由伸展非局部时间算子所支配的分数M/M/1队列
A Fractional M/M/1 Queue Governed by Stretched Non-Local Time Operators
AI总结：
本文提出了一种由伸展非局部时间算子支配的分数M/M/1队列模型，通过引入记忆效应和非局部时间动态，扩展了经典队列理论，验证了其在捕捉长记忆尾动态方面的有效性。
摘要：
我们通过将扩展的非局部时间动态纳入柯尔莫哥洛夫前向方程中，引入了经典的M/M/1队列的非马尔可夫泛化。我们通过将标准时间导数替换为扩展的Caputo型算子来获得该模型。该模型保留了标准队列的出生-死亡转移结构，同时在时间演进中引入了记忆效应。我们通过拉普拉斯变换技术，推导出瞬态状态概率的显式表示，以Kilbas-Saigo函数形式出现，该函数自然地作为与伸展算子相关的松弛核。我们构建了一个时间变化的解释，并表明分数队列可以被视为一个经典M/M/1过程在非递减随机时间处的分布。观察到分数队列可以被视为一个经典M/M/1过程在非递减随机时间处的分布。我们证明在标准稳定性条件ρ<1下，稳态分布保持几何分布，并与经典队列的分布一致，同时证明伸展分数参数显著影响暂态 regime中的收敛速率。基于蒙特卡洛模拟的数值示例突显了参数(α,γ)对空状态分布、尾长分布以及平均尾演化的效应，并验证了所提出框架在捕捉长记忆尾动态方面的灵活性。
英文摘要：
We introduce a non-Markovian generalization of the classical M/M/1 queue by incorporating extended nonlocal time dynamics into Kolmogorov forward equations. We obtain the model by replacing the standard time derivative with an extended Caputo-type operator. It preserves the birth-death transition structure of the standard queue while introducing memory effects into the temporal evolution. We derive explicit representations for transient state probabilities in terms of the Kilbas-Saigo function, which naturally emerges as the relaxation kernel associated with the stretched operator, using Laplace transform techniques. We construct a time-varying interpretation and show that the fractional queue can be viewed as a distribution of a classical M/M/1 process evaluated at a non-decreasing random time. It is observed that the fractional queue can be viewed as a distribution of a classical M/M/1 process evaluated at a non-decreasing random time. We prove that under the standard stability condition $ρ<1$, the steady-state distribution remains geometric and coincides with the distribution of the classical queue, whilst we prove that the stretched fractional parameters significantly affect the convergence rate in the transient regime. Numerical examples based on Monte Carlo simulations highlight the effect of the parameters $(α,γ)$ on the distribution of empty states, tail length distributions, and the average tail evolution, and validate the flexibility of the proposed framework in capturing long-memory tail dynamics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
