Regulated Random Walks and the LCFS Backlog Probability: Analysis and Application
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Bibliographic record
Abstract
Random walks have been used extensively within operations research models such as inventory systems and single-server queues to estimate performance measures. In this paper, we use sample-path analysis to express the steady-state probability of a one-sided regulated random walk to increase and be above a threshold, referred to as the last-come-first-serve (LCFS) backlog probability. We approximate the LCFS backlog probability under mild assumptions on the distribution of the random walk's steps and provide its exact expression when the steps are exponentially distributed, and a closed-form approximation when the steps are normally distributed. In our numerical experiments, the average relative gap between the approximated LCFS backlog probabilities and their simulated values is 5.13%. We further show that the LCFS backlog probability is an upper bound on the loss probability—the probability that a two-sided regulated random walk is at a boundary. This bound is tighter than the backlog probability—the probability that a random walk ever crosses a threshold—that also bounds the loss probability. In an inventory application, we demonstrate that using the LCFS backlog probability rather than the backlog probability reduces the inventory level required to satisfy a service-level constraint on the percentage of orders backlogged. In our examples, this reduction leads to cost savings of 31% on average.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.003 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it