Massive Internet of Things (IoT) is foreseen to introduce plethora of applications for a fully connected world. Heterogeneous traffic is envisaged, where packets generated at each IoT device should be differentiated and served according to their priority. This paper develops a novel priority-aware spatiotemporal mathematical model to characterize massive IoT networks with uplink prioritized multistream traffic (PMT). Particularly, stochastic geometry is utilized to account for the macroscopic network wide mutual interference between the coexisting IoT devices. Discrete time Markov chains (DTMCs) are employed to track the microscopic evolution of packets within each priority stream at each device. To alleviate the curse of dimensionality, we decompose the prioritized queueing model at each device to a single-queue system with server vacation. To this end, the IoT network with PMT is modeled as spatially interacting vacation queues. Interactions between queues, in terms of the packet departure probabilities, occur due to mutual interference. Service vacations occur to lower priority packets to address higher priority packets. Based on the proposed model, dedicated and shared channel access strategies for different priority classes are presented and compared. The results show that shared access provides better performance when considering the transmission success probability, queues overflow probability and latency.