## Appendix A  
In this Appendix we present for the case of a vector normal process the results of the LCR of a curve \(\partial D\) bounding a domain \(D\). In accordance to the shadowing process we assume that the process represents the logarithm of the received signal in \(N\) links. The domain \(D\) corresponds to the domain of reliable communication for the two cases of an \(N\)-tuple diversity scheme with SC and a regenerative \(N\)-hop relay link as shown in Figures 1 and 2 for the two-dimensional case. The LCR is calculated for out crossings of the domain \(D\), i.e., incrossings of domain \(D^{C}\), when the system shifts to outage. The N-tuple diversity system is in outage when all \(N\) signals fall below the required SNR threshold. This is equivalent to \(X_{i}\leq-FM_{i}\) for the corresponding shadowing value and fade margin. According to [46] the outcrossing rate of a domain \(D\) with boundary \(\partial D\) is given by the relation  
\[\mu^{+}=\int\limits_{\partial D}E\big{[}(\mathbf{n}^{T}\mathbf{X}_{t}{}^{ \prime})^{+}\big{]}\mathbf{X}_{t}=\mathbf{b}\big{]}p(\mathbf{b})dl(\mathbf{b}) \tag{1}\]  
Figure 8: Average hand-off rate for a mobile receiver moving between two base stations. Because the process is assumed normal its derivatives are independent of the process itself, so (A1) becomes  
\[\mu^{+}=\int\limits_{\partial D}E\Big{[}\big{(}\mathbf{n}^{T}\mathbf{X}_{t}{}^{ \prime}\big{)}^{+}\Big{]}p(\mathbf{b})dl(\mathbf{b})\] (A2)  
We now consider the random function \(g(\omega)=\mathbf{n}^{T}\mathbf{X}_{t}{}^{\prime}=f(\dot{X}_{1}(\omega),\dots, \dot{X}_{N}(\omega))\) and the event in the basic probability space \(A=\{\omega\in\Omega:g(\omega)\in[0,+\infty)\}\). Then  
\[E\Big{[}\big{(}\mathbf{n}^{T}\mathbf{X}_{t}{}^{\prime}\big{)}^{+} \Big{]}=E(g(\omega)1_{A}(\omega))=\int\limits_{A}g(\omega)dP(\omega)\] (A3) \[=\int\limits_{A}f(\dot{X}_{1}(\omega),\dots,\dot{X}_{N}(\omega)) dP(\omega)=\int\limits_{f^{-1}([0,+\infty))}f(\dot{x}_{1},\dots,\dot{x}_{N})dP_{ \dot{x}_{1}\dots\dot{x}_{N}}\] \[=\int\limits_{f^{-1}([0,+\infty))}f(\dot{x}_{1},\dots,\dot{x}_{N })p(\dot{x}_{1},\dots,\dot{x}_{N})d\dot{x}_{1}\dots d\dot{x}_{N}\]  
In the case of the N-tuple diversity scheme with SC the hypersurface \(\partial D\) consists of \(N\) hyperplanes. Thus the total LCR is equal to the sum of the LCR for each of the \(N\) hyperplanes. For hyperplane \(1\) corresponding to link \(1\) the normal vector is \(\mathbf{n}_{1}=(-1,0,\dots,0)\), so \(f(\dot{x}_{1},\dots,\dot{x}_{N})=-\dot{x}_{1}\) and  
\[\mu_{1}^{+}=\int\limits_{-\infty}^{0}(-\dot{x}_{1})p(\dot{x}_{1})d\dot{x}_{1} \underbrace{\int\limits_{-\infty}^{-FM_{2}}\int\limits_{-\infty}^{-FM_{3}} \cdots\int\limits_{-\infty}^{-FM_{N}}p(-FM_{1},x_{2},\dots,x_{N})dx_{2}\cdots dx _{N}}_{N-1}\] (A4)  
Similarly for the \(k\)-th hyperplane corresponding to link \(k\)  
\[ \begin{split}&\mu_{k}^{+}=\int\limits_{-\infty}^{0}(-\dot{x}_{k})p (\dot{x}_{k})d\dot{x}_{k}\cdot l_{k}\\ & l_{k}=\underbrace{\int\limits_{-\infty}^{-FM_{1}}\cdots\int \limits_{-\infty}^{-FM_{k-1}}\int\limits_{-\infty}^{-FM_{k+1}}\cdots\int \limits_{-\infty}^{-FM_{N}}p(x_{1},\dots,x_{k-1},-FM_{k},x_{k+1},\dots,x_{N} )dx_{1}\cdots dx_{k-1}dx_{k+1}\cdots dx_{N}}_{N-1}\end{split} \] (A5)  
Finally the outcrossing rate is given by  
\[\mu^{+}=\sum\limits_{k}\mu_{k}^{+}\] (A6)  
Equation (A5) can be further simplified by noting that  
\[\int\limits_{-\infty}^{0}(-\dot{x}_{k})p(\dot{x}_{k})d\dot{x}_{k}=\frac{ \sigma_{\dot{x}_{k}}}{\sqrt{2\pi}}\] (A7)  
for the case of a zero mean normal r.v. Furthermore by using the conditional probability densities the integral \(l_{k}\) can be calculated as  
\[l_{k}=p_{x_{k}}(-FM_{k})Q(-FM_{1},\dots,-FM_{k-1},-FM_{k+1},\dots,-FM_{N})\] (A8)  
where \(Q(x_{1},\dots,x_{k-1},x_{k+1},\dots,x_{N})\) is the cumulative distribution function of the \(N-1\)-variate conditional pdf, which can be calculated by use of numerical integration techniques or standard software packages.