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In this paper we consider the Gerber-Shiu (G-S) discounted penalty function of a risk model with two classes of claims, random income and all system parameters controlled by independent Markov-Modulated (MM) environments. The model is motivated by the flexibility in modeling the two different claim arrival processes, for instance, fatal and nonfatal conditions in health insurance or from the four wheelers and two wheelers in vehicle insurance. Also the states of MM processes describe, for example, the epidemic types in health insurance or weather conditions in vehicle insurance. It is natural to consider that the controlling environment conditions are different for the discount rates, risk arrivals and income arrivals. We establish the system of integral equations satisfied by the G-S function, given the initial environment states for the risk model. Assuming that the random income size is exponentially distributed, explicit expressions for Laplace transform of the G-S function are derived. As an illustration, explicit results are obtained for the ruin probabilities when claim sizes are also exponentially distributed and some other numerical results are also presented.

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