On the Finite Capacity Shortest Queue Problem
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Bibliographic record
Abstract
We consider two parallel queues. There is one server tending to each queue and the capacity of each queue is K . The network is fed by a single Poisson arrival stream of rate λ, and the two servers are identical exponential servers working at rate µ. A new arrival is routed to the queue with the smaller number of customers. If both have the same number of customers then the arrival is routed randomly, with the probability of joining either queue being 1/2. If there are more than 2 K customers in the system, further arrivals are turned away and lost. We let ρ = λ/µ and take K →∞, and consider the cases ρ 2 and ρ − 2 = O ( K − 1 ). We shall obtain asymptotic approximations to the joint steady state distribution of finding m customers in the first queue and n in the second. The asymptotic approximations are shown to be quite accurate numerically. We shall identify precisely for what ranges of m and n can the finite capacity model be approximated by the infinite capacity one. We will also show that the marginal distribution of finding n customers in the second queue undergoes a transition when ρ = 4. Key words: Shortest queue problem; Finite capacity; Poisson arrival stream; Analytical approximations
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| 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