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Sliding Block Martingale based Multi-hop Delay QoS Analysis

Yuchao Dang, Xuefen Chi

TL;DR

A multi-hop delay QoS analysis framework based on the sliding block martingale, addressing the loose boundary issue of prior methods that rely on service process martingales and min-plus transformations and quantifying the upper bound of event occurrence frequency to constrain the solution space of $\theta$.

Abstract

With the growing density of wireless networks and demand for multi-hop transmissions, precise delay Quality of Service (QoS) analysis has become a critical challenge. This paper introduces a multi-hop delay QoS analysis framework based on the sliding block martingale, addressing the loose boundary issue of prior methods that rely on service process martingales and min-plus transformations. By constructing a sliding block martingale with a window, we capture both long-term trends and short-term fluctuations in the backlog, eliminating the reliance on the generalized incremental property. The framework redefines delay unreliability events using cascading attributes, deriving a more compact Delay Unreliability Probability Boundary (DUPB). To improve the efficiency of solving the key parameter $θ$, we propose a Micrometric Intervals based Supermartingale Upcrossing Estimate Theorem, quantifying the upper bound of event occurrence frequency to constrain the solution space of $θ$. Simulations based on the 3GPP UMa/UMi channel model validate the framework's effectiveness. Results show that in 2-7 hop scenarios, the maximum deviation between theoretical boundaries and Monte Carlo simulations is $4.116 \times 10^{-5}$, with a lower RMSE than existing methods. Iteration count and CPU time for solving $θ$ are reduced by $59\%-72\%$ and $60.6\%-70.5\%$, respectively, improving analysis efficiency. Furthermore, the derived minimum service rate for multi-hop queues offers a valuable reference for resource allocation. The framework demonstrates high accuracy, scalability, and practicality in complex multi-hop networks.

Sliding Block Martingale based Multi-hop Delay QoS Analysis

TL;DR

A multi-hop delay QoS analysis framework based on the sliding block martingale, addressing the loose boundary issue of prior methods that rely on service process martingales and min-plus transformations and quantifying the upper bound of event occurrence frequency to constrain the solution space of .

Abstract

With the growing density of wireless networks and demand for multi-hop transmissions, precise delay Quality of Service (QoS) analysis has become a critical challenge. This paper introduces a multi-hop delay QoS analysis framework based on the sliding block martingale, addressing the loose boundary issue of prior methods that rely on service process martingales and min-plus transformations. By constructing a sliding block martingale with a window, we capture both long-term trends and short-term fluctuations in the backlog, eliminating the reliance on the generalized incremental property. The framework redefines delay unreliability events using cascading attributes, deriving a more compact Delay Unreliability Probability Boundary (DUPB). To improve the efficiency of solving the key parameter , we propose a Micrometric Intervals based Supermartingale Upcrossing Estimate Theorem, quantifying the upper bound of event occurrence frequency to constrain the solution space of . Simulations based on the 3GPP UMa/UMi channel model validate the framework's effectiveness. Results show that in 2-7 hop scenarios, the maximum deviation between theoretical boundaries and Monte Carlo simulations is , with a lower RMSE than existing methods. Iteration count and CPU time for solving are reduced by and , respectively, improving analysis efficiency. Furthermore, the derived minimum service rate for multi-hop queues offers a valuable reference for resource allocation. The framework demonstrates high accuracy, scalability, and practicality in complex multi-hop networks.

Paper Structure

This paper contains 10 sections, 4 theorems, 36 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The Sliding Block Martingale based Multi-hop Delay QoS Analysis Framework
  3. Simulation and Numerical Analysis
  4. Conclusion
  5. Proof of correctness of constructing Martingales based on backlog
  6. Proof of correctness of constructing Sliding Block Martingale
  7. Proof of Theorem \ref{['theorem1']}
  8. Proof of the Lemma \ref{['lemma1']}
  9. Proof of Theorem \ref{['theorem2']}
  10. Proof of Corollary \ref{['corollary4']}

Key Result

Theorem 1

The ${hop}$-hop network with the backlogs $\left \{ Q_i\left(t\right)\mid i\in \left[1,{hop}\right] \right \}$ and the arrival $A_1\left(t\right)$ admit sliding block martingales ${M_{SB}}\left(Q_i,\theta,{W}^b\right)$ and ${M_{SB}}\left(A_1,\theta,{W}^b\right)$, respectively. Then, the multi-hop De

Figures (4)

  • Figure 1: Multi-hop Transmission
  • Figure 2: Multi-hop Transmission, Uma and Umi
  • Figure 3: Comparison of Delay unreliability probability bound compactness for Theorem \ref{['theorem1']}, rr24, rr7, rr6. In multi-hop scenarios of 2 to 4 hops.
  • Figure 4: Comparison of Delay unreliability probability bound compactness for Theorem \ref{['theorem1']}, rr24, rr7, rr6. In multi-hop scenarios of 5 to 7 hops.

Theorems & Definitions (6)

  • Definition 1: Backlog-Martingale
  • Definition 2: Sliding Block Martingale
  • Theorem 1
  • Lemma 2
  • Theorem 3
  • Corollary 1