The Internet of Things (IoT) is large-scale by nature. This is not only manifested by the large number of connected devices, but also by the high volumes of traffic that must be accommodated. Cellular networks are indeed a natural candidate for the data tsunami the IoT is expected to generate in conjunction with legacy human-type traffic. However, the random access process for scheduling request represents a major bottleneck to support IoT via LTE cellular networks. Accordingly, this paper develops a mathematical framework to model and study the random access channel (RACH) scalability to accommodate IoT traffic. The developed model is based on stochastic geometry and discrete time Markov chains (DTMC) to account for different access strategies and possible sources of inter-cell and intra-cell interferences. To this end, the developed model is utilized to assess and compare three different access strategies, which incorporate a combination of transmission persistency, back-off, and power ramping. The analysis and the results showcased herewith clearly illustrate the vulnerability of the random access procedure as the IoT intensity grows. Finally, the paper offers insights into effective scenarios for each transmission strategy in terms of IoT intensity and RACH detection thresholds.