Proof and Application of Little’s Formula in Optimizing Bulk-service Multi-server Queue Systems
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Bibliographic record
Abstract
This paper presents closed-form expressions for the steady-state mean queue length in a complex bulk-service, multi-server queueing system denoted as GI/M a,b /c. In this continuous system, customers arrive according to an arbitrary distribution, and ′′ c ′′ servers operate independently, each serving a group of customers with a size between ′′ a ′′ and ′′ b ′′ . We demonstrate the applicability of Little’s formula to this queueing system, establishing a foundational relationship between the average queue length and the average time customers spend in the system. Our findings provide valuable insights into the system’s performance, and essential for assessing efficiency and optimizing service strategies. Additionally, We formulate an objective function and compute the optimal values of the decision variables (a, b) that maximize the expected profit.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| 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