Care services within the patient rooms are the most critical and time consuming processes in patient care deliveries in emergency department, clinics, and other healthcare facilities. In this paper, we introduce a Markov chain model to study such processes. A closed, parallel, and reentrant network with limited resources is used to model the process. Formulas to evaluate the patient length of stay and staff utilizations are developed. System-theoretic properties are discussed. The extension to non-Markovian scenarios is also investigated. Such a model provides a quantitative tool for healthcare professionals to study and improve patient flow in care deliveries.