Quality of service (QoS) is a crucial requirement in distributed applications. Internet of Things architectures have become a widely used approach in many application domains, from Industry 4.0 to smart agriculture; thus, it is crucial to develop appropriate methodologies for managing QoS in such contexts. In an overcrowded spectrum scenario, cognitive radio technology could be an effective methodology for improving QoS requirements. In order to evaluate QoS in the context of a cognitive radio Internet of Things network, we propose a Petri net-based model that evaluates the cognitive radio environment and operates in a 200 kHz GSM/EDGE transponder band. The model is quite flexible as it considers several circuit and packet switching primary user network loads and configurations and several secondary user types of services (that involve semantic transparency or time transparency); furthermore, it is able to take into account mistakes of the spectrum sensing algorithm used by secondary users. Specifically, we derive the distribution of the response time perceived by the secondary users, where it is then possible to obtain an estimation of both the maximum throughput and jitter. The proposed cognitive radio scenario considers a secondary user synchronized access to the channel when using the GSM/EDGE frame structure.