Parisian ruin in the dual model with applications to the G/M/1 queue

The dual risk model describes the capital of a company with fixed expense rate and occasional income inflows of random size, called innovations. Parisian ruin occurs once the process stays continuously below zero for a given period. We consider the dual risk model where ruin is declared either at the first time that the reserve stays continuously below zero for an exponentially distributed time, or once it reaches a given negative threshold. We obtain the Laplace transform of the time to ruin and the Laplace transform of the time period that the process is negative. Applying a duality relationship between our risk model and the queueing model, we derive quantities related to the G/M/1 busy period, idle period and cycle maximum.