Admission control and service rate speedup may be used during periods of congestion to minimize customer waiting in different service settings. In a healthcare setting, this can mean sending patients to alternative care facilities that may take more time and/or provide less ideal treatment. While waiting can be detrimental to patient outcomes, strategies used to control congestion can also be costly. In this work, we examine a multi-server queueing system that considers both admission control and speedup. We use dynamic programming to characterize properties of the optimal control and find that in some instances the optimal policy has a simple form of a threshold policy. Leveraging this insight, we examine a queueing system where speedup is used when the number of customers (patients) in the system exceeds some threshold and admission control is used when that number exceeds some (potentially different) threshold. Using a fluid model and a stochastic loss model, we develop a methodology to derive approximations for the probability that speedup will be applied, the probability that admission control will be applied and the expected queue length customers experience. We use the approximations as the basis for a greedy heuristic to derive a near optimal solution to the original stochastic optimization problem. We use simulation to demonstrate the quality of these approximations and find that they can be quite accurate and robust. This analysis can provide insight to managers deciding how to balance admission control and speedup in service settings: when and to what extent to use each.