Semi-open queuing networks (SOQNs) are widely applied to measure the performance of manufacturing, logistics, communications, restaurant, and health care systems. Many of these systems observe variability in the customer arrival rate. Therefore, solution methods, which are developed for SOQNs with time-homogeneous arrival rate, are insufficient to evaluate the performance of systems which observe time-varying arrivals. This paper presents an efficient solution approach for SOQNs with time-varying arrivals. We use a Markov-modulated Poisson Process to characterize variability in the arrival rate and develop a matrix-geometric method (MGM)-based approach to solve the network. The solution method is validated through extensive numerical experiments. Further, we develop a stochastic model of the landside operations at an automated container terminal with time-varying truck arrivals and evaluate using the MGM-based approach. Results show that commonly used time-homogeneous approximation of time-varying truck arrivals is inaccurate (error is more than 15% in expected waiting time and expected number of trucks waiting outside the terminal) for performance evaluation of the landside operations. The application results are insightful in resource planning, demand leveling, and regulating the number of trucks permitted inside the terminal.