A Simple and Extended Computational Analysis of <i>M/G</i><sub><i>j</i></sub><sup>(<i>a,b</i>)</sup>/1 and <i>M/G</i><sub><i>j</i></sub><sup>(<i>a,b</i>)</sup>/1/(<i>B</i> + <i>b</i>) Queues Using Roots
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Bibliographic record
Abstract
A recent article on bulk-service queues gives a generating function for the steady-state probabilities of the embedded Markov chain for a single-server infinite-space system in which, the customers arrive according to a Poisson process and are served in batches with quorum a and capacity b, and the service time follows a general distribution dependent on batch size. This system is equivalent to the bulk queueing model M/Gj(a,b)/1, whose general solution requires finding the roots of the denominator of the underlying generating function. The article claims that the use of roots may result in numerical inaccuracies, especially for large values of b. Hence it only solves for the finite-space model M/Gj(a,b)/1/(B + b) using (B+1) simultaneous linear equations. We present a simple way to obtain the probability distribution of queue length at post-departure epochs for the infinite-space model M/Gj(a,b)/1 using roots, then an alternative method to solve the finite-space queue M/Gj(a,b)/1/(B+b). We derive, for the first time, closed-form formulas for the queue-length distribution of models with deterministic service time for both infinite (M/Dj(a,b)/1) and finite-space (M/Dj(a,b)/1/(B+b)) systems. We also show that the queue-length distribution of (M/Dj(a,b)/1 can be approximated by a Poisson distribution when the traffic intensity p is low. Numerical results are both tabulated and graphed.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.005 | 0.005 |
| Science and technology studies | 0.003 | 0.002 |
| Scholarly communication | 0.004 | 0.018 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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