Math & Numbers

M/M/1 Queue Calculator (L, Lq, W, Wq)

Steady-state M/M/1 metrics: ρ, P0, L, Lq, W, Wq, state probabilities, and Little's-law cross-checks; unstable λ ≥ μ inputs are rejected.

text· Text resultOpen tool

Quick start

Call this tool from your code in three languages.

cURL

curl -X POST 'https://api.elysiatools.com/en/api/tools/queuing-theory-mm1' \
  -H 'Content-Type: application/json' \
  -d '{"arrivalRate":4,"serviceRate":6,"decimalPlaces":4}'

API reference

Send a POST request with your inputs as JSON. File parameters require a separate upload first.

Endpoint

HTTP

POST https://api.elysiatools.com/en/api/tools/queuing-theory-mm1

Parameters

Name	Type	Required	Description
arrivalRate	number	Yes	Average arrivals per time unit (Poisson process). Must be below the service rate μ.
serviceRate	number	Yes	Average services per time unit (exponential distribution). Must exceed the arrival rate λ.
decimalPlaces	number	No	—

Response format

Text result

JSON

{
  "result": "Processed text content",
  "error": "Error message (optional)",
  "message": "Notification message (optional)",
  "metadata": {
    "key": "value"
  }
}

MCP integration

Add this tool to your Model Context Protocol server so AI agents can list and call it.

Server configuration

Add this block to your MCP client configuration:

mcp.json

{
  "mcpServers": {
    "elysiatools-queuing-theory-mm1": {
      "name": "queuing-theory-mm1",
      "description": "Steady-state M/M/1 metrics: ρ, P0, L, Lq, W, Wq, state probabilities, and Little's-law cross-checks; unstable λ ≥ μ inputs are rejected.",
      "baseUrl": "https://api.elysiatools.com/mcp/sse?toolId=queuing-theory-mm1",
      "command": "",
      "args": [],
      "env": {},
      "isActive": true,
      "type": "sse"
    }
  }
}

List available tools

After connecting to the SSE endpoint, list the exposed tools:

tools/list

{
  "jsonrpc": "2.0",
  "id": 1,
  "method": "tools/list"
}

Call this tool

Invoke the tool by its id, passing arguments built from its parameters:

tools/call

{
  "jsonrpc": "2.0",
  "id": 2,
  "method": "tools/call",
  "params": {
    "name": "queuing-theory-mm1",
    "arguments": {
  "arrivalRate": 4,
  "serviceRate": 6,
  "decimalPlaces": 4
}
  }
}

Chain multiple tools in one session with a comma-separated toolId list, e.g. /mcp/sse?toolId=png-to-webp,jpg-to-webp,gif-to-webp (max 20).

Questions or issues? Contact [email protected]

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Name

Type

Required

Description

arrivalRate

number

Yes

Average arrivals per time unit (Poisson process). Must be below the service rate μ.

serviceRate

number

Yes

Average services per time unit (exponential distribution). Must exceed the arrival rate λ.

decimalPlaces

number

—

{ "mcpServers": { "elysiatools-queuing-theory-mm1": { "name": "queuing-theory-mm1", "description": "Steady-state M/M/1 metrics: ρ, P0, L, Lq, W, Wq, state probabilities, and Little's-law cross-checks; unstable λ ≥ μ inputs are rejected.", "baseUrl": "https://api.elysiatools.com/mcp/sse?toolId=queuing-theory-mm1", "command": "", "args": [], "env": {}, "isActive": true, "type": "sse" } } }