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.