An analysis similar to the van-Driest damping factor model for Newtonian fluids results in a viscoelastic damping factor for turbulent pipe flow of dilute polymer solutions. The turbulent relaxation time λ_t which must be used in the viscoelastic damping factor model is defined and correlated empirically. The present analysis is tested with the measured velocity distribution in the unsaturated drag reduction region, and friction factor in the unsaturated and saturated drag reduction regions. A comparison of the present model with two other well-known models, the elastic sublayer model of Virk et al. and the thickened laminar sublayer model of Seyer and Metzner, is made at the maximum drag reduction asymptote. From these comparisons it is concluded that the present model is valid to predict the friction factor and the velocity distribution in both the unsaturated and the saturated drag reduction regions.