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

Transition behavior of the waiting time distribution in a jumping model with the internal state