{ "openapi": "3.1.1", "info": { "title": "Twinworks Queueing Check API", "version": "1.0.0", "summary": "M/M/1, M/M/c, M/D/1, M/G/1 and Kingman waiting times — optionally verified by simulation.", "description": "Every endpoint performs the same calculation as the human page and returns inputs, result, units, assumptions, sources, warnings and computedAt. MCP endpoint: /mcp." }, "servers": [ { "url": "https://twinworksqueueformulas.usefulweb.site" } ], "paths": { "/api/v1/calculate": { "post": { "operationId": "calculate", "summary": "M/M/1, M/M/c, M/D/1, M/G/1 and Kingman waiting times — optionally verified by simulation.", "description": "Closed-form queueing results for a single workstation: utilization, mean waiting time, queue length and time in system for M/M/1, M/M/c (Erlang C), M/D/1, M/G/1 (Pollaczek–Khinchine) and the Kingman G/G/1 approximation. Optionally runs the Twinworks simulation of the same station and reports whether the theory lies inside the simulated confidence interval.\n\nMethodology:\n- M/M/1: Wq = ρ/(μ−λ). M/M/c: Wq = C(c, a)/(cμ−λ) with the Erlang C probability C computed by a stable recursion. M/G/1: Wq = λE[S²]/(2(1−ρ)) (Pollaczek–Khinchine), M/D/1 is the special case SCV = 0. Kingman: Wq ≈ ((ca²+cs²)/2)·(ρ/(1−ρ))·te.\n- L = λW and Lq = λWq (Little’s law).\n- Verification builds a source → unlimited queue → station → sink model, simulates long replications (≈ 20 000 arrivals each, 5 % warm-up) and checks whether the theoretical Wq lies inside the Student-t interval of the simulated mean wait.\n\nFormulas assume steady state, infinite queue capacity and FIFO service. Kingman is an approximation.", "tags": [ "apps" ], "x-mcp-tool": "twinworks_queue_formulas", "requestBody": { "required": true, "content": { "application/json": { "schema": { "$ref": "#/components/schemas/TwinworksQueueFormulasInput" }, "examples": { "example1": { "summary": "M/M/1 at 80 % utilization, verified by simulation", "value": { "queue": "mm1", "arrivalsPerHour": 60, "serviceTime": 48, "verify": true, "replications": 10 } }, "example2": { "summary": "Two operators (M/M/2)", "value": { "queue": "mmc", "arrivalsPerHour": 100, "serviceTime": 60, "servers": 2 } }, "example3": { "summary": "Kingman: low arrival variability, high process variability", "value": { "queue": "gg1", "arrivalsPerHour": 50, "serviceTime": 60, "arrivalScv": 0.25, "serviceScv": 1.5 } } } } } }, "responses": { "200": { "description": "Calculation result envelope.", "content": { "application/json": { "schema": { "$ref": "#/components/schemas/TwinworksQueueFormulasResult" } } } }, "400": { "description": "Malformed JSON body.", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "422": { "description": "Invalid input (see errors[]).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "429": { "description": "Rate limit exceeded (see Retry-After).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } } } } }, "/api/v1/tool": { "get": { "operationId": "getToolDefinition", "summary": "Inputs, output schema and examples for this tool", "responses": { "200": { "description": "Tool definition.", "content": { "application/json": { "schema": { "type": "object" } } } } } } } }, "components": { "schemas": { "Problem": { "type": "object", "description": "RFC 9457 problem details. 422 responses include per-field `errors`.", "required": [ "type", "title", "status", "detail" ], "properties": { "type": { "type": "string" }, "title": { "type": "string" }, "status": { "type": "integer" }, "detail": { "type": "string" }, "code": { "type": "string" }, "errors": { "type": "array", "items": { "type": "object", "required": [ "field", "message" ], "properties": { "field": { "type": "string" }, "message": { "type": "string" } } } } } }, "TwinworksQueueFormulasInput": { "type": "object", "additionalProperties": false, "properties": { "queue": { "title": "Queue model", "type": "string", "enum": [ "mm1", "mmc", "md1", "mg1", "gg1" ], "default": "mm1" }, "arrivalsPerHour": { "title": "Arrival rate", "type": "number", "minimum": 0.0001, "maximum": 10000000, "default": 60, "x-unit": "parts/h" }, "serviceTime": { "title": "Mean service (cycle) time", "type": "number", "minimum": 0.0001, "maximum": 10000000, "default": 48, "x-unit": "s" }, "servers": { "title": "Servers (M/M/c)", "type": "integer", "minimum": 1, "maximum": 200, "default": 2 }, "serviceScv": { "title": "Service SCV (variance ÷ mean²)", "type": "number", "minimum": 0, "maximum": 100, "default": 1 }, "arrivalScv": { "title": "Inter-arrival SCV (G/G/1)", "type": "number", "minimum": 0, "maximum": 100, "default": 1 }, "verify": { "title": "Verify by simulation", "description": "Runs the same station in Twinworks (M/M/1, M/M/c, M/D/1 only).", "type": "boolean", "default": false }, "replications": { "title": "Replications for the check", "type": "integer", "minimum": 3, "maximum": 20, "default": 10 } }, "required": [ "arrivalsPerHour", "serviceTime" ] }, "TwinworksQueueFormulasResult": { "type": "object", "additionalProperties": false, "required": [ "schemaVersion", "tool", "computedAt", "inputs", "summary", "result", "units", "assumptions", "sources", "warnings", "presentation", "links" ], "properties": { "schemaVersion": { "type": "string", "const": "1.0" }, "tool": { "type": "object", "additionalProperties": false, "required": [ "id", "name", "version" ], "properties": { "id": { "type": "string" }, "name": { "type": "string" }, "version": { "type": "string" } } }, "computedAt": { "type": "string", "format": "date-time" }, "inputs": { "type": "object", "description": "Normalised inputs actually used, including applied defaults." }, "summary": { "type": "string" }, "result": { "type": "object", "required": [ "model", "rho", "stable", "Wq", "W", "Lq", "L", "exact", "verification" ], "properties": { "model": { "type": "string" }, "rho": { "type": "number" }, "stable": { "type": "boolean" }, "Wq": { "type": [ "number", "null" ] }, "W": { "type": [ "number", "null" ] }, "Lq": { "type": [ "number", "null" ] }, "L": { "type": [ "number", "null" ] }, "pWait": { "type": "number" }, "exact": { "type": "boolean" }, "verification": { "type": [ "object", "null" ], "properties": { "WqSim": { "type": "object", "required": [ "mean", "lo", "hi", "halfWidth", "n" ], "properties": { "mean": { "type": "number" }, "lo": { "type": "number" }, "hi": { "type": "number" }, "halfWidth": { "type": "number" }, "n": { "type": "integer" } } }, "utilizationSim": { "type": "object", "required": [ "mean", "lo", "hi", "halfWidth", "n" ], "properties": { "mean": { "type": "number" }, "lo": { "type": "number" }, "hi": { "type": "number" }, "halfWidth": { "type": "number" }, "n": { "type": "integer" } } }, "insideCi": { "type": "boolean" }, "replications": { "type": "integer" }, "horizonSeconds": { "type": "number" } } } } }, "units": { "type": "object", "additionalProperties": { "type": "string" } }, "assumptions": { "type": "array", "items": { "type": "string" } }, "sources": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "title" ], "properties": { "title": { "type": "string" }, "url": { "type": "string" }, "publisher": { "type": "string" }, "retrievedAt": { "type": "string" }, "note": { "type": "string" } } } }, "warnings": { "type": "array", "items": { "type": "string" } }, "presentation": { "type": "array", "items": { "type": "object" }, "description": "Display blocks used by the human page (same data)." }, "links": { "type": "object", "additionalProperties": false, "required": [ "human", "api", "openapi", "methodology" ], "properties": { "human": { "type": "string", "format": "uri" }, "api": { "type": "string", "format": "uri" }, "openapi": { "type": "string", "format": "uri" }, "methodology": { "type": "string", "format": "uri" } } } } } } } }