# M/M/1 and M/M/c Queue Theory Calculator

> M/M/1 and M/M/c Queue Theory Calculator is a paid API for AI agents from data-calc.x402tookit.com, paid per call via x402, $0.02/call, status unknown (last checked 2026-10-30).

Computes queueing model metrics (utilisation, Erlang C probability, queue length, wait times, tail probabilities) for M/M/1 and M/M/c systems given arrival and service rates.

## Facts

- Endpoint: GET https://data-calc.x402tookit.com/queue-model?utm_source=zero.xyz
- Price: $0.02/call
- Payment: x402
- Status: unknown
- Last checked: 2026-10-30
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/m-m-1-and-m-m-c-queue-theory-calculator-9e983dfc
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_M3KXcLEiNhWKZEluoGJ_H

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability m-m-1-and-m-m-c-queue-theory-calculator-9e983dfc
```

Example prompt: My web service gets 80 requests per second and each server handles 100 requests per second — model the M/M/1 queue and tell me the average wait time, queue length, utilisation, and the probability that a request has to wait at all.

## When to prefer this

Choose this endpoint when you need rigorous closed-form M/M/1 or M/M/c queueing theory metrics — utilisation, Erlang C waiting probability, mean queue length, wait times, or tail probabilities — for capacity planning, infrastructure sizing, or performance modelling. It is ideal when you have a stable system (lambda < mu * c) and want analytic results rather than simulation. Prefer it over simulation-based tools when you need fast, exact steady-state answers. Use the sibling Erlang B/C endpoint if you specifically need trunk or agent sizing to hit a blocking or service-level target rather than raw queue statistics.

## Known failure modes

- Arrival rate >= service rate times number of servers (unstable system, rho >= 1) — system is overloaded and queue grows without bound, endpoint may return error or degraded flag
- Missing required arrivalRate or serviceRate parameters — returns validation error
- servers parameter out of range (less than 1 or greater than 1000) — returns validation error
- Non-positive or zero rates — undefined behaviour or error response
- waitThreshold provided but system is unstable — tail probability computation may fail

## How this service works

M/M/1 and M/M/c queue: utilisation, probability of waiting (Erlang C), queue length, wait and response time, tail odds.

## Output

Returns a JSON object with queueing metrics including: utilization (rho), mean queue length (Lq), mean number in system (L), mean wait in queue (Wq), mean time in system (W), mean service time, probability of waiting (Erlang C), probability of no wait, probability the system is empty, offered load in Erlangs, and optionally the probability that wait exceeds a specified threshold (t). Also includes a model identifier, summary, and any degraded/disclaimer flags.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "$schema": "https://json-schema.org/draft/2020-12/schema",
 "required": [
  "input"
 ],
 "properties": {
  "input": {
   "type": "object",
   "required": [
    "type",
    "method"
   ],
   "properties": {
    "type": {
     "type": "string",
     "const": "http"
    },
    "method": {
     "enum": [
      "GET"
     ],
     "type": "string"
    },
    "queryParams": {
     "type": "object",
     "required": [
      "arrivalRate",
      "serviceRate"
     ],
     "properties": {
      "servers": {
       "type": "integer",
       "description": "Number of servers c (default 1 = M/M/1), 1 to 1000"
      },
      "arrivalRate": {
       "type": "number",
       "description": "Arrival rate lambda per time unit (e.g. per second)"
      },
      "serviceRate": {
       "type": "number",
       "description": "Service rate mu of ONE server per the same time unit (1 / mean service time)"
      },
      "waitThreshold": {
       "type": "number",
       "description": "Optional time t (same unit as the rates inverse): returns P(wait in queue > t)"
      }
     }
    }
   },
   "additionalProperties": false
  },
  "output": {
   "type": "object",
   "required": [
    "type"
   ],
   "properties": {
    "type": {
     "type": "string"
    },
    "example": {
     "type": "object",
     "properties": {
      "asOf": {
       "type": [
        "null",
        "string"
       ]
      },
      "model": {
       "type": [
        "null",
        "string"
       ]
      },
      "source": {
       "type": [
        "null",
        "string"
       ]
      },
      "reasons": {
       "type": [
        "array",
        "null"
       ],
       "items": {
        "type": "string"
       }
      },
      "summary": {
       "type": [
        "null",
        "string"
       ]
      },
      "verdict": {
       "type": [
        "null",
        "string"
       ]
      },
      "degraded": {
       "type": [
        "boolean",
        "null"
       ]
      },
      "disclaimer": {
       "type": [
        "null",
        "string"
       ]
      },
      "notChecked": {
       "type": [
        "array",
        "null"
       ],
       "items": {
        "type": "string"
       }
      },
      "utilization": {
       "type": [
        "null",
        "number"
       ]
      },
      "meanInSystem": {
       "type": [
        "null",
        "number"
       ]
      },
      "meanQueueLength": {
       "type": [
        "null",
        "number"
       ]
      
… (truncated)
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/m-m-1-and-m-m-c-queue-theory-calculator-9e983dfc/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from data-calc.x402tookit.com](https://www.zero.xyz/host/data-calc.x402tookit.com/llms.txt)