variable ${\cal I}_{s}(t)$. Note that the notation ${\cal L}_{{{\cal I}_{s}(t)}} (\cdot)$ is a terminology that is a slight abuse of subscript ${{\cal I}_{s}(t)}$.

The following lemma characterizes the LT of aggregate interference $\mathcal{I}_s(t)$.

\[lem:LT\_interference\_expression\] For the origin RRH, the LT of its received aggregate interference from active IoT devices associated with it is given by $$\label{eq:LT_interference_expression}
        {\cal L}_{{{\cal I}_{s}(t)}} \left(\varpi_s\right) = {\frac{{1 + \varpi_s{\rho _o}}}{{{{\left( {1 + \alpha_s \varpi_s{\rho _o}/\left( {1 + \varpi_s{\rho _o}} \right)} \right)}^{3.5}}}} - \frac{{1 + \varpi_s{\rho _o}}}{{{{\left( {1 + \alpha_s } \right)}^{3.5}}}}}$$ where $\varpi_s = \frac{\theta_s^{th}}{\rho_o}$, $\alpha_s  = \frac{{{P_{nr,s}}(t){P_{ne,s}}(t){\lambda _{{s}}^I}}}{{3.5{\lambda _R}{\xi F_s}}}$, for all $s \in {\mathcal{S}^I}$.

Please refer to Appendix A.

With the conclusion in Lemma \[lem:LT\_interference\_expression\], we can then obtain the mathematical expression of the RA success probability of a randomly selected IoT device at $t$ in the following corollary.

\[lem:QoS\_prob\_analysis\] For a randomly selected IoT device in a mIoT slice $s \in {\cal S}^I$, its RA success probability at minislot $t$ is given by $$\label{eq:QoS_prob_analysis}
       {P_s}(t) =  {\frac{{(1 + \varpi_s{\rho _o})}{{e^{-\varpi_s{\sigma ^2}}}}}{{{{\left( {1 + \alpha_s \varpi_s{\rho _o}/\left( {1 + \varpi_s{\rho _o}} \right)} \right)}^{3.5}}}} - \frac{{(1 + \varpi_s{\rho _o})}{{e^{-\varpi_s{\sigma ^2}}}}}{{{{\left( {1 + \alpha_s } \right)}^{3.5}}}}}$$

By substituting (\[eq:LT\_interference\_expression\]) into (\[eq:QoS\_analysis\]), we can obtain (\[eq:QoS\_prob\_analysis\]).

