We develop a unifying framework to obtain efficient index policies for restless multi-armed bandit problems with birth-and-death state evolution. This is a broad class of stochastic resource allocation problems whose objective is to determine efficient policies to share resources among competing projects. In a seminal work, Whittle developed a methodology to derive well-performing Whittle’s index policies that are obtained by solving a relaxed version of the original problem. Our first main contribution is the derivation of a closed-form expression for Whittle’s index as a function of the steady-state probabilities. In some particular cases, qualitative insights can be obtained from its expression; nevertheless, it requires several technical conditions to be verified. We, therefore, formulate a fluid version of the relaxed optimization problem, and in our second main contribution, we develop a fluid index policy. The latter does provide qualitative insights and it is equivalent to Whittle’s index policy in the light-traffic regime. The applicability of our approach is illustrated by two important problems: optimal class selection and optimal load balancing. Allowing state-dependent capacities, we can model important phenomena, e.g., power-aware server-farms and opportunistic scheduling in wireless systems. Whittle’s index and our fluid index policy show remarkably good performance in numerical simulations.