We consider the transmission of variable bit rate video over a network offering a guaranteed service such as ATM VBR or the guaranteed service of the IETF. The guaranteed service requires that the flow accepted by the network has to be conforming with a traffic envelope; in return, it receives a service guarantee expressed by a network service curve. In order to satisfy the traffic envelope constraint, the output of the encoder is fed to a smoother, possibly with some look-ahead. The resulting stream is transported by the network; at the destination, the decoder waits for an initial playback delay and reads the stream from the receiver buffer. First, we consider the problem of whether there exists one optimal strategy at the smoother which minimizes the playback delay and the receiver buffer size, given the traffic envelope, the service curve and the fixed encoder output. We show that there does exist such an optimal smoothing, and give an explicit representation for it. Next, we show there exists other encoder outputs that require the same playback delay and receive buffer size when optimally smoothed with the same traffic envelope and service curve. We provide the theoretical bounds on the cummulative encoding rate such that the optimal smoothing solution meets both the channel and the client constraints. Experimental results exhibit significant improvements in terms of total average distortion compared to the smoothing of a fixed coder output, under equivalent traffic parameters and decoding constraints.