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Routing Protocol Is Taking A Vital Role In The Modern Internet Era. A Routing Protocol Determines How The Routers Communicate With Each Other To Forward The Packets By Taking The Optimal Path To Travel From A Source Node To A Destination Node. In This Paper We Have Explored Two Eminent Protocols Namely, Enhanced Interior Gateway Routing Protocol (EIGRP) And Open Shortest Path First (OSPF) Protocols. Evaluation Of These Routing Protocols Is Performed Based On The Quantitative Metrics Such As Convergence Time, Jitter, End-to- End Delay, Throughput And Packet Loss Through The Simulated Network Models. The Evaluation Results Show That EIGRP Routing Protocol Provides A Better Performance Than OSPF Routing Protocol For Real Time Applications. Through Network Simulations We Have Proved That EIGRP Is More CPU Intensive Than OSPF And Hence Uses A Lot Of System Power. Therefore EIGRP Is A Greener Routing Protocol And Provides For Greener Internetworking.