M/M/1 queue wait API

Solves an M/M/1 queue from arrival rate λ and service rate μ, returning utilization, average number in system/queue, and average wait in system/queue, with a stability check. Answers 'How long will requests wait in this queue?', 'What's the average backlog at this load?'.

Price$0.03per request

MethodPOST

Route/v1/scheduling/mm1-queue

StatusLive

MIME typeapplication/json

Rate limit120/minute

Cache0s public

schedulingcapacityqueuemm1queueing-theorywait-timeutilizationthroughput

API URLhttps://x402.hexl.dev/v1/scheduling/mm1-queue

Integration docs

Example request

{
  "arrivalRate": 6,
  "serviceRate": 8
}

Example response

{
  "utilization": 0.75,
  "avgInSystem": 3,
  "avgInQueue": 2.25,
  "avgWaitInSystem": 0.5,
  "avgWaitInQueue": 0.375,
  "units": {
    "wait": "hour",
    "rates": "per hour"
  },
  "stable": true,
  "interpretation": "ρ=75% busy; an arrival waits ~0.375 hour in queue, 0.5 hour total."
}

Input schema

{
  "type": "object",
  "required": [
    "arrivalRate",
    "serviceRate"
  ],
  "properties": {
    "arrivalRate": {
      "type": "number",
      "examples": [
        6
      ]
    },
    "serviceRate": {
      "type": "number",
      "examples": [
        8
      ]
    },
    "timeUnit": {
      "type": "string",
      "examples": [
        "hour"
      ]
    }
  }
}

Output schema

{
  "type": "object",
  "additionalProperties": true
}