The Online Pause and Resume Problem: Optimal Algorithms and An Application to Carbon-Aware Load Shifting

TL;DR

This work studies the online pause and resume problem (OPR), introducing two deterministic threshold-based algorithms, DTPR-min for the minimization setting and DTPR-max for the maximization setting, each achieving the best possible competitive ratio under switching costs. The key innovation is a double-threshold design: two state-dependent thresholds that hedge against worst-case price paths while accounting for the fixed switching cost $\beta$, with thresholds explicitly characterized by $\alpha$ and $\omega$ through balancing equations. The authors prove these ratios are tight via lower-bound constructions and recover the classic $k$-search results when $\beta=0$, showing the framework generalizes known online results. They further validate the approach empirically on real carbon-trace data for carbon-aware temporal workload shifting, demonstrating substantial improvements over carbon-agnostic and switching-cost-agnostic baselines. This work provides both theoretical optimality and practical guidance for energy-aware scheduling under uncertain prices and switching penalties.

Abstract

We introduce and study the online pause and resume problem. In this problem, a player attempts to find the $k$ lowest (alternatively, highest) prices in a sequence of fixed length $T$, which is revealed sequentially. At each time step, the player is presented with a price and decides whether to accept or reject it. The player incurs a switching cost whenever their decision changes in consecutive time steps, i.e., whenever they pause or resume purchasing. This online problem is motivated by the goal of carbon-aware load shifting, where a workload may be paused during periods of high carbon intensity and resumed during periods of low carbon intensity and incurs a cost when saving or restoring its state. It has strong connections to existing problems studied in the literature on online optimization, though it introduces unique technical challenges that prevent the direct application of existing algorithms. Extending prior work on threshold-based algorithms, we introduce double-threshold algorithms for both the minimization and maximization variants of this problem. We further show that the competitive ratios achieved by these algorithms are the best achievable by any deterministic online algorithm. Finally, we empirically validate our proposed algorithm through case studies on the application of carbon-aware load shifting using real carbon trace data and existing baseline algorithms.

Figures (46)

• Figure 1: DTPR-min thresholds $\ell_i$ and $u_i$ for $i \in~[1,k]$ plotted using example parameters ($k = 10$).
• Figure 2: DTPR-max thresholds $u_i$ and $\ell_i$ for $i \in~[1,k]$ plotted using example parameters ($k = 10$).
• Figure 3: DTPR-min: Plotting actual values of competitive ratio $\alpha$ for fixed $k \geq 1$, fixed $U > L$, and varying values for $L$ and $\beta$ (switching cost). Color represents the order of $\alpha$ for a given setting of $\theta$ and $\beta$.
• Figure 4: DTPR-max: Plotting actual values of competitive ratio $\omega$ for fixed $k \geq 1$, fixed $U > L$, and varying values for $L$ and $\beta$ (switching cost). Color represents the order of $\omega$ for a given setting of $\theta$ and $\beta$.
• Figure 5: Experiments for three distinct slack values, where $T \in \{48, 72, 96\}$. (a): Ontario, Canada carbon trace, with $\theta = 12.0\bar{6}$ (b): U.S. Pacific Northwest carbon trace, with $\theta = 36$ (c): New Zealand carbon trace, with $\theta = 3.0\bar{5}$

Theorems & Definitions (18)

• Definition 1: DTPR-min Threshold Values
• Definition 2: DTPR-max Threshold Values
• Theorem 4
• Theorem 5
• Corollary 6
• Corollary 7
• Theorem 8
• Theorem 9
• proof : Proof of Theorem \ref{['thm:compPRmin']}
• Lemma 10