In this paper, the uplink sum-rate of the multi-cell multi-user massive MIMO system is studied under correlated Rayleigh fading channels. First, by considering pilot contamination effect, the equivalent system model of the massive MIMO is given with MMSE channel estimation. Then, the lower bound of the sum-rate is derived, and its asymptotical performance is studied when the base station antenna number goes to infinity. The result is general and is very accurate under the physical channel models.