Abstract We consider two different continuous-time Markov chain models recently studied in Göbel et al.[8], which were created to model the interactions between a small pool of miners, and a larger collection of miners, within the Bitcoin network. The first model we discuss represents the case where all miners behave honestly and follow the Bitcoin protocol, while the second model represents the case where the smaller pool of miners use the Selfish Mining strategy of Eyal and Sirer[3]. We give a new derivation of the stationary distribution of the process in the honest mining case and further build on the results of Göbel et al.[8] by showing that the normalizing constant can be expressed in closed-form. We also use similar techniques to derive expressions for the Laplace transforms of the transition functions. We then illustrate how these techniques yield similar expressions for the stationary distribution of the process when the smaller pool implements Selfish Mining: the Laplace transforms of the transition functions can be calculated as well. Lastly, we briefly explain how our methods can be extended to more general models of a similar type.

中文翻译：

---

Göbel 等人对一些马尔可夫比特币模型的进一步研究。

摘要 我们考虑了最近在 Göbel 等人 [8] 中研究的两种不同的连续时间马尔可夫链模型，它们被创建来模拟比特币网络中小矿工池和更大矿工集合之间的交互。我们讨论的第一个模型代表所有矿工行为诚实并遵循比特币协议的情况，而第二个模型代表较小矿工池使用 Eyal 和 Sirer [3] 的自私挖矿策略的情况。我们给出了诚实采矿案例中过程平稳分布的新推导，并进一步建立在 Göbel 等人的结果的基础上。 [8] 通过证明归一化常数可以用封闭形式表示。我们还使用类似的技术来推导出转换函数的拉普拉斯变换的表达式。然后，我们将说明当较小的矿池实施自私挖掘时，这些技术如何为过程的平稳分布产生类似的表达式：也可以计算转换函数的拉普拉斯变换。最后，我们简要解释了我们的方法如何扩展到类似类型的更一般的模型。