## 4 Coverage Probability of MHCP-MHCP Network  
We assume the open access strategy, i.e., the tagged MUE, choose to access the BS which can deliver the maximum received SIR. That is to say, the tagged MUE at the origin is in coverage if: \(\max\limits_{x\in\Phi_{k}}SIR(x)>\beta_{k}\ (k=1,2)\), where \(\beta_{k}\) is the SIR threshold for the \(k\)-th tier. It has been proved that, in the PPP-modelled network, if \(\beta_{k}>1\) (0 dB), the tagged MUE can only access one base station at most [4]. On the basis of these existing conclusions, the coverage probability for tagged MUE in the MHCP-MHCP network in open access can be derived as follows:  
\[ \begin{array}{l}\mathrm{P}_{c}(\{\lambda_{k-MHCP}\},\{\beta_{k}\},\{P_{Tk} \})=P\Bigg{(}\underset{k=1,2x_{k}\in\[PERSON]_{k}^{HPCP}}{\cup}SIR(x_{k})>\beta_{k} \Bigg{)}=\mathbb{E}\Bigg{(}1\Bigg{(}\underset{k=1,2x_{k}\in\[PERSON]_{k}^{MHCP}}{ \cup}SIR(x_{k})>\beta_{k}\Bigg{)}\Bigg{)}\\ \overset{(a)}{\approx}\mathbb{E}\Bigg{(}1\Bigg{(}\underset{k=1,2x_{k}\in\[PERSON]_ {k}^{PPP}}{\cup}SIR(x_{k})>\beta_{k}/G_{k}\Bigg{)}\Bigg{)}\overset{(b)}{=} \underset{k=1}{\overset{2}{\sum}}\mathbb{E}\Bigg{[}\underset{k=0}{\overset{2} {\sum}}\underset{k=0}{\overset{2}{\sum}}\mathbb{E}\Bigg{[}\underset{k=1}{ \overset{2}{\sum}}\underset{k=0}{\overset{2}{\sum}}\mathbb{E}\Bigg{[}1(SIR( x_{k})>\beta_{k}/G_{k})\Bigg{]}\Bigg{]}\\ \overset{(c)}{=}\underset{k=1}{\overset{2}{\sum}}\lambda_{k-MHCP}\int_{ \mathbb{R}^{2}}P\bigg{(}\frac{P_{Tk}n_{k}\|x_{k}\|^{-\alpha}}{\underset{T_{k}} {\underset{T_{k}}{\underset{T_{k}}{\underset{T_{k}}{\underset{T_{k}}{\underset{T _{k}}{\underset{T_{k}}{\underset{T_{k}}{\underset{T_{k}}{\underset{T_{k}}{ \underset{T}}{\underset{T_{k}}{\underset{T}}{\underset{T_{k}}{\underset{T}}{ \underset{T_{k}}{\underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{ \underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{ \underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{ \underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{ \underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{ \underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{ \underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{ \underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{\underset{T}}{ \underset{T}}{\underset{T}}{\underset{T}}}\Bigg{)}dx_{k}\end{array}  \tag{7}\]  
In step (a), ASAPPP method is used, by scaling the SIR threshold of networks based on PPP models, we can get the approximate SIR distribution based on the MHCP model, where \(G_{k}\) is the SIR scaling factor of the \(k\)-th tier based on the MHCP model. However, due to the mutual exclusion between BSs in the MHCP modelled network, such processing will lead to increased interference and reduce the coverage of the results. Thus, the interference from \(\Phi_{k}^{MHCP}\) is upper bounded by the interference from a PPP with the same density. So, the coverage provided by step (a) will be slightly lower than the actual coverage. Step (b) follows from the assumption that \(\beta_{k}/G_{k}>1\forall k\)[4]. Step (c) follows from the [PERSON] theorem [44]. The derivation of Step (d) is based on the Rayleigh fading channel. Here \(\mathcal{L}_{I_{k_{k}}}(\cdot)\) is the Laplace transform of the cumulative interference from all the tiers when the tagged MUE access to a BS in k-th tier. Due to the stability of the point process, the interference is independent of the location \(x_{k}\). Therefore, we denote \(\mathcal{L}_{I_{k_{k}}}(\cdot)\) by \(\mathcal{L}_{I}(\cdot)\). In the PPP-PPP network, \(\mathcal{L}_{I}(s)\) is derived as [4]  
\[\mathcal{L}_{I}(s)=\exp\Bigg{(}-s^{2/\alpha}C(\alpha)\overset{2}{\underset{i =1}{\overset{2}{\sum}}}\lambda_{i}P_{Ti}^{2/\alpha}\Bigg{)} \tag{8}\]  
where \(C(\alpha)=2\pi^{2}\csc(2\pi/\alpha)\alpha^{-1}\), \(\lambda_{i}\) represents the density of the BSs in k-th tier in PPP-PPP network.