Calculations are presented of the transit times of guided and tunnelling rays in graded-index optical fibres. It is shown that 25% of the power launched by an incoherent source into a parabolic-index fibre is carried by leaky rays, which may increase the r.m.s. width of the impulse response by up to 41%. The profile which most effectively equalises the combined transit times of both guided and tunnelling rays is given by α=2−4δ