We consider two bivariate models with two-way interactions in context of risk and queueing theory. The two entities interact with each other by providing assistance but otherwise evolve independently. We focus on certain random quantities underlying the joint survival probability and the joint stationary workload, and show that these admit stochastic decomposition. Each one can be seen as an independent sum of respective quantities for the two models with one-way interaction. Additionally, we discuss a rather general method of establishing decompositions from a given kernel equation by identifying two independent random variables from their difference, which may be useful for other models. Finally, we point out that the same decomposition is true for uncorrelated Brownian motion reflected to stay in an orthant, and it concerns the face measures appearing in the basic adjoint relationship.

中文翻译：

---

互助双变量风险和排队模型中的随机分解

我们在风险和排队理论的背景下考虑具有双向交互作用的两个双变量模型。这两个实体通过提供帮助相互交互，但在其他方面独立发展。我们关注联合生存概率和联合平稳工作量背后的某些随机量，并表明这些允许随机分解。每一个都可以看作是具有单向交互作用的两个模型各自数量的独立总和。此外，我们讨论了一种相当通用的方法，通过从它们的差异中识别两个独立的随机变量，从给定的核方程建立分解，这可能对其他模型有用。最后，我们指出，对于反射停留在正交中的不相关布朗运动，同样的分解是正确的，