The worldwide outbreak of the coronavirus was first identified in 2019 in Wuhan, China. Since then, the disease has spread worldwide. As it is cur- rently 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 health- care 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 by flattening the curve.In this thesis, we use queueing theoretic methods to analyze the time evo- lution 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 time 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. The single server queue is one of the most basic queueing systems to model stochastic waiting dynamics. Most work involving the single server queue only analyzes the customer or agent behavior. However, in this work, we are in- spired by COVID-19 applications and are interested in the interaction between customers and more specifically, the time that adjacent customers overlap in the queue. To this end, we derive a new recursion for this overlap time and study the steady state behavior of the overlap time via simulation and proba- bilistic analysis. We find that the overlap time between adjacent customers in the M/M/1 queue has a conditional distribution that is given by an exponential distribution, however, as the distance between customers grows, the probabil- ity of a non-negative overlap time decreases geometrically. We also find via simulation that the exponential distribution still holds when the distribution is non-exponential, hinting at a more general result. Additionally, in this thesis we attempt to learn the Lindley’s recursion, one of the most important formula’s in queueing theory and applied probability. In this thesis, we leverage stochastic simulation and current machine learn- ing methods to learn the Lindley recursion directly from waiting time data of the G/G/1 queue. To this end, we use methods such as Gaussian Processes, k-Nearest Neighbors and Deep neural networks to learn the Lindley recur- sion. We also analyze specific parameter regimes for the M/M/1 to understand where learning the Lindley recursion may be easy or hard. Finally, we com- pare the machine learning methods to see how well we can predict the Lindley recursion multiple steps into the future with missing data. Moreover, we also attempt to learn the tandem version of Lindley’s recur- sion directly from data. We combine stochastic simulation with current machine learning methods such as Gaussian Processes, K-Nearest Neighbors, Linear Re- gression, Deep Neural Networks, and Gradient Boosted Trees to learn the tan- dem network Lindley recursion. We uncover specific parameter regimes where learning the tandem network Lindley recursion is easy or hard.