# On Edge Pruning of Communication Networks under an Age-of-Information Framework
## 3 Proposed Model
### Phase-Type Distribution  
In this subsection, we introduce the Phase-type distribution. A phase-type distribution is a probability distribution that we may construct by a mixture of independent exponential distributions. It arises out in a natural way from a setting of sequence inter-related Poisson processes. One of the standard applications in which Phase-type distribution appears is when we model a representative random variable related to a Markov chain with one absorbing state. This random variable describes the total elapsed time until absorption. Each one of the states in the Markov process is represented by one phase [31]. Furthermore, an important characteristic of phase-type distributions is that they are dense on the field of analytical probability distributions. This means that this distribution closely approximates all positive-valued analytical distributions. It is worth mentioning that in our context here, we make use of a continuous-time Markov process with exactly \(m+1\) states. All states are transient states except state 0. State 0 is the sole absorbing state of the Markov process. To start defining the parameters of the distribution, consider that the Markov process has an initial probability distribution to start at any one of the different \(m+1\) phases. This distribution of probabilities can be represented as the vector \((\alpha_{0},\mathbf{\alpha})\) such that \(\mathbf{\alpha}\) is a \(1\times m\) vector. From there, The phase-type distribution is used for a random variable that represents the time elapsed from the starting of such a process until the end, i.e., the absorption state 0. To this end, The transition rate matrix of the Markov process is:  
\[Q= \begin{bmatrix}0&\mathbf{0}\\ \mathbf{S}^{0}&\mathbf{S}\end{bmatrix}  \tag{3}\]  
such that the size of \(\mathbf{S}\) is \(m\times m\), and that \(S^{0}=\mathbf{S}\times\mathbf{1}\), where \(\mathbf{1}\) is the \(m\times 1\) vector of unity elements. Now, we define the random variable \(Z\) which has the probability distribution of the elapsed time the Markov process takes until absorption. The phase-type distributed to this end is typically denoted as \(PH(\mathbf{\alpha},\mathbf{S})\)[32]. The cumulative distribution function (CDF) of the random variable \(Z\) is given by \(F(z)=1-\mathbf{\alpha}\exp(Sz)\mathbf{1}\). The probability density function is given by \(f(z)=\mathbf{\alpha}\exp(Sz)\mathbf{S}^{0}\) for all \(x>0\), where \(\exp(\cdot)\) is the matrix exponential operation. This operation is defined as \(exp(\mathbf{X})=\Sigma_{k=0}^{\infty}\frac{1}{H}X^{k}\). Typically, the trivial case of starting at the absorption state is discarded by setting the probability of starting there to zero (setting \(\alpha_{0}=0\)). Furthermore, We may derive the moments of the distribution by \(E[X^{n}]=(-1)^{n}n!\mathbf{\alpha}S^{-n}\mathbf{1}\). The Laplace transform of the distribution can be derived as well by \(M(s)=\alpha_{0}+\mathbf{\alpha}(sI-S)^{-1}\mathbf{S}^{0}\) where \(\mathbf{I}\) is the identity matrix with the proper dimensions. Now, we introduce the first example of the phase-type distribution, which is the hyperexponential distribution. This distribution is a mixture of exponential distributions too that arises when you add up independent exponential distributions with different parameters \(\lambda_{1},\lambda_{2},\dots,\lambda_{n}\). We may represent this distribution as a phase-type one with:  
\[\mathbf{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{n}),\sum_{i=1}^{n}\alpha_{i}=1 \tag{4}\]\[\mathbf{S}= \begin{bmatrix}-\lambda_{1}&0&\dots&0\\ 0&-\lambda_{2}&\dots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\dots&-\lambda_{n}\end{bmatrix}  \tag{5}\]  
Hence, this mixture of independent exponentially distributed random variables can be characterized through \(f(x)=\sum_{i=1}^{n}\alpha_{i}\lambda_{i}e^{-\lambda_{i}x}=\sum_{i=1}^{n}a_{f}x _{i}(x)\). The CDF would be \(F(x)=1-\sum_{i=1}^{n}\alpha_{i}e^{-\lambda_{i}x}=\sum_{i=1}^{n}\alpha_{i}F_{X_{ i}}(x)\), where \(X_{i}\)\(\sim\)\(exp(\lambda_{i})\).