The time to ruin and the number of claims until ruin for phase-type claims

We consider a renewal risk model with phase-type claims, and obtain an explicit expression for the joint transform of the time to ruin and the number of claims until ruin, with a penalty function applied to the deficit at ruin. The approach is via the duality between a risk model with phase-type claims and a particular single server queueing model with phase-type customer interarrival times; see Frostig (2004). This result specializes to one for the probability generating function of the number of claims until ruin. We obtain explicit expressions for the distribution of the number of claims until ruin for exponentially distributed claims when the inter-claim times have an Erlang-n distribution.