This paper considers a continuous-time quasi birth-death (qbd) process, which informally can be seen as a birth-death process of which the parameters are modulated by an external continuous-time Markov chain. The aim is to numerically approximate the time-dependent distribution of the resulting bivariate Markov process in an accurate and efficient way. An approach based on the Erlangization principle is proposed and formally justified. Its performance is investigated and compared with two existing approaches: one based on numerical evaluation of the matrix exponential underlying the qbd process, and one based on the uniformization technique. It is shown that in many settings the approach based on Erlangization is faster than the other approaches, while still being highly accurate. In the last part of the paper, we demonstrate the use of the developed technique in the context of the evaluation of the likelihood pertaining to a time series, which can then be optimized over its parameters to obtain the maximum likelihood estimator. More specifically, through a series of examples with simulated and real-life data, we show how it can be deployed in model selection problems that involve the choice between a qbd and its non-modulated counterpart.

本文考虑了一个连续时间准生死（qbd）过程，非正式地可以将其视为参数由外部连续时间马尔可夫链调制的生死过程。目的是以准确和有效的方式在数值上近似所得二元马尔可夫过程的时间相关分布。提出并正式证明了一种基于 Erlangization 原则的方法。对其性能进行了研究，并与两种现有方法进行了比较：一种基于qbd基础矩阵指数的数值评估过程，一种基于均匀化技术。结果表明，在许多设置中，基于 Erlangization 的方法比其他方法更快，同时仍然非常准确。在论文的最后一部分，我们演示了在评估与时间序列有关的似然性的上下文中使用开发的技术，然后可以对其参数进行优化以获得最大似然估计量。更具体地说，通过一系列带有模拟和真实数据的示例，我们展示了如何将其部署在涉及在qbd与其非调制对应物之间进行选择的模型选择问题中。