Mathematical Problems in Engineering

Volume 2015, Article ID 456485, 5 pages

http://dx.doi.org/10.1155/2015/456485

## Stochastic Simulation on System Reliability and Component Probabilistic Importance of Road Network

^{1}School of Civil Engineering, Zhengzhou University, Zhengzhou 450001, China^{2}SEU-Monash Joint Graduate School, Southeast University, Suzhou 215123, China

Received 29 September 2014; Revised 4 January 2015; Accepted 11 January 2015

Academic Editor: Wai Yuen Szeto

Copyright © 2015 Xiangyu Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Because of the combination explosion problem, it is difficult to use probability analytical method to calculate the system reliability of large networks. The paper develops a stochastic simulation (Monte Carlo-based) method to study the system reliability and component probabilistic importance of the road network. The proposed method considers the characteristics of the practical road network as follows: both link (roadway segment) and node (intersection) components are emphasized in the road network; the reliability for a link or node component may be at the in-between state; namely, its reliability value is between 0 and 1. The method is then implemented using the object-oriented programming language C++ and integrated into a RARN-MGG (reliability analysis of road network using Monte Carlo, GIS, and grid) system. Finally, two numerical examples based on a simple road network and a large real road network, respectively, are carried out to characterize the feasibility and to demonstrate the strength of the stochastic simulation method.

#### 1. Introduction

Reliability evaluation of a road network has been extensively studied in the literature. While the initial impetus appears to have been derived from the study of major natural events—such as earthquakes [1]—affecting the connectivity of a road network, it has had wider impacts on the way of thinking in which less severe but more frequency-occurring events may affect the operation of a road network [2]. A reliable road network should consider everyday disturbances, including minor accidents, on-street parking violations, snow, flooding, road maintenance, and traffic signal failures, all of which would lead to the performance deterioration of certain roadway segments or intersections in the network. Reliability of the road network reflects the quality of service that it would normally provide. Thus, the importance of a reliable road network cannot be overemphasized.

Although there are some calculating methods [3–5] about the system reliability of the road network, the component probabilistic importance measure, which implies probabilistic contribution of improving component reliability to the system reliability [6], is neglected. Besides, most of the existing studies deal with the connectivity, capacity, or travel time reliability of the link, while node (intersection in the road network) reliability is not taken into account.

Extending the definition of connectivity reliability, system reliability of a road network in this study is defined as a measure of network reliability, given the reliability values of each roadway (link) and intersection (node) component. Reliability describes the ability of a system or component to function under stated conditions for a specified period of time. A reliability value is theoretically defined as the probability of an item to perform its required or intended function under stated conditions for a specific period of time. The reliability of the road network reflects the ability of the road network in completing the traffic carrying function in specific time and conditions. Generally, its probability is used to measure the degree, known as road network system reliability.

Objective of this paper is to develop a stochastic simulation (Monte Carlo-based) method to study the system reliability and component probabilistic importance of the road network. First of all, it introduces the concept of system reliability and the Monte Carlo-based algorithm to calculate the extended connectivity reliability for an O-D pair and the system reliability for the road network. Then the paper proposes the method to calculate the component probabilistic importance. Finally, two numerical examples (a small network and a large real network) are analyzed to illustrate the accuracy and feasibility of the proposed algorithms. The final section gives some concluding remarks and discussion of future research.

#### 2. System Reliability Measurement

Consider a road network with links and nodes. Each link or node is regarded as a component of the road network. Let be the reliability value of a component, either a link or a node. is the number of components in the road network. The state or reliability value of a component can be considered as the capacity under conditions of degradation because of certain disturbances.

Let denote a random vector to group the random numbers. is a random variable with uniform distribution between 0 and 1. To calculate the system reliability, we first use the Monte Carlo simulation approach to generate realizations of random variable , denoted by . If , component fails; otherwise, component is reliable. Since is uniformly distributed, the larger the is, the smaller the probability that is, which indicates that there is a smaller probability that component fails. Hence, larger denotes a more reliable component. This is the judgment criterion for further using Monte Carlo simulation to calculate the system reliability. According to the failure state of each component, a path search algorithm could be used to judge whether the O-D pair is still connected [7].

This process is repeated times. In each test, a set of random numbers is generated and the connectivity for each O-D pair is judged. is the number of O-D pairs in the road network. Denoting the number of times that the O-D pair is connected by , the connectivity reliability can be calculated as follows: when the sample size approaches infinity. Mak et al. gave statistical analysis results on the relationship between quality of the approximated solution and sample size [8]. These results can guide us to take an appropriate sample size for a given instance.

Then the system reliability of the road network could be defined as where is the weighting value for O-D pair in the road network. The weighting value could be determined according to the degree of importance of each O-D pair and .

#### 3. Component Probabilistic Importance Analysis

In the road network system, contribution of each component to the system reliability is different. Variation of the reliability value of some components may significantly affect the system reliability; while some others may just have small or even few impacts. The component probabilistic importance could be used to evaluate probabilistic contribution of improving component reliability to the system reliability. This measure might be helpful for better understanding the mechanism of system reliability of road networks and thus gives more hints such as the upgrading of a roadway and the strengthening of traffic management to minimize the performance deterioration.

Denote the component probabilistic importance of component by . is the variation of the reliability value of component ; is the variation of the system reliability due to the variation of the reliability value of component .

Therefore, can be calculated using the Monte Carlo simulation. Assume the reliability value of component is 1, and the new road network under this condition is denoted by . Then where is the connectivity of O-D pair when the reliability value of component is 1; is the connectivity of O-D pair in the original network .

is defined as follows: where , if the O-D pair is connected in the test in network ; otherwise, . Similarly, , if the O-D pair is connected in network ; otherwise, . Both and are binary variables.

Besides, in the test, component in the network is reliable, because the reliability value has been assumed to be 1. Assume is the sample network for in the test, and is the sample network for in the test. Then the difference between networks and only depends on component . Hence, if, in the test, component in network is reliable, and will become identical: And if is connected in the test in network , is certainly connected in the test in network : if is disconnected in the test in network and is disconnected in the test in network : if is disconnected in the test in network while is connected in the test in network : According to (5)–(8), we can have According to (4) and (9), where denotes the number of times when component in network fails, is connected in network , and is disconnected in network .

Besides, under the condition that , there has been no possibility for reliability improvement of component ; namely, . Moreover, is the reliability value of a component (i.e., a link or an intersection) in the road network. It is an indicator to measure ability of component to complete the traffic carrying function in specific time and conditions, which is the result of many traffic supply and demand factors. Hence, rounding the component reliability to three or four decimals is always accurate enough in practical research, which ensures that formulation (3) is valid.

#### 4. Numerical Examples

Based on the proposed Monte Carlo-based algorithms for calculating the system reliability and component probabilistic importance, a simulation system called RARN-MGG (reliability analysis of road network using Monte Carlo, GIS, and grid) is developed. The system uses the object-oriented programming language C++ and includes three main modules: preprocessing module, calculation module, and postprocessing module (shown in Figure 1). It applies to any road network for calculating the system reliability and component probabilistic importance. The calculation module, which is the core of the system, is based on the proposed Monte Carlo-based algorithms and the grid computing technology.