In this paper, we present an analytical tool for understanding the performance of structured overlay networks under churn based on the master-equation approach of physics. We motivate and derive an equation for the average number of hops taken by lookups during churn, for the Chord network. We analyse this equation in detail to understand the behaviour with and without churn. We then use this understanding to predict how lookups will scale for varying peer population as well as varying the sizes of the routing tables. We also consider a change in the maintenance algorithm of the overlay, from periodic stabilisation to a reactive one which corrects fingers only when a change is detected. We generalise our earlier analysis to understand how the reactive strategy compares with the periodic one.