M/M/1 Queue Simulation Server

""" MCP Schema for M/M/1 Queue Simulation System This schema defines the structured context that will be provided to LLMs for generating M/M/1 queue simulation code. """ MM1_SCHEMA = { "schema_version": "1.0", "system_type": "M/M/1 Queue", "description": "Single-server queuing system with Poisson arrivals and exponential service times", "parameters": { "arrival_rate": { "symbol": "λ", "description": "Customer arrival rate (customers per unit time)", "type": "float", "constraints": { "min": 0.0, "exclusive_min": True }, "example": 5.0 }, "service_rate": { "symbol": "μ", "description": "Service rate (customers per unit time)", "type": "float", "constraints": { "min": 0.0, "exclusive_min": True }, "example": 8.0 }, "simulation_time": { "description": "Duration of simulation run (in time units)", "type": "float", "default": 1000.0, "constraints": { "min": 0.0, "exclusive_min": True } }, "random_seed": { "description": "Random seed for reproducibility", "type": "integer", "optional": True, "example": 42 } }, "system_characteristics": { "queue_capacity": "infinite", "num_servers": 1, "service_discipline": "FIFO", "arrival_process": { "type": "Poisson", "distribution": "Exponential inter-arrival times" }, "service_process": { "type": "Exponential", "distribution": "Exponential service times" } }, "performance_metrics": { "required": [ { "name": "server_utilization", "symbol": "ρ", "description": "Fraction of time server is busy", "formula": "λ / μ", "type": "float", "range": [0.0, 1.0] }, { "name": "average_queue_length", "symbol": "L_q", "description": "Average number of customers in queue (not including service)", "formula": "ρ² / (1 - ρ)", "type": "float" }, { "name": "average_waiting_time", "symbol": "W_q", "description": "Average time customer spends waiting in queue", "formula": "ρ / (μ * (1 - ρ))", "type": "float" }, { "name": "average_system_time", "symbol": "W", "description": "Average total time customer spends in system", "formula": "1 / (μ * (1 - ρ))", "type": "float" } ], "optional": [ { "name": "customers_served", "description": "Total number of customers served during simulation", "type": "integer" }, { "name": "queue_length_over_time", "description": "Time series of queue length", "type": "list[tuple[float, int]]" } ] }, "output_requirements": { "code_language": "Python", "framework": "SimPy", "version": ">=4.0", "include_visualization": False, "include_validation": True, "include_theoretical_comparison": True }, "implementation_guidelines": { "required_imports": [ "simpy", "numpy" ], "class_structure": { "config_class": { "name": "MM1Config", "type": "dataclass", "fields": ["arrival_rate", "service_rate", "simulation_time", "random_seed"] }, "metrics_class": { "name": "PerformanceMetrics", "type": "dataclass", "fields": ["average_queue_length", "average_waiting_time", "average_system_time", "server_utilization"] }, "simulation_class": { "name": "MM1Queue", "methods": [ { "name": "__init__", "parameters": ["config: MM1Config"] }, { "name": "customer_arrival", "parameters": ["customer_id: int"], "description": "Process for handling a single customer" }, { "name": "arrival_process", "description": "Generate arrivals following Poisson process" }, { "name": "run", "returns": "PerformanceMetrics", "description": "Execute simulation and return results" } ] } }, "validation_rules": [ "arrival_rate must be positive", "service_rate must be positive", "arrival_rate must be less than service_rate (stability condition: ρ < 1)", "simulation_time must be positive" ], "error_handling": [ "Raise ValueError for invalid parameters", "Check stability condition before running simulation" ] }, "theoretical_validation": { "stability_condition": { "formula": "ρ = λ / μ < 1", "description": "System must be stable (arrival rate < service rate)" }, "little_law": { "formula": "L = λ * W", "description": "Average number in system equals arrival rate times average time in system" }, "formulas": { "utilization": "λ / μ", "avg_queue_length": "ρ² / (1 - ρ)", "avg_num_in_system": "ρ / (1 - ρ)", "avg_waiting_time": "ρ / (μ * (1 - ρ))", "avg_system_time": "1 / (μ * (1 - ρ))" } }, "example_use_cases": [ { "scenario": "Low utilization system", "parameters": { "arrival_rate": 3.0, "service_rate": 8.0, "simulation_time": 5000.0 }, "expected_utilization": 0.375 }, { "scenario": "Medium utilization system", "parameters": { "arrival_rate": 5.0, "service_rate": 8.0, "simulation_time": 10000.0 }, "expected_utilization": 0.625 }, { "scenario": "High utilization system", "parameters": { "arrival_rate": 7.0, "service_rate": 8.0, "simulation_time": 20000.0 }, "expected_utilization": 0.875 } ] }