In this paper, we consider electric vehicle charging facilities with limited space and power resources. We assume the facility offers a finite selection of levels, i.e., charging rates, for varying prices. Users arrive at the facility randomly, requiring a random amount of charge and possessing a random impatience factor dictating their value of time. Each user then chooses a charging rate that minimizes their total cost that includes an opportunity cost for the time required to charge associated with their impatience factor. Knowing the probability distribution of user charging demands, user impatience factors, and the number of arrivals at a charging facility, we present high-confidence bounds on the total number of active users and aggregate power use of all active users at any given time. We present a case study to illustrate the results.