Stochastic models for performance analysis, optimization and control of queues hinge on a multitude of alternatives for input point processes. In case of bursty traffic, one very popular model is the \textit{Markov Modulated Poisson Process} (MMPP), however it is not the only option. Here, we introduce an alternative that we call \textit{Markovian transition counting process} (MTCP). The latter is a point process counting the number of transitions of a finite continuous-time Markov chain. For a given MTCP one can establish an MMPP with the same first and second moments of counts. In this paper, we show the other direction by establishing a duality in terms of first and second moments of counts between MTCPs and a rich class of MMPPs which we refer to as slow MMPPs (modulation is slower than the events). Such a duality confirms the applicability of the MTCP as an alternative to the MMPP which is superior when it comes to moment matching and finding the important measures of the inter-event process. We illustrate the use of such equivalence in a simple queueing example, showing that the MTCP is a comparable and competitive model for performance analysis.
翻译:用于绩效分析、优化和控制队列的斯托孔模型取决于输入点过程的多种替代模式。 在爆炸性交通中,一个非常流行的模式是 \ textit{ Markov Modated Poisson Process} (MMPP),但这不是唯一的选项。在这里,我们引入了一个我们称之为 textit{ Markovian 过渡计数进程} (MTCP) 的替代模式。 后者是一个点数过程, 计算一个有限的连续时间 Markov 链的过渡次数。 对于一个特定的 MTPP, 一个特定的 MTPP 能够以相同的第一和第二时刻建立 MMPPP 。 在本文中, 我们通过在MTCP 和 一个丰富的 MMPPP 类 之间的点数计数的第一和第二时刻建立双轨, 来显示另一个方向。 我们称之为慢 MMPPP ( 调慢于事件) 。 这种双重性确认 MCP 作为MP 的替代品的替代程序的适用性, 当它到达时, 当它到达时, 和 找到事件间进程的重要措施时, 我们举例说明了这种等同的模型, 在简单的排队列中, 示范性分析中, 显示一种可比较性 。