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Network-Calculus Service Curves of the Interleaved Regulator

Ludovic Thomas, Jean-Yves Le Boudec

TL;DR

The paper addresses whether network-calculus service curves can capture the Interleaved Regulator (IR) used in TSN. It shows that the IR can cause unbounded latencies when placed after a non-FIFO system and proves there is no context-agnostic service-curve model that explains the shaping-for-free property of the IR, though non-trivial, weak service-curve options exist. It establishes a strict service-curve for the aggregate and upper bounds for individual and long-run aggregate service curves, revealing a fundamental limit: for aggregates with at least four flows, the long-run service rate cannot exceed three times the per-flow contract rate. Collectively, these results delimit the applicability of service-curve-based analysis to the IR, guiding regulator modeling and end-to-end latency reasoning in TSN. The work highlights that while NC remains powerful for many TSN elements, the IR’s context-sensitive behavior resists a general, context-agnostic service-curve description, especially in non-FIFO contexts.

Abstract

The interleaved regulator (implemented by IEEE TSN Asynchronous Traffic Shaping) is used in time-sensitive networks for reshaping the flows with per-flow contracts. When applied to an aggregate of flows that come from a FIFO system, an interleaved regulator that reshapes the flows with their initial contracts does not increase the worst-case delay of the aggregate. This shaping-for-free property supports the computation of end-to-end latency bounds and the validation of the network's timing requirements. A common method to establish the properties of a network element is to obtain a network-calculus service-curve model. The existence of such a model for the interleaved regulator remains an open question. If a service-curve model were found for the interleaved regulator, then the analysis of this mechanism would no longer be limited to the situations where the shaping-for-free holds, which would widen its use in time-sensitive networks. In this paper, we investigate if network-calculus service curves can capture the behavior of the interleaved regulator. We find that an interleaved regulator placed outside of the shaping-for-free requirements (after a non-FIFO system) can yield unbounded latencies. Consequently, we prove that no network-calculus service curve exists to explain the interleaved regulator's behavior. It is still possible to find non-trivial service curves for the interleaved regulator. However, their long-term rate cannot be large enough to provide any guarantee (specifically, we prove that for the regulators that process at least four flows with the same contract, the long-term rate of any service curve is upper bounded by three times the rate of the per-flow contract).

Network-Calculus Service Curves of the Interleaved Regulator

TL;DR

The paper addresses whether network-calculus service curves can capture the Interleaved Regulator (IR) used in TSN. It shows that the IR can cause unbounded latencies when placed after a non-FIFO system and proves there is no context-agnostic service-curve model that explains the shaping-for-free property of the IR, though non-trivial, weak service-curve options exist. It establishes a strict service-curve for the aggregate and upper bounds for individual and long-run aggregate service curves, revealing a fundamental limit: for aggregates with at least four flows, the long-run service rate cannot exceed three times the per-flow contract rate. Collectively, these results delimit the applicability of service-curve-based analysis to the IR, guiding regulator modeling and end-to-end latency reasoning in TSN. The work highlights that while NC remains powerful for many TSN elements, the IR’s context-sensitive behavior resists a general, context-agnostic service-curve description, especially in non-FIFO contexts.

Abstract

The interleaved regulator (implemented by IEEE TSN Asynchronous Traffic Shaping) is used in time-sensitive networks for reshaping the flows with per-flow contracts. When applied to an aggregate of flows that come from a FIFO system, an interleaved regulator that reshapes the flows with their initial contracts does not increase the worst-case delay of the aggregate. This shaping-for-free property supports the computation of end-to-end latency bounds and the validation of the network's timing requirements. A common method to establish the properties of a network element is to obtain a network-calculus service-curve model. The existence of such a model for the interleaved regulator remains an open question. If a service-curve model were found for the interleaved regulator, then the analysis of this mechanism would no longer be limited to the situations where the shaping-for-free holds, which would widen its use in time-sensitive networks. In this paper, we investigate if network-calculus service curves can capture the behavior of the interleaved regulator. We find that an interleaved regulator placed outside of the shaping-for-free requirements (after a non-FIFO system) can yield unbounded latencies. Consequently, we prove that no network-calculus service curve exists to explain the interleaved regulator's behavior. It is still possible to find non-trivial service curves for the interleaved regulator. However, their long-term rate cannot be large enough to provide any guarantee (specifically, we prove that for the regulators that process at least four flows with the same contract, the long-term rate of any service curve is upper bounded by three times the rate of the per-flow contract).
Paper Structure (31 sections, 14 theorems, 63 equations, 13 figures, 4 tables)

This paper contains 31 sections, 14 theorems, 63 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Network-Calculus Service Curves
  4. The Packetizer and Fluid Service Curves
  5. Individual Service Curve for a Flow
  6. Traffic Regulators and their Shaping-For-Free Properties
  7. Related Work on the Modeling of Traffic Regulators
  8. System Model and Notations
  9. Limits of the Shaping-For-Free Property for the Interleaved Regulator
  10. Instability of the IR when Placed after a Non-FIFO System
  11. The Shaping-for-Free Property of the Interleaved Regulator Cannot be Explained by a Service Curve
  12. Service Curves of The Interleaved Regulator
  13. A Strict Service Curve for the Aggregate
  14. Upper-Bound on the Individual Service Curve
  15. Limits on the Aggregate Service Curve
  16. ...and 16 more sections

Key Result

Theorem 1

Consider a flow $f$ with input arrival curve $\alpha_f$ that crosses in sequence a causal system $S$ followed by a PFR (Figure fig:related-work:shaping-for-free:pfr). If the PFR is configured with $\sigma_f\ge \alpha_f$ and if $S$ is FIFO for $f$, then the worst-case delay $D_{f,S+\ac{PFR}}$ of $f$

Figures (13)

  • Figure 1: Notations of Definition \ref{['def:fluid-service-curve']}. $Z$ offers the fluid service curve$\beta$ if it can be modelled as the concatenation of $Z'$ followed by a packetizer, where $Z'$ offers the service curve $\beta$.
  • Figure 2: Two flavors of traffic regulators. With PFR, we need one PFR per flow. In contrast, the IR uses a single FIFO queue to shape several flows.
  • Figure 3: Shaping-for-free properties of the traffic regulators. For the PFR, the system $S$ only needs to be FIFO-per-flow. For the IR, $S$ must be FIFO for the aggregate.
  • Figure 4: Input [resp., output] observation point $\mathtt{B}$ [resp., $\mathtt{D}$] for a regulator.
  • Figure 5: The network $\mathcal{N}_1$ and the Spring-generated Trajectory 1 that yields unbounded latencies in the IR when $S_1$ is not assumed FIFO.
  • ...and 8 more figures

Theorems & Definitions (28)

  • Definition 1: Fluid service curve
  • Theorem 1: Shaping-for-free property of the PFR leboudecTheoryTrafficRegulators2018
  • Theorem 2: Shaping-for-free property of the IR leboudecTheoryTrafficRegulators2018
  • Theorem 3: Instability of the IR after a non-FIFO system
  • Definition 2: A curve explains the shaping-for-free property
  • Proposition 1
  • Theorem 4
  • Lemma 1
  • Theorem 5: IR strict service curve
  • Proposition 2: Upper Bound on the Strict Service Curve
  • ...and 18 more