In this paper we present results for bivariate exponential distributions which are represented by phase type distributions. The paper extends results from previous publications [5, 14] on this topic by introducing new representations that require a smaller number of phases to reach some correlation coefficient and introduces different ways to describe correlation between exponentially distributed random variables. Furthermore, it is shown how Markovian Arrival Processes (MAPs) with exponential marginal distribution can be generated from the phase type representations of exponential distributions and how the results for exponential distributions can be applied to define correlated hyperexponential or Erlang distributions. As application examples we analyze two queueing models with correlated inter-arrival and service times.

中文翻译：

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用相型分布表示相关指数分布

在本文中，我们展示了由相位类型分布表示的双变量指数分布的结果。该论文通过引入新的表示法来扩展之前关于该主题的出版物 [5, 14] 的结果，这些表示法需要较少数量的相位来达到某个相关系数，并介绍了描述指数分布随机变量之间相关性的不同方法。此外，还展示了如何从指数分布的相位类型表示生成具有指数边际分布的马尔可夫到达过程 (MAP)，以及如何应用指数分布的结果来定义相关的超指数或 Erlang 分布。作为应用示例，我们分析了两个具有相关到达间隔和服务时间的排队模型。