Network function virtualization is a key enabling technology for the customization of network services in next-generation networks to support diverse applications. Most enterprise and network services contain specific network functions that are stitched together in a predefined sequence to form a service function chain. The deployment and scheduling of a service function chain onto the substrate network play a vital role in deciding the efficiency of resource utilization and the performance of network management. For a delay-sensitive network service request traversing a service function chain, the end-to-end packet delay is a crucial parameter that indicates the deployment performance. Transmission, propagation, processing, edge queueing, and virtualization delays all impact the order in which virtual network functions execute. Service level agreement violations and incorrect schedules are produced when the controller does not take edge queueing and virtualization delays into account. In this work, we propose a service function chain scheduling problem for the optimization of the end-to-end delay while considering transmission, propagation, queueing, virtualization, and processing delays. Then, we propose a scheduling approach based on the earliest finish times of the physical machines to minimize the end-to-end delay of the service function chain. The performance of the proposed service function chain scheduling approach using the earliest finish time is evaluated in terms of end-to-end delay, service level agreement violation ratio, resource utilization, and acceptance ratio. We compare our proposed algorithm with four existing approaches from the literature. Simulation results show that our proposed approach outperforms existing approaches in terms of end-to-end delay, service level agreement violation ratio, resource utilization, and acceptance ratio.