Dynamic Temperature Scheduling

• Dynamic temperature scheduling is a strategy that adapts thermal parameters over time to meet system performance and energy efficiency goals.
• It employs methodologies such as fluid-based online heuristics, mixed-integer programming, and adaptive ML tuning to manage temperature dynamics.
• Applications span data centers, HVAC systems, embedded hardware, and machine learning, yielding measurable benefits like reduced peak temperatures and energy savings.

A dynamic temperature schedule is a control or optimization strategy that adapts one or more temperature-related variables over time in response to evolving system states, constraints, or environmental conditions. Such schedules are crucial in a wide range of domains, including real-time computing, embedded systems, large-scale data centers, HVAC operations, communication networks, and knowledge distillation in machine learning. The primary goal is to ensure operational objectives (e.g., deadline adherence, energy minimization, equipment protection, or improved learning) while respecting hard or soft thermal constraints and capitalizing on temporal flexibility.

1. Theoretical Foundations and Motivation

The impetus for dynamic temperature scheduling arises from the nonlinear and time-varying relationship between heat generation, cooling capacity, and system performance. Static thermal limits or setpoints do not account for the stochasticity of workloads, temporal load imbalances, equipment heterogeneity, environmental uncertainty, or predictive information. By dynamically modulating thermal thresholds, setpoints, or surrogate parameters (e.g., softmax temperature in neural nets), it is possible to (a) reduce peak or average operating temperatures, (b) avoid unnecessary thermal margins, (c) trade off performance and reliability, and (d) operate closer to optimal conditions, thus improving energy efficiency, throughput, or learning generalization (Dowling et al., 2024, Tian et al., 2021, Ilager et al., 2020, Adegbija et al., 2016, Rostami et al., 2023).

Several fundamental models underpin dynamic temperature scheduling:

• Lumped RC or POD-based thermal models for hardware; these govern the evolution of thermal states under variable applied power or load (e.g., $dT/dt = -\alpha(T-T_{\text{amb}}) + \beta P(t)$).
• Linear or mixed-integer linear models for zone/building temperature dynamics in HVAC or data centers.
• Convex optimization or fluid scheduling theory for analytic tractability and performance guarantees.
• Adaptive temperature modulation in the softmax operation for machine learning models.

2. Algorithmic Schemes and Formulations

A spectrum of algorithmic strategies exists for realizing dynamic temperature schedules, tailored to the domain and constraints.

2.1 Real-Time and Embedded Systems

In hardware platforms (e.g., multiprocessors or embedded controllers), variable-threshold scheduling can be achieved via online algorithms such as VTF-TAS (Dowling et al., 2024) or phase-based multi-objective tuning (TaPT) (Adegbija et al., 2016). VTF-TAS updates a running temperature threshold $T_H(t)$ for each scheduling interval by applying a fluid-scheduling–inspired heuristic that continuously compares actual work progress to an idealized (fluid) trajectory. This yields the threshold-update rule:

• Compute fluid utilization $U_F$ and actual utilization $U_σ$
• $H = \tfrac{1}{2}[(U_σ/U_F) + (U_σ - U_F)] - 0.5$
• Scale and dampen $H$; increment or decrement $T_H$ proportional to $H$ with dead-zone and direction-damping for stability.

Task assignment respects the latest computed $T_H$ and system state, while online temperature prediction is performed with a reduced-order POD (or HotSpot/FEM) model.

2.2 Data Centers and Cloud Workload Management

Thermal-aware dynamic scheduling in large-scale systems deploys policies such as non-preemptive dynamic window (NPDW) scheduling (Michel et al., 2022), switching-aware MIP or MPC-based schemes (Rostami et al., 2023), and ML-augmented VM placement (Ilager et al., 2020). These frameworks maintain time-varying system-level thermal constraints, either directly (as schedule parameters) or indirectly (as weights/performance metrics in window-sizing, or in switching cost minimization).

For example, in NPDW, the accumulation window $\Delta t_{\text{next}}$ is adapted after each execution window according to throughput and core-temperature performance across multiple time scales, as tabulated below:

Metric	Definition	Scheduling Role
Window Performance	$WP(C) = \alpha_2 UT(C)+\alpha_1 OT(C)$	Adjusts window length
Matching Performance	$$MP(M,t) = \frac{1}{|M|}\sum_{(u,v)\in M}(t-u_{arrival})$$	Balances latency
Core/Task Preference	Weighted by core temperature, utilization	Guides task-to-core matching

2.3 HVAC and Building Systems

Dynamic temperature scheduling in HVAC typically involves modulation of environmental setpoints or chilled-water temperatures as a function of predictive models. In (Tian et al., 2021), building temperature setpoints are scheduled with robust MILP subject to dynamic and uncertain exogenous data (ambient temperature, load), using distributionally robust optimization (DRO) with a Wasserstein ambiguity set. Conversely, in integrated simulation frameworks for TES-coupled cooling systems (Oh et al., 16 Jan 2026), the chilled-water charge temperature $T_{TES,0}$ is dynamically optimized based on multistage prediction (humidity, cooling load, coil, and TES models) to exploit thermal headroom while maintaining comfort constraints.

2.4 Communication Systems

For energy harvesting transmitters, the optimal transmit power $p^*(t)$ schedule is obtained by solving a convex functional program under time-varying temperature and energy causality constraints (Ozel et al., 2015). The optimal solution structure is piecewise monotone-decreasing power, with flat segments when thermal caps are active.

2.5 Machine Learning and Knowledge Distillation

Dynamic temperature schedules for softmax or knowledge distillation introduce per-sample or per-iteration temperature parameters (e.g., $\tau_i$ or $T_t$, $T_s$), typically modulated by input difficulty metrics (naturalness score, sharpness, or sample hardness) (Kim et al., 20 May 2025). Here, curriculum learning or KD proceeds in stages, activating dynamic temperature scaling only during advanced training epochs, producing greater robustness and generalization relative to static-temperature baselines.

3. Formal Models and Mathematical Analysis

The essence of dynamic temperature scheduling is the formulation of the schedule as an optimal control, convex program, receding-horizon MPC, or feedback-control loop. Common mathematical components include:

• Thermal evolution: ODEs ($dT/dt = -\alpha (T-T_{\text{amb}}) + \beta P(t)$) or their discrete-time, reduced-order (POD) or RC-circuit analogs.
• Schedule updates: Heuristic (fluid-based), linear program (e.g., minimize sum of completion times in temperature-bounded job scheduling (Lambers et al., 2023)), or MILP (DRO-based HVAC (Tian et al., 2021), workload distribution (Rostami et al., 2023)).
• Predictive modeling: ML-based regression (XGBoost for temperature/humidity/coil performance (Ilager et al., 2020, Oh et al., 16 Jan 2026))
• Objective trade-offs: Multi-objective minimization (time, energy, peak temperature), with Pareto-optimal set calculations (TaPT (Adegbija et al., 2016)).
• Dual decomposition and Lagrangian KKT analysis to derive optimal offline power/temperature policies under causality and cap constraints (Ozel et al., 2015).

4. Performance Evaluation and Comparative Results

Evaluation of dynamic temperature schedules utilizes trace-driven simulation, benchmarks (e.g., COMBS or EEMBC), and real-data deployments. Notable findings include:

• VTF-TAS achieves lower peak CPU temperatures than static-threshold baselines (e.g., 76.9°C vs. 78.4°C in COMBS scenarios) without the need for offline search (Dowling et al., 2024).
• Non-preemptive dynamic window scheduling (NPDW) yields tighter core temperature spread and improved thermal/throughput fairness at the cost of marginally lower utilization (Michel et al., 2022).
• Distributionally robust MILP-based HVAC scheduling cuts violation frequency and cost simultaneously, with up to 6.6% lower energy cost compared to robust and scenario-based methods (Tian et al., 2021).
• In buildings with TES-coupled cooling, dynamic charging schedules allow a +2.55°C increase in charge temperature, producing a 5–10% improvement in heat pump COP with indoor comfort maintained (Oh et al., 16 Jan 2026).
• Machine learning tasks with naturalness-aware dynamic temperature achieve a 23% relative reduction in EER on speech spoofing tasks, without modifying the network architecture (Kim et al., 20 May 2025).
• In energy-harvesting communications, the optimal dynamic (power, temperature) schedule is explicitly characterized, with monotonicity and threshold-activation properties guaranteeing feasibility and efficiency (Ozel et al., 2015).

5. Practical Implementation and Tuning Guidelines

Practical deployment of dynamic temperature schedules requires calibration of time-scales, threshold or window parameters, ML model retraining, and integration with existing system control hierarchies.

Key implementation considerations:

• Initial thresholds (e.g., $T_H$ in VTF-TAS) should be estimated from long-term steady-state operation.
• Dead-zone and damping parameters should be tuned to trade off reactivity versus instability in threshold adaptation (Dowling et al., 2024).
• In data centers, aggregation window size and matching algorithms must be adapted to the workload arrival distribution to stabilize temperature trajectories (Michel et al., 2022).
• Robust and predictive control frameworks (MILP, MPC) call for real-time data pipelines, scenario generation, and receding-horizon execution (Tian et al., 2021, Rostami et al., 2023).
• ML-based thermal prediction should receive regular retraining and be paired with fallback physical models for abnormal sensor scenarios (Ilager et al., 2020).
• Hardware implementations (e.g., cache/frequency tuning in TaPT) require negligible (<5%) die area overhead and can amortize optimization cost across phase boundaries (Adegbija et al., 2016).

6. Limitations, Open Challenges, and Future Directions

While dynamic temperature schedules reliably lower peak and average system temperatures, domain-specific challenges remain:

• In hardware, strong NP-hardness results for certain scheduling objectives (unit-length jobs) preclude efficient exact offline algorithms, though effective heuristics and competitive online algorithms exist (0801.4238).
• Distributionally robust optimization and ML-based scheduling can be limited by model mismatch, incomplete input data, or non-convexity in physical constraints (Tian et al., 2021, Rostami et al., 2023).
• Predictive scheduling in building systems remains sensitive to forecast error and model granularity; modular integrated simulation frameworks mitigate but do not eliminate these sensitivities (Oh et al., 16 Jan 2026).
• In distributed or networked systems, spatial temperature gradients and inter-component thermal coupling demand more sophisticated, often decentralized, predictive control strategies.
• A plausible implication is that further advances in interpretable, physics-informed ML models, as well as stochastic and robust control formulations, will increase the reliability and scope of dynamic temperature scheduling in infrastructure and learning domains.

7. Domain-Specific Case Studies and Exemplars

The following table summarizes canonical dynamic temperature scheduling strategies by domain and their reported achievements:

Domain	Dynamic Schedule Mechanism	Core Benefit/Result
Real-Time TAS	Variable thermal thresholds (fluid-based heuristic)	↓1.5°C peak, no search overhead
Data Center	ML-driven VM migration + window/matching optimization	-6.5°C peak, 34.5% energy savings
HVAC	MILP w/ robust/stochastic ambiguity sets	6.6% energy cost savings
Building-TES	Integrated simulation for optimal chilled-water temp	+2.55°C charge temp, 5–10% COP gain
Embedded CPU	Phase-based Pareto tuning (cache + frequency)	-21–25% temp, -31–34% EDP
Energy Harvest	KKT-driven monotonic optimal power/temperature law	Piecewise-decreasing power, constraint satisfaction
ML/SDD	Naturalness-aware dynamic softmax temp	23% EER reduction, more robust generalization

These results establish dynamic temperature schedules as a foundational component in multidisciplinary systems optimization, with application-specific tuning aligning the dynamic policy with operational priorities and domain constraints (Dowling et al., 2024, Ilager et al., 2020, Tian et al., 2021, Michel et al., 2022, Kim et al., 20 May 2025, Rostami et al., 2023, Adegbija et al., 2016, Oh et al., 16 Jan 2026, Lambers et al., 2023, 0801.4238, Ozel et al., 2015).