  Continuous time random walks combining diffusive and ballistic regimes are
introduced to describe a class of L\'evy walks on lattices. By including
exponentially-distributed waiting times separating the successive jump events
of a walker, we are led to a description of such L\'evy walks in terms of
multistate processes whose time-evolution is shown to obey a set of coupled
delay differential equations. Using simple arguments, we obtain asymptotic
solutions to these equations and rederive the scaling laws for the mean squared
displacement of such processes. Our calculation includes the computation of all
relevant transport coefficients in terms of the parameters of the models.
