\mathcal{S}^I$, the probability that its queue is not empty can be defined as $$\label{eq:non_empty_prob}
    P_{ne,s}{(t)} = {\mathbb P}\{N_{a,s}(t) > 0\}, \forall s \in {\mathcal{S}^I}$$

(\[eq:non\_empty\_prob\]) implicitly reflects that new arrival packets at $t$ will not be sent out immediately. According to the evolution of $N_{a,s}(t)$, it can be observed that $P_{ne,s}(t)$ is determined by the probability distribution of $N_{a,s}(t-1)$ and the RA success probability. Since these probabilities and their correlations are unknown, the derivation of the explicit expression of $P_{ne,s}(t)$ is difficult.

Next, we describe the packet departure process combined with a frame and minislot structure for mIoT packets transmission. As mentioned above, partly because of the limitation on the frame and minislot structure, NB-IoT and LTE-M can only admit 50,000 devices. For NB-IoT, only one physical resource block (PRB) with a bandwidth of $180$ KHz in the frequency domain is allocated for IoT transmission, and each physical channel occupies the whole PRB. For LTE-M, although the physical channels are time and frequency multiplexed, it only reserves six in-band PRBs with a total bandwidth of $1.08$ MHz in the frequency domain for IoT data transmission. Therefore, the frame and minislot structure for mIoT transmission should be revisited if more RA 