We present a systematic theoretical investigation of the influence of p-doping on the gain characteristics of strained In0.35Ga0.65As/GaAs multiple-quantum-well (MQW) lasers, and compare the results with those obtained experimentally from devices with record 30 GHz modulation bandwidths. Experimentally, the combination of p-doping and strain has been found to lead to only a small increase in the differential gain, partial derivative g/dn but a large decrease in the non-linear gain coefficient, epsilon; this behaviour has been theoretically accounted for by a doping-induced decrease in the intraband relaxation time, tau(in). The theoretical investigations reveal that the assumption of a constant intraband relaxation time is not sufficient to describe the role of p-doping in the above devices, and highlight the importance of utilizing an appropriate lineshape function for the modelling of high speed laser modulation behaviour.