  The multi-armed restless bandit framework allows to model a wide variety of
decision-making problems in areas as diverse as industrial engineering,
computer communication, operations research, financial engineering,
communication networks etc. 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. However, the computation of
Whittle's index itself is a difficult problem and hence researchers focused on
calculating Whittle's index numerically or with a problem dependent approach.
  In our main contribution we derive an analytical expression for Whittle's
index for any Markovian bandit with both finite and infinite transition rates.
We derive sufficient conditions for the optimal solution of the relaxed problem
to be of threshold type, and obtain conditions for the bandit to be indexable,
a property assuring the existence of Whittle's index. Our solution approach
provides a unifying expression for Whittle's index, which we highlight by
retrieving known indices from literature as particular cases. The applicability
of finite rates is illustrated with the machine repairmen problem, and that of
infinite rates by an example of communication networks where transmission rates
react instantaneously to packet losses.
