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Probabilistic Nets-within-Nets

Michael Köhler-Bußmeier

TL;DR

This paper studies Hornets extended with firing probabilities, a Nets-within-Nets formalism where the tokens are Petri nets again, and uses the model to analyse self-modifying systems quantitatively.

Abstract

In this paper we study Hornets extended with firing probabilities. Hornets are a Nets-within-Nets formalism, i.e., a Petri net formalism where the tokens are Petri nets again. Each of these net-tokens has its own firing rate, independent from the rates of other net-tokens. Hornets provide algebraic operations to modify net-tokens during the firing. For our stochastic extension these operators could also modify the net-token's firing rate. We use our model to analyse self-modifying systems quantitatively. Hornets are very well suited to model self-adaptive systems performing a MAPE-like loop (monitoring-analyse-plan-execute). Here, the system net describes the loop, and the net-tokens describe the adapted model elements.

Probabilistic Nets-within-Nets

TL;DR

This paper studies Hornets extended with firing probabilities, a Nets-within-Nets formalism where the tokens are Petri nets again, and uses the model to analyse self-modifying systems quantitatively.

Abstract

In this paper we study Hornets extended with firing probabilities. Hornets are a Nets-within-Nets formalism, i.e., a Petri net formalism where the tokens are Petri nets again. Each of these net-tokens has its own firing rate, independent from the rates of other net-tokens. Hornets provide algebraic operations to modify net-tokens during the firing. For our stochastic extension these operators could also modify the net-token's firing rate. We use our model to analyse self-modifying systems quantitatively. Hornets are very well suited to model self-adaptive systems performing a MAPE-like loop (monitoring-analyse-plan-execute). Here, the system net describes the loop, and the net-tokens describe the adapted model elements.

Paper Structure

This paper contains 21 sections, 1 theorem, 19 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Structure
  4. Algebraic Nets-within-Nets: Hornets
  5. Multisets and P/T Nets
  6. Net-Algebras
  7. Object Nets and Net-Algebras
  8. Finite Models
  9. Nested Markings and Synchronisation
  10. Nested Markings
  11. Projections
  12. Elementary Hornets
  13. Events and Firing Rule
  14. Events
  15. Firing Rule
  16. ...and 6 more sections

Key Result

proposition 1

For each $k \in K$ the cardinality of each net universe $\mathcal{U}_k$ is bound as follows: $|\mathcal{U}_k | \leq 2^{\left( 2^{4|P_k|} \right)}$.

Figures (5)

  • Figure 1: Nets within Nets: Nets as Tokens
  • Figure 2: Modification of the Net-Token's Structure.
  • Figure 3: A Stochastic eHornet
  • Figure 4: The eHornet: System-Net containing the Battle-of-Sexes Interaction (right) and the Structural Adaption Logic (left)
  • Figure 5: A Sample Run: The dynamics of the Probabilities of Options $a_{0,1}$

Theorems & Definitions (5)

  • proposition 1: Lemma 2.1 in Koehler13-fi-hornets
  • definition 1: Elementary Hornet, eHornet
  • definition 2: Firing Rule
  • definition 3
  • definition 4