channel chosen by the randomly selected IoT device, $f_o = f_m$ indicates that the randomly selected IoT device and the $m$-th IoT device select the same preamble and channel. ${\mathbbm 1}(\cdot)$ is the indicator function that equals to one if the statement ${\mathbbm 1}(\cdot)$ is ture; otherwise, it equals to zero. Just like [@zhang2015resource], in (\[eq:I\_intra\]), co-channel inter-cell interference is assumed as a part of thermal noise mainly because of the severe wall penetration loss.

Then, for the randomly selected IoT device in $s \in {\mathcal{S}^I}$, we can rewrite (\[eq:preamble\_trans\_suc\_prob\]) as the following form with (\[eq:I\_intra\]) $$\label{eq:QoS_analysis}
    \begin{array}{l}
P_s(t) = {\mathbb P}\{ SINR_s(t) \ge \theta _s^{th}\} \\
\qquad \text{ } = {\mathbb P}\{ {h_o} \ge \frac{{\theta _s^{th}}}{{{\rho _o}}}({\sigma ^2} + {{\cal I}_{s}(t)})\} \\
\qquad \text{ } \mathop  = \limits^{(a)} {{\mathbb E}}\left[ {\exp \left\{ { - \frac{{\theta _s^{th}}}{{{\rho _o}}}({\sigma ^2} + {{\cal I}_{s}(t)})} \right\}} \right]\\
\qquad \text{ } =  \exp \left\{ { - \frac{{\theta _s^{th}}}{{{\rho _o}}}{\sigma ^2}} \right\}{\cal L}_{{{\mathcal I}_{s}(t)}}\left(\frac{\theta_s^{th}}{\rho_o}\right), \forall s \in {\mathcal{S}^I}
\end{array}$$ where $\theta_s^{th} = {2^{\gamma _s^{th}/a}} - 1$. (a) follows from the full probability law over ${\cal I}_{s}(t)$, and ${\cal L}_{{{\cal I}_{s}(t)}} (\cdot)$ denotes the Laplace transform (LT) of the probability density function (PDF) of the random 