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Future wireless networks are envisioned to serve massive Internet of things (mIoT) via some radio access technologies, where the random access channel (RACH) procedure should be exploited for IoT devices to access the networks. However, the theoretical analysis of the RACH procedure for massive IoT devices is challenging. To address this challenge, we first correlate the RACH request of an IoT device with the status of its maintained queue and analyze the evolution of the queue status. Based on the analysis result, we then derive the closed-form expression of the random access (RA) success probability, which is a significant indicator characterizing the RACH procedure of the device. Besides, considering the agreement on converging different services onto a shared infrastructure, we investigate the RAN slicing for mIoT and bursty ultra-reliable and low latency communications (URLLC) service multiplexing. Specifically, we formulate the RAN slicing problem as an optimization one to maximize the total RA success probabilities of all IoT devices and provide URLLC services for URLLC devices in an energy-efficient way. A slice resource optimization (SRO) algorithm exploiting relaxation and approximation with provable tightness and error bound is then proposed to mitigate the optimization problem. Simulation results demonstrate that the proposed SRO algorithm can effectively implement the service multiplexing of mIoT and bursty URLLC traffic.