Minimum Feedback for Collision-Free Scheduling in Massive Random Access
Bibliographic record
Abstract
Consider a massive random access scenario in which a small set of$k$active users out of a large number of$n$potential users need to be scheduled in$b\ge k$slots. What is the minimum common feedback to the users needed to ensure that scheduling is collision-free? Instead of a naive scheme of listing the indices of the$k$active users in the order in which they should transmit, at a cost of$k\log (n)$bits, this paper shows that for the case of$b=k$, the rate of the minimum fixed-length common feedback code scales only as$k \log (e)$bits, plus an additive term that scales in$n$as$\Theta (\log \log (n))$for fixed$k$. If a variable-length code can be used, assuming uniform activity among the users, the minimum average common feedback rate still requires$k \log (e)$bits, but the dependence on$n$can be reduced to$O(1)$. When$b>k$, the number of feedback bits needed for collision-free scheduling can be significantly further reduced. Moreover, a similar scaling on the minimum feedback rate is derived for the case of scheduling$m$users per slot, when$k \le mb$. The problem of constructing a minimum collision-free feedback scheduling code is connected to that of constructing a perfect hashing family, which allows practical feedback scheduling codes to be constructed from perfect hashing algorithms.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.025 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".