lambda&0&0\\ 0&0&0&-\lambda&\lambda&0\\ 0&0&0&0&-\lambda&\lambda\\ 0&0&0&0&0&-\lambda\end{bmatrix}  \tag{7}\]  
Another special case of the phase-type distribution is the hypoexponential distribution which generalizes the Erlang distribution by having different rates for each one of the transitions. So, considering the mixture of Erlang distributions makes us see that the mixture of two Erlang distributions with parameters \(Erl(3,\beta_{1}),E(3,\beta_{2})\) and \((\alpha_{1},\alpha_{2})\), for example, can be written as a phase-type distribution with [33]:  
\[\boldsymbol{\alpha}=(\alpha_{1},0,0,\alpha_{2},0,0) \tag{8}\]  
and  
\[\mathbf{S}= \begin{bmatrix}-\beta_{1}&\beta_{1}&0&0&0&0\\ 0&-\beta_{1}&\beta_{1}&0&0&0\\ 0&0&-\beta_{1}&0&0&0\\ 0&0&0&-\beta_{2}&\beta_{2}&0\\ 0&0&0&0&-\beta_{2}&\beta_{2}\\ 0&0&0&0&0&-\beta_{2}\end{bmatrix}  \tag{9}\]  
Finally, the last example is the Coxian distribution, which generalizes the Erlang distribution even more. In this distribution, the Markov process allows entering the absorbing state from any phase. The phase-type distribution format of it is given by:  
\[\mathbf{S}= \begin{bmatrix}-\lambda_{1}&p_{1}\lambda_{1}&0&\dots&0&0\\ 0&-\lambda_{2}&p_{2}\lambda_{2}&\ddots&0&0\\ \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&0&\ddots&-\lambda_{k-2}&p_{k-2}\lambda_{k-2}&0\\ 0&0&\dots&0&-\lambda_{k-1}&p_{k-1}\lambda_{k-1}\\ 0&0&\dots&0&0&-\lambda_{k}\end{bmatrix}  \tag{10}\]  
and  
\[\boldsymbol{\alpha}=(1,0,\dots,0) \tag{11}\]  
such that \(p_{1},\dots,p_{k-1}1\) are positive and less than one. A special case is having all \(p_{i}=1\), which reduces to the Erlang distribution. We now need to introduce the Kronecker product and the Kronecker sum to simplify the presentation of the properties of the phase-type distribution. Those Kronecker compositions of matrices are often used when manipulating matrices. Let \(\mathbf{A}=(a_{ij})\) be a \(n_{1}\times m_{1}\) matrix, and let \(\mathbf{B}=(b_{ij})\) be a \(n_{2}\times m_{2}\) matrix. The Kronecker product of \(\mathbf{A}\) and \(\mathbf{B}\) is the \((n_{1}n_{2})\times(m_{1}m_{2})\) matrix:  
\[\mathbf{A}\otimes\mathbf{B}= \begin{bmatrix}a_{11}B&a_{12}B&\dots&a_{1m_{1}}B\\ a_{21}B&a_{22}B&\dots&a_{2m_{1}}B\\ \vdots&\vdots&\ddots&\vdots\\ a_{n_{1}1}B&a_{n_{1}2}B&\dots&a_{n_{1}m_{1}}B\end{bmatrix}  \tag{12}\]  
If \(\mathbf{A}\) and \(\mathbf{B}\) are square matrices, i.e., \(n_{k}=m_{k}\) for \(k=1,2\), and \(\mathbf{I}_{n}\) denotes the \(n\times n\) identity matrix, then the Kronecker sum, which we will use in the continuous setting, is defined as the \((n_{1}n_{2})\times(m_{1}m_{2})\) matrix:  
\[\mathbf{A}\oplus\mathbf{B}=\mathbf{A}\otimes\mathbf{I}_{n_{2}}+\mathbf{I}_{n_{1}}\otimes\mathbf{B} \tag{13}\]  
We now introduce the two most important properties of the phase-type distribution for our paper [34]: