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        "description": "The scheduling results use the paper’s machine, dependency and task-duration model. They bound specified list-scheduling behavior and illustrate anomalies; they do not establish that adding resources always worsens performance or solve every scheduling objective optimally. Changes in precedence, communication cost and processor heterogeneity need separate analysis.",
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      "statement": "# Graham: why a greedy schedule can slow down after apparent improvements\n\nGraham bounded how much a fixed-list schedule can worsen when task priorities, durations, precedence constraints or the number of identical processors are changed.\n\nAdding processors sounds like an automatic way to finish work faster. A greedy scheduler can behave differently: starting a ready task earlier may put later tasks in an unfavorable order. Graham modeled tasks with fixed lengths and dependencies, assigned without interruption to identical processors. He gave a bound on the finishing-time ratio after changing the priority list, shortening tasks, removing dependencies or changing the processor count. For equal processor counts the bound becomes 2 minus the reciprocal of that count. The comparison is about a specified scheduling rule, not the best possible use of extra hardware. This account does not reproduce every original diagram or supply a certified formula transcription. The central ratio was separately reconstructed from total work and an idle-covering dependency chain; this does not certify every original sharpness construction.\n\n## Scientific scope\n\nThe scheduling results use the paper’s machine, dependency and task-duration model. They bound specified list-scheduling behavior and illustrate anomalies; they do not establish that adding resources always worsens performance or solve every scheduling objective optimally. Changes in precedence, communication cost and processor heterogeneity need separate analysis.\n\n## People and systems\n\nR. L. Graham\n\n\n\n## Original sources\n\n- [R. L. Graham, Bounds for Certain Multiprocessing Anomalies, author-hosted original scan](https://www.math.ucsd.edu/~ronspubs/66_04_multiprocessing.pdf)\n- [Publisher Crossref deposit for Graham1966 scheduling-anomaly paper](https://api.crossref.org/works/10.1002%2Fj.1538-7305.1966.tb01709.x)\n",
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