On Matrix-analytic Methods and on Collective Risk in Life Insurance

Jakob Riishede Möller

Abstract:: Two of the classical topics in applied probability, queues and insurance models, are investigated. The queueing models considered are matrix-analytic models and the problem we consider in insurance mathematics is the tail behavior of the ruin probability in life insurance models.
: Typical models and key results of matrix-analytic methods are presented. Some of these models describe arrival processes and service time distributions, respectively, and some models are generalizations of queue length processes/waiting time processes.
: One of the results of the thesis deals with an integral equation for the waiting time distribution in many-server queues. A complete solution and numerical examples are given.
: Some results describe tail asymptotics for distributions of the queue length in some matrix-analytic models. Models where the increments of the queue length have subexponential tails and light tails, respectively, are described.
: Finally, the ruin probability in some life insurance models is characterized by means of the adjustment coefficient, that is, the tail behavior of the ruin probability is investigated. In particular, the effect of the premium principle used is investigated. Numerical examples are given.
Key words:: queues, insurance mathematics, matrix-analytic methods, subexponential distribution, exponential change of measure, many server queues, life insurance, adjustment coefficient, premium principle.