The main goal of this paper's extensive analysis of a client-server fog computing network is to increase system availability. Fog computing, which extends cloud computing to the network's edge, necessitates a robust and reliable architecture to handle distributed computational tasks effectively. To achieve this, the paper introduces a sophisticated architecture comprising five distinct subsystems: A, B, C, D, and E. Each subsystem plays a critical role in ensuring the seamless operation of the network. Subsystem A represents the clients, the devices or applications that generate computational tasks. Subsystem B consists of fog nodes strategically placed closer to the clients to process data with minimal latency. Subsystem C includes the servers, which provide more substantial computational power and storage capacity. Subsystems D and E function as first- and second-level load balancers to distribute the workload efficiently across the network. The arrangement of these subsystems is meticulously designed to enhance the overall performance and availability of the network. The system can distribute and manage computational tasks more effectively by organizing clients, fog nodes, servers, and load balancers in a series-parallel configuration. This setup allows optimal resource utilization and ensures the network can handle varying loads without compromising availability. To model the availability dynamics of the network, the study employs differential-difference equations and a transition diagram. These mathematical tools help understand the system's long-run availability under different conditions. The analysis involves conducting numerical experiments thoroughly documented using tables and graphs. These visual aids effectively illustrate how various network parameters influence the optimization of system availability. The findings from these experiments underscore the vital role of load balancers and fog nodes configured in a series-parallel arrangement. This configuration not only facilitates optimal task distribution but also significantly boosts the overall availability of the system. The study concludes by emphasizing the effectiveness of this approach, highlighting it as a strategic method to enhance system availability in client-server fog computing networks. The results of this study provide valuable insights for researchers, system administrators, and network architects. By demonstrating the benefits of a series-parallel configuration of fog nodes and load balancers, the paper offers practical guidance for improving the performance and reliability of fog computing environments. These findings can help stakeholders design more resilient and efficient networks, ultimately advancing the field of fog computing.