It is known that the all-fiber self-organizing laser cavity can be formed through inserting an unpumped erbium-doped fiber into an erbium-doped fiber ring cavity. The steady state model of this self-organizing laser cavity is established through introducing the nonlinear coupling equations of the saturable absorption grating into the standby equations of the multimode ring laser, and the single-longitudinal-mode solution with flat side mode spectrum is obtained. The influences of the position of the output coupler, output ratio and pump power on the characteristics of the saturable absorption grating are numerically analyzed. The numerical results show that the grating in the prepositive output coupler structure has a higher longitudinal-mode side mode suppression ratio (LM-SMSR) than in the postpositive output coupler structure with the same pump level and output ratio; and in both structures, the LM-SMSRs of the gratings increase with the output ratio but decrease with the pump power. The optimal range of the output ratio for the prepositive coupler structure is 60%~80%. The numerical results are confirmed by the measured stability maps of the lasers with both structures.