Traditional wireless sensor networks (WSNs) working in the license-free spectrum suffer from uncontrolled interference as the license-free spectrum becomes increasingly crowded. Designing a WSN based on cognitive radios can be promising in the near future as the quality of service requirement for data transmissions increases. In this paper we design and analyze performance of a cognitive radio sensor network (CRSN), which opportunistically accesses spectrum of licensed spectrum unused by other networks and supports both real-time constant-bit-rate (CBR) traffic and best effort (BE) traffic. We consider two different policies for prioritizing the resource allocations, develop analytical models to find delay and capacity performance for the CBR traffic and amount of resources for best effort (BE) data transmissions. The analysis is verified by computer simulations. Our results indicate that satisfactory real-time performance can be achieved in the CRSN. Depending on the service policy used, the amount of resources for serving the BE traffic is different.