ON MULTISERVER RETRIAL QUEUES: HISTORY, OKUBO-TYPE HYPERGEOMETRIC SYSTEMS AND MATRIX CONTINUED-FRACTIONS
Bibliographic record
Abstract
In this paper, we study two families of QBD processes with linear rates: (a) the multiserver retrial queue and its easier relative; and (b) the multiserver M/M/∞ Markov modulated queue. The linear rates imply that the stationary probabilities satisfy a recurrence with linear coefficients; as known from previous work, they yield a"minimal/nondominant" solution of this recurrence, which may be computed numerically by matrix continued-fraction methods. Furthermore, the generating function of the stationary probabilities satisfies a linear differential system with polynomial coefficients, which calls for the venerable but still developing theory of holonomic (or D-finite) linear differential systems. We provide a differential system for our generating function that unifies problems (a) and (b), and we also include some additional features and observe that in at least one particular case we get a special "Okubo-type hypergeometric system", a family that recently spurred considerable interest. The differential system should allow further study of the Taylor coefficients of the expansion of the generating function at three points of interest: (i) the irregular singularity at 0; (ii) the dominant regular singularity, which yields asymptotic series via classic methods like the Frobenius vector expansion; and (iii) the point 1, whose Taylor series coefficients are the factorial moments.
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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".