Design a distributed edge computing system for managing thousands of IoT devices with varying computational requirements. Your architecture should include: (1) hierarchical structure with fog and edge layers; (2) intelligent task distribution algorithm considering device capabilities, network conditions, and task requirements; (3) data caching and processing decisions based on freshness requirements and network constraints; (4) energy-efficient operation strategies for battery-powered devices; (5) security framework for device authentication and secure communication; (6) failure detection and recovery mechanisms; (7) dynamic adaptation to changing conditions; (8) performance optimization for latency-sensitive applications; (9) model for cost-benefit analysis of edge vs. cloud processing; (10) management dashboard for operators. Include algorithms with complexity analysis and simulation results demonstrating performance characteristics.