This paper presents a large-scale traffic signal optimization system, OSaaS, that leverages low penetration rate vehicle trajectory data to optimize traffic signals without relying on physical road infrastructure. The system uses a probabilistic time-space diagram to connect a stochastic point-queue model and vehicle trajectories under Newellian coordinates, enabling the reconstruction of recurrent spatial-temporal traffic states from historical data. Optimization algorithms are developed to update traffic signal parameters, reducing delay and the number of stops at intersections by up to 20% and 30%, respectively. A real-world test in Birmingham, Michigan, demonstrated the effectiveness of the system, showing significant improvements in traffic flow. OSaaS provides a scalable, sustainable, and efficient solution for traffic light optimization, potentially applicable to every fixed-time signalized intersection globally. The system's closed-loop nature allows for dynamic optimization every few weeks, compared to the current practice of 3-5 years. The paper also discusses the method of moments estimator for traffic parameter estimation and the diagnosis and optimization methods for traffic signal timing parameters, including pair-wise coordination diagnosis.This paper presents a large-scale traffic signal optimization system, OSaaS, that leverages low penetration rate vehicle trajectory data to optimize traffic signals without relying on physical road infrastructure. The system uses a probabilistic time-space diagram to connect a stochastic point-queue model and vehicle trajectories under Newellian coordinates, enabling the reconstruction of recurrent spatial-temporal traffic states from historical data. Optimization algorithms are developed to update traffic signal parameters, reducing delay and the number of stops at intersections by up to 20% and 30%, respectively. A real-world test in Birmingham, Michigan, demonstrated the effectiveness of the system, showing significant improvements in traffic flow. OSaaS provides a scalable, sustainable, and efficient solution for traffic light optimization, potentially applicable to every fixed-time signalized intersection globally. The system's closed-loop nature allows for dynamic optimization every few weeks, compared to the current practice of 3-5 years. The paper also discusses the method of moments estimator for traffic parameter estimation and the diagnosis and optimization methods for traffic signal timing parameters, including pair-wise coordination diagnosis.