# The qs Inequality: Quantifying the Double Penalty of Mixture-of-Experts at Inference

arXiv: https://arxiv.org/abs/2603.08960 · HTML: https://mlsanity.com/paper/2603.08960/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in MoE theory and gating functions, one of 12 directions in Mixture-of-experts.
- Builds on 10 of its 16 references that are in the map.
- First posted on arXiv 9 Mar 2026.

## Placed in

- Mixture-of-experts / [MoE theory and gating functions](https://mlsanity.com/moe/moe-theory-and-gating-functions/)

## Builds on (most cited first, 10 of 10)

- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [GShard: Scaling Giant Models with Conditional Computation and Automatic Sharding](https://mlsanity.com/paper/2006.16668/) (2020)
- [GLaM: Efficient Scaling of Language Models with Mixture-of-Experts](https://mlsanity.com/paper/2112.06905/) (2021)
- [DeepSpeed-MoE: Advancing Mixture-of-Experts Inference and Training to Power Next-Generation AI Scale](https://mlsanity.com/paper/2201.05596/) (2022)
- [Context Parallelism for Scalable Million-Token Inference](https://mlsanity.com/paper/2411.01783/) (2024)
- [Helix Parallelism: Rethinking Sharding Strategies for Interactive Multi-Million-Token LLM Decoding](https://mlsanity.com/paper/2507.07120/) (2025)

## Abstract (arXiv)

Mixture-of-Experts (MoE) models deliver high quality at low training FLOPs, but this efficiency often vanishes at inference. We identify a double penalty that structurally disadvantages MoE architectures during decoding: first, expert routing fragments microbatches and reduces weight reuse; second, massive resident expert pools reduce high-bandwidth memory (HBM) headroom for the KV cache. This phenomenon, formalized as reuse fragmentation, pushes feed-forward networks (FFNs) into a bandwidth-bound regime, especially at long context lengths. We introduce the qs inequality, a predictive criterion that identifies when MoE is structurally disadvantaged relative to a quality-matched dense model. This criterion unifies sparsity (s), the fraction of parameters activated per token, and the quality-equivalence factor (q), the size multiplier required for a dense model to match MoE performance. Our evaluation across frontier models including DeepSeek-V3, Qwen3-235B, Grok-1, and Switch-C demonstrates that this fragmentation is a general architectural phenomenon. For DeepSeek-V3 at 128k context, this results in a 4.5x throughput advantage for a quality-matched dense baseline. Crucially, massive architectures like Switch-C can become infeasible on cluster sizes where a quality-matched dense model remains viable. Our results suggest that training-time FLOP efficiency is an incomplete proxy for inference-time performance in long-context serving. They also indicate that MoE may be best viewed as a training-time optimization, with distillation into dense models as a possible path toward inference-efficient deployment.