Constant Job‐Allowance Policies for Appointment Scheduling: Performance Bounds and Numerical Analysis
Bibliographic record
Abstract
We consider the classical appointment scheduling problem that does not have a session length constraint. The decision is to determine job allowances to minimize total expected weighted costs of waiting times of jobs and idle times of the service provider. In particular, we study a simple yet effective scheduling policy—the so‐called “constant" policy, which allocates a constant job allowance for each appointment. Prior studies on appointment scheduling suggest a “dome”‐shaped structure for the optimal job allowance over the planning horizon. This implies that job allowance does not vary significantly in the middle of the schedule sequence, but varies at the beginning as well as the end of the optimal schedule. We show that an even simple scheduling policy—the constant policy, is asymptotically optimal thus provide theoretical justification for such a widely used policy. Using a dynamic programming formulation, we express each job's waiting time as the maximum of a random walk, which allows us to bound the performance gap between the constant policy and the optimal schedule based on classical results on D/ G/1 queues. We derive an explicit upper bound for the performance gap when either of the following conditions holds: (i) the server idling cost is relatively small compared to the job waiting cost, and (ii) the number of appointments is sufficiently large. We also extend this result to a more general setting with multiple service types. Numerical experiments show that the constant policy is near optimal even when the number of appointments is small or when the server idling cost is moderately large, which complements our theoretical results. Our result provides a justification and strong support for the constant policy under certain mild conditions. Moreover, with minor modifications, the constant policy can be adapted to more general scenarios with patient no‐shows, or with heterogeneous appointment types.
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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 | 0.005 | 0.028 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 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".