Dealing with Network Address Translators (NATs) is a central problem in many peer-to-peer applications on the Internet today. However, most analytical models of overlay networks assume the underlying network to be a complete graph, an assumption that might hold in evaluation environments such as PlanetLab but turns out to be simplistic in practice. In this work we introduce an analytical network model where a fraction of the communication links are unavailable due to NATs. We investigate how the topology induced by the model affects the performance of ring based DHTs. We quantify two main performance issues induced by NATs namely large lookup inconsistencies and increased break-up probability, and suggest how theses issues can be addressed. The model is evaluated using discrete based simulation for a wide range of parameters.