带相位型跳变的阈值依赖破产模型与排队模型的瞬态行为
Transient Behavior of Threshold-Dependent Ruin and Queueing Models with Phase-Type Jumps
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中文总结 AI 辅助
该研究针对带相位型跳变的阈值依赖Cramér–Lundberg风险模型与M/G/1型排队模型，利用谱正Lévy过程波动理论，刻画指数时刻前的破产概率与工作量拉普拉斯-斯蒂尔杰斯变换，给出相位型分布下辅助量的显式表示。
中文摘要 AI 辅助
我们研究两类阈值依赖随机模型的瞬态行为：Cramér–Lundberg风险模型与M/G/1型排队模型。两类模型的动态均由两个带漂移的复合泊松过程驱动，驱动过程取决于当前状态是否低于固定阈值。对于风险模型，我们刻画了指数分布时刻前的破产概率；对于排队模型，我们刻画了指数分布时刻下工作量的拉普拉斯-斯蒂尔杰斯变换。这些量通过谱正Lévy过程的波动理论结果刻画，风险模型的拉普拉斯变换针对时间取，排队模型的拉普拉斯变换则针对状态与时间取。我们考虑一般的索赔额与作业量分布，并在这些分布为相位型时给出相关辅助量的显式表示。
英文摘要
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.
