Randomised rumour spreading scheme is an extremely simple and powerful model to represent a propagation of information within members of a network. We studied a particular variation of the rumour spreading scheme where the informed individuals are periodically absent from the spreading activity with a certain rate called the switching rate. We quantified the running time of the scheme and showed that a switching rate threshold function exists in the sense that the expected delay time is negligible if and only if the switching rate is significantly smaller than the threshold function.