# On the Local and String Stability Analysis of Traffic Collision Risk
## 6 Conclusions  
Conventional traffic stability analysis focuses on the evolution of the disturbance of vehicle motions, while it is still unknown how the safety condition (collision risk) evolves over time and space. In this study, the traffic stability analysis is extended to the collision risk analysis. A commonly used surrogate safety measure (SSM), gap time (GP), is adopted to quantify the collision risk. The linear analysis technique is employed to investigate small disturbances in the context of collision risk, specifically around an equilibrium state. Then, inspired by the conventional traffic stability analysis, risk local and string stability are defined, and the boundary conditions for risk local and string stability are formulated for a general car-following model. The study shows that risk local stability exhibits consistency with traffic local stability within a rational driving constraint. However, the risk string stability differs from the traffic string stability in terms of boundary conditions. In the specific rational range, the traffic string stability criterion is a sufficient but unnecessary condition for the risk string stability criterion. Numerical solutions are found and microscopic simulations with various car-following models are performed to validate the theoretical derivation and the applicability of the proposed risk stability analysis. Results show that both traffic stability analysis and risk stability demonstrate high consistency between theoretical derivation and simulation when the disturbance magnitude is small. This study has substantial implications for the enhancement of transportation safety. For example, by integrating the identified risk stability conditions into traffic management systems, a more adaptive response to risk disturbances could be developed. And understanding of risk propagation dynamics may engender safer driving behaviors, which could be further promoted in driver education initiatives. This work can be expanded in several directions. Firstly, to advance our understanding of traffic safety dynamics, future research should consider incorporating nonlinear methods for a more comprehensive understanding of car-following behavior under a wider range of conditions. Secondly, it would be interesting to incorporate vehicle heterogeneity, along with considering cases where \(\omega>1\), aligning with the driver characteristics outlined in [29; 32], to provide a more robust and comprehensive traffic safety analysis. Furthermore, this paper investigates safety from a theoretical perspective. In future work, applying real-world empirical data to validate the proposed risk stability analysis is an interesting direction. Such an application would not only enhance the practical relevance of the theoretical findings but also provide a deeper understanding of real-world traffic safety conditions and lead to more accurate prediction of collision risks. Conceptualization, T.D., [PERSON] ([PERSON]) and F.Z. ; methodology, T.D., [PERSON] ([PERSON]) and F.Z. ; software, T.D. and J.Z. ([PERSON]); validation, T.D., [PERSON] ([PERSON]), [PERSON] ([PERSON]), B.L. and F.Z. ; torsional analysis, T.D., [PERSON] ([PERSON]), [PERSON] ([PERSON]), B.L. and F.Z. ; investigation, T.D., [PERSON] ([PERSON]), [PERSON] ([PERSON]), B.L. and F.Z. ; writing--original draft preparation, T.D. and J.Z. ([PERSON]); writing--review and editing, [PERSON] ([PERSON]), B.L. and F.Z. ; visualization, T.D. and J.Z. ([PERSON]); supervision, F.Z. ; project administration, B.L. and F.Z. ; funding acquisition, B.L. and F.Z. All authors have read and agreed to the published version of the manuscript. This research was funded by the Singapore Ministry of Education Academic Research Fund Tier 1 RG79/21 and, in part, by the National Natural Science Foundation of China 62203123. The data presented in this study are available on request from the corresponding author.