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.

翻译：本文通过将扩展的非局部时间动力学引入Kolmogorov前向方程，提出了经典M/M/1队列的一种非马尔可夫推广模型。我们通过用扩展的Caputo型算子替换标准时间导数来建立该模型。该模型保留了标准队列的生灭转移结构，同时在时间演化中引入了记忆效应。利用拉普拉斯变换技术，我们以Kilbas-Saigo函数的形式推导了瞬态状态概率的显式表示，该函数自然地作为与拉伸算子相关的松弛核出现。我们构建了一个时变解释，并证明分数阶队列可视为经典M/M/1过程在一个非递减随机时间上取值的分布。研究表明，在标准稳定性条件$ρ<1$下，稳态分布仍保持几何形式，并与经典队列的分布一致，同时我们证明了拉伸分数阶参数会显著影响瞬态区域的收敛速率。基于蒙特卡洛模拟的数值算例突出了参数$(α,γ)$对空态分布、尾长分布及平均尾部演化的影响，验证了所提框架在捕捉长记忆尾部动力学方面的灵活性。