In this paper, we consider a risk model in which two types of individual
claims, main claims and by-claims, are defined. Every by-claim is induced by the main
claim randomly and may be delayed for one time period with a certain probability.
The dividend policy that certain amount of dividends will be paid as long as the
surplus is greater than a constant dividend barrier is also introduced into this delayed
claims risk model. By means of the probability generating functions, formulae for the
expected present value of total dividend payments prior to ruin are obtained for
discrete-type individual claims. Explicit expressions for the corresponding results are
derived for $K_n$ claim amount distributions. Numerical illustrations are also given.
