{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "little-stationary-queue-occupancy-proof-1961",
    "title": "Little: a general stationary-queue proof linking occupancy, arrivals and time in system",
    "summary": "Little proved L=λW for consistently defined queue occupancy, arrival rate and sojourn time under finite-mean strict stationarity and an ergodic interarrival process, extending earlier heuristic and special-model uses of the relation.\n\nHow many requests are usually inside a system? How long do they stay? Little’s relation connects these quantities: average occupancy equals the arrival rate multiplied by average time in the system. His1961 article proves this under explicit stationary stochastic-process assumptions, without requiring Poisson arrivals, a particular service distribution or first-come-first-served order. The key counting picture is the area under occupancy over time: it equals customers’ accumulated time inside, with carry-over corrections at the ends of an observation window. Ergodic arguments control the long-run averages and corrections. The same system boundary must be used for all three quantities. If some offered arrivals never enter, combining the offered arrival rate with the mean stay of admitted customers gives the wrong result. One can instead use the admitted rate, or include zero stays for rejected arrivals. The formula was already known heuristically and for particular models; this record recognizes the broader proof. Little explicitly credited H.Newton Garber with the key occupancy definition and its implications.",
    "boundary": "The identity relates long-run occupancy, arrival rate and time in system under the paper’s stationarity/existence and accounting assumptions. It is not a pointwise identity for every finite interval or a stability proof when averages do not exist. System boundaries and customer accounting must agree. Later transient and generalized formulations are separate.",
    "area": "Mathematics and statistics",
    "topic": "Queueing theory and stochastic systems",
    "year": 1961,
    "lane": "HUMAN",
    "result": "THEORY",
    "package_id": "little-stationary-queue-occupancy-proof-1961",
    "target": {
      "record_id": "urn:uuid:af81ffde-1084-426d-a289-35e22a6d2f99",
      "version": 1,
      "content_hash": "sha256:b952a3eeb3caa5e84f684e96c60e8bbcb84ba64390cd31ba5207c4e46b33beae"
    },
    "humans": [
      "John D. C. Little"
    ],
    "authors_complete": true,
    "human_role_note": "Source-listed contributors to this scoped work. Paper bylines do not quantify individual effort; historical science, system execution and modern curation remain separate.",
    "agents": [],
    "related": [],
    "subresults": [],
    "scientific_assessment": {
      "source_review": "SOURCE_CURATED",
      "scientific_review": "NOT_REVIEWED_BY_WRS",
      "formal_replay": "NOT_RUN",
      "provider_attestation": "NONE"
    },
    "sources": [
      {
        "id": "wrs24-little-stationary-queue-occupancy-proof-1961-article",
        "title": "Little: a general stationary-queue proof linking occupancy, arrivals and time in system — complete assessed article text",
        "url": "https://br.k21.global/wp-content/uploads/2025/07/Littles-Law-Lei-de-Little-Artigo-Original.pdf"
      },
      {
        "id": "wrs24-little-stationary-queue-occupancy-proof-1961-publisher",
        "title": "INFORMS original publication metadata and abstract: A Proof for the Queuing Formula",
        "url": "https://pubsonline.informs.org/doi/abs/10.1287/opre.9.3.383"
      }
    ],
    "points": "160000",
    "published_grade": {
      "id": "little-stationary-queue-occupancy-proof-1961",
      "entry_id": "little-stationary-queue-occupancy-proof-1961",
      "band": "A",
      "modifier": null,
      "target": {
        "record_id": "urn:uuid:af81ffde-1084-426d-a289-35e22a6d2f99",
        "version": 1,
        "content_hash": "sha256:b952a3eeb3caa5e84f684e96c60e8bbcb84ba64390cd31ba5207c4e46b33beae"
      },
      "what_changed": "Little proved L=λW for consistently defined queue occupancy, arrival rate and sojourn time under finite-mean strict stationarity and an ergodic interarrival process, extending earlier heuristic and special-model uses of the relation.",
      "scope": "Little proved L=λW for consistently defined queue occupancy, arrival rate and sojourn time under finite-mean strict stationarity and an ergodic interarrival process, extending earlier heuristic and special-model uses of the relation. The identity relates long-run occupancy, arrival rate and time in system under the paper’s stationarity/existence and accounting assumptions. It is not a pointwise identity for every finite interval or a stability proof when averages do not exist. System boundaries and customer accounting must agree. Later transient and generalized formulations are separate.",
      "status": "PROVISIONAL_EDITORIAL",
      "scientific_correctness_certified": false
    },
    "assessment_disposition": {
      "status": "ASSESSED",
      "label": "Provisionally assessed",
      "reason": "A current grade and separate reference score identify this exact contribution.",
      "target": {
        "record_id": "urn:uuid:af81ffde-1084-426d-a289-35e22a6d2f99",
        "version": 1,
        "content_hash": "sha256:b952a3eeb3caa5e84f684e96c60e8bbcb84ba64390cd31ba5207c4e46b33beae"
      },
      "edition": "ratings-e039106f30dec177"
    }
  },
  "version": {
    "kind": "PUBLIC_RECORD_VIEW",
    "previous": null,
    "published_at": "2026-10-03T00:00:00Z",
    "publisher": {
      "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33",
      "role": "WRS_EDITORIAL_PUBLICATION"
    },
    "record": {
      "contributors": [
        {
          "attribution_basis": "SOURCE_CITED",
          "credit": "historical_scientific",
          "entity_kind": "historical_person",
          "name": "John D. C. Little",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "formulator"
          ],
          "source": {
            "citation": "Little: a general stationary-queue proof linking occupancy, arrivals and time in system — complete assessed article text",
            "url": "https://br.k21.global/wp-content/uploads/2025/07/Littles-Law-Lei-de-Little-Artigo-Original.pdf"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "Little: a general stationary-queue proof linking occupancy, arrivals and time in system — complete assessed article text",
            "url": "https://br.k21.global/wp-content/uploads/2025/07/Littles-Law-Lei-de-Little-Artigo-Original.pdf"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-1",
          "kind": "source_reference",
          "source": {
            "citation": "INFORMS original publication metadata and abstract: A Proof for the Queuing Formula",
            "url": "https://pubsonline.informs.org/doi/abs/10.1287/opre.9.3.383"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "The identity relates long-run occupancy, arrival rate and time in system under the paper’s stationarity/existence and accounting assumptions. It is not a pointwise identity for every finite interval or a stability proof when averages do not exist. System boundaries and customer accounting must agree. Later transient and generalized formulations are separate.",
        "domain": "Mathematics and statistics"
      },
      "sources": [
        {
          "citation": "Little: a general stationary-queue proof linking occupancy, arrivals and time in system — complete assessed article text",
          "url": "https://br.k21.global/wp-content/uploads/2025/07/Littles-Law-Lei-de-Little-Artigo-Original.pdf"
        },
        {
          "citation": "INFORMS original publication metadata and abstract: A Proof for the Queuing Formula",
          "url": "https://pubsonline.informs.org/doi/abs/10.1287/opre.9.3.383"
        }
      ],
      "statement": "# Little: a general stationary-queue proof linking occupancy, arrivals and time in system\n\nLittle proved L=λW for consistently defined queue occupancy, arrival rate and sojourn time under finite-mean strict stationarity and an ergodic interarrival process, extending earlier heuristic and special-model uses of the relation.\n\nHow many requests are usually inside a system? How long do they stay? Little’s relation connects these quantities: average occupancy equals the arrival rate multiplied by average time in the system. His1961 article proves this under explicit stationary stochastic-process assumptions, without requiring Poisson arrivals, a particular service distribution or first-come-first-served order. The key counting picture is the area under occupancy over time: it equals customers’ accumulated time inside, with carry-over corrections at the ends of an observation window. Ergodic arguments control the long-run averages and corrections. The same system boundary must be used for all three quantities. If some offered arrivals never enter, combining the offered arrival rate with the mean stay of admitted customers gives the wrong result. One can instead use the admitted rate, or include zero stays for rejected arrivals. The formula was already known heuristically and for particular models; this record recognizes the broader proof. Little explicitly credited H.Newton Garber with the key occupancy definition and its implications.\n\n## Scientific scope\n\nThe identity relates long-run occupancy, arrival rate and time in system under the paper’s stationarity/existence and accounting assumptions. It is not a pointwise identity for every finite interval or a stability proof when averages do not exist. System boundaries and customer accounting must agree. Later transient and generalized formulations are separate.\n\n## People and systems\n\nJohn D. C. Little\n\n\n\n## Original sources\n\n- [Little: a general stationary-queue proof linking occupancy, arrivals and time in system — complete assessed article text](https://br.k21.global/wp-content/uploads/2025/07/Littles-Law-Lei-de-Little-Artigo-Original.pdf)\n- [INFORMS original publication metadata and abstract: A Proof for the Queuing Formula](https://pubsonline.informs.org/doi/abs/10.1287/opre.9.3.383)\n",
      "summary": "Little proved L=λW for consistently defined queue occupancy, arrival rate and sojourn time under finite-mean strict stationarity and an ergodic interarrival process, extending earlier heuristic and special-model uses of the relation.\n\nHow many requests are usually inside a system? How long do they stay? Little’s relation connects these quantities: average occupancy equals the arrival rate multiplied by average time in the system. His1961 article proves this under explicit stationary stochastic-process assumptions, without requiring Poisson arrivals, a particular service distribution or first-come-first-served order. The key counting picture is the area under occupancy over time: it equals customers’ accumulated time inside, with carry-over corrections at the ends of an observation window. Ergodic arguments control the long-run averages and corrections. The same system boundary must be used for all three quantities. If some offered arrivals never enter, combining the offered arrival rate with the mean stay of admitted customers gives the wrong result. One can instead use the admitted rate, or include zero stays for rejected arrivals. The formula was already known heuristically and for particular models; this record recognizes the broader proof. Little explicitly credited H.Newton Garber with the key occupancy definition and its implications.",
      "title": "Little: a general stationary-queue proof linking occupancy, arrivals and time in system"
    },
    "record_id": "urn:uuid:af81ffde-1084-426d-a289-35e22a6d2f99",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:af81ffde-1084-426d-a289-35e22a6d2f99",
      "version": 1,
      "content_hash": "sha256:b952a3eeb3caa5e84f684e96c60e8bbcb84ba64390cd31ba5207c4e46b33beae"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:c796ceaa8b07ffc5ccf16778d55b9a64b35814f6e2f77d008082f90b72998b59"
  },
  "target": {
    "record_id": "urn:uuid:af81ffde-1084-426d-a289-35e22a6d2f99",
    "version": 1,
    "content_hash": "sha256:b952a3eeb3caa5e84f684e96c60e8bbcb84ba64390cd31ba5207c4e46b33beae"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "little-stationary-queue-occupancy-proof-1961",
    "entry_id": "little-stationary-queue-occupancy-proof-1961",
    "revision": 1,
    "title": "Little: a general stationary-queue proof linking occupancy, arrivals and time in system",
    "context": "Mathematics and statistics",
    "track": "Mathematics and statistics",
    "band": "A",
    "modifier": null,
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-04",
    "evidence_cutoff": "2026-10-04",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "Little proved L=λW for consistently defined queue occupancy, arrival rate and sojourn time under finite-mean strict stationarity and an ergodic interarrival process, extending earlier heuristic and special-model uses of the relation.",
    "scope": "Little proved L=λW for consistently defined queue occupancy, arrival rate and sojourn time under finite-mean strict stationarity and an ergodic interarrival process, extending earlier heuristic and special-model uses of the relation. The identity relates long-run occupancy, arrival rate and time in system under the paper’s stationarity/existence and accounting assumptions. It is not a pointwise identity for every finite interval or a stability proof when averages do not exist. System boundaries and customer accounting must agree. Later transient and generalized formulations are separate.",
    "evidence_summary": "Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.",
    "references": [
      {
        "title": "Little: a general stationary-queue proof linking occupancy, arrivals and time in system — complete assessed article text",
        "url": "https://br.k21.global/wp-content/uploads/2025/07/Littles-Law-Lei-de-Little-Artigo-Original.pdf",
        "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits."
      },
      {
        "title": "INFORMS original publication metadata and abstract: A Proof for the Queuing Formula",
        "url": "https://pubsonline.informs.org/doi/abs/10.1287/opre.9.3.383",
        "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits."
      }
    ],
    "roles": [
      {
        "who": "John D. C. Little",
        "did": "Sole original byline author and general stationary-queue proof.",
        "credit": "SCIENCE"
      },
      {
        "who": "H. Newton Garber",
        "did": "Original acknowledgment explicitly credits equation2 and its definitional implications plus proof suggestions; not a byline coauthor.",
        "credit": "SCIENCE"
      }
    ],
    "target": {
      "record_id": "urn:uuid:af81ffde-1084-426d-a289-35e22a6d2f99",
      "version": 1,
      "content_hash": "sha256:b952a3eeb3caa5e84f684e96c60e8bbcb84ba64390cd31ba5207c4e46b33beae"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "160000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": null,
  "reading_status": "SOURCE_LINKED_NOTE"
}