Optimal smoothing for guaranteed service

We consider a scenario where multimedia data is sent over a network offering a guaranteed service such as ATM VER or the guaranteed service of the IETF. A smoothing device writes the stream into a networking device for transmission, possibly with some pre-fetching; at the destination, the decoder waits for an initial playback delay and reads the stream from the receive buffer. We consider the problem of whether there exists a smoothing which minimizes the playback delay and the receive buffer size over all possible strategies, given that we know a service curve property for the flow in the network. We show that there does exist such an optimal smoothing. It can be expressed using the deconvolution operator of min-plus algebra. We obtain the smallest playback delay which can be achieved by smoothing, provided that the information about the network is reduced to its service curve beta. We also give a constructive expression for the deconvolution operator, using a time inversion transform, introduced in the paper. We illustrate on an example the difference with optimal shaping, a smoothing strategy which aims at minimizing buffer and delay on the sender side but does not allow pre-fetching.