# Highway Travel-Time Forecasting with Greenshields Model-Based Cascaded Fuzzy Logic Systems
## 1 Introduction  
An increased traffic flow on highways inevitably leads to longer periods of congestion and a higher risk of accidents. A greater vehicle volume creates bottlenecks and worsens existing congestion, particularly on multi-lane roads where accident rates tend to rise as traffic increases. According to statistics from the Freeway Bureau of Taiwan [1], daily vehicle usage on Taiwan's highways surpassed 90 million vehicle kilometers (MVKs) in 2024, representing a 3.4% increase compared to the previous year. Traffic congestion between cities has become a major concern for drivers and commuters, driven by rapid population growth and accelerated urbanization in modern cities. Addressing challenges related to highway development, transportation efficiency, congestion, road safety, and environmental pollution requires careful planning and innovative solutions. Intelligent Transportation Systems (ITSs), which integrate vehicles, roadside equipment, and traffic data centers, enable the incorporation of advanced technologies--such as artificial intelligence (AI), the Internet of Things (IoT), and computer networking--into transportation infrastructure and vehicles. These systems collect, analyze, process, and distribute traffic data to support better traffic management, route planning, and safety measures [2, 3, 4, 5]. Intelligence-poweredtechnologies and management strategies within ITS enhance road safety, improve traffic efficiency, and reduce environmental impact by enabling smarter route planning, optimizing traffic signal timings, and supporting autonomous driving. To fully realize the benefits of these technologies, accurate prediction of key traffic parameters is essential. Among these, reliable travel-time predictions play a critical role in ensuring timely arrivals, minimizing congestion, and improving the overall travel costs. Precise travel-time forecasts not only help drivers make informed route choices but also support traffic management efforts to reduce highway congestion [6; 7; 8]. For individual drivers, vehicle speed is the most convenient parameter for estimating travel time, as it can be predicted for different highway segments based on various traffic data collected from detection devices. Information related to vehicle speed enables ITS to provide drivers with real-time route guidance, helping them avoid congestion and optimize travel time. Traffic prediction schemes can be developed using either parametric or non-parametric modeling approaches [9; 10]. A traffic model based on highway data relies on both historical and real-time information to predict and manage traffic flow. The Greenshields, Greenberg, Underwood, and other nonlinear models are widely employed for predicting traffic conditions and supporting traffic management and control on highways [11; 12; 13; 14; 15; 16]. These models consider vehicle speed, traffic flow, and vehicle density as the key parameters influencing highway traffic dynamics. The Greenshields model establishes a fundamental relationship among speed, flow, and density, offering a foundational understanding of traffic behavior. The Greenberg and Underwood models, as nonlinear extensions of [PERSON] work, are designed to capture more complex and specialized traffic flow patterns [12; 13; 17]. These models offer valuable insights into the interactions among vehicle speed, flow, and density, supporting the development of intelligent traffic management systems. Other common parametric models for vehicle speed prediction and travel-time estimation employ statistical or regression analysis techniques. By capturing key characteristics of traffic conditions within a defined problem scope, these models can predict vehicle speed or other traffic status parameters through simulations or computations based on collected data [18; 19; 20]. In certain scenarios, hybrid algorithms that incorporate the autoregressive integrated moving average (ARIMA) time-series model have demonstrated a superior prediction performance [21; 22; 23].