Operation regimes of a two section monolithic quantum dot (QD) mode-locked laser are studied experimentally with InGaAs lasers and theoretically, using a model that takes into account carrier exchange between QD ground state and two-dimensional reservoir of a QD-in-a-well structure. It is shown analytically and numerically that, when the absorber section is long enough, the laser exhibits bistability between laser off state and different mode-locking regimes. The Q-switching instability leading to slow modulation of the mode-locked pulse peak intensity is completely eliminated in this case. When, on the contrary, the absorber length is rather short, in addition to usual Q-switched mode-locking, pure Q-switching regimes are predicted theoretically and observed experimentally.