  This paper proposes a general framework of multi-armed bandit (MAB) processes
by introducing a type of restrictions on the switches among arms evolving in
continuous time.
  The Gittins index process is constructed for any single arm subject to the
restrictions on switches and then the optimality of the corresponding Gittins
index rule is established. The Gittins indices defined in this paper are
consistent with the ones for MAB processes in continuous time, integer time,
semi-Markovian setting as well as general discrete time setting, so that the
new theory covers the classical models as special cases and also applies to
many other situations that have not yet been touched in the literature. While
the proof of the optimality of Gittins index policies benefits from ideas in
the existing theory of MAB processes in continuous time, new techniques are
introduced which drastically simplify the proof.
