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Extending Network Calculus To Deal With Partially Negative And Decreasing Service Curves

Anja Hamscher, Vlad-Cristian Constantin, Jens B. Schmitt

TL;DR

NC is extended to deal with systems only offering aggregate min-plus service curves to multiple flows, and the key to this extension is the exploitation of minimal arrival curves, i.e., lower bounds on the arrival process.

Abstract

Network Calculus (NC) is a versatile analytical methodology to efficiently compute performance bounds in networked systems. The arrival and service curve abstractions allow to model diverse and heterogeneous distributed systems. The operations to compute residual service curves and to concatenate sequences of systems enable an efficient and accurate calculation of per-flow timing guarantees. Yet, in some scenarios involving multiple concurrent flows at a system, the central notion of so-called min-plus service curves is too weak to still be able to compute a meaningful residual service curve. In these cases, one usually resorts to so-called strict service curves that enable the computation of per-flow bounds. However, strict service curves are restrictive: (1) there are service elements for which only min-plus service curves can be provided but not strict ones and (2) strict service curves generally have no concatenation property, i.e., a sequence of two strict systems does not yield a strict service curve. In this report, we extend NC to deal with systems only offering aggregate min-plus service curves to multiple flows. The key to this extension is the exploitation of minimal arrival curves, i.e., lower bounds on the arrival process. Technically speaking, we provide basic performance bounds (backlog and delay) for the case of negative service curves. We also discuss their accuracy and show them to be tight. In order to illustrate their usefulness we also present patterns of application of these new results for: (1) heterogeneous systems involving computation and communication resources and (2) finite buffers that are shared between multiple flows.

Extending Network Calculus To Deal With Partially Negative And Decreasing Service Curves

TL;DR

NC is extended to deal with systems only offering aggregate min-plus service curves to multiple flows, and the key to this extension is the exploitation of minimal arrival curves, i.e., lower bounds on the arrival process.

Abstract

Network Calculus (NC) is a versatile analytical methodology to efficiently compute performance bounds in networked systems. The arrival and service curve abstractions allow to model diverse and heterogeneous distributed systems. The operations to compute residual service curves and to concatenate sequences of systems enable an efficient and accurate calculation of per-flow timing guarantees. Yet, in some scenarios involving multiple concurrent flows at a system, the central notion of so-called min-plus service curves is too weak to still be able to compute a meaningful residual service curve. In these cases, one usually resorts to so-called strict service curves that enable the computation of per-flow bounds. However, strict service curves are restrictive: (1) there are service elements for which only min-plus service curves can be provided but not strict ones and (2) strict service curves generally have no concatenation property, i.e., a sequence of two strict systems does not yield a strict service curve. In this report, we extend NC to deal with systems only offering aggregate min-plus service curves to multiple flows. The key to this extension is the exploitation of minimal arrival curves, i.e., lower bounds on the arrival process. Technically speaking, we provide basic performance bounds (backlog and delay) for the case of negative service curves. We also discuss their accuracy and show them to be tight. In order to illustrate their usefulness we also present patterns of application of these new results for: (1) heterogeneous systems involving computation and communication resources and (2) finite buffers that are shared between multiple flows.
Paper Structure (11 sections, 11 theorems, 82 equations, 8 figures)

This paper contains 11 sections, 11 theorems, 82 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Some Mathematical Background
  4. Network Calculus Background
  5. Extension of NC Performance Bounds for Negative Service Curves
  6. Generalizing the Delay Bound
  7. Backlog Bound
  8. Patterns of Application
  9. Computation-Communication Systems
  10. Finite Shared Buffers
  11. Conclusion

Key Result

Proposition 6

Let $f,g,h \in \mathcal{F}:$

Figures (8)

  • Figure 1: Problem statement and solution.
  • Figure 2: Different cases of the generalized delay bound theorem.
  • Figure 3: Vertical deviation in case of a negative service curve.
  • Figure 4: Graphical illustration of different cases in Thm. \ref{['thm:backlog-bound-tightness']}.
  • Figure 5: General form of a C/C system with three flows crossing each C/C component.
  • ...and 3 more figures

Theorems & Definitions (31)

  • Definition 1
  • Definition 2: Pseudo-inverse
  • Definition 3: Shift Function
  • Definition 4: Operators bouillard2018deterministic
  • Remark 5: Isotonicity of $\otimes$le2001network
  • Proposition 6: Composition of $\oslash$ and $\otimes$
  • Definition 7: Backlog at Time $t$
  • Definition 8: Virtual Delay at Time $t$
  • Definition 9: Maximal and Minimal Arrival Curve
  • Definition 10: Service Curve (SC)
  • ...and 21 more