In this thesis, we are concerned with the finite-time ruin probabilities in two alternative dependent risk models, the insurance risk model and the dual risk model, including the numerical evaluation of the explicit expressions for these quantities and the application of the probabilistic results obtained. We first investigate the numerical properties of the formulas for the finite-time ruin probability derived by Ignatov and Kaishev (2000, 2004) and Ignatov et al. (2001) for a generalized insurance risk model allowing dependence. Efficient numerical algorithms are proposed for computing the ruin probability with a prescribed accuracy in order to facilitate the following studies. We then propose a new definition of alarm time in the insurance risk model, which generalizes that of Das and Kratz (2012), expressed in terms of the joint distribution of the time to ruin and the deficit at ruin. The alarm time is devised to warn that the future ruin probability within a finite-time window has reached a pre-specified critical level and capital injection is required. Due to our definition, the implementation of the alarm time highly relies on the computation of the finite-time ruin probability, which utilizes the previous results on computing the ruin probability with a prescribed accuracy. The results of the ruin probability and the alarm time are then transferred nicely to a generalized dual risk model, whose name stems from its duality to the insurance risk model, through an enlightening link established between the two risk models. Finally, based on the two alternative risk models, we introduce a framework for analyzing the risk of systems failure based on estimating the failure probability, and illustrate how the probabilistic models and results obtained can be applied as risk analytic tools in various practical risk assessment situations, such as systems reliability, inventory management, flood control via dam management, infection disease spread and financial insolvency.