A Fractional M/M/1 Queue Governed by Stretched Non-Local Time Operators

TL;DR

This work extends the classical M/M/1 queue by inserting a stretched nonlocal time operator $\mathcal{D}_t^{(\alpha,\gamma)}$ to model memory in the forward equations, yielding a non-Markovian queue with preserved birth–death structure. The authors derive explicit transient state probabilities in terms of the Kilbas–Saigo function via Laplace transforms and provide a time-change interpretation through inverse subordinators. Under $\rho<1$, the stationary distribution remains geometric and identical to the classical queue, while the extended parameters $\alpha$ and $\\gamma$ modulate convergence rates in the transient regime. Numerical simulations illustrate how $(\alpha,\gamma)$ shape empty-state probabilities, queue-length tails, and mean evolution, demonstrating the framework's ability to capture long-memory tail dynamics in queueing systems.

Abstract

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.

Figures (3)

• Figure 1: Evolution of the empty-system probability $p_0(t)$ with respect to different combinations of $(\alpha,\gamma)$ for given arrival and service rates $\lambda$ and $\mu$. The dashed horizontal line depicts the stationary value $1-\rho$.
• Figure 2: Queue-length distribution $p_n(t^\ast)$ at fixed time $t^\ast=8$ for different $(\alpha,\gamma)$. Fractional dynamics yield a broader spread than the classical M/M/1 case. .
• Figure 3: The evolution of the expected queue length $m(t)$ for different fractional parameter pairs $(\alpha,\gamma)$. The dashed line indicates the steady-state mean $\rho/(1-\rho)$.

Theorems & Definitions (12)

• Definition 1: Fractional M/M/1 queue
• Lemma 1: Positivity and normalization
• proof
• Theorem 1: Time-change representation
• proof
• Theorem 2
• proof
• Corollary 1: Non-exponential relaxation
• Theorem 3: Stationary distribution under stretched fractional dynamics
• proof