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The logistic queue model: theoretical properties and performance evaluation

Franco Coltraro, Marc Ruiz, Luis Velasco

TL;DR

The paper introduces the logistic queue model, a smooth fluid-flow queueing framework for telecom networks that yields q'(t) = X(t) - [ μ + e^{-α q(t)}(min{μ,X(t)}-μ) ]. It provides formal proofs of key properties (positivity, FIFO) and extends the base model to finite buffers, variable service, multiple servers, and priority/separation of flows. Through numerical validation against discrete-event simulation using realistic video traffic, the authors demonstrate near-equivalent accuracy with orders-of-magnitude faster run times, enabling near real-time KPI estimation in Digital Twins. A DT use case demonstrates practical KPI estimation (latency) under diverse traffic, underscoring the approach's scalability and applicability to B5G/6G network management. Overall, the work delivers a rigorous, extensible, and computationally efficient tool for DT-driven network performance analysis.

Abstract

The advent of digital twins (DT) for the control and management of communication networks requires accurate and fast methods to estimate key performance indicators (KPI) needed for autonomous decision-making. Among several alternatives, queuing theory can be applied to model a real network as a queue system that propagates entities representing network traffic. By using fluid flow queue simulation and numerical methods, a good trade-off between accuracy and execution time can be obtained. In this work, we present the formal derivation and mathematical properties of a continuous fluid flow queuing model called the logistic queue model. We give novel proofs showing that this queue model has all the theoretical properties one should expect such as positivity of the queue and first-in first-out (FIFO) property. Moreover, extensions are presented in order to model different characteristics of telecommunication networks, including finite buffer sizes and propagation of flows with different priorities. Numerical results are presented to validate the accuracy and improved performance of our approach in contrast to traditional discrete event simulation, using synthetic traffic generated with the characteristics of real captured network traffic. Finally, we evaluate a DT built using a queue system based on the logistic queue model and demonstrate its applicability to estimate KPIs of an emulated real network under different traffic conditions.

The logistic queue model: theoretical properties and performance evaluation

TL;DR

The paper introduces the logistic queue model, a smooth fluid-flow queueing framework for telecom networks that yields q'(t) = X(t) - [ μ + e^{-α q(t)}(min{μ,X(t)}-μ) ]. It provides formal proofs of key properties (positivity, FIFO) and extends the base model to finite buffers, variable service, multiple servers, and priority/separation of flows. Through numerical validation against discrete-event simulation using realistic video traffic, the authors demonstrate near-equivalent accuracy with orders-of-magnitude faster run times, enabling near real-time KPI estimation in Digital Twins. A DT use case demonstrates practical KPI estimation (latency) under diverse traffic, underscoring the approach's scalability and applicability to B5G/6G network management. Overall, the work delivers a rigorous, extensible, and computationally efficient tool for DT-driven network performance analysis.

Abstract

The advent of digital twins (DT) for the control and management of communication networks requires accurate and fast methods to estimate key performance indicators (KPI) needed for autonomous decision-making. Among several alternatives, queuing theory can be applied to model a real network as a queue system that propagates entities representing network traffic. By using fluid flow queue simulation and numerical methods, a good trade-off between accuracy and execution time can be obtained. In this work, we present the formal derivation and mathematical properties of a continuous fluid flow queuing model called the logistic queue model. We give novel proofs showing that this queue model has all the theoretical properties one should expect such as positivity of the queue and first-in first-out (FIFO) property. Moreover, extensions are presented in order to model different characteristics of telecommunication networks, including finite buffer sizes and propagation of flows with different priorities. Numerical results are presented to validate the accuracy and improved performance of our approach in contrast to traditional discrete event simulation, using synthetic traffic generated with the characteristics of real captured network traffic. Finally, we evaluate a DT built using a queue system based on the logistic queue model and demonstrate its applicability to estimate KPIs of an emulated real network under different traffic conditions.
Paper Structure (36 sections, 4 theorems, 52 equations, 14 figures, 2 tables)

This paper contains 36 sections, 4 theorems, 52 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Organization
  5. Notations
  6. Derivation of the logistic queue model
  7. Outflow.
  8. Mathematical properties of the logistic queue model
  9. Logistic queue model extensions
  10. Finite queue
  11. Variable service times
  12. Multiple servers
  13. Separation of flows
  14. Priority queues
  15. Evaluation methodology
  16. ...and 21 more sections

Key Result

Proposition 4.1

Given an initial condition $q(t_0)=q_0\geq 0$, a continuous and positive inflow function $X\geq0$ and a maximum outflow rate $\mu>0$, we have that there exists a unique continuous differentiable solution $q(t)$ to the system (modlog) defined in an interval $[t_0,T_{max})$ for $T_{max}\leq+\infty$. M

Figures (14)

  • Figure 1: Example of a system with a queue and a server of speed $\mu$.
  • Figure 2: Simulation of packets generated by a video user during two hours. Each blue line represents a packet.
  • Figure 3: Inflow to the system. Its standard deviation is 2.53 Mb/s.
  • Figure 4: Queue of the discrete simulator.
  • Figure 5: Queue of the logistic model.
  • ...and 9 more figures

Theorems & Definitions (9)

  • Definition 3.2: Logistic queue model
  • Proposition 4.1: Existence, Uniqueness and Positivity
  • proof
  • Proposition 4.3: Asymptotical behaviour
  • proof
  • Proposition 4.6: FIFO property
  • proof
  • Theorem 5.1
  • proof