# User Association Performance Trade-Offs in Integrated RF/mmWave/THz Communications
## 4 Coverage Probability Analysis  
To calculate the coverage probability for a given scenario with three types of users and three types of base stations, let us consider the subscript \(u\) and \(k\) for any user and base station indices in general and \(z\) for the subcarrier index. The coverage probability for a given user \(u\) associated with base station \(k\) and served by subcarrier \(z\), denoted as (the same equation is applicable to all three bands) \(P_{c}(\gamma_{k,u}^{(z)})\), can be expressed as [20]:  
\[P_{c}(\gamma_{k,u}^{(z)})=\frac{\sqrt{\sigma}}{2}N^{2}E_{c}^{2} \lambda_{k}\int_{\mathbb{R}^{2}}e^{\left[\frac{\left(\frac{\zeta_{k}^{2}}{P_{ k}^{2}|k_{\mathrm{ex}}|^{-2k}}\right)^{z}}{2}-\left(\mu+e^{2}\right)\left(\frac{ \zeta_{k}}{P_{k}^{2}|k_{\mathrm{ex}}|^{-2k}}\right)\right]}{\mathrm{d}( \mathbf{d}_{\mathbf{k},\mathbf{u}})} \tag{12}\]  
where \(\lambda_{k}\) is the density of base station \(k\) (BSs per unit area), \(E_{c}^{2}\) is the signal power considering the PN sequence, \(\zeta_{k}\) is the SINR threshold for coverage for user \(u\) associated with BS \(k\), \(P_{T_{k}}\) is the transmit power of BS \(k\), \(\mu\) is the mean of the interference distribution, \(\sigma^{2}\) is the variance of the interference distribution, \(R\) is the cell radius, \(|d_{k,u}|\) is the distance from the BS \(k\) to the user \(u\), and \(\alpha\) is the path loss exponent. Since there are three different types of users (radio frequency, millimeter wave, and terahertz), each with its own SINR threshold and path loss exponent, we need to calculate the coverage probability separately for each user type and then combine the results. Let us denote the three types of users as \(u_{1}\) (radio frequency), \(u_{2}\) (millimeter wave), and \(u_{3}\) (terahertz), and their corresponding base stations as \(k_{1}\), \(k_{2}\), and \(k_{3}\), respectively. The total coverage probability, denoted as \(P_{c,\text{total}}\), is the probability that any of the three users achieves the required SINR threshold. Therefore, it can be expressed as:  
\[P_{c,\text{total}}=1-\prod_{i=1}^{3}(1-P_{c}(\gamma_{k_{i},u_{i}}^{(z)})) \tag{13}\]  
Here, \(P_{c}(\gamma_{k_{i},u_{i}}^{(z)})\) represents the coverage probability for user \(u_{i}\) associated with base station \(k_{i}\) and served by subcarrier \(z\). The expression above accounts for the fact that the overall coverage probability is determined by the probability that at least one of the three users achieves coverage. To evaluate \(P_{c,\text{total}}\), we need to calculate the individual coverage probabilities \(P_{c}(\gamma_{k_{i},u_{i}}^{(z)})\) for each user type based on their respective SINR thresholds, path loss exponents, and other parameters. The coverage probability for a randomly located UE can be written as:  
\[P_{c}=\mathbb{P}(\bigcup_{k\in\mathcal{K}}\max_{d_{k}\in\Phi_{k}}\gamma_{k}> \zeta_{k}) \tag{14}\]  
where \(\gamma_{k}\) is the SINR from the \(k\)th BS, \(\zeta_{k}\) is the SINR threshold, and \(\phi_{k}\) is the set of BSs in the \(k\)th tier. Using the Law of Total Probability and Union Bound, we can rewrite the coverage probability for an RF user as:  
\[P_{c}=\sum_{k\in K}\lambda_{k}N^{2}E_{c}^{2}\int_{\mathbb{R}^{2}}\mathbb{P}( \gamma_{k}>\zeta_{k})d(d_{k}) \tag{15}\]  
where \(\lambda_{k}\) is the density of the BSs in each band, \(\ell(\mathbf{d_{k,u}})=\|\mathbf{d_{k,u}}\|^{-\alpha}\), \(\mathrm{d}(\cdot)\) is the derivative operator, and \(d_{k}\) is the measure of the distance from the UE to the serving BS. The probability term can be expressed using the Laplace transform of interference \(\mathcal{L}_{I}(.