Particle model for the reservoirs in the simple symmetric exclusion process

In this paper, we will study the long time behavior of the simple symmetric exclusion process in the ``channel'' $\Lambda_N=[1,N]\cap \mathbb{N}$ with reservoirs at the boundaries. These reservoirs are also systems of particles which can be exchanged with the particles in the channel. The size $M$ of each reservoir is much larger than the one of the channel, i.e. $M=N^{1+\alpha}$ for a fixed number $\alpha>0$. Based on the size of the channel and the holding time at each reservoir, we will investigate some types of rescaling time.