Assessing collision risk with unpredictable obstacles on transportation infrastructures

This paper presents a novel theoretical framework for quantifying vehicle collision risk with unpredictable obstacles on transport infrastructures (roadways and railways). While existing models primarily rely on empirical crash data or deterministic approaches, this study introduces a stochastic formulation integrating Poisson-based vehicle arrivals, normally distributed vehicle speeds, and braking dynamics influenced by reaction time and deceleration capabilities. The temporal occurrence of obstacles is modeled as a Poisson process, while spatial probability depends on obstacle geometry and infrastructure longitudinal and transversal exposure. The framework calculates both impact probability and overall collision risk, considering also road user vulnerability. A real use case application shows the framework capa- bilities by assessing the risk of falling road signs on the Florence-Pisa-Livorno (FIPILI) highway, demonstrating its effectiveness and reliability in quantifying collision likelihood under realistic conditions. This research supports transportation engineers and policymakers in developing mitigation strategies for exposed transport infrastructure, while also providing forensic engineers with a transparent and objective method for assessing accidents involving unpredictable obstacles, such as landslides, debris flows, rockfalls, animal crossings, and various other sudden disruptions that may interfere with infrastructure.

Stochastic Modeling of Collision Risk with Unpredictable Obstacles in Transportation Systems

Transportation safety remains a global concern, with vehicle collisions resulting in significant human and economic losses. According to the World Health Organization, road crashes account for approximately 1.35 million fatalities annually (World Health Organisation, 2018). As for railways, in the last report of the European Union Agency for Railways (ERA, 2024), in 2023 more than 1,500 significant railway accidents occurred in the EU, resulting in 1410 fatal or injury people.

While many accidents stem from well-documented factors such as speeding and distraction, a significant yet less studied risk involves unpredictable obstacles that suddenly appear on roadways and railways. These obstacles – ranging from fallen objects, rockfalls, debris flows, landslides, wildlife crossings, infrastructure failures, and many others – pose immediate hazards that leave drivers with little reaction time, increasing the likelihood of severe crashes (Cova & Conger, 2004; Fernández & Vitoriano, 2004).

Traditional risk assessment methodologies often struggle to adequately model these unpredictable events. Conventional approaches rely on empirical crash data and deterministic models that assume fixed traffic exposure and road conditions (Nicolet et al., 2016). However, these methods fail to capture the stochastic nature of vehicle arrivals, speed variability, and driver reaction times. Moreover, reliability-based methods, such as event-tree analysis, have been employed in specific contexts (e.g., landslides or rockfalls), where analysts define sequences of events leading to collisions (Lu et al., 2020). These methods introduce probability estimates for each branch of events, improving risk quantification. However, they typically assume fixed traffic densities and exposure probabilities, neglecting the stochastic nature of vehicle arrivals.

In contrast to deterministic approaches, stochastic traffic flow models provide a more robust way to evaluate collision risks by incorporating randomness in vehicle movements and interactions. Under low-to-moderate traffic conditions, vehicle arrivals on roadways are well described by a Poisson distribution, which captures the random yet statistically predictable spacing of vehicles (Lu et al., 2020).

Similarly, vehicle speed distributions play a critical role in collision risk assessment. Empirical studies show that under free-flow conditions, speeds tend to follow a normal (Gaussian) distribution, which accounts for natural variability due to driver behaviour and vehicle performance (Elvik, 2019). Incorporating speed distributions allows for a probabilistic treatment of stopping distances and reaction times, which are key in evaluating the likelihood of an impact when an obstacle suddenly appears.

Finally, obstacle occurrences on roadways are inherently rare and random, and can be modelled using Poisson processes for temporal probability (Jaiswal et al., 2010), whereas spatial probability can be determined by relying on obstacle size and roadway exposure (Nicolet et al., 2016). Unlike traditional models, which assume that vehicles are either always or never exposed to hazards, probabilistic formulations allow for a more dynamic representation of collision likelihood as a function of time and space.

To address these limitations, this study proposes a closed-form model integrating stochastic traffic modelling and probabilistic obstacle occurrence. The model considers key traffic characteristics, including Poisson-distributed vehicle arrivals, normally distributed vehicle speeds, and braking dynamics affected by driver reaction time and deceleration capabilities of vehicles. Furthermore, it incorporates temporal and spatial probabilities of obstacle occur- rence modelled as Poisson Process (temporal) and through geometrical definition of the shape of the obstacle and exposed areas (spatial), providing a comprehensive assessment of colli- sion risk with unpredictable obstacles on transport infrastructure.

A case study evaluating the risk of falling flag-type steel road sign portals along the FIPILI highway illustrates the applicability of the model and demonstrates how the framework can quantify impact probability and overall risk, offering valuable insights for transportation safety assessment.

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IF CRASC ’25 ha posto al centro del confronto tecnico ingegneria forense, crolli, affidabilità e consolidamento strutturale, riunendo a Napoli esperti del settore per analizzare cause dei dissesti, responsabilità tecniche e soluzioni avanzate per la sicurezza del costruito, tra ricerca, pratica professionale e ambito giudiziario. All'interno interviste e video delle relazioni.
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Methodology

Vehicular Collision Probability with an Unpredictable Obstacle

Vehicle arrivals are modeled as a Poisson process with an intensity rate 𝜆𝑣 (vehicles per sec- ond). The probability of observing 𝑁 = 𝑛 vehicles in a time interval 𝑇 is given by Equation 1:

Vehicle speeds are assumed to follow a probability density function 𝑓𝑣(𝑣), with mean speed 𝜇𝑣. The stopping distance 𝑑𝑠𝑡𝑜𝑝 of a vehicle is computed as in Equation 2:

Where 𝑡𝑟 is the reaction time of drivers and 𝑎𝑚𝑎𝑥 is the maximum deceleration capability of the vehicle. A collision occurs when the obstacle appears and a vehicle with speed equal to 𝑣 has a distance from the obstacle lower than 𝑑𝑠𝑡𝑜𝑝(𝑣). Given a Poisson traffic flow with in- tensity 𝜆𝑣 and mean speed 𝜇𝑣, the spatial density of vehicles at a location can be easily com- puted as in Equation 3:

It is important to note that the Poisson process inherently assumes a stochastic, random arrival pattern of vehicles, and in this scenario, the traffic flow is considered to be stationary, mean- ing that the rate of vehicle arrivals and their average speeds remain constant over time, pro- vided that 𝜆𝑣 and 𝜇𝑣 are not subject to temporal variations. Moreover, the traffic flow is assumed to be uniform in space on average, even though there may be small local variations due to the inherent variability in vehicle speeds and inter-arrival times. Therefore, the prob- ability of a vehicle being at a distance 𝑑 from the obstacle is proportional to the density 𝜌𝑣, which is constant on average under these assumptions. As stated before, a vehicle fails to stop in time if 𝑑𝑠𝑡𝑜𝑝(𝑣) > 𝑑.

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