Complete REST API with FastAPI for simulation, scenarios, and analytics

API Endpoints - Simulation (/api/simulation):
- POST /start: Start new simulation with custom config
  - Accepts SimulationConfigModel (Pydantic validation)
  - Returns session_id for result retrieval
  - Runs simulation synchronously

- GET /results/{session_id}: Get simulation results
  - Returns complete SimulationResultsModel
  - Includes coordinator and server statistics
  - Stability assessment

- DELETE /results/{session_id}: Clean up session
- GET /sessions: List all active sessions

API Endpoints - Scenarios (/api/scenarios):
- GET /: List all 5 predefined scenarios
  - Returns scenario metadata (id, name, description)

- GET /{scenario_id}: Get scenario configuration
  - Returns full SimulationConfigModel
  - Ready to use or modify

- GET /scenario_5/variations: Get parameter sensitivity variations
  - Returns all variations for scenario 5

- POST /{scenario_id}/run: Execute predefined scenario
  - Runs scenario and stores results
  - Returns session_id for retrieval

API Endpoints - Analytics (/api/analytics):
- POST /jackson: Analyze with Jackson's theorem
  - Input: SimulationConfigModel
  - Output: Analytical results (L, W, ρ, stability)
  - No simulation required

- GET /compare/{session_id}: Compare analytical vs simulation
  - Requires existing simulation session
  - Returns detailed comparison with percentage differences
  - Per-queue and system-wide metrics

- POST /stability: Quick stability check
  - Input: Configuration
  - Output: Stability conditions for all queues
  - Utilization calculations

Architecture:
- Modular router design (simulation, scenarios, analytics)
- Pydantic models for validation and serialization
- In-memory session storage (could use Redis in production)
- Comprehensive error handling
- CORS enabled for frontend integration

Testing:
- test_api.py: Complete API test suite
  - Tests all endpoints
  - End-to-end workflow
  - Includes examples of usage

Documentation:
- Auto-generated OpenAPI docs at /docs
- ReDoc documentation at /redoc
- Comprehensive endpoint descriptions

Features:
- Type-safe API with Pydantic validation
- Automatic request/response serialization
- Built-in API documentation
- CORS support for React frontend
- Session management for results

Phase 5 Complete ✓
Next: Phase 6 - Frontend React dashboard

Implement analytical analysis using Jackson's theorem for open networks

Jackson's Theorem Implementation:
- analytics/jackson.py: Complete Jackson theorem analyzer
  - Calculate effective arrival rates (λᵢ)
  - Calculate utilizations (ρᵢ = λᵢ/μᵢ)
  - Stability check (ρᵢ < 1 for all queues)
  - M/M/1 formulas: L = ρ/(1-ρ), W = L/λ
  - System-wide metrics using Little's Law
  - Handles both stable and unstable queues

Mathematical Formulas:
- Effective arrival rates for network topology
- Utilization: ρᵢ = λᵢ / μᵢ
- Average customers: Lᵢ = ρᵢ / (1 - ρᵢ)
- Average time: Wᵢ = Lᵢ / λᵢ (Little's Law)
- Average wait: Wq = W - 1/μ
- Total system: L_total = Σ Lᵢ, W_total = L_total / λ₀

Comparison Module:
- analytics/comparison.py: Compare analytical vs simulation
  - QueueComparison for per-queue metrics
  - NetworkComparison for system-wide comparison
  - Percentage difference calculations
  - Formatted comparison reports

Testing & Validation:
- tests/test_analytics.py: 7 tests validating Jackson's theorem
  - Simple M/M/1 queue (analytical vs theory)
  - Unstable queue detection (ρ > 1)
  - Network with multiple servers
  - Little's Law validation (L = λW)
  - Probability conservation
  - Effective arrival rates
  - Multi-server stability

Demo & Examples:
- demo_analytical.py: Compare analytical vs simulation for scenarios
  - Runs scenarios 1-3
  - Shows theoretical predictions
  - Shows simulation results
  - Detailed comparison tables
  - Percentage differences

Results:
- ✅ All 7 analytical tests pass
- ✅ Little's Law validated for all queues
- ✅ Stability detection working correctly
- ✅ Comparison reveals expected statistical variation

Observations:
- Analytical predictions are accurate for stable systems
- Simulation shows statistical variation (finite sample)
- Unstable queues detected correctly (ρ ≥ 1)
- Differences increase with higher utilization

Phase 4 Complete ✓
Next: Phase 5 - Backend API implementation

Add Pydantic models for API validation and 5 project scenarios

Pydantic Models:
- models/config.py: SimulationConfigModel with full validation
  - ServerConfig for each server in the network
  - Probability conservation validation
  - Conversion to internal SimulationConfig

- models/results.py: Complete results models for API responses
  - QueueStatisticsModel per queue
  - TimeSeriesDataModel for evolution tracking
  - HistogramDataModel for processing time distribution
  - SimulationResultsModel with all metrics

Predefined Scenarios:
- scenarios.py: 5 scenarios from project requirements
  - Scenario 1: 1 fast server (120ms) - instability test
  - Scenario 2: 1 fast + 1 slow server (120ms/240ms)
  - Scenario 3: 3 slow servers (240ms each)
  - Scenario 4: 1 fast + 1 medium (120ms/190ms) - compare with scenario 3
  - Scenario 5: Parameter sensitivity (vary λ and p)

- Theoretical utilization calculations for each scenario
- Scenario registry for easy access
- list_scenarios() function for API

Testing:
- test_all_scenarios.py: Comprehensive test of all scenarios
- Runs all 5 scenarios with variations
- Compares theoretical vs simulation results
- Summary table for performance comparison

Results Analysis:
- All scenarios execute successfully
- Stable systems show ρ < 1 as expected
- Some scenarios show slight instability (ρ ≈ 1.0) due to high load
- Parameter sensitivity variations demonstrate impact of λ and p

Phase 3 Complete ✓
Next: Phase 4 - Analytical module (Jackson's theorem)

Implement event-driven discrete simulation for queueing networks

Core Components:
- events.py: Event and EventType for discrete event simulation
  - Priority queue ordering by time
  - ARRIVAL and SERVICE_END event types

- request.py: Request entity with full visit tracking
  - QueueVisit records for each queue visited
  - Automatic calculation of wait/service/system times
  - Journey tracking through network

- random_utils.py: Random number generation
  - Exponential distribution (inverse transform method)
  - Probabilistic choice for routing decisions
  - Seed control for reproducibility

- queues.py: M/M/1 Queue implementation
  - FIFO discipline with single server
  - Complete statistics collection (utilization, avg times)
  - Exponential service times

- router.py: Probabilistic routing logic
  - Route from coordinator: exit (p) or server (qi)
  - Route from server: always return to coordinator
  - Probability conservation validation

- simulation.py: Main event-driven simulator
  - Priority queue (heapq) for event scheduling
  - Warmup period to reach steady state
  - Complete statistics generation
  - Support for multiple servers

Testing:
- test_random_utils.py: Validate exponential distribution (mean, reproducibility)
- test_simulation.py: End-to-end simulation tests
  - Stable and unstable systems
  - Multiple servers
  - Reproducibility with seeds

Demo:
- demo_simulation.py: Scenario 1 demonstration (1 fast server)
- Theoretical vs simulation comparison
- All 14 tests passing ✓

Phase 2 Complete ✓
Next: Phase 3 - Statistics collection and multi-server scenarios

Initialize Turborepo monorepo with Python backend and React frontend

- Setup Turborepo configuration with workspaces
- Configure Python FastAPI backend
  - Create project structure (core, analytics, api, models)
  - Add requirements.txt with FastAPI, Uvicorn, Pydantic, etc.
  - Basic FastAPI app with health endpoints
  - CORS middleware for frontend integration
- Configure React + TypeScript + Vite frontend
  - Install dependencies (Chart.js, D3.js, Zustand, Axios)
  - Setup Tailwind CSS with PostCSS
  - Create component directory structure
  - Basic landing page with Tailwind styling
- Add comprehensive README files
- Configure .gitignore for Python and Node.js

Phase 1 Complete ✓
Next: Phase 2 - Core simulation engine implementation