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In this paper, a theoretical model based on rate equations is used to investigate static and dynamic behaviors of InAs-InP (113)B quantum-dot (QD) lasers emitting at 1.55 mum. More particularly, it is shown that two modelling approaches are required to explain the origin of the double laser emission occurring in QD lasers grown on both, GaAs and InP substrates. Numerical results are compared to experimental ones by using either a cascade or a direct relaxation channel model. The comparison demonstrates that when a direct relaxation channel is taken into account, the numerical results match very well the experimental ones and lead to a qualitative understanding of InAs-InP (113)B QD lasers. Numerical calculations for the turn-on delay are also presented. A relaxation oscillation frequency as high as 10 GHz is predicted which is very promising for the realization of directly modulated QD lasers for high-speed transmissions.