Most microorganisms regulate their cell size through multiple rounds of growth and division, but how they perform such processes remains highly elusive. Current understanding suggests that cell division its mostly dependent on size. However, recent experiments suggest that cells experiment in a silence period in which division machinery is turned off. In this paper, we revisit a proposed mathematical framework from the literature and we extended to include such a silence period. We present results on the stability of the size statistical moments of such proposed systems.