The RCSJ (resistor - capacitor shunted junction) model provides us with an analytical formula for the conversion efficiency for a resonant Josephson effect mixer. An external rf signal is applied to a long Josephson junction operating in the fluxon oscillating regime. The sine - Gordon equation is used to investigate the dynamics of the phase-locking of the fluxon oscillations and the external signal. We establish that it is possible to have conversion efficiency greater than unity.