Light-Tailed Behavior in QBD Processes with Countably Many Phases
Bibliographic record
Abstract
Generally speaking, analysis of tail asymptotics in two-dimensional queueing systems is very challenging. Earlier work based on complex analysis led to determinations of exact forms of tail asymptotics. Ideas of large deviations, a powerful tool for characterizing light-tailed decay rates or analysis of rough tail asymptotics, have been utilized recently to develop probabilistic methods to do exact tail asymptotic analysis. Another promising approach to do tail asymptotics analysis, both exact and rough, is the matrix-analytic method. In this article, we combine the matrix-analytic method with techniques from probability and analysis to characterize tail asymptotics in a QBD process with infinitely many phases. The main results include conditions on: (1) exact geometric decay; (2) light-tailed behavior without an exact geometric decay, which in general is not the focus of the large deviations method; and (3) upper and lower bounds for stationary probabilities. We apply the main results to two two-dimensional queueing systems, including a polling system and a gated random-order server queue to characterize their light-tailed behavior of the queue length processes.
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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 | 0.003 | 0.025 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".