The ACB scheme can relieve slice congestion to some extent by reducing RACH requests of active IoT devices.

With the introduced access control schemes, we can define the non-restriction probability of a randomly selected IoT device in $s$ as follows.

\[mydef:unrestricted\_prob\] At minislot $t$, for a randomly selected IoT device in slice $s \in \mathcal{S}^I$, the probability that its RACH request is not restricted is defined as $$\label{eq:non_restriction_prob}
    P_{nr,s}{(t)} = {\mathbb P}\{{\rm Unrestricted} \text{ } {\rm RACH} \text{ } {\rm requests}\}, \forall s \in {\mathcal{S}^I}$$

For all $s \in \mathcal{S}^I$ at any minislot $t$, we have $P_{nr,s}(t) = 1$ for the unrestricted scheme and $P_{nr,s}(t) = P_{ACB}$ for the ACB scheme.

### Analysis of RA success probability

For an RRH, two significant reasons may lead to an error preamble decoding i) the achieved preamble transfer rate at the RRH is less than a preset threshold; ii) the RRH simultaneously decodes at least two similar co-channel preambles, and thus preamble collision occurs. The research of the mitigation of preamble collision has been well conducted in [@jang2019versatile; @jiang2018collision]. Just like [@jiang2018random], we focus on the exploration of enabling successful single preamble transmission that is discussed in detail as follows.

We utilize a 