arXiv subjects

Daniël Rutgers

Publications and source records attributed to Daniël Rutgers.

3 recordsLinked to original sources

arXiv2026

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

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.

math.PR↗

arXiv2026

Exact analysis of transient behavior of finite-capacity MAP-driven queues

This paper studies the workload distribution of a finite-capacity queue driven by a spectrally one-sided Markov additive process (MAP). Our main result provides the Laplace-Stieltjes transform of the workload at an exponentially distributed time, thereby uniquely characterizing its transient distribution. The proposed approach combines several decompositions with established fluctuation-theoretic results for spectrally one-sided Lévy processes. For the special case of Markov-modulated compound Poisson input, we additionally derive results for the idle time and the cumulative amount of lost work. We conclude this paper with a series of numerical experiments.

math.PR↗

arXiv2025

Risk theory in a finite customer-pool setting

This paper investigates an insurance model with a finite number of major clients and a large number of small clients, where the dynamics of the latter group are modeled by a spectrally positive Lévy process. We begin by analyzing this general model, in which the inter-arrival times are exponentially distributed (though not identically), and derive the closed-form Laplace transform of the ruin probability. Next, we examine a simplified version of the model involving only the major clients, and explore the tail asymptotics of the ruin probability, focusing on the cases where the claim sizes follow phase-type or regularly varying distributions. Finally, we derive the distribution of the overshoot over an exponentially distributed initial reserve, expressed in terms of its Laplace-Stieltjes transform.

math.PR↗