  It has been noticed that when the waiting time distribution exhibits a
transition from an intermediate time power law decay to a long-time exponential
decay in the continuous time random walk model, a transition from anomalous
diffusion to normal diffusion can be observed at the population level. However,
the mechanism behind the transition of waiting time distribution is rarely
studied. In this paper, we provide one possible mechanism to explain the origin
of such transition. A jump model terminated by a state-dependent Poisson clock
is studied by a formal asymptotic analysis for the time evolutionary equation
of its probability density function. The waiting time behavior under a more
relaxed setting can be rigorously characterized by probability tools. Both
approaches show the transition phenomenon of the waiting time $T$, which is
further verified numerically by particle simulations. Our results indicate that
a small drift and strong noise in the state equation and a stiff response in
the Poisson rate are crucial to the transitional phenomena.
