MCP Server for M/M/1 and M/M/c Queue Simulation

A Model Context Protocol server that provides comprehensive resources, tools, and prompts for M/M/1 and M/M/c queuing system simulation and analysis.

New in v0.2.0: Full M/M/c support with comparison tools!

What is M/M/1 and M/M/c?

M/M/1 and M/M/c are fundamental queuing models in operations research:

• First M: Markovian (Poisson) arrivals

• Second M: Markovian (exponential) service times

• 1 or c: Number of servers (1 for M/M/1, c for M/M/c)

This MCP server enables LLMs like Claude to:

• Simulate both M/M/1 and M/M/c systems

• Compare separate queues vs pooled queue strategies

• Validate simulation parameters

• Calculate theoretical performance metrics

• Generate and execute SimPy simulations

• Analyze real-world scenarios (supermarkets, banks, etc.)

Related MCP server: M/M/1 Queue Simulation MCP Server

Features

📚 Resources (7)

• mm1://schema - Complete M/M/1 system schema

• mm1://parameters - Parameter definitions with constraints

• mm1://metrics - Performance metrics catalog

• mm1://formulas - Theoretical formulas

• mm1://guidelines - Implementation best practices

• mm1://examples - Pre-configured scenarios

• mm1://literature - References and citations

🔧 Tools (8)

• validate_config - Validate M/M/1 parameters and check stability

• calculate_metrics - Compute theoretical performance metrics

• run_simulation - Execute M/M/1 simulation (legacy)

• run_mmc_simulation - Execute M/M/c simulation (NEW!)

• compare_mm1_vs_mmc - Compare separate vs pooled queues (NEW!)

• analyze_cashier_problem - Supermarket scenario analysis (NEW!)

• compare_results - Analyze simulation accuracy

• recommend_parameters - Suggest optimal configuration

💬 Prompts (4)

• generate_simulation_code - Create production-ready SimPy code

• explain_mm1_theory - Educational content on M/M/1 theory

• analyze_results - Interpret simulation outcomes

• debug_simulation - Troubleshoot common issues

Installation

Option 1: Using uvx (Recommended)

uvx mcp-server-mm1

Option 2: Using pip

pip install mcp-server-mm1
mcp-server-mm1

Option 3: From Source

git clone https://github.com/yourusername/mcp-server-mm1.git
cd mcp-server-mm1
uv pip install -e .
mcp-server-mm1

Usage with Claude Desktop

Add to your claude_desktop_config.json:

macOS

Location: ~/Library/Application Support/Claude/claude_desktop_config.json

Windows

Location: %APPDATA%\Claude\claude_desktop_config.json

Configuration

{
  "mcpServers": {
    "mm1-simulation": {
      "command": "uvx",
      "args": ["mcp-server-mm1"]
    }
  }
}

Restart Claude Desktop and the server will be available!

Example Usage in Claude

1. Get M/M/1 Schema

User: Show me the M/M/1 queue schema

Claude uses: mm1://schema resource

2. Validate Configuration

User: Is λ=5, μ=8 a valid M/M/1 configuration?

Claude uses: validate_config tool
Result: ✓ Valid, ρ=0.625

3. Calculate Theoretical Metrics

User: Calculate theoretical metrics for λ=5, μ=8

Claude uses: calculate_metrics tool
Result:
- Utilization: 0.625
- Avg queue length: 1.0417
- Avg waiting time: 0.2083
- Avg system time: 0.3333

4. Run Simulation

User: Run a simulation with λ=5, μ=8 for 10,000 time units

Claude uses: run_simulation tool
Result: Simulation metrics + theoretical comparison + accuracy analysis

5. Generate Code

User: Generate SimPy code for λ=3, μ=10

Claude uses: generate_simulation_code prompt
Result: Complete, production-ready Python code

Development

Setup Development Environment

# Clone repository
git clone https://github.com/yourusername/mcp-server-mm1.git
cd mcp-server-mm1

# Install dependencies
uv pip install -e ".[dev]"

Run Tests

pytest

Code Quality

# Format code
black src/ tests/

# Lint
ruff check src/ tests/

# Type check
mypy src/

Local Testing

Use the MCP Inspector to test the server locally:

# Install MCP inspector (if not already installed)
npm install -g @modelcontextprotocol/inspector

# Run server with inspector
mcp dev src/mcp_server_mm1/server.py

This opens a web interface where you can:

• Browse available resources

• Test tool invocations

• Try prompt templates

• Inspect JSON-RPC communication

Architecture

src/mcp_server_mm1/
├── server.py          # FastMCP server with resources/tools/prompts
├── schemas/
│   └── mm1_schema.py  # M/M/1 system schema definition
├── simulations/
│   └── mm1_queue.py   # SimPy simulation implementation
└── utils/
    └── metrics.py     # Theoretical calculations

M/M/1 Theory Quick Reference

Key Formulas

Given arrival rate λ and service rate μ:

• Utilization: ρ = λ/μ

• Avg Queue Length: L_q = ρ²/(1-ρ)

• Avg Time in Queue: W_q = ρ/(μ(1-ρ))

• Avg Time in System: W = 1/(μ(1-ρ))

Stability Condition

System must satisfy ρ < 1 (λ < μ)

If ρ ≥ 1, the queue grows unbounded!

Contributing

Contributions are welcome! Please:

1. Fork the repository

2. Create a feature branch

3. Make your changes with tests

4. Submit a pull request

License

MIT License - see LICENSE file for details.

Related Work

This MCP server was developed as part of research on LLM-assisted simulation code generation for the Winter Simulation Conference (WSC) 2025.

References

• Model Context Protocol Documentation

• SimPy Documentation

• M/M/1 Queue - Wikipedia

Support

• Issues: GitHub Issues

• Discussions: GitHub Discussions

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