Compute-Bound Analytical Model

• Compute-bound analytical models are formal approaches that express execution time as a function of algorithmic and hardware parameters under compute limitations.
• They utilize closed-form representations and resource attribution to identify pipeline bottlenecks and optimize both static analysis and codesign strategies.
• Recent advances blend analytical methods with machine learning to improve throughput predictions and guide optimizations in quantized deep learning and microarchitecture design.

A compute-bound analytical model is a formal approach designed to characterize and predict the performance of computations under the assumption that performance is limited primarily by available computational resources (i.e., arithmetic throughput or ALU execution) rather than by external factors such as memory bandwidth, I/O, or latency. These models provide tractable, often closed-form, representations of performance on modern microarchitectures, encapsulating either the algorithmic structure, hardware pipeline bottlenecks, or specific resource constraints. The following sections highlight the foundational methodologies, technical mechanisms, evaluation metrics, and research implications of compute-bound analytical models, synthesizing recent progress and use cases across static analysis, software optimization, hardware-aware modeling, and deep learning quantization.

1. Model Construction and Formalism

Compute-bound analytical models are typically constructed by expressing execution time, throughput, or cost as explicit functions of algorithmic parameters (loop bounds, operations per task), hardware features (core count, vector width), and compiler transformation choices. For instance, analytical models for tiled stencil codes express time as

$$T_{\mathrm{ideal}} = \frac{S_1 S_2 T}{n_{SM} n_V C_{iter}}$$

where $S_1, S_2, T$ are the problem sizes, $n_{SM}$ and $n_V$ are architectural parameters, and $C_{iter}$ denotes the compute-intensive time per iteration, assuming data locality on-chip and negligible memory bottlenecks (Prajapati et al., 2018). Variants for pipelined hardware, such as in bounded pipelines, model total latency as

$$T_q(p, n) = [p + n - 1 + (p-q)^{+} \left\lfloor \frac{n-1}{q} \right\rfloor] \cdot (t_p/p + t_o)$$

where $p$ is pipeline depth, $q$ the number of available function units, $n$ the data volume, $t_p$ stage delay, and $t_o$ overhead per latch (Husainov, 2018).

In processor-centric models, particularly for basic-block prediction, throughput is analytically bounded by the slowest pipeline component:

$$TP = \max\{\text{Predecode}, \text{Decode}, \text{Issue}, \text{Ports}, \text{Precedence}\}$$

with each component analytically (or algorithmically) determined for the microarchitecture in question (Abel et al., 2023).

2. Resource Attribution and Performance Boundaries

The compute-bound paradigm explicitly assumes in-memory data and focuses on resource attribution—the identification of the limiting (bottleneck) computational unit or pipeline stage. Modern analytical models typically perform a maximum-of-components composition, where the overall bound is set by the slowest unit among competing resources (decoder width, issue width, ALU pipelines, etc.).

In models such as Facile, individual predictors (predecode, decoder, issue stage, execution ports, dependency analysis) are run independently and then composed via a max operator, yielding both an overall performance estimate and direct attribution of the bottleneck (Abel et al., 2023). This compositionality is mirrored in more advanced CPU modeling, where, for example, Concorde computes per-component throughput across a trace and aggregates them into empirical distributions for further fusion (Nasr-Esfahany et al., 29 Mar 2025).

Resource attribution in accelerator and code design is further enabled by parameterizing the cost models in terms of tile size, wavefront scheduling, and mapping to hardware functional units, allowing both fine-grained and global optimization for compute intensity.

3. Analytical Models in Software and Systems Optimization

Practical application of compute-bound analytical models spans static code analysis, compiler optimization, and codesign. For example, scalable static bound analysis abstracts programs into lossy vector addition systems (VASS) and employs lexicographic ranking functions to establish precise amortized complexity bounds, which are especially crucial for nested or data-dependent loops (Sinn et al., 2014). The explicit resource-bound formulas enable scalable, predictable analyses without the need for heavy-weight abstract interpreters.

In the domain of hardware-software codesign, models parameterized by both software (tiling, unrolling) and hardware (SM count, vector width, shared memory size) allow co-optimization across execution time, silicon area, and energy metrics. The integrated optimization problem can be expressed as

$$\min_{\vec{P}, \vec{S}, \vec{A}} M_T(\vec{P}, \vec{S}, \vec{A}) \quad \text{subject to } \mathcal{F}_T(\vec{P}, \vec{S}, \vec{A})$$

with $M_T$ representing the time model over program, strategy, and architecture parameters and $\mathcal{F}_T$ expressing feasible sets (Prajapati et al., 2018).

4. Analytical Model Evaluation, Validation, and Efficiency

Compute-bound analytical models are validated both experimentally and through cross-method comparison. For prediction of basic-block throughput, baseline models such as

$TP_{baseline} = \max(n/4, m_r/2, m_w)$

by instruction count, memory reads, and writes are surprisingly effective, yielding average error margins competitive with simulation-based tools (Abel et al., 2021). More sophisticated analytical models are demonstrated to reach mean absolute percentage errors near or below 1% across microarchitectures (Abel et al., 2023), while simulation-augmented approaches such as Concorde achieve average CPI errors of 2–3% with speedups exceeding $10^5$ versus cycle-level simulation (Nasr-Esfahany et al., 29 Mar 2025).

Key evaluation metrics include prediction error (MAPE), Kendall’s tau rank correlation (for optimization orderings), empirical resource utilization, scaling with thread count, and quantitative metrics such as speedup and memory footprint reduction in quantized machine learning inference (Zhang et al., 28 Feb 2024).

5. Extensions to Deep Learning and Quantized Compute

Compute-bound analytical reasoning appears prominently in recent advances on LLM inference and quantized deep learning. Techniques such as FlattenQuant overcome the compute-bound limitations of large matrix multiplications by flattening tensors to reduce value ranges and enable efficient INT4/INT8 computation, minimizing reliance on slower FP16 operations common in per-channel quantization (Zhang et al., 28 Feb 2024). The standard quantization equation

$$X_k = \text{round}(X_{\text{fp16}} / s),\quad s = \max |X_{\text{fp16}}| / (2^{k-1} - 1)$$

is adapted to flattened tensors to allow up to 48.29% of linear layers to use 4-bit arithmetic, yielding up to $2\times$ speed and $2.3\times$ memory savings with minimal accuracy degradation.

The methodology relies on per-tensor (not per-channel) quantization accompanied by aggressive smoothing and channel expansion strategies (controlled via parameters $\beta$ and $\gamma$, for truncation and KL divergence ratio), making direct use of hardware accelerator units practical in the compute-bound regime.

6. Machine Learning Fusion and Large-Scale Microarchitecture Modeling

A contemporary trend is the integration of analytical models with learning-based components, exemplified by the Concorde system (Nasr-Esfahany et al., 29 Mar 2025). In this approach, analytical models estimate per-resource performance limits across short trace windows, which are distilled into fixed-size performance distributions (percentiles or CDF summaries) and then input into a shallow neural network predictor. This compositional analytical-ML fusion enables rapid prediction across vast microarchitectural parameter spaces (20+ dimensions, up to $2.2 \times 10^{23}$ combinations) and supports both performance prediction and sensitivity or Shapley value–based attribution analyses, with evaluation rates exceeding 100 million CPI queries per hour and prediction error at the 2% level.

This methodology supports microarchitecture design, performance debugging, and iterative optimization at scales previously inaccessible due to the prohibitive run time of cycle-accurate simulators.

7. Research Implications and Future Directions

Compute-bound analytical modeling is now an essential methodology across computer systems, from static resource-bound analysis and software codesign to hardware simulation and LLM inference optimization. The adoption of compositional, interpretable, and hybrid analytical-ML frameworks has dramatically increased scalability and accuracy for performance estimation in compute-dominated regimes. These approaches have enabled the efficient exploration of massive design spaces, quantization strategies for large models, and detailed attribution of performance limits in practical processor design and code optimization scenarios.

Future research directions include further automating parameter selection in hybrid quantization methods, extending analytical-ML fusion to more aggressive code transformations, and incorporating compute-bound reasoning into end-to-end optimization of heterogeneous and exascale platforms.