To investigate the properties of quantum-dot semiconductor optical amplifiers (QD-SOA), we establish a three-energy-system model in the conduction band of the QD-SOA. Optimize and compare the rate equations with those in references. Temporal solutions and distribution of electronic states derived by numerical calculation certify the discrete nature of the energy levels of the quantum-dot. Using the fact that the occupation probabilities of electrons and holes are linear in the ground state, we solve the rate equations at steady state to obtain the gain characteristics of QD-SOA and its relation to the occupation probabilities of electrons of the ground state. The results show that the QD-SOA has larger saturated optical gain, higher differential gain and lower operation current. The QD-SOA has more merits than the bulk or quantum-well optical amplifiers, which provide the instructions on designing optical amplifiers.