The nonlinearity of Mach-Zehnder modulator greatly deteriorates the performance of on-board microwave photonics system. A dual-tone modulation model is presented including optical source, Mach-Zehnder modulator and photodiode, and an exact analytical solution is derived with the method of Fourier expansion, Fourier transform and Gegenbauer addition theory. According to the analytical expression, it can precisely predict the nonlinearity characteristics of the on-board microwave photonics system under different modulation and optimize the system performance. Analytical results indicate that the third-order inter-modulation distortion ratio and the third-order inter-modulation intercept can be approximately calculated with the case when the second-order terms of modulator output signal are included, and the third-order inter-modulation distortion ratio increases and then decreases as the modulation index increases. When the modulation index is less than 1.4674, the third-order inter-modulation distortion ratio of single sideband modulation is at least 6 dB less than that of double sideband modulation and it is suitable for the application of on-board microwave photonics system.