  We consider a version of the continuous-time multi-armed bandit problem where
decision opportunities arrive at Poisson arrival times, and study its Gittins
index policy. When driven by spectrally one-sided L\'evy processes, the Gittins
index can be written explicitly in terms of the scale function, and is shown to
converge to that in the classical L\'evy bandit of Kaspi and Mandelbaum (1995).
