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Transient Evaluation of Non-Markovian Models by Stochastic State Classes and Simulation

Gabriel Dengler, Laura Carnevali, Carlos E. Budde, Enrico Vicario

TL;DR

This work enumerates SSCs near the root of the state-space tree and relies on simulation to reach the target, affording transient evaluation of models for which the method of SSCs is not viable while reducing computational time and variance of the estimator of transient probabilities with respect to simulation.

Abstract

Non-Markovian models have great expressive power, at the cost of complex analysis of the stochastic process. The method of Stochastic State Classes (SSCs) derives closed-form analytical expressions for the joint Probability Density Functions (PDFs) of the active timers with marginal expolynomial PDF, though being hindered by the number of concurrent non-exponential timers and of discrete events between regenerations. Simulation is an alternative capable of handling the large class of PDFs samplable via inverse transform, which however suffers from rare events. We combine these approaches to analyze time-bounded transient properties of non-Markovian models. We enumerate SSCs near the root of the state-space tree and then rely on simulation to reach the target, affording transient evaluation of models for which the method of SSCs is not viable while reducing computational time and variance of the estimator of transient probabilities with respect to simulation. Promising results are observed in the estimation of rare event probabilities.

Transient Evaluation of Non-Markovian Models by Stochastic State Classes and Simulation

TL;DR

This work enumerates SSCs near the root of the state-space tree and relies on simulation to reach the target, affording transient evaluation of models for which the method of SSCs is not viable while reducing computational time and variance of the estimator of transient probabilities with respect to simulation.

Abstract

Non-Markovian models have great expressive power, at the cost of complex analysis of the stochastic process. The method of Stochastic State Classes (SSCs) derives closed-form analytical expressions for the joint Probability Density Functions (PDFs) of the active timers with marginal expolynomial PDF, though being hindered by the number of concurrent non-exponential timers and of discrete events between regenerations. Simulation is an alternative capable of handling the large class of PDFs samplable via inverse transform, which however suffers from rare events. We combine these approaches to analyze time-bounded transient properties of non-Markovian models. We enumerate SSCs near the root of the state-space tree and then rely on simulation to reach the target, affording transient evaluation of models for which the method of SSCs is not viable while reducing computational time and variance of the estimator of transient probabilities with respect to simulation. Promising results are observed in the estimation of rare event probabilities.

Paper Structure

This paper contains 23 sections, 1 theorem, 7 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motivation.
  3. Related Work.
  4. Contributions.
  5. Background
  6. Stochastic Time Petri Nets
  7. Transient Evaluation by the Method of Stochastic State Classes
  8. Monte Carlo Simulation and Importance Sampling
  9. Switch from Stochastic State Classes to Simulation
  10. Approach Overview
  11. Conditioning Stochastic State Classes on Fired Transition
  12. Sampling Methods
  13. Metropolis-Hastings Algorithm.
  14. Importance Sampling.
  15. Collecting Results and Obtaining Confidence Intervals
  16. ...and 8 more sections

Key Result

theorem thmcountertheorem

The $1 - \alpha$CI of the mean $\mu$ for the estimator in eq:mean_mixed_analysis can be calculated with where $z_{\alpha/2}$ corresponds to the $\alpha/2$-quantile for the standard normal distribution $\mathcal{N}(0, 1)$ and $\sigma^2_\text{mixed}$ is defined as:

Figures (6)

  • Figure 1: STPN modeling four parallel overlapping activities
  • Figure 2: Illustration of the approach with analytical expansion depth $d=1$
  • Figure 3: Transient probabilities for four overlapping activities with $95\%$CI
  • Figure 4: Repairable DFT\ref{['ftm:extendedDFT']}
  • Figure 5: Transient probabilities for repairable DFT example with $95\%$CI
  • ...and 1 more figures

Theorems & Definitions (4)

  • definition thmcounterdefinition: SSC
  • definition thmcounterdefinition: Succession relation
  • theorem thmcountertheorem: CI for composite analysis
  • proof