A generalisation of the waiting model GiG/l is considered: The first order in every busy period suffers a random delay with distribution function Cbefore its service commences. As according to [10] the stationary waiting time distribution W' results from the stationary waiting time W of the fundamental model G/G/1 by W' = W ∗ G, connections between C and G are studied. By making a special assumption the characteristic function corresponding to G is analytical or rrtioncl if and only if the characteristic function corresponaing to C is analytical or rational. I n addition to that G is a mixture of negtive-exponential random variables if and only if C is a mixture of this manner.