We consider coding for sources that output the symbols according to Poisson process from the viewpoint of real-time transmission. In order to reduce the transmission delay we avoid using input buffers. However, the lack of buffer causes overflow error. The theoretical relation between the transmission rate and the error probability is clarified. It is shown that the optimal code that minimizes the probability of error differs from the code that minimizes the expected codeword length. We also investigate the case of block coding as one of the applications of buffers.