Abstract: In this paper, we apply Delphic-semigroups theory that is initiated by D. G. Kendall to delayed renewal sequences. First, we discuss some fundamental properties of delayed renewal sequences and prove that delayed renewal sequences constitute a semigroup. Secondly, we define delayed renewal sequence element and derive a necessary and sufficient condition for positive infinitely divisible delayed renewal sequence element. Finally, we prove that semigroup of positive delayed renewal sequence elements is a Delphic-semigroup.