$N_{a,s}(t)$ of a randomly selected IoT device in slice $s$ at $t$ is determined by the accumulated number of packets and the number of new arrivals at $t - 1$ and whether the preamble of the device can be successfully decoded by its associated RRH. Table \[table:queue\_evolution\] shows the evolution of accumulated packets in an IoT device. In this table, $x_s = \gamma_s^{th}/L$ packets at the head of the queue will be popped out if the corresponding RA succeeds, where $L$ denotes the IoT packet length; otherwise, they will be kept in the queue and wait for the opportunity of re-transmission at the next minislot. The operation $[x]^+ = \max(x,0)$.

       Value                              Success                                          Failure
  ---------------- ------------------------------------------------------ -----------------------------------------
    $N_{a,s}(1)$                            $0$                                              $0$
    $N_{a,s}(2)$                 $[N_{w,s}{(1)}$ - $x_s]^+$                            $N_{w,s}{(1)}$
   $N_{a,s}{(3)}$       $[N_{a,s}{(2)}$ + $N_{w,s}{(2)}$ - $x_s]^+$            $N_{a,s}{(2)}$ + $N_{w,s}{(2)}$
      $\ldots$                            $\ldots$                                        $\ldots$
   $N_{a,s}{(t)}$   $ [N_{a,s}(t$ - $1)$ + $N_{w,s}(t$ - $1)$ - $x_s]^+$   $N_{a,s}(t$ - $1)$ + $N_{w,s}(t$ - $1)$

  : Accumulated packets evolution in an IoT device[]{data-label="table:queue_evolution"}

With the evolution of accumulated packets, we can define the non-empty probability of the queue of an IoT device in $s$ as the following.

\[def:non\_empty\_prob\] At minislot $t$, for a randomly selected IoT device in slice $s \in 