Patients entering an emergency care center in a hospital usually visit medical processes in different orders depending on the urgency level and the medical treatments required. We formulate the patient flows among diverse processes in an emergency care center using the Jackson network, which is one of the queueing networks, in order to evaluate the system performances such as the expected queue length and the expected waiting time. We present a case study based on actual data collected from an emergency care center in a hospital, in order to prove the validity of applying the Jackson network model in practice. After assessing the current system performances, we provide operational strategies to reduce waiting at the bottleneck processes and evaluate the impact of those strategies on the entire system.