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A probabilistic analysis on general probabilistic scheduling problems

Daiki Suruga

TL;DR

This paper develops a general probabilistic framework for analyzing scheduling problems under non-i.i.d. job distributions, with a focus on uniform-machines and makespan. It introduces a discarding-based performance measure COST(φn,Sn) and a corresponding achievable-rate notion Ē that characterizes asymptotic behavior across general distributions, including i.i.d., Markov, and mixtures. The authors prove that the optimal-rate equals Ē, and provide explicit forms for IID, mixture, and Markov cases, along with a precise second-order expansion for IID inputs. By connecting to the information-spectrum approach from Information Theory, the work offers both theoretical insight and practical implications for robust scheduling under realistic, correlated workloads.

Abstract

The scheduling problem is a key class of optimization problems and has various kinds of applications both in practical and theoretical scenarios. In the scheduling problem, probabilistic analysis is a basic tool for investigating performance of scheduling algorithms, and therefore has been carried out by plenty amount of prior works. However, probabilistic analysis has several potential problems. For example, current research interest in the scheduling problem is limited to i.i.d. scenarios, due to its simplicity for analysis. This paper provides a new framework for probabilistic analysis in the scheduling problem and aims to deal with such problems. As a consequence, we obtain several theorems including a theoretical limit of the scheduling problem which can be applied to \emph{general, non-i.i.d. probability distributions}. Several information theoretic techniques, such as \emph{information-spectrum method}, turned out to be useful to prove our results. Since the scheduling problem has relations to many other research fields, our framework hopefully yields other interesting applications in the future.

A probabilistic analysis on general probabilistic scheduling problems

TL;DR

This paper develops a general probabilistic framework for analyzing scheduling problems under non-i.i.d. job distributions, with a focus on uniform-machines and makespan. It introduces a discarding-based performance measure COST(φn,Sn) and a corresponding achievable-rate notion Ē that characterizes asymptotic behavior across general distributions, including i.i.d., Markov, and mixtures. The authors prove that the optimal-rate equals Ē, and provide explicit forms for IID, mixture, and Markov cases, along with a precise second-order expansion for IID inputs. By connecting to the information-spectrum approach from Information Theory, the work offers both theoretical insight and practical implications for robust scheduling under realistic, correlated workloads.

Abstract

The scheduling problem is a key class of optimization problems and has various kinds of applications both in practical and theoretical scenarios. In the scheduling problem, probabilistic analysis is a basic tool for investigating performance of scheduling algorithms, and therefore has been carried out by plenty amount of prior works. However, probabilistic analysis has several potential problems. For example, current research interest in the scheduling problem is limited to i.i.d. scenarios, due to its simplicity for analysis. This paper provides a new framework for probabilistic analysis in the scheduling problem and aims to deal with such problems. As a consequence, we obtain several theorems including a theoretical limit of the scheduling problem which can be applied to \emph{general, non-i.i.d. probability distributions}. Several information theoretic techniques, such as \emph{information-spectrum method}, turned out to be useful to prove our results. Since the scheduling problem has relations to many other research fields, our framework hopefully yields other interesting applications in the future.
Paper Structure (25 sections, 14 theorems, 62 equations)

This paper contains 25 sections, 14 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Scheduling problems
  4. Probabilistic analysis
  5. Our contributions
  6. First result
  7. Second result
  8. Third result
  9. Our framework and Information theory
  10. Preliminaries
  11. Lemmas
  12. The average case analysis
  13. The i.i.d. scenario:
  14. Markov distributions:
  15. Characterization of optimal rate
  16. ...and 10 more sections

Key Result

Theorem 1

For any $m$-uniform machines scheduling problem, holds where $\bar{E} := \inf\{\alpha \geq 0 \mid \lim_{n \to \infty} P_{\mathcal{J}^n}\{j^n \mid \frac{1}{n \cdot v_\mathrm{sum}} \sum_{i \leq n}T(j_i) > \alpha\} = 0\}$, $v_\mathrm{sum}$ is the sum of speeds of all machines and $T(j_i)$ denotes the processing time of a job $j_i$ at unit speed.

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 1
  • proof
  • Corollary 1
  • ...and 17 more