The burstiness of compressed video complicates the provisioning of network resources for emerging multimedia services. For stored video applications, the server can smooth the variable-bit-rate stream by prefetching frames into the client playback buffer in advanced of each burst. Drawing on a priori knowledge of the frame lengths and client buffer size, such bandwidth smoothing techniques can minimize the peak and variability of the rate requirements while avoiding underflow and overflow of the playback buffer. However, in an internetworking environment, a single service provider typically does not control the entire path from the stored-video server to the client buffer. To develop efficient techniques for transmitting variable-bit- rate video across a portion of the route, we investigate bandwidth smoothing across a tandem of nodes, which may or may not include the server and client sites. We show that it is possible to compute an optimal transmission schedule for the tandem system by solving a collection of independent single-link problems. To develop efficient techniques for minimizing the network bandwidth requirements, we characterize how the peak rate varies as a function of the buffer allocation and the playback delay. Simulation experiments illustrate the subtle interplay between buffer space, playback delay, and bandwidth requirements for a collection of full-length video traces. These analytic and empirical results suggest effective guidelines for provisioning network services for the transmission of compressed video.