The worldwide outbreak of the coronavirus was first identified in 2019 in Wuhan, China. Since then, the disease has spread worldwide. As it is currently spreading in the United States, policy makers, public health officials and citizens are racing to understand the impact of this virus on the United States healthcare system. They fear a rapid influx of patients overwhelming the healthcare system and leading to unnecessary fatalities. Most countries and states in America have introduced mitigation strategies, such as using social distancing to decrease the rate of newly infected people. This is what is usually meant byflattening the curve. In this paper, we use queueing theoretic methods to analyze the time evolution of the number of people hospitalized due to the coronavirus. Given that the rate of new infections varies over time as the pandemic evolves, we model the number of coronavirus patients as a dynamical system based on the theory of infinite server queues with time inhomogeneous Poisson arrival rates. With this model we are able to quantify how flattening the curve affects the peak demand for hospital resources. This allows us to characterize how aggressive societal policy needs to be to avoid overwhelming the capacity of healthcare system. We also demonstrate how curve flattening impacts the elapsed lag between the times of the peak rate of hospitalizations and the peak demand for the hospital resources. Finally, we present empirical evidence from Italy and the United States that supports the insights from our model analysis.