Photon pumping in Laser

Let’s consider a ring laser where the laser must pass through the gain material before it is sent toward a partially reflective surface $ R=1-T $. The other mirrors are perfect reflectors with $ R_1=R_2=1 $. Furthermore the output irradiance of the setup is give as $ I_{out} = I_{sat}(gamma – gamma_{th})L$ where $I_{sat}$ is the saturation irradiance, $gamma_{th} $ is the threshold gain coefficient, and $gamma $ is the small-signal gain coefficient. Now I am trying to understand what needs to happen in order to have the scenario when every pump event results in an output photon. That is to say a theoretical one to one correspondence, every photon I put in I will get right back. (Neglect spontaneous emission). Does this happen when $gamma > gamma_{th}$ ? or when they are equal?