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The authors model the emergence processes of road obstacles, such as fallen objects on roads, the deformation and destruction of pavements, and the damage and destruction of road facilities, as counting processes. Especially, in order to take into account the heterogeneity of the emergence risk of a variety of road obstacles, the authors model a mixture Poisson process in which the arrival rate of road obstacles is subject to a probability distribution. In detail, the authors formulate a Poisson-Gamma model expressing the heterogeneity of the arrival rate as a Gamma distribution and formulate the management indicator of the emergence risk of road obstacles. Then, a methodology is developed in order to design a road patrol policy that can minimize the road obstacle risk with a limited amount of budget. Furthermore, the authors empirically verify that it is possible to design road patrol policy based on the emergence risk of actual road obstacles with the proposed methodology, by studying the cases of the application of the methodology to general national roads.

Road administrators must make efforts to keep roads in a sound condition. Particularly, fallen objects on roads and the deformation or destruction of pavements have the risk of inducing a vehicle-damaging accident. Accordingly, road administrators are required to patrol roads constantly and remove fallen objects on roads and repair the deformation of pavements. On the other hand, amid the retrenched finance due to the dwindling birthrate and aging society, it is imperative to streamline administrative tasks also in the field of the maintenance and repair of road facilities. In the road administration costs, the proportion of road patrol costs is not low, and so it is necessary to discuss road patrol methods while considering efficiency as well as safety.

In general, road patrol is carried out at certain intervals. Therefore, the costs for road patrol are fixed, no matter whether or not there are fallen objects or damages on roads. On the other hand, the probability of discovering a fallen object or damage on roads in a unit period varies significantly according to road sections. As the frequency of road patrol is increased, it is possible to respond more swiftly to the emergence of an event that would degrade the safety and traffic flow on roads, and the risk of leaving a road obstacle on a road for a long time. Meanwhile, frequent road patrol results in the increase in patrol expenditures and the augmentation of social costs. Like this, there is a trade-off relation between the obstacle emergence risk and the road patrol costs, and so road administrators need to set up a goal for controlling the obstacle emergence risk and then design road patrol policies so as to minimize the road patrol costs.

The emergence processes of road obstacles can be modeled as counting processes in which each event occurs randomly. In general, the emergence of fallen objects and the deformation and damage of pavements can be modeled as Poisson processes. However, Poisson processes are restricted by the assumption that the mean value is equal to its variance [

Road patrol is carried out with the purpose of removing obstacles left on roads and swiftly responding to the damage or destruction of road facilities and so forth. The emergence processes of such road obstacles are random phenomena, and such emergence processes can be modeled as probabilistic processes. Also in the field of civil engineering, there have been many researches that modeled random arrival events using Poisson processes [

The research into mixture Poisson processes were pioneered by Fisher [

The road patrol frequency is closely related to the obstacle emergence risk. By decreasing the road patrol frequency, it is possible to reduce the patrol costs. On the other hand, when road obstacles are ignored for a long time, the obstacle emergence risk augments, causing traffic accidents. Like this, there exists a trade-off relation between the patrol costs and the obstacle emergence risk, via the road patrol frequency. In this circumstance, this paper models the emergence processes of road obstacles as mixture Poisson processes and proposes a methodology for designing road patrol policy with the purpose of curtailing the patrol costs effectively. As far as the authors know, there have been no researches that discussed rational road patrol policy based on empirical measurements of road obstacles, except this study. In addition, the Poisson-Gamma model used in this study is the same as that proposed by previous studies [

Let us model the emergence processes of road obstacles. Firstly, a target road is divided into several road sections. Then, suppose that the time-series data about the frequency of the road obstacle emergence in each road section is available. As shown in Figure

Emergence process of road obstacles.

Here, suppose that a road obstacle follows the Poisson arrival with a certain arrival rate

In Poisson processes, it is assumed that one kind of event occurs repeatedly with the same arrival rate. However, road obstacles include various kinds of fallen objects, road deformation, and damage and destruction of road facilities, and it is hard to believe that all of these obstacles emerge with the same arrival rate. It is rather appropriate to consider a phenomenon in which a variety of road obstacles emerge randomly. Suppose that many types of road obstacles emerge with different arrival rates and the arrival rate in a certain period is subject to a probability distribution. Namely, suppose that the arrival rates of road obstacles are subject to a probability distribution for each target road section. At the same time, suppose that road obstacles emerge in accordance with Poisson processes for each road section. The Poisson process in which arrival rates are subject to a probability distribution is called a “mixture Poisson process.” By utilizing mixture Poisson processes, it is possible to remove the constraint condition that expectation is equal to variance in Poisson processes. Accordingly, it is possible to model a more flexible counting process. With such a mixture Poisson process, the heterogeneity of arrival rates is expressed by a Gamma distribution, and the event emergence is described by a Poisson process model. The Poisson-Gamma model has the simplest model structure among mixture Poisson models, and this model can be expressed theoretically. Furthermore, this model has a characteristic that the number of road obstacles emerging in a certain period of observation can be expressed by a negative binomial distribution. Therefore, it is possible to derive readily the indicator for controlling the road obstacle risk.

In the Poisson-Gamma model, the number of road obstacles observed in a certain unit period is expressed by a probability distribution. However, in the data obtained after road patrol, observation period varies according to road sections. Moreover, in order to discuss road patrol frequency, it is necessary to model the effects of patrol cycle on the obstacle emergence risk. In this circumstance, the authors propose a Poisson-Gamma model that explicitly considers patrol cycle (observation period). In addition, in order to secure the operability of the risk control indicators, the arrival rate distribution is expressed by a Gamma distribution with mean 1.

Assume that the arrival rate of a road obstacle in a road section

By conducting road patrol, it is possible to obtain the information on road obstacles. Suppose that a total of

In the Poisson-Gamma model (

As the indicator for risk management, the authors propose the number of road obstacles. The risk management indicator can be defined for each road section or each route. Here, the risk management indicators are defined for each road section

Suppose that patrol is conducted on the road section

Distribution of road obstacle emergence.

Suppose that road patrol is carried out

As mentioned above, in the road section having the characteristic

In this study, the authors analyze the database of road patrol conducted on general national road, Route A (target section length: 82.6 km), Route B (88.3 km), and Route C (93.6 km). This database has accumulated the data of daily and night-time patrols from April 1, 2009 to March 31, 2010. The daily logs of road patrol record not only the type and number of road obstacles but also the routes and road sections where road obstacles were discovered and the patrol times. As tabulated in Table

Route data.

Route | A1 and A2 | B1 and B2 | C1 and C2 |
---|---|---|---|

Route length [km] | 82.6 | 88.3 | 93.6 |

Number of sample | 3,820 | 3,296 | 4,140 |

Number of fallen objects | 3,501 | 3,079 | 3,817 |

Number of average fallen objects | 8.48 | 6.97 | 8.16 |

In order to model the emergence processes of road obstacles, the Poisson-Gamma model was estimated. For this estimation, 11,256 sample data of normal patrol were available. As explanatory variables, the following parameters were adopted. Namely, the estimation equation can be expressed as follows:

Estimation results of the Poisson-Gamma model.

Poisson-Gamma model | ||||||

Maximum likelihood estimate | −4.82 | 0.11 | 3.20 | |||

( | (−15.51) | (14.56) | (17.25) | (4.22) | (3.99) | (20.22) |

Log likelihood | −18,325 | |||||

AIC | 36,663 |

Let us obtain the cumulative discovery probability of road obstacles in target area of Routes A, B, and C, using the above-estimated Poisson-Gamma model. By analyzing the relation between the cumulative discovery probability and the patrol cycle, it is possible to analyze how the probability of the discovery of road obstacles in the route or road section concerned changes due to the patrol cycle. The cumulative discovery probability can be obtained by calculating the probability of the discovery of one or more road obstacles under a given patrol cycle. This is equivalent to the subtraction of the probability that road obstacles do not emerge from the total probability 1. Namely, the cumulative probability of the discovery of road obstacles can be defined as follows:

Cumulative discovery probability in the Poisson-Gamma model.

In order to determine the optimum patrol cycle of the target route for road patrol, the risk management indicator for the patrol cycle mentioned and the optimum patrol cycle for the risk management limit are calculated for all road sections.

Firstly, the VaR index regarding the number of road obstacles

Figures

Risk management limit (number of road obstacles) and the minimum patrol cycle (confidence level

Risk management limit (number of road obstacles) and the minimum patrol cycle (confidence level

The above passages discussed the minimum patrol cycle that satisfies the risk management limit for each road section, for the proposed risk management indicator. It can be inferred that which is adopted as a risk management indicator depends on the situation of the roads managed by the road administrator concerned, but it can be considered that the proposed indicator is practical for gauging the risk management limit for road obstacles. In addition, even if another indicator is adopted, the risk management limit can be defined by adopting the same idea.

In this study, the authors have proposed a method for evaluating the risk of emergence of road obstacles, including fallen objects, road surface deformation, damages to road-attached facilities, and also a methodology for designing a road patrol policy that can curtail the patrol costs effectively. Through this study, the authors also proposed a method of expressing the risk of emergence of road obstacles based on the Poisson-Gamma model. In addition, for the risk management of road obstacles, the authors pointed out that the proposed indicator is important. Furthermore, the authors studied the case of application to national routes and empirically clarified that it is effective to adopt a mixture Poisson process model that takes into account the heterogeneity in the arrival rates of road obstacles, in order to describe the emergence processes of actual road obstacles. In addition, the authors designed a patrol policy that can appropriately control the obstacle emergence risk with a limited amount of budget, using the obstacle emergence risk management model proposed in this study.

The methodology proposed in this study is highly practical, but there still remain the following problems to be solved. The first problem is that the analysis target in this study was limited to specific national routes. In order to study a variety of road characteristic variables, it is essential to collect patrol data of a broad range of routes and accumulate the cases of application of the proposed methodology. The second problem is that this study assumed that the length of patrol cycle does not influence the probability distribution of arrival rates. There is a possibility that the emergence risk of road obstacles will depend on road patrol policy according to patrol cycle and road characteristics. One possible method for coping with such problems is to estimate the variance parameter

For conducting this study, the authors received a great deal of support; for example, some data was provided by the Road Administration Section, Kinki Regional Development Bureau, the Ministry of Land, Infrastructure and Transport.