Transient Behavior of Threshold-Dependent Ruin and Queueing Models with Phase-Type Jumps

Abstract

We study the transient behavior of two threshold-dependent stochastic models: a Cramér--Lundberg risk model and an M/G/1-type queueing model. In both models, the dynamics are driven by two different compound Poisson processes with drift, with the governing process depending on whether the current state is below or above a fixed threshold. For the risk model, we characterize the probability of ruin before an exponentially distributed epoch, while for the queueing model, we characterize the Laplace--Stieltjes transform of the workload at an exponentially distributed epoch. These quantities are characterized using fluctuation-theoretic results for spectrally positive Lévy processes, with Laplace transforms taken with respect to time for the risk model and with respect to both state and time for the queueing model. We consider general claim-size and job-size distributions and provide explicit representations of the relevant auxiliary quantities when these distributions are of phase-type.