Caching is undoubtedly one of the most popular solution that easily scales up with a world-wide deployment of resources. Records in Domain Name System (DNS) caches are kept for a pre-set duration (time-to-live or TTL) to avoid becoming outdated. Modern caches are those that set locally the TTL regardless of what authoritative servers say. In this paper, we introduce analytic models to study the modern DNS cache behavior based on renewal arguments. For tree cache networks, we derive the cache performance metrics, characterize at each cache the miss process and the aggregate request process. We address the problem of the optimal caching duration and find that constant TTL is the best only if if inter-request times have a concave CDF. We validate our theoretical findings using real DNS traces (single cache case) and via event-driven simulations (network case). Our models are very robust as the relative error between empirical and analytic values stays within 1% in the former case and less than 5% in the latter case.