User:Mengying Cui

Vulnerability, resilience, robustness and reliability

Introduction
Transport network is one of the lifelines to provide transportation service for both for person and goods for the society as a whole. The ability of transport network to deal with the failure of certain nodes, links or subnetworks in the network, such as car accidents, road maintenance, or serious road congestions, shows its importance for the efficient operation of the whole transportation system. And for transport network, the conceptions of vulnerability, resilience, robustness and reliability could all presents such an ability.

Introduction of reliability of road network
For transport network, the reliability represents the possibility that people or goods could move from one place to another successfully. The importance of reliability for a road network was started to be concerned because of the recent earthquakes, like the earthquakes happened in Turkey in the year of 1999 and the Kobe earthquake in 1995 in Japan. Such natural disasters could damage the connections of the roadways and make the transport system collapse. For the earthquake in Kobe, the mobility of the network was completely lost since the disruption of the roadways. With the development of transportation network in cities, the reliability researches are not only for dealing with the natural disasters, but for analyzing the unpredictable variations caused by the uncertainties or turbulences happened on the road network. Such uncertainties could be resulted from the changes in the demand for transport services and the supply of that. A reliable transport network should have the ability to deal with these turbulences to guarantee an acceptable service level. And a higher reliability for transport networks could be considered as a higher quality of transportation systems.

Based on the previous studies, connectivity reliability, travel time reliability and capacity reliability are three main measures to analyze the reliability of transport network. The details are showing as follows,

Connectivity reliability
Connectivity reliability could be defined as the probability for the nodes in the networks to maintain connected. Connectivity is the simplest measure for reliability, and it reflects whether links are open or not or whether origins and destinations are connected or not for each OD pairs in the matrix. And terminal reliability is a special case in connectivity reliability, which also considers the paths between OD pairs. Terminal reliability could reflect the redundancy of road network that alternative routes could be used although the connection of certain links is damaged.

In a functional expression, for a certain link, its connectivity could be expressed as a binary variable , (1) An example of road network shows in Figure 1, which contain two nodes (A and B) and 5 links connected to these nodes. And no direction limitation is considered in the network. Four minimal paths could reach to node B from A, which are Hence, for such a network, X is 1 based on equation (1). And if some links are shutdown, the network still has the ability to maintain the connectivity between node A and B. For example, if link 1 does not work, path2 and path4 could be used to connect given nodes. And if both link1 and link2 are disabled, path2 still could connect node A and B.

However, some cut on the links could make the nodes disconnected. For instance, if both link1 and link3 are cut, X would be 0, which stands for the disconnection of given nodes. Questions: Can you find any other cuts?

The reliability of the whole network instead of only two nodes could be expressed as the structure functionφ(x), in which x is a vector of link Xa. For a series network, φ(x) is the produce of all the X, which is φ(x)=∏_i▒X_i. While for a parallel network, the structure function isφ(x)=1-∏_i▒〖〖(1-X〗_i)〗. The structure function for more complicated network could be derived by the minimal path sets, as equation 2, and the minimal cut set, as equation 3. And the value of reliability is the expected value ofφ(x).

Travel time reliability
Travel time reliability reflects the uncertainties of travel time. It is defined as “the probability that a trip can reach the destination within a specified time interval”. Travel time reliability concerns more about the turbulence on daily traffic conditions rather than the extreme ones as connectivity reliability, such as natural disasters. And it specified on the turbulences of travel time based on the traffic flow variations. Figure 2 shows travel time on turbulences of a segment of Northbound I-5 in 2005.

Travel time reliability shows its importance for the users of all traffic modes in transportation system, such as autos and transit, and for different purposes of trips, such as working, traveling, shopping or picking up kids. Hence, the improvement of travel time reliability is necessary to provide a higher quality of transportation system.

Traffic management used both average travel time and travel time reliability to optimize the operation of the system. The before and after comparison results shows the travel time reliability is a better measure than the simple average of daily travel time, which are showed in Figure 3. Figure 2 Figure 3 There are some simple measures, which could evaluate the travel time reliability. 90th or 95th percentile travel time stands for the longest travel time happened on specific routes, which reflects the worst delay with the heaviest traffic flow. Buffer index is the extra time that needed to arrive the destinations based on the average travel time with 95 percent certainty. While planning time index represents the total travel time to ensure a 95 percent certainty to reach the destinations. The relationship between this three measures is showed in Figure 4.

Figure 4

To measure the travel time reliability, the methods are more complicated. Most of the measurements consider the variation of travel time based on normal daily observations of travel time. For each link, the travel time is assumed to follow a normal distribution. The distribution of travel time for an entire path could be computed by combing the distribution of every link of this path. And the reliability will be the probability that the travel time plus uncertainty is less than an expected value of travel time.

Capacity reliability
Capacity reliability is defined as “the probability that the road network can accommodate a certain level of traffic demand” based on the reserved capacity of road network. Hence, capacity reliability could be computed as the probability that the actual travel demand is less than the maximum flow capacity or the highest possible travel demand.

The maximum flow capacity is the tricky here, since the many factors could affect the capacity, such as physical design of the roads or the travel behavior on route choice, which should be considered in the estimation of it. Network reserve capacity was used to calculate the maximum flow capacity, which could be expressed as the largest multiplier for the OD demand assigned to a transport network without exceeding the limitation of capacity,. The functional expression is showed as below, Where va is the traffic flow of link a on equilibrium with the OD demand of q and capacity of Ca.

And the capacity reliability to satisfy the required demand level Ur could be showed as follows, For calculation the capacity reliability, Chen use the Monte Carlo simulation to estimate the distribution of the Ur.

Introduction of vulnerability of road network
The vulnerability of road network could reflect the effect of a failure node, link or network on the quality of transportation network as a whole. And the changes of some important variables for network could evaluate the vulnerability, which are showed as variations. Hence, in a simple way, the vulnerability could be viewed as the inverse of the conception of reliability, which means that the higher reliability stands for the lower vulnerability for transportation network.

For vulnerability, many different definitions were proposed and no one has been generally accepted. Berdica defined vulnerability for transportation network as the susceptibility to incidents, which could cause significant reductions of serviceability. Such a definition could be presented as a wheel showed in Figure 5 Figure 5. While D’ Este and Tylor used accessibility as the measure to evaluate the consequences of failures in road network to show the vulnerability, which is showed as the following definitions.

=== Case study: an accessibility approach in assessing regional road network vulnerability ===

This case study chose the Green Triangle Region in Australia, which shows a great economical development potential because of its forestry and tourism industry, to analyze the vulnerability on the regional road network level. The reducing of the vulnerability for the road network in the Green Triangle Region could attract more demand. The region is located at the southern end of the border and between two metropolitan cities, Adelaide and Melbourne, which is showed in Figure 7. Figure 7

Glenelg Highway is an important road for the green Triangle region, and four critical links in the Glenelg Highway were chosen to analyze its vulnerability, which are Glenelg Highway near Dunkeld, Glenelg Highway near Hamilton, Glenelg Highway near Casterton and Glenelg Highway near Tarpeena. These links were emphasized on Figure 7.

To measure the vulnerability on the regional road network level, ARIA was used as the index to evaluate the accessibility, which is short for the Accessibility/Remoteness index of Australia. ARIA could be defined as the accessibility to the services centers, such as schools, shopping or hospitals. The service centers has the requirement that the population that for them to serve is larger than 1,000. And it is classified into 5 categories based on the population, which is showed in Table 1.

Table 1 ARIA service center categories

The ARIA index could be expressed as, where xiL is the distance from point i to the nearest service centers, while the xL stands for the average value of distance to the nearest service centers. Since the vulnerability shows the changes of accessibility for the before-and –after scenario, Tylor presents the relative accessibility changes as, where Ai0 is the accessibility for the complete network, and Ai1 is that for the network with failure links.

The results of vulnerability for the fours links chosen on the Glenelg Highway by using the ARIA index are showed on Figure 8. Figure 8

The vulnerability of each link shows its differences based on Figure 8. The degradation of Glenelg Highway near Dunkeld has significant effects on the accessibility of Dunkeld, and the changes of ARIA index has increased more than 57%. While the degradations of both Glenelg Highway near Hamilton and Glenelg Highway near Casterton have great effect on the accessibility of Casterton, and ARIA indices change around 20%. Combing with the changes of ARIA indices on other service centers, the Glenelg Highway near Dunkeld could be treated as the most critical link in those four

Resilience of road network
In transportation network, the conception of resilience has some definitions with respect to different aspects. Murry-Tuite proposed that “resilience is a characteristic that indicates system performance under unusual conditions, recovery speed, and the amount of outside assistance required for restoration to its original functional state”. And according to Heaslip et al.’s definition, “resilience is the ability for the system to maintain its demonstrated level of service or to restore itself to that level of service in a specified timeframe”.

The concept of the resiliency cycle was also introduced by Heaslip, which contains four stages: normality, breakdown, self annealing and recovery. The resiliency cycle is showed in Figure 9. Figure 9 The normality stands for the network conditions without the effect of any disruptions and the network performs as a normal operational system. In this stage, the network has the maximum efficiency. The breakdown stage starts because of the happening of disruptions. It describes the whole process of efficiency reduction suddenly or gradually depends on the characteristics of the disruptions. The efficiency drops to the minimum at the end of the breakdown stage. Self-Annealing begins after the breakdown stage, and both network and network users themselves try to deal with the conditions of the breakdown. Emergency management practices to ease the conditions of the breakdown, and network users may change into alternative routes or traffic modes. The efficiency of the network starts to rise, but in a modest way. Recovery is the stage that the disruptions caused in the breakdown stage are getting repaired, such as remove the obstructions or restore the damaged facilities. The speed for recovery, which is defined as rapidity, depends on the technologies or resources that are used on recovery. And after that, the system could reach a new stage of normality. And its efficiency may or may not equal to the original efficiency.

A graphical diagram of the resiliency cycle is showed in Figure 10. Figure 10 The graph shows the changes of network performance from the old normality to the new normality due to the disruptions happened in the breakdown stage. And the disruptions could be divided into gradual event and sudden event based on the time period that the breakdown stage experienced. The slope of the recovery line is the rapidity we mentioned before. And after recovery, the system could have three conditions: normal better than old, normal same as old and normal worse than old. The area between the solid and dash lines stands for the total performance loss because of the disruptions.

Considering different indices could be used to evaluate the network performance, the total performance loss could have different expressions respect to those indices, such as capacity, accessibility and so on. An example network respecting to capacity, which includes four nodes and five links, are showed in Figure 11. (a) Figure 20

We assume that, in the old normality, the capacity for each link is the value showed besides the link in Figure 20. In this stage, the network could maintain its performance with a relative higher capacity to satisfy the traffic demand. Imaging a disruption happened in Link AD, such as car accident or road maintenance, which decreases its capacity to 5. The network could be treated as in the self-annealing stage when some drivers choose Link AC and Link CD to go from Node A to Node D instead of choosing Link AD directly, or the network management starts to deal with this disruption. And the network could recover itself after the event would have been solved. Moreover, if the event happened to maintain the road condition, the new normality could reach a higher performance than the old one.

Robustness of road network
For road network, robustness is used to evaluate its ability to cope with disturbances happened in the network. Unlike reliability, robustness is the property for the network itself rather than for the users. But a road network with higher robustness could provide users a higher reliability.

Many measures of the network could affect the road network robustness, such as redundancy. Redundancy could be defined as the existence of alternatives for certain travel demand, such as alternative traffic modes or alternative paths. When disruptions happened, an efficient corresponding alternative could guarantee an accepted condition of the network. While the effect of interdependency on road network robustness could be showed by that the cascading disruptions could happen in a large part of the road network because of the failure of a critical node or link for a road network with relative higher interdependency. Hence, maintaining a reasonable hierarchy to minimize the interdependency is important to improve the robustness for a road network.

A. Nagurney and Q.Qiang used the unified network performance to measure the robustness of road network. And for a given road network G and the vector of the equilibrium demand, the network performance measure could be expressed as , Where dw is the demand in the equilibrium condition, while nw is the total number of OD pairs in the network. is defined as the equilibrium travel disutility, and it equals to the minimal cost of the path if the equilibrium traffic flow is not 0. The cost of a path is the function of its traffic flow. Considering the road connectivity, the cost function of path P could be expressed as , Where ca is the cost of link a, while is a binary function and it equals to 1 is link a is contained in the path. The importance of a node or a link could be evaluated by the following expression, G-g stands for the new network that g has been removed.

Considering the capacity degradation rather than the connections only, the capacity could be reflected in the cost function, which is showed as Ua in the following equation, And the network robustness measure is Where r is the degradation ratio of the capacity.