Bufferless and single-buffer queueing systems have recently been shown to be effective in coping with escalated Age of Information (AoI) figures arising in single-source status update systems with large buffers and FCFS scheduling. In this paper, for the single-source scenario, we propose a numerical algorithm for obtaining the exact distributions of both the AoI and the peak AoI (PAoI) in (i) the bufferless PH/PH/1/1/P(p) queue with probabilistic preemption with preemption probability p, 0 ≤ p ≤ 1, and (ii) the single buffer M/P H/1/2/R(r) queue with probabilistic replacement of the packet in the queue by the new arrival with replacement probability r, 0 ≤ r ≤ 1. The proposed exact models are based on the well-established theory of Markov Fluid Queues (MFQ) and the numerical algorithms are matrix-analytical and they rely on numerically stable and efficient vector-matrix operations. Moreover, the obtained exact distributions are in matrix exponential form, making it amenable to calculate the tail probabilities and the associated moments straightforwardly. Firstly, we validate the accuracy of the proposed method with simulations, and for sume sub-cases, with existing closed-form results. We then comparatively study the AoI performance of the queueing systems of interest under varying traffic parameters.

中文翻译：

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查找 PH/PH/1/1 和 M/PH/1/2 类型队列的信息（峰值）年龄的精确分布


最近，无缓冲区和单缓冲区排队系统已被证明可以有效应对具有大缓冲区和 FCFS 调度的单源状态更新系统中出现的信息时代 (AoI) 数字的上升。在本文中，针对单源场景，我们提出了一种数值算法，用于获得 (i) 无缓冲 PH/PH/1/1/P(p) 中 AoI 和峰值 AoI (PAoI) 的精确分布具有抢占概率 p 的概率抢占队列，0 ≤ p ≤ 1，以及 (ii) 单缓冲区 M/PH/1/2/R(r) 队列，通过新到达的替换概率替换队列中的数据包概率 r，0 ≤ r ≤ 1。所提出的精确模型基于完善的马尔可夫流体队列 (MFQ) 理论，数值算法是矩阵分析的，并且依赖于数值稳定且高效的向量矩阵运算。此外，获得的精确分布是矩阵指数形式，使得可以直接计算尾部概率和相关矩。首先，我们通过模拟验证所提出方法的准确性，并针对某些子情况，使用现有的封闭形式结果。然后，我们比较研究了不同流量参数下感兴趣的排队系统的 AoI 性能。