will analyze an IoT device queue evolution model, with the analysis of which the RA success probability of the IoT device will be derived.

### Queue evolution model

The queue evolution process consists of the packet arrival process, packet accumulate process, and packet departure process.

During minislot $t$, a Poisson distribution with intensity (or arrival rate) $\epsilon_{w,s}(t)$ is exploited to model the random, mutually independent endogenous packet arrivals in an IoT device in slice $s$. Then during minislot $t$ with a duration $\tau$, the arrival intensity of new packets can be expressed as $\mu_{w,s}(t) = \epsilon_{w,s}(t)\tau$. Once arrived, new packets will not be sent out immediately in general and will enter a queue, which is modelled as an $M/M/k$ queue with unlimited capacity, to wait for their scheduling. In the $M/M/k$ queue, packets will be scheduled according to the first-come, first-served (FCFS) basis. The unlimited queue capacity indicates that the age of information [@kaul2011minimizing; @kaul2012real] of new arrivals will not be considered, and packets will not be dropped before sending out. Besides, owing to the RA behavior of a slotted-ALOHA protocol, new arrivals during $t$ will only be counted at minislot $t + 1$. Thus, the accumulated number of packets 