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Stochastic Network Calculus with Localized Application of Martingales

Anne Bouillard

TL;DR

The paper addresses the challenge of loose stochastic SNC bounds caused by union bounds by introducing a localized martingale approach to tandem networks. It shows how to apply the martingale analysis at a single server while leveraging Pay-Multiplexing-Only-Once-based end-to-end bounding for the rest, enabling tighter backlog and delay guarantees. The main theoretical contribution is a Martingale-based bound for tandem networks that can be combined with existing MGF-based SNC bounds, along with a transformation that removes a server to separate the martingale and PMOO components. Numerical experiments on small networks demonstrate substantial improvements over PMOO, validating the practical impact for network design and dimensioning, while also highlighting limitations for larger topologies and suggesting avenues for extending the method. The work also provides critical discussion of previous martingale approaches and offers a rigorous set of proofs in the accompanying sections.

Abstract

Stochastic Network Calculus is a probabilistic method to compute performance bounds in networks, such as end-to-end delays. It relies on the analysis of stochastic processes using formalism of (Deterministic) Network Calculus. However, unlike the deterministic theory, the computed bounds are usually very loose compared to the simulation. This is mainly due to the intensive use of the Boole's inequality. On the other hand, analyses based on martingales can achieve tight bounds, but until now, they have not been applied to sequences of servers. In this paper, we improve the accuracy of Stochastic Network Calculus by combining this martingale analysis with a recent Stochastic Network Calculus results based on the Pay-Multiplexing-Only-Once property, well-known from the Deterministic Network calculus. We exhibit a non-trivial class of networks that can benefit from this analysis and compare our bounds with simulation.

Stochastic Network Calculus with Localized Application of Martingales

TL;DR

The paper addresses the challenge of loose stochastic SNC bounds caused by union bounds by introducing a localized martingale approach to tandem networks. It shows how to apply the martingale analysis at a single server while leveraging Pay-Multiplexing-Only-Once-based end-to-end bounding for the rest, enabling tighter backlog and delay guarantees. The main theoretical contribution is a Martingale-based bound for tandem networks that can be combined with existing MGF-based SNC bounds, along with a transformation that removes a server to separate the martingale and PMOO components. Numerical experiments on small networks demonstrate substantial improvements over PMOO, validating the practical impact for network design and dimensioning, while also highlighting limitations for larger topologies and suggesting avenues for extending the method. The work also provides critical discussion of previous martingale approaches and offers a rigorous set of proofs in the accompanying sections.

Abstract

Stochastic Network Calculus is a probabilistic method to compute performance bounds in networks, such as end-to-end delays. It relies on the analysis of stochastic processes using formalism of (Deterministic) Network Calculus. However, unlike the deterministic theory, the computed bounds are usually very loose compared to the simulation. This is mainly due to the intensive use of the Boole's inequality. On the other hand, analyses based on martingales can achieve tight bounds, but until now, they have not been applied to sequences of servers. In this paper, we improve the accuracy of Stochastic Network Calculus by combining this martingale analysis with a recent Stochastic Network Calculus results based on the Pay-Multiplexing-Only-Once property, well-known from the Deterministic Network calculus. We exhibit a non-trivial class of networks that can benefit from this analysis and compare our bounds with simulation.
Paper Structure (50 sections, 9 theorems, 73 equations, 9 figures, 1 table)

This paper contains 50 sections, 9 theorems, 73 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Several existing SNC analyses.
  3. Contribution of the paper.
  4. Organization of the paper.
  5. Stochastic Network Calculus framework
  6. Network Calculus formalism
  7. Arrival processes.
  8. $S$-servers.
  9. Performance bounds.
  10. Markov modulated processes and SNC
  11. MGF-based SNC and $(\sigma(\theta), \rho(\theta))$-representations
  12. Markov modulated processes
  13. $(\sigma(\theta), \rho(\theta))$-characterization of a process
  14. An analytic combinatorics interpretation of the $(\sigma(\theta), \rho(\theta))$- representation
  15. Processes without $(\sigma(\theta), \rho(\theta))$ representation
  16. ...and 35 more sections

Key Result

Theorem 1

Let $A$ be a bivariate process crossing an $S$-dynamic server. Then

Figures (9)

  • Figure 1: Example of MMOO process.
  • Figure 2: Examples of network: (a) two-server tandem; (b) interleaved tandem.
  • Figure 3: Network obtained from the network in Figure \ref{['fig:tandem']}(b) after removal of server 2.
  • Figure 4: Illustration of the erroneous bound for $P_j$ in AK11.
  • Figure 5: Two-server tandem network with Bernoulli service process. (left) violation probability in function of the target delay; (right) delay bound in in function of the service probability of server 1.
  • ...and 4 more figures

Theorems & Definitions (14)

  • Theorem 1: Performance bounds Chang2000Fid06
  • Remark 1
  • Example 1: Markov-modulated On-Off process
  • Example 2: MMOO process (continued)
  • Theorem 2: BNS22a
  • Theorem 3: End-to-end dynamic server BNS21
  • Example 3
  • Theorem 4: End-to-end service bgf BNS22a
  • Example 4
  • Lemma 1
  • ...and 4 more