The particular behavior you mention can be modeled by a Semi-Markov Model. In these, the time you stay at a given state can be an arbitrary fixed distribution.

The particular behavior you mention can be modeled by a Semi-Markov Model. In these, the time you stay at a given state can be an arbitrary fixed distribution. In a regular chain this time is geometric, so memoryless. But other distributions, with Gaussian like tails or power law tails would produce the effect you want.