# Highway Travel-Time Forecasting with Greenshields Model-Based Cascaded Fuzzy Logic Systems
## 1 Introduction
Neural networks, particularly deep learning models, consist of multiple layers of interconnected artificial neurons that simulate aspects of human cognitive processing. These models learn from large datasets through optimization algorithms such as backpropagation and gradient descent. They are highly effective for capturing nonlinear relationships in complex data and have shown an exceptional performance in real-time traffic prediction, incident detection, and vehicle behavior analysis. However, neural networks often require extensive labeled data and high computational power and are often limited in interpretability. In contrast, fuzzy logic provides a rule-based, transparent, and human-readable framework for handling uncertainty and imprecision in data. It employs linguistic variables and incorporates expert knowledge through if-then rules, making it especially suitable for systems that are difficult to model mathematically. Fuzzy logic is particularly advantageous in ITS applications where explainability and system transparency are critical. A fuzzy logic-based model is generally easier to implement and maintain than traditional mathematical models. This paper proposes a highway travel-time prediction method based on cascaded fuzzy logic systems. In the proposed approach, each subsystem corresponds to a specific highway segment and is developed using the Greenshields model for short-range vehicle speed prediction. Two input parameters--traffic flow and density--are used in the fuzzy rules to determine the output variable, vehicle speed, which serves as the basis for evaluating the travel time. Fuzzy rules derived from Greenshields theory define the relationships between these inputs and outputs. During defuzzification, the outputs from activated rules are combined and converted into a single crisp value representing the predicted speed or travel time. Since a highway route typically consists of multiple segments, the proposed system establishes a Greenshields-model-based fuzzy logic subsystem for each segment, allowing it to dynamically adapt to real-time conditions segment by segment. The vehicle speed for each segment can thus be predicted in advance, enabling the estimation of the required travel time. Each fuzzy logic subsystem predicts the travel time for its corresponding segment and passes the result to the next subsystem. By cascading these subsystems sequentially along a route, the total travel time for the entire highway route can be calculated. The experimental evaluation compares the performance of the proposed fuzzy logic system with regression models using data collected from roadside sensors. This comparison highlights the advantages of the cascaded fuzzy logic system, particularly its ability to handle long-range highway routes and provide a more practical solution for travel-time prediction. Our main contributions are summarized as follows:  
1. We propose a cascaded fuzzy logic system for predicting travel times along planned highway routes. The system consists of multiple fuzzy logic subsystems, each built on the Greenshields model to predict the vehicle speed for a specific road segment. The predicted vehicle speed from each subsystem is dynamically updated based on current traffic conditions on that segment. By cascading these subsystems sequentially--one for each designated segment--and summing the predicted travel times, the total travel time for the entire route can be calculated and continuously updated according to the real-time traffic status on each segment. 2. The proposed fuzzy logic subsystem operates in two modes: congested and non-congested. It uses traffic flow and density as input membership functions in the respective modes. In each mode, the input variables are mapped to fuzzy sets defined over specified ranges that reflect realistic traffic conditions. For each highway segment, the Greenshields model, which describes the relationships among traffic density, flow, and vehicle speed, serves as the rule base to infer and generate fuzzy outputs for vehicle speed prediction.