The Internet of Things (IoT) application in agriculture provides rural poultry farmers with automated tools and decision support systems to increase productivity and profit. The integration of Edge, Fog, and Cloud Computing technologies in this scenario enables real-time data collection, processing, and analysis. This, in turn, empowers poultry farmers to make informed decisions, enhancing production efficiency and ensuring poultry quality. However, when implementing IoT devices, the challenge arises in dealing with a large volume of generated data. Delays in data transfer between the Edge, Fog, and Cloud layers can impact the system's real-time responsiveness. Slowness in the system's responsiveness can compromise performance, especially in poultry monitoring systems, which are time-sensitive, and an error could result in losses. This work proposes a Stochastic Petri Net (SPN) model to analyze the performance of an integrated smart poultry house with Edge, Fog, and Cloud Computing. The proposed model covers the analysis of metrics such as average response time, resource utilization, discard probability, and system throughput. Furthermore, the highly flexible model allows developers and system administrators to calibrate various parameters.