## 3 Analysis of the Proposed Analytical Model  
This section describes the impact of (\(m\)) values on the \(E(N)\) according to Equations (7), (10) and (14), and the cost function \(C(m)\) according to Equations (8), (11) and (15). Figure 6a-c plot the relation between the \(E(N)\) versus the value of \(m\) for the multiple arrival rate (\(\lambda\)), where the residence time is distributed (exponential, hypo exponential and gamma) with means \(\mu^{-1}\), \(\mu_{1}{}^{-1}+\mu_{2}{}^{-1}\), and \(\mu^{-1}\), respectively. It is obvious the \(E(N)\) is a decreasing function of \(m\) and the plotted points are closed to each other. After a while \(m\geq 10\), \(E(N)\) is insignificantly reduced by increasing the value of \(m\). Figure 6: _Cont._On the other hand, the function \(\Psi\) (\(n\), \(m\)) had a different behavior with respect to \(m\), for the fixed ratio \(\gamma\). Figure 7a-f plot the probability density function \(\Psi\) (\(n\), \(m\)) when the number of SAPs was \(5\leq m\leq 20\), for various residence-time distributions. Notice that the behavior of \(\Psi\) (\(n\), \(m\)) was similar after a specified number of \(n\); for \(n\geq 6\), the plotted points were closed to each other. This observation was consistent with Figure 6, i.e., the \(E(N)\) value was the same for the large (\(m\)) value, and the increasing value of \(m\) did not improve the \(E(N)\) value. Figure 6: Effect of SAPs on the expected LAPs when the residence time is distributed as in (**a–c**). (**a**) Exponentially distributed residence time with mean \(\mu^{-1}\). (**b**) Hypo exponential distributed residence time with mean \(\mu_{1}^{-1}+\mu_{2}^{-1}\). (**c**) Gamma distributed residence time, when \(\lambda=20\mu\). Figure 7: \(Cont.\)  
Figure 8a-c show the effect of \(m\) values on the trend of the cost function \(\mathrm{C}(m)\) for fixed \(\alpha\), \(\beta\), and \(\lambda\). The trend of the plots is the same for various residence time distribution, all plots obviously show that there is a critical value (\(m\)), which is minimizing the cost function, and after this point, the \(\mathrm{C}(m)\) is rapidly increased. Also, the \(\mathrm{C}(m)\) values are significantly increased with the increasing of the (\(\lambda\)) values. These results are proportionate with goal of the direct authentication between the Pi and SD\(j\), that if there are more SAPs, then more authentication keys (PS\(ij\)) should be derived by the Pi and SD\(j\). Figure 8: The cost function \(C(m)\) when \(\alpha=\beta=1\), with different residence time distributions as in (**a–c**). (**a**) The residence time is exponential distributed. (**b**) The residence time is hypo exponential distributed. (**c**) The residence time is gamma distributed. Figure 7: With different residence/request time distributions as in (**a–e**). (**a**) Exponentially distributed residence time, mean \(\mu^{-1}\), when \(\lambda~{}=~{}10\mu\). (**b**) Hypo-exponential distributed residence time, when \(\lambda=2\mu_{1}=20\mu_{2}\). (**c**) Hypo-exponential distributed residence time, when \(\lambda=10\mu_{1}=2\mu_{2}\). (**d**) Gamma-distributed residence time is when \(\kappa=1\), and \(\lambda=10\mu\). (**e**) Gamma-distributed residence time, when \(\kappa=2\), and \(\lambda=20\mu\). (**f**) Gamma-distributed residence time, when \(\kappa=3\), and \(\lambda=30\mu\). Figures 6-8 show that applying various distributions (gamma, hypo-exponential and exponential) as residence times did not change the trend of \(\Psi\) (\(n\), \(m\)), \(E(N)\) or C(\(m\)) significantly. Therefore, studying the extent of the influence of one of these probability distributions was sufficient. Where the exponential distribution was good in the mean and dealt with all the trends was a special case of the gamma and hypo-exponential distributions.