We present a one-dimensional model of early transients in pipeline restarts of waxy crude oils, which are modelled as compressible yield stress fluids. We show that restart transients are effectively controlled by three dimensionless numbers: a Reynolds number, Re, a compressibility number chi*, and the Bingham number B. The first two of these combine to define three different dimensionless timescales, for compressive pressure diffusion, for the propagation of acoustic waves and for viscous damping. The Bingham number governs whether or not the pipeline restarts. We illustrate each qualitatively different regime computationally, and make favourable comparisons against the fully two-dimensional model in [G. Vinay, A. Wachs, J.-F. Agassant, Numerical simulation of weakly compressible Bingham flows: the restart of pipeline flows of waxy crude oils, J. Non-Newtonian Fluid Mech. 136 (2006) 93-105]. Our model is able to explain certain counter-intuitive observations, such as the fact that a pipeline full of more compressible fluid may in certain circumstances restart earlier than the same pipeline filled with a less compressible fluid, see e.g. [M.G. Cawkwell, M.E. Charles, An improved model for start-up of pipelines containing gelled crude oil, J. Pipelines 7 (1987) 41-52]. (C) 2007 Elsevier B.V. All rights reserved.