Fault-tolerant data transmission models based on the redundant residue number system are proposed in this paper; they can transmit data correctly between two ends unless the residue errors exceed the error-correction capability. The expressions for the probability of error are presented when the channel noise is additive Gaussian noise and each branch is M-ary orthogonal signaling modulation. The expressions of the probabilities of undetected and uncorrected error are also obtained when the redundant residue number system are single error-checking and single error-correcting model, respectively.