A Note on the Waiting-Time Distribution in an Infinite-Buffer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>G</mml:mi><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:mi>C</mml:mi><mml:mtext>-</mml:mtext><mml:mi>M</mml:mi><mml:mi>S</mml:mi><mml:mi>P</mml:mi><mml:mo>/</mml:mo><mml:mn fontstyle="italic">1</mml:mn></mml:math> Queueing System
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
This paper deals with a batch arrival infinite-buffer single server queue. The interbatch arrival times are generally distributed and arrivals are occurring in batches of random size. The service process is correlated and its structure is presented through a continuous-time Markovian service process (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>C</mml:mi><mml:mtext>-</mml:mtext><mml:mi>M</mml:mi><mml:mi>S</mml:mi><mml:mi>P</mml:mi></mml:math>). We obtain the probability density function (p.d.f.) of actual waiting time for the first and an arbitrary customer of an arrival batch. The proposed analysis is based on the roots of the characteristic equations involved in the Laplace-Stieltjes transform (LST) of waiting times in the system for the first, an arbitrary, and the last customer of an arrival batch. The corresponding mean sojourn times in the system may be obtained using these probability density functions or the above LSTs. Numerical results for some variants of the interbatch arrival distribution (Pareto and phase-type) have been presented to show the influence of model parameters on the waiting-time distribution. Finally, a simple computational procedure (through solving a set of simultaneous linear equations) is proposed to obtain the “<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi mathvariant="bold">R</mml:mi></mml:mrow></mml:math>” matrix of the corresponding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>G</mml:mi><mml:mi>I</mml:mi><mml:mo>/</mml:mo><mml:mi>M</mml:mi><mml:mo>/</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:math>-type Markov chain embedded at a prearrival epoch of a batch.
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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 | 0.010 | 0.010 |
| Meta-epidemiology (narrow) | 0.003 | 0.006 |
| Meta-epidemiology (broad) | 0.001 | 0.005 |
| Bibliometrics | 0.002 | 0.005 |
| Science and technology studies | 0.006 | 0.006 |
| Scholarly communication | 0.007 | 0.006 |
| Open science | 0.007 | 0.006 |
| Research integrity | 0.006 | 0.006 |
| Insufficient payload (model declined to judge) | 0.253 | 0.004 |
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