A significant challenge of IoT networks is to offer Quality of Service (QoS) and meet deadline requirements when packets from a massive number of IoT devices are forwarded to an IoT gateway. Many IoT devices tend to report their data to their wired or wireless network gateways at closely correlated instants of time, leading to congestion known as the MassiveAccess Problem (MAP), which increases the probability that the IoT data will not meet its required deadlines. Since IoT data loses much of its value if it arrives to destination beyond a required deadline, MAP has been extensively studied in the literature. Thus we first take a queueing theoretic view of the problem, and also use a Diffusion Approximation to gain insight into the IoT traffic statistics that affect MAP. Then we introduce the Quasi-Deterministic Transmission Policy (QDTP) which significantly alleviates MAP when the average traffic rate grows beyond a given level and substantially reduces the probability that IoT data deadlines are missed. The results are validated using real IoT data which has been placed in IP packets for transmission.