# Stochastic Network Calculus-Aided Delay Analysis of Wireless-Power Line Mixed Networks
## 3 Delay Analysis  
Recall that the wireless-PLC mixed system in this paper is a discrete-time, fluid-flow queuing system and that the arrival and service process can be defined as \(A(t)\) and \(S(t)\). The arrival and service increments are denoted by \(a\) and \(s\), respectively. To make use of SNC in the SNR domain, we convert these two factors into the exponential domain as \(\mathcal{A}(t)=e^{A(t)}\) and \(\mathcal{S}(t)=e^{S(t)}\). We can easily obtain the probabilistic performance bounds of any non-negative random variable (RV) \(X\) with the well-known [PERSON]'s bound \(\Pr\{X(\tau,t)\geq x\}\leq x^{-\theta}\mathcal{M}_{X}(1+\theta,\tau,t)\), where \(\mathcal{M}_{X}(\theta,\tau,t)=E[(X(\tau,t))^{\theta-1}]\) is the Mellin transform of RV \(X\). With the Chernoff's inequality and the Mellin transform of the arrival and service incremental process, the delay bound can be easily calculated via SNC in the SNR domain ([16], Theorem 1). In this work, we consider a Poisson arrival process for the data flow model. Provided that the arrival increment is a Poisson random variable with an average \(\delta\), it can be equivalently characterized in the SNR domain as  
\[\mathcal{M}_{a}(\theta)=\sum_{n=0}^{\infty}e^{n(\theta-1)}\frac{\delta^{n}}{n!}e^{-\delta}=e^{\delta(e^{\theta-1}-1)}. \tag{5}\]  
Then, we proceed to evaluate the Mellin transform of the service incremental process of both the wireless link and PLC link. We denote \(\gamma_{p}^{1}\) and \(\gamma_{p}^{2}\) as the instantaneous SNRs. In the situation where impulsive noise is absent or present in the PLC link, respectively, we have  
\[\gamma_{p}=\left\{ \begin{array}{ll}\gamma_{p}^{1}=\frac{|h_{p}|^{2}P_{p}}{ \sigma_{g}^{2}},\text{ only background noise},\\ \gamma_{p}^{2}=\frac{|h_{p}|^{2}P_{p}}{\sigma_{g}^{2}(1+K)},\text{ with impulsive noise}.\end{array} \right. \tag{6}\]It is noteworthy that these two RVs in Equation (6) are also log-normal RVs, and their pdfs are given by  
\[f_{\gamma\hat{p}}(\gamma) =\frac{1}{\gamma\sqrt{2\pi 4\sigma_{{{{B}}_{p}}}^{2}}}\exp\left(- \frac{(\ln\gamma-\ln\frac{P_{p}}{\sigma_{{{{B}}_{p}}^{2}}}-2\mu_{{{{B}}_{p}}})^ {2}}{8\sigma_{{{{B}}_{p}}}^{2}}\right), \tag{7}\] \[f_{\gamma\hat{p}}(\gamma) =\frac{1}{\gamma\sqrt{2\pi 4\sigma_{{{{B}}_{p}}}^{2}}}\exp\left(- \frac{(\ln\gamma-\ln\frac{P_{p}}{\sigma_{{{{B}}_{p}}^{2}}(K+1)}-2\mu_{{{{B}}_{ p}}})^{2}}{8\sigma_{{{{B}}_{p}}}^{2}}\right).\]  
With the above pdfs of the SNRs of the PLC link and the definition of the Mellin transform, we can further derive the Mellin transform of the service increments of the PLC link with Shannon capacity as  
\[\mathcal{M}_{{{{S}}_{p}}}(\theta)=E\bigg{[}(1+\gamma)^{\frac{B(\theta-1)}{\ln 2 }}\bigg{]}=\int_{0}^{\infty}(1+x)^{\frac{B(\theta-1)}{\ln 2}}f_{\gamma_{p}}(x) \mathbf{d}x, \tag{8}\]  
where \(B\) is the bandwidth of the wireless channel. Inserting Equation (7) into Equation (8), the Mellin transform of the service incremental process of the PLC link is given by  
\[\mathcal{M}_{{{{S}}_{p}}}=\left\{ \begin{array}{rl}\mathcal{M}_{{{{S}}_{p}}}^ {1}&=\frac{1}{\sqrt{2\pi 2\sigma_{{{{B}}_{p}}}}}\int_{0}^{\infty}(1+x)^{\frac{B( \theta-1)}{\ln 2}}x^{-1}\exp\left(-\frac{(\ln x-\ln\frac{P_{p}}{\sigma_{{{{B}}_{p}} ^{2}}}-2\mu_{{{{B}}_{p}}})^{2}}{8\sigma_{{{{B}}_{p}}}^{2}}\right)\mathbf{d}x, \\ &\quad\text{only background noise},\\ \mathcal{M}_{{{{S}}_{p}}}^{2}&=\frac{1}{\sqrt{2\pi 2\sigma_{{{{B}}_{p}}}}} \int_{0}^{\infty}(1+x)^{\frac{B(\theta-1)}{\ln 2}}x^{-1}\exp\left(-\frac{(\ln x-\ln \frac{P_{p}}{\sigma_{{{{B}}_{p}}^{2}}(K+1)}-2\mu_{{{{B}}_{p}}})^{2}}{8\sigma_ {{{{B}}_{p}}}^{2}}\right)\mathbf{d}x,\\ &\quad\text{with impulsive noise}.\end{array} \right.\]  
With the independent property of the Mellin transform, the analytical average delay bound for the PLC link is expressed as  
\[\mathcal{M}_{{{{S}}_{p}}}=(1-\rho)\mathcal{M}_{{{{S}}_{p}}}^{1}+\rho\mathcal{ M}_{{{{S}}_{p}}}^{2}. \tag{9}\]  
Similarly, for the Rayleigh fading wireless link, the service incremental process can be calculated with the Shannon capacity.