The spread of the Coronavirus (COVID-19) pandemic across countries all over the world urges governments to revolutionize the traditional medical hospitals/centers to provide sustainable and trustworthy medical services to patients under the pressure of the huge overload on the computing systems of wireless sensor networks (WSNs) for medical monitoring as well as treatment services of medical professionals. Uncertain malfunctions in any part of the medical computing infrastructure, from its power system in a remote area to the local computing systems at a smart hospital, can cause critical failures in medical monitoring services, which could lead to a fatal loss of human life in the worst case. Therefore, early design in the medical computing infrastructure's power and computing systems needs to carefully consider the dependability characteristics, including the reliability and availability of the WSNs in smart hospitals under an uncertain outage of any part of the energy resources or failures of computing servers, especially due to software aging. In that regard, we propose reliability and availability models adopting stochastic Petri net (SPN) to quantify the impact of energy resources and server rejuvenation on the dependability of medical sensor networks. Three different availability models (A, B, and C) are developed in accordance with various operational configurations of a smart hospital's computing infrastructure to assimilate the impact of energy resource redundancy and server rejuvenation techniques for high availability. Moreover, a comprehensive sensitivity analysis is performed to investigate the components that impose the greatest impact on the system availability. The analysis results indicate different impacts of the considered configurations on the WSN's operational availability in smart hospitals, particularly 99.40%, 99.53%, and 99.64% for the configurations A, B, and C, respectively. This result highlights the difference of 21 h of downtime per year when comparing the worst with the best case. This study can help leverage the early design of smart hospitals considering its wireless medical sensor networks' dependability in quality of service to cope with overloading medical services in world-wide virus pandemics.