We consider a scenario where multimedia data is sent o v er a network o ering a guaranteed service such a s A TM VBR 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 bu er. We consider the problem of whether there exists a smoothing which minimizes the playback delay and the receive bu er

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... ize over all possible strategies, given that we know a service curve property for the ow 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 . We also give a constructive expression for the deconvolution operator, using a time inversion transform, introduced in the paper. We illustrate on some examples the di erence with optimal shaping, a smoothing strategy which aims at minimizing bu er and delay on the sender side but does not allow pre-fetching. We apply the theory to the determination of the minimum T-SPEC required to support a given ow with admissible playback delay or decoding bu er size constraints. Smoother decoder B(t) R(t) R'(t) R*(t) R(t) guaranteed service network decoding buffer encoder Figure 1 : The prefetching scenario. The scenario is illustrated on Figure 1 . A multimedia stream is encoded, and then input into a smoothing device. The smoothing device writes the stream into a networking device for transmission; its output R 0 is constrained by a speci ed arrival curve . For example, a ow conforming to the IETF speci cation for integrated service 5 , with maximum packet size M, peak rate P, sustainable rate r and burst tolerance b, has an arrival curve de ned by t = minM + P t ; b + rt for t 0 and t = 0 if t 0. A similar de nition holds for ATM variable bit rate services. We call Rt the cumulative numberofbits observed on the encoded ow, starting from an arbitrary point in time, and R 0 t the output of the smoother. The smoothing constraint is expressed as