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

This paper introduces a non-Markovian generalization of the M/M/1 queue using extended nonlocal time operators, demonstrating that while the steady-state distribution remains geometric under standard stability conditions, the fractional parameters significantly alter transient convergence rates and long-memory tail dynamics.

Original authors: Mehmet Sıddık Çadırcı

Published 2026-02-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mehmet Sıddık Çadırcı

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a busy coffee shop with a single barista. This is the classic M/M/1 queue, a mathematical model used to understand lines, waiting times, and congestion. In the "classic" version of this story, time moves like a smooth, steady river. If a customer arrives, the chance of the next one arriving is the same every second, regardless of what happened five minutes ago. The system has no memory; it lives entirely in the present moment.

However, the real world isn't always a smooth river. Sometimes, time feels "sticky" or "stretched." Maybe the barista gets distracted, or customers arrive in unpredictable bursts that seem to linger. This is where the paper by Mehmet Sıddık Çadırcı comes in.

Here is an explanation of the paper's core ideas using simple analogies:

1. The New "Sticky" Clock

The authors propose a new version of the coffee shop queue. Instead of a smooth river, they imagine time is governed by a "stretched non-local time operator."

The Analogy: Think of the classic queue as a train moving on a perfect track at a constant speed. The new model is like that same train, but now it's moving through a thick, sticky substance like honey.
What it does: In this "honey," time doesn't pass uniformly. The system remembers the past. If a long line formed earlier, that "memory" affects how the line behaves right now. The paper replaces the standard math for "speed of change" (the derivative) with a more complex tool called the stretched fractional operator. This tool allows the model to capture "long memory" and "persistence."

2. The Magic Function: The Kilbas-Saigo

To solve the math for this sticky, memory-filled queue, the authors use a special mathematical tool called the Kilbas-Saigo function.

The Analogy: In the classic model, the math uses simple exponential curves (like a ball rolling down a hill and stopping). In this new model, the ball rolls down a hill that changes shape as it goes. The Kilbas-Saigo function is the specific shape of that changing hill.
Why it matters: This function is a "super-version" of older math tools (like the Mittag-Leffler function). It allows the model to describe a much wider variety of "relaxation" behaviors—how quickly the line settles down after a rush.

3. The "Ghost" Time Traveler

One of the most fascinating findings is how this new queue relates to the old one. The paper proves that the behavior of this complex, memory-filled queue is actually just a classic queue running on a random, distorted clock.

The Analogy: Imagine two identical coffee shops.
- Shop A (Classic): Time ticks normally.
- Shop B (Fractional): The barista is the same, the customers are the same, but the clock on the wall is broken. It speeds up and slows down randomly based on a specific rule.
- The Result: If you watch Shop B, it looks like the line is moving slower or behaving strangely, but if you could "rewind" Shop B's broken clock to match Shop A's normal time, the two lines would look exactly the same.
The Claim: The paper proves mathematically that the fractional queue is just a classical queue evaluated at a "non-decreasing random time."

4. The Big Surprise: The End Result Doesn't Change

You might think that if time is sticky and memory is heavy, the final state of the line would be totally different. The paper shows this is not the case.

The Analogy: Whether you walk to the store at a brisk pace or wade through deep mud, if you start at the same place and follow the same path, you will eventually arrive at the same destination.
The Claim: As long as the shop isn't overwhelmed (the arrival rate is lower than the service rate), the final, steady-state distribution of customers in the line is exactly the same as the classic model. The "sticky time" changes how fast the system gets there, but not where it ends up.

5. The Real Difference: The Journey, Not the Destination

While the final destination is the same, the journey is very different.

The Analogy: In the classic model, if the line gets long, it shrinks back to normal very quickly (exponentially fast). In the new "sticky" model, the line shrinks much more slowly. It has "long tails."
The Claim: The parameters $\alpha$ $α$ and $\gamma$ $γ$ (which control the "stickiness" and "stretching" of time) significantly affect the transient regime (the temporary period before things settle).
- Small $\alpha$ or large $\gamma$ : The system takes much longer to calm down. It holds onto the "memory" of a busy period for a long time.
- Large $\alpha$ : The system behaves more like the classic, fast-moving model.

Summary

The paper introduces a more flexible way to model waiting lines that accounts for "memory" and "stickiness" in time.

It changes the rules of time: It replaces standard time with a "stretched" version that remembers the past.
It uses a new math tool: The Kilbas-Saigo function to describe how the system relaxes.
It connects to the old: It shows this new system is just an old system running on a weird, random clock.
It keeps the same ending: The long-term average number of people in line stays the same as the classic model.
It slows the start: The main difference is that the system takes much longer to settle down after a rush, creating "heavy tails" where the line stays longer than expected before returning to normal.

The authors validated this with computer simulations (Monte Carlo), showing that by tweaking the "stickiness" parameters, they could model systems that relax much slower than traditional models allow.

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

Problem Statement
The classical M/M/1 queue is a Markovian system governed by linear Kolmogorov forward equations with ordinary time derivatives. While effective for systems with memoryless inter-event times, this framework fails to capture the non-local temporal effects, persistence, and anomalous relaxation often observed in complex real-world systems. Existing fractional generalizations typically rely on the standard Caputo derivative, which introduces memory but lacks the flexibility to model more complex temporal scaling behaviors. This paper addresses the gap in systematically integrating extended non-local time operators into the transient analysis of queueing models to better characterize long-memory tail dynamics.

Methodology
The authors propose a non-Markovian generalization of the M/M/1 queue by replacing the standard first-order time derivative in the Kolmogorov forward equations with an extended stretched fractional operator, denoted as $D^{(\alpha, \gamma)}_t$ . This operator is defined as $t^{-\gamma} {}^C D^\alpha_t$ , where ${}^C D^\alpha_t$ is the Caputo fractional derivative of order $0 < \alpha < 1$ , and $\gamma \ge 0$ is a stretching parameter satisfying $\alpha + \gamma \le 1$ .

The analysis proceeds through the following steps:

Model Formulation: The state probabilities $p_n(t)$ are governed by a system of fractional difference-differential equations where the time derivative is substituted by $D^{(\alpha, \gamma)}_t$ .
Analytical Solution: The authors employ Laplace transform techniques to solve the governing equations. The solution structure relies on the Kilbas–Saigo function ( $E_{a,m,l}$ ), a special function that generalizes the Mittag-Leffler family and serves as the eigenfunction for the stretched fractional operator.
Stochastic Representation: A time-change interpretation is constructed, demonstrating that the fractional queue process $\{N_{\alpha,\gamma}(t)\}$ is distributionally equivalent to a classical M/M/1 process evaluated at a non-decreasing random time $A_{\alpha,\gamma}(t)$ . This random time is the inverse of a subordinator associated with the stretched operator.
Numerical Validation: Monte Carlo simulations are conducted using a time-varying simulation scheme. The classical M/M/1 paths are generated via an event-driven algorithm and evaluated at random operational times derived from the stretched fractional dynamics to estimate transient performance metrics.

Key Contributions
The paper makes four primary contributions:

Extended Model Proposal: It introduces a fractional M/M/1 model incorporating a non-local time operator with an additional stretching parameter ( $\gamma$ ), allowing for a richer spectrum of decay behaviors than standard fractional models.
Analytical Derivation: It derives explicit transient state probabilities expressed in terms of the Kilbas–Saigo function, providing a closed-form representation for the system's evolution.
Limiting Regimes: It rigorously defines limiting cases that recover both the standard classical M/M/1 queue (as $\alpha \to 1, \gamma \to 0$ ) and the standard time-fractional M/M/1 queue (as $\gamma \to 0$ ), ensuring consistency with existing literature.
Stochastic Interpretation: It establishes a probabilistic link between the fractional queue and a time-changed classical process, clarifying the mechanism of memory introduction.

Results

Transient Behavior: The fractional parameters $(\alpha, \gamma)$ significantly alter the transient regime. Specifically, the system exhibits non-exponential relaxation and slower convergence to equilibrium compared to the classical case. The probability of the system being empty ( $p_0(t)$ ) and the mean queue length ( $m(t)$ ) converge to steady-state values more slowly as the memory effects (controlled by $\alpha$ and $\gamma$ ) increase.
Steady-State Distribution: Under the standard stability condition $\rho = \lambda/\mu < 1$ , the paper proves that the steady-state distribution remains geometric and coincides exactly with that of the classical M/M/1 queue ( $\pi_n = (1-\rho)\rho^n$ ). The stretched fractional parameters affect the rate of convergence but do not alter the invariant measure of the system.
Tail Dynamics: Numerical examples indicate that fractional dynamics result in a broader spread of the queue-length distribution at intermediate times, highlighting persistence effects caused by the non-local time operator.

Significance and Claims
The paper claims to provide a flexible and mathematically consistent framework for modeling queueing systems with memory and non-local temporal effects. By utilizing the Kilbas–Saigo function, the model bridges the gap between classical dynamics and complex fractional dynamics, offering a tool to capture long-memory tail dynamics that standard Markovian or simple fractional models cannot. The authors emphasize that while the stationary distribution is invariant under these time changes, the transient performance metrics are highly sensitive to the fractional parameters, making the model particularly relevant for systems where the speed of convergence and transient congestion patterns are critical. The work validates that the extended framework preserves the linear birth-death structure of the queue while introducing memory through the temporal evolution.