In this paper, we consider two dependent classes of insurance business with heavy-tailed claims. The dependence comes from the assumption that claim arrivals of the two classes are governed by a common renewal counting process. We study two types of ruin in the two-dimensional framework. For each type of ruin, we establish an asymptotic formula for the finite-time ruin probability. These formulae possess a certain uniformity feature in the time horizon. Copyright © 2010 John Wiley & Sons, Ltd.