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Dawn of the Dead(line Misses): Impact of Job Dismiss on the Deadline Miss Rate

Jian-Jia Chen, Mario Günzel, Peter Bella, Georg von der Brüggen, Kuan-Hsun Chen

TL;DR

This work addresses the problem of quantifying the long-run deadline miss rate (DMR) for an arbitrary-deadline periodic soft real-time task on a uniprocessor, allowing a missed deadline to be processed up to a dismiss point $δ$. It develops a Markov-chain model of the task's execution under greedy processing with supply functions or supply bound functions, and proves convergence to a stationary distribution under ergodicity, enabling exact DMR computation from the chain's stationary mass on miss states. The authors provide constructive algorithms to build finite Markov chains for deterministic and bound supply scenarios, and show how to obtain the DMR from the stationary distribution, thus broadening the analysis beyond traditional CBS or non-preemptive abort-after-deadline models. The paper's novelty lies in enabling a task-specific dismiss point and offering scalable analysis frameworks for various scheduling policies, with potential extensions to probabilistic WCET and more complex workload models.

Abstract

Occasional deadline misses are acceptable for soft real-time systems. Quantifying probabilistic and deterministic characteristics of deadline misses is therefore essential to ensure that deadline misses indeed happen only occasionally. This is supported by recent research activities on probabilistic worst-case execution time, worst-case deadline failure probability, the maximum number of deadline misses, upper bounds on the deadline miss probability, and the deadline miss rate. This paper focuses on the deadline miss rate of a periodic soft real-time task in the long run. Our model assumes that this soft real-time task has an arbitrary relative deadline and that a job can still be executed after a deadline-miss until a dismiss point. This model generalizes the existing models that either dismiss a job immediately after its deadline miss or never dismiss a job. We provide mathematical notation on the convergence of the deadline miss rate in the long run and essential properties to calculate the deadline miss rate. Specifically, we use a Markov chain to model the execution behavior of a periodic soft real-time task. We present the required ergodicity property to ensure that the deadline miss rate in the long run is described by a stationary distribution.

Dawn of the Dead(line Misses): Impact of Job Dismiss on the Deadline Miss Rate

TL;DR

This work addresses the problem of quantifying the long-run deadline miss rate (DMR) for an arbitrary-deadline periodic soft real-time task on a uniprocessor, allowing a missed deadline to be processed up to a dismiss point . It develops a Markov-chain model of the task's execution under greedy processing with supply functions or supply bound functions, and proves convergence to a stationary distribution under ergodicity, enabling exact DMR computation from the chain's stationary mass on miss states. The authors provide constructive algorithms to build finite Markov chains for deterministic and bound supply scenarios, and show how to obtain the DMR from the stationary distribution, thus broadening the analysis beyond traditional CBS or non-preemptive abort-after-deadline models. The paper's novelty lies in enabling a task-specific dismiss point and offering scalable analysis frameworks for various scheduling policies, with potential extensions to probabilistic WCET and more complex workload models.

Abstract

Occasional deadline misses are acceptable for soft real-time systems. Quantifying probabilistic and deterministic characteristics of deadline misses is therefore essential to ensure that deadline misses indeed happen only occasionally. This is supported by recent research activities on probabilistic worst-case execution time, worst-case deadline failure probability, the maximum number of deadline misses, upper bounds on the deadline miss probability, and the deadline miss rate. This paper focuses on the deadline miss rate of a periodic soft real-time task in the long run. Our model assumes that this soft real-time task has an arbitrary relative deadline and that a job can still be executed after a deadline-miss until a dismiss point. This model generalizes the existing models that either dismiss a job immediately after its deadline miss or never dismiss a job. We provide mathematical notation on the convergence of the deadline miss rate in the long run and essential properties to calculate the deadline miss rate. Specifically, we use a Markov chain to model the execution behavior of a periodic soft real-time task. We present the required ergodicity property to ensure that the deadline miss rate in the long run is described by a stationary distribution.
Paper Structure (15 sections, 8 theorems, 25 equations, 10 figures, 1 algorithm)

This paper contains 15 sections, 8 theorems, 25 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model
  3. Supply Bound Functions
  4. Problem Definition
  5. Deadline Miss Rate of the First $N$ Jobs
  6. Deadline Miss Rate in the Long Run
  7. DMR in the Literature
  8. State Reduction and Markov Chains
  9. Convergence and Ergodicity
  10. DMR under Supply Functions
  11. Algorithm to Construct $X_{\bullet}$
  12. Analysis of the Constructed Markov Chain
  13. DMR under Supply Bound Functions
  14. Remarks
  15. Conclusion

Key Result

Theorem 10

Consider a Markov chain $X_{\bullet} = (X_n)_{n\in \Nbb}$ with transition matrix $P$. Let $f: S \to \Rbb$ be a bounded function and $\lambda$ any initial distribution. If $X_{\bullet}$ is irreducible and positive recurrent, then a unique invariant distribution $\pi$ exists, and

Figures (10)

  • Figure 1: DMR of the first three jobs in Example \ref{['example:DMR-N-variable']}.
  • Figure 2: Probability distribution of $\mathrm{DMR}_N$ of the first $N$ jobs. We observe that the distribution converges to a constant value.
  • Figure 3: Markov chain of Figure \ref{['fig:DMR_after_3']} for Example \ref{['example:DMR-N-variable']}.
  • Figure 4: Finite Markov chain of an infinite amount of jobs for Example \ref{['example:DMR-N-variable']}.
  • Figure 5: DMR of $\tau$ when $\Pbb(C_j = 2) = p$ and $\Pbb(C_j = 3) = 1-p$, $\forall j \in \Nbb$ in Example \ref{['example:missrate-ergodic-stationary-different-probability']}.
  • ...and 5 more figures

Theorems & Definitions (27)

  • Example 1: Supply Function
  • Example 2: More Complex Supply Function
  • Example 3: Supply Bound Functions
  • Example 4: More Complex Supply Bound Functions
  • Example 5
  • Example 6
  • Definition 7: Irreducible
  • Definition 8: Positive recurrent
  • Definition 9: Stationary distribution
  • Theorem 10: Ergodic Theorem. Reformulated from norris_1997
  • ...and 17 more