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This paper analyses the Gerber-Shiu penalty function of a Markov modulated risk model with delayed by-claims and random incomes. It is assumed that each main claim will also generate a by-claim and the occurrence of the by-claim may be delayed depending on associated main claim amount. We derive the system of integral equations satisfied by the penalty function of the model. Further, assuming that the premium size is exponentially distributed, an explicit expression for the Laplace transform of the expected discounted penalty function is derived. For a two-state model with exponential claim sizes, we present the explicit formula for the probability of ruin. Finally we numerically illustrate the influence of the initial capital on the ruin probabilities of the risk model using a specific example. An example for the risk model without any external environment is also provided with numerical results.

Gerber Shiu Function of Markov Modulated Delayed By-Claim Type Risk Model with Random Incomes