A capacity related reliability for transportation networks

Journal of Advanced Transportation, Vol. 33, No, 2, pp. 183-200

A Capacity Related Reliability For Transportation Networks

Anthony Chen Hai Yang

Hong K. Lo Wilson H. Tang

A capacity related reliability for transportation networks with random link capacity is introduced. It is defined as the probability that the road network can accommodate a certain level of traffic demand, and is built on the concept of network reserve capacity. Network reserve capacity is defined as the largest multiplier applied to an existing origin-destination demand matrix that can be allocated to a transportation network in a user-optimal way without violating the link capacities. Due to large variability associated with link capacities, a probabilistic approach is adopted to model the different physical and operational factors that often degrade the capacity of roadways. A Monte Carlo simulation procedure is developed to estimate the capacity related reliability measure. Numerical results are provided to demonstrate the feasibility of the approach.

Introduction

With increasing demands for better and reliable service, many systems (e.g. electrical power systems, water distributions, communication networks, etc.) have incorporated reliability analysis as an integral part in their planning, design, and operation (Ang and Tang, 1990). However, reliability analysis has received little or not attention in the study of transportation systems. A recent article by Nicholson and Du (1997) highlights the importance of studying the reliability of a degradable transportation system as a lifeline. Without a reliable costs

Anthony Chen, Hai Yang, Hong K. Lo, and Wilson H. Tang are at the Department of Civil Engineering, Hong Kong University of Science & Technology, Clear Water Bay, Hong Kong, China. Received December 1998; Accepted March 1999.

184 Anthony Chen, Hai Yang, Hong K. Lo and Wilson H. Tang

of repairs to these lifeline systems, but also the indirect costs which can be quite substantial. They also emphasize that transportation system reliability should not be limited to just severe disasters. It should also consider everyday disturbances (e.g., traffic congestion, accidents, and bad weather) to guarantee an acceptable level of service, even when the capacity of certain roadways are degraded by various physical and operational problems.

Previous studies on transportation network reliability mainly focused on two aspects: connectivity and travel time reliability (see Bell and Iida, 1997, for further details). Connectivity reliability is concerned with the probability that the network nodes remain connected. A special case of the connectivity reliability is the terminal reliability, which concerns with path availability of a specific origin-destination (OD) pair (Iida and Wakabayashi, 1989; Wakabayashi and Iida, 1992). For each given node pair, the network is considered successful if at least one of the paths is operational. A path consists of a set of components (e.g. roadways or links) which are characterized by zero-one variables to denote state of each link: operating or failed. When a path is operational, it assumes that any required amount of flow can pass through it. Capacity constraints on the links are not accounted for when finding the connectivity reliability. This assumption often leads to over-estimation of the reliability and under-estimation of the potential risk for the overall network. Efforts to incorporate capacity into connectivity reliability have been made for communication networks (Aggarwal, 1985), but remain as a future research topic for transportation networks. One difficulty in directly adapting the approach used in communication networks to transportation networks is that it ignores route choice behavior when calculating the maximum network flow. This negligence essentially assumes that drivers have no preference in choosing paths and can be assigned to take paths that lead to maximum flow in the network. This is clearly not feasible for transportation networks where route choice behavior must be considered. There is also a conceptual problem in travel demand analysis, as the connectivity analysis approach violates common basic assumptions about route choice. Another deficiency of the connectivity reliability is that it only allows for two operating states: operating at full capacity or completely failed with zero capacity. This binary state approach limits the application to handle everyday situations where links are operating in-betwee7, these two extremes.

The other measure is the travel time reliability. It is concerned with the probability that a trip between a given OD pair can be successfully

A Capacity Related Reliabili ty... 185

made within a specified interval of time. This measure is used to evaluate network performance under normal traffic conditions subject to daily flow variations. Asakura (1996) further extends the travel time reliability to consider capacity degradation due to deteriorated roads. He defines the travel time reliability as a function of the ratio of travel times under the degraded and non-degraded states. The ratio is exogenously supplied, which can be used a criterion to define the level of service that should be maintained despite certain links in the network are deteriorated. When the ratio is close to unity, it is essentially operating at ideal capacity. On the other hand, when it approaches infinity, i t implies that the destination is not reachable within a finite amount of time due to certain links are so severely degraded, which effectively remove the connectivity of those damaged links from the network. This definition of travel time reliability is consistent with the connectivity reliability and includes it as a special case.

Recently, Du and Nicholson (1997) propose an integrated equilibrium model to analyze a degradable transportation system whose components (roadway capacities) are subject to a range of levels of degradation. System surplus is used as a performance measure to assess the socio-economic impacts of system degradation. The reliability of interest here is the probability that the reduction in flow of the system is less than a certain threshold, as a result of capacity degradation of the network.

The different reliability measures discussed above are useful to assess different impacts concerning the reliability of transportation networks. However, none of the above measures is suitable to assess the capacity reliability of the network - the probability that the network capacity can accommodate a certain demand level. One natural question arises is how to calculate the maximum network capacity while accounting for route choice behavior. Conventional maximum flow model is not appropriate here since it assumes drivers must cooperate in choosing their paths to obtain the maximum possible flow in the network. In this study, we introduce the concept of network reserve capacity to estimate the maximum flow capacity of a transportation network that is consistent with the Wardropian route choice behavior. Link capacity is treated as a random variable to allow operating at multiple discrete states or infinite continuous states ranging from zero to one. This extension allows the reliability measure to model everyday events without restricting to just major disasters, as in the connectivity reliability model. A side product of this capacity related reliability is the


travel time reliability, which can be obtained directly when solving the traffic equilibrium problem to obtain the maximum network reserve capacity.

The next section defines the maximum flow capacity of a transportation network using the concept of network reserve capacity and discusses the treatment of link capacity as a random variable to allow for various levels of capacity degradation. Section 3 presents the Monte Carlo simulation procedure to estimate the capacity reIated reliability. Section 4 illustrates the approach with a small but realistic network. Conclusions and future research issues are provided in Section 5.

Maximum Flow Capacity Of A Transportation Network - A Probabilistic Treatment

Conventionally in communication networks, the maximum flow capacity is estimated using a maximum flow model. This method is not suitable for transportation networks where route choice of drivers must be explicitly considered. Here we apply the concept of network reserve capacity, introduced by Wong and Yang (1997) for a general signal- controlled road network, to estimate the maximum flow capacity of a transportation network with a route choice. Network reserve capacity is defined as the largest multiplier applied to an existing origin-destination (OD) demand matrix that can be allocated to a transportation network without violating the link capacities or exceeding a prespecified volume to capacity ratio (level of service). Mathematically stated, it is to find the maximum OD matrix multiplier p subject to the link flows resulting from the user equilibrium route choice model not exceeding the link capacities:

subject to: v, 019)s C,, tl a E A (2)

where v, 01s) is the equilibrium flow on link a with the demands of all OD pairs being uniformly scaled by p times the based OD demands q , and C, is the capacity of link a. Route choice behavior is explicitly considered in the equilibrium constraints, which bound the equilibrium link flows obtained by solving traffic equilibrium problem below their

A Capacity Related Reliability.. . 187

corresponding capacities. As the scaled demand approaches the network capacity, the equilibrium constraints will have a substantial effect on the distribution of flow as well as on the network reserve capacity. This approach of finding the maximum flow capacity is markedly different from the conventional maximum flow model.

Given the network topology with known link capacities and OD demands, the above process of finding the maximum OD matrix multiplier is essentially deterministic. In real world situations, however, capacity of a link (roadway section) is influenced by a wide variety of physical and operational characteristics (e.g. roadway type, geometric configuration, mixture of vehicle types, weather conditions, etc.). These factors often degrade the capacity of the transportation networks. For some extreme cases, the destruction to the links may be so severely disrupted by natural disasters (e.g. earthquakes), which effectively remove its connectivity from the network, reducing the availability of routes for the users as well as the capacity of the transportation network. Needless to say, roadway capacity is not deterministic in real life situations. It is stochastic in nature. A reasonable way to capture the stochasticity associated with roadway capacity and its impact on the maximum flow capacity of a transportation network is to model it as a random variable with defined distributions. Both discrete and continuous distributions can be used here, as long as the probability sums up to unity.

To assess whether the network capacity is sufficient to accommodate the demand, a probabilistic approach is adopted to estimate the largest OD demand matrix multiplier. Let C, be a nonnegative random variable that represents the capacity of a link. Since the largest OD matrix multiplier p depends on link capacities, p is also a random variable and it can be computed subject to error. However, it is difficult, if not impossible, to derive an analytical expression for the distribution of the maximum OD matrix multiplier for a given distribution of link capacity. Since there exists no closed form solution for p given a distribution of link capacity, a Monte Carlo simulation is used to estimate the distribution of p , which is also the distribution for the capacity related reliability of the transportation network.

This study examines the impact on the transportation system reliability, with different percentages of the network links subject to degradation at different ranges of capacity reduction. Hence, we are interested in estimating the probability that the network reserve capacity is greater than or equal to the required demand for different levels of

188 Anthony Chen, Hui Yung, Hong K. Lo and Wilson H. Tang

capacity degradation. This probability is also the capacity related reliability, which predicts how reliable the existing network capacity can accommodate the required demand despite certain links in the network are degraded. Let p, be the required demand level, the capacity related reliability satisfying F, is given as:

When there is no demand, the capacity related reliability is Rk,, = 0)= 1. On the other hand, when demand approaches infinity, Rk, = =)= 0 since there is only finite capacity for the transportation network. It should be noted when the link capacity takes only binary values of zero or one; it is the connectivity reliability measure. Thus, the capacity related reliability proposed here includes the connectivity reliability as a special case.

As mentioned before, a by-product of the capacity related reliability is the travel time reliability. The travel time reliability is concerned with the probability that a trip between an OD pair can be successfully made within a specified time. Two definitions are given in Section 1 - one is subject to demand variations and the other is subject to capacity variations. For this study, we are more interested in the latter definition even though both travel time reliability definitions can be estimated with the Monte Carlo simulation procedure described in Section 3. For the travel time reliability subject to degraded capacity, let C and C, be the vectors of degraded and non-degraded states and the corresponding travel times between origin r and destination s of these two states be denoted as u, (C) and u, (C,) . The travel time reliability is defined as the probability T, (0) of whether the ratio of u, (C) to u, (C,) is kept within an acceptable level 0 . The value 0 can be interpreted as the level of service that should be maintained even when capacity degradation has occurred on certain finks in the network.

Monte Carlo Simulation Procedure

With the advancement of computer technology, it opens the possibility to efficiently analyze many problems using simulation methods. Particularly, there has been increased interest in using Monte


Carlo simulation to estimate the reliability measures by simulating the random behavior of the system (Billinton and Li, 1994). In fact, Monte Carlo methods are often preferred when complex operating conditions are involved andfor the number of component states is large (e.g., many combinations resulting from different levels of capacity degradation). In addition, it can calculate not only the reliability measures in the form of expected values from the random variables, but also the distributions of these measures, which in general cannot be done in the analytical techniques.

Using the Monte Carlo simulation method, we develop a procedure to estimate the distribution for the largest OD demand multiplier given a set of random link capacity samples. The procedure, as shown in Figure 1, can be described as follows:

Step 0: Initialize the multiplier value p = po . Step 1: et sample number k := 1. Step 2: Generate a vector of link capacity C, = {C, ;--, C,} Step 3: Perform the traffic assignment (TA) with the scaled demand

Step 4: Collect statistics to construct reliability distribution function Step 5: If sample number k is less than the required sample size k,, ,

then increment sample number k := k + 1 and go to Step 2. Otherwise, go to Step 6.

Step 6: If the current valuep is the largest multiplier, then stop. Otherwise, increase the value p by 6 p and go to Step 1.

In Step 2, capacity of each link is randomly generated with a defined distribution. In the present study, capacity degradation of each network link is treated independently. In realistic situations, capacity degradation

of network links may need to be simulated jointly to account for dependency among links. The main steps can be summarized as follows:

p q and link capacity vector C, .

Step 2.1 :Generate U, from a standard uniformly distributed random number to determine whether the link is subject to capacity degradation or not.

190

b

Anthony Chen, Hai Yang, Hong K. Lo and Wilson H. Tang

Set k = 1

). Generate C,

Perform TA with and c,

Collect Statistics

Yes, k = k +I *

Figure 1. Monte Carlo simulation procedure


Step 2.2: If U , 5 d where d is the percentage of network links subject to capacity degradation, then generate U, to determine the range of capacity reduction. Otherwise, the link is kept at its non-degraded capacity. Repeat Steps 2.1 and 2.2 until all links are completed. Step 2.3:

The range of capacity reduction in Step 2.2 follows a predefined distribution. Here, we used two distributions to test the performance of the network: (1) Maximum degradation is half of its link capacity (This ensures that

all the OD pairs will have at least one functional path in the degraded network if the existing non-degraded network has one).

(2) Maximum degradation is all of its link capacity (Since all OD pairs in the ideal network have more than one alternative path, the chance of totally disconnecting an OD pair is low. Here we test which links once completely removed will have a greater impact on the OD travel times as well as the total system travel time). For each sample with the scaled demand p q and link capacity

vector C, in Step 3, the traffic assignment (TA) is essentially deterministic. Thus, a variety of algorithms can be used to solve TA. These algorithms can be broadly classified into two groups - link-based and path-based algorithms. Link-based algorithms operate in the link space and yield only link-flow solution; path-based algorithms operate in the path space and produce the aggregate link-flow solution as well as the individual path-flow solution. We choose to use a path-based algorithm for two reasons: (1) It has been shown that path-based algorithm, such as the gradient

projection algorithm, converges much faster than the commonly used Frank-Wolfe algorithm which is a link-based algorithm (see Jayakrishnan et al., 1994, for details of the gradient projection algorithm).

(2) Path-based algorithms have good reoptimization capabilities with respect to changes in both demands and network topology. The equilibrium path set from the previous run can be used as an initial path set for the next run in the Monte Carlo simulation. This can significantly reduce the computation for the subsequent runs. It should be noted that the equilibrium link-flow solution resulting from a link-based algorithm does not share the same advantage as the equilibrium path set. Since capacity of certain links can vary widely


in consecutive runs, using the equilibrium link-flow solution from the previous run may give an erroneous initial solution and can actually retard the convergence. This is especially true in link-based algorithms of the Frank-Wolfe types when an incorrect flow is assigned in the early iterations (Janson and Gorostiza, 1987). In Step 4, we collect the count of successful samples (volume-

capacity ratio less than 1.0) to construct the distribution of the capacity related reliability. Depending on the purpose of the study, other statistics could be collected resulting from TA. For example, OD travel times, link utilization (average volume-capacity ratio or average number of link use for all successful samples), total system travel time, and others.

Numerical Results

Tuen Mun Road Tuen ,Mun

A

A c2

Castle Peak Road

A

A A Castle Peak Road

Figure 2. Tuen Mun network

A c 3

Kowloon

To demonstrate the Monte Carlo simulation procedure, we applied it to the Tuen Mun Road corridor network, as shown in Figure 2. This is a highly aggregate, but realistic network which connects Tuen Mun and Kowloon in Hong Kong. The network consists of 3 zones, 4 nodes, and 10 links. The link travel time function used for this study is as follows:

A Capacity Related Reliability.. .

1 3 4 Attraction

I93

1 3 4 Generation 220 4952 . 5 172

313 127 440 4492 78 4570 4805 298 5079 10182

where v, , a,, andC, are the flow, free-flow travel time, and capacity of link a, respectively. p, and y, are the calibrated parameters of link a. The values of the link characteristics (free-flow travel time and link capacity) from actual observations are given in Table 1. Table 2 provides the base origin-destination (OD) demands.

Table 1. Link data of the Tuen Mun Road network

Table 2. Origin-destination matrix demand of the Tuen Mun Road network

4. I Preliminaries

Before performing the Monte Carlo simulation, convergence tests on the sample sizes and seed numbers are determined first. Three simulations of sample sizes of 1,000, 5,000, and 10,000 were


1.2

1 .o

.- 2 0.8 B

0.6

0 0.4 n

0.2

0.0

- .- 5, c) .-

0

I I . . , , , , , - *

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Demand Level

Figure 3. Effect of sample sizes on the capacity reliability

examined. It was found that in Figure 3 the capacity related reliability distribution using 1,000 samples was statistically different from the ones using 5,000 and 10,000 samples, whereas the capacity related reliability distributions for the two larger samples showed no significant difference. Thus, a sample size of 5,000 was used for all subsequent simulations. Three seed numbers were randomly selected to initialize the random number generator to test the impact of different initial conditions. As shown in Figure 4, all three seed numbers produced similar probability distribution functions with no significant differences. This showed that the different initial seed numbers had no influence on the estimation when the sample size was chosen sufficiently large.

To determine the upper bound of the largest OD demand multiplier, a deterministic traffic assignment was performed with the non-degraded link capacities by incrementally increasing the multiplier to the base OD demand matrix until at least one of the equilibrium link flows exceeds its link capacity. The largest multiplier of p* = 0.9 was found for the test network. This implies that the current network was overloaded by lOO(1- p* ) percent of the base OD matrix q . Note that the magnitude of the OD demand is not important here. It is the distribution pattern of the OD matrix that is of concern.


-m- seed 97531

.r" 0.8

0.6 .-

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Demand Level

Figure 4. Effect of different seed numbers on the capacity reliability .

4.2 Capacity Reliubility

To study how the reliability of the network is affected by the increase in traffic demand, the capacity reliability is examined by increasing the percentages of network links subject to different ranges of capacity degradation. Specifically, we randomly selected 25, 50, 75, and 100 percentage of the network links for two ranges of capacity degradation: a maximum capacity reduction of 50 and 100 percent (see Step 2 of the Monte Carlo simulation procedure). The 0 percentage is the upper bound of the largest OD demand multiplier, which corresponds to the ideal case where all links are operating at full capacity. The traffic demand is incrementally and uniformly increased for all OD pairs and the corresponding capacity related reliability are plotted. Figures 5 and 6 show the two cases with a maximum of 50 and 100 percent capacity reduction, respectively. Both figures indicate that the capacity reliability degrades as demand increases. The reliability degrades even further when higher percentage of network links and larger capacity reduction are specified. A noticeable difference between Figures 5 and 6 is that the network remains 100 percent reliable for low demand levels in the maximum of 50 percent capacity reduction case, whereas for the


maximum of 100 percent capacity reduction case the network may be so severely disrupted that the 100 percent reliability is not guaranteed even for low demand levels.

1 1 .2 ,

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Demand Level

Figure 5. Effect of increasing the percentage of degradable links at a maximum of 50 percent capacity reduction on the capacity reliability.

1.2

1 .o

.- 5 0.8 n

= 0.6 K *

- .- - * -

0.4 4 0

0.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Demand Level

Figure 6. Effect of increasing the percentage of degradable links at a maximum of 100 percent capacity reduction on the capacity reliability.

4.3 Travel Time Reliability


* 1.0

d

CI .- - 5 .- m 0.8 -

i! O6 i= 5 0.4

e l- 0.2

0.0

>

-OD (3.1)

-OD (3.4)

-OD (4.1)

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.0C

e

Figure 7. Origin-destination travel time reliability for all OD pairs at demand level 0.7.

A by-product of the capacity related reliability is the travel time reliability. Here we show the travel time reliability subject to capacity variations as discussed in Section 2. Figure 7 depicts the travel time reliability of each individual OD pair as a function IJLlfor a demand level of 0.7. As mentioned before, 0 can be used as an indicator to measure the level of service of the network. A lower value of 8' implies that the network is operating at a higher level of service. As the value of 8 increases, the travel time reliability approaches unity for all OD pairs. In the worst case as shown here, travel time can go up to three times over the non-degraded travel time in order to guarantee 100% reliability. In Figure 8, we plot the source-terminal travel time reliability for OD pair 2 by fixing the demand level of all other OD pairs at 0.7 and varying the demand on OD pair 2 from 0.3 to 0.9. As expected, the source-terminal travel time reliability for the four different demand levels is monotonically non-decreasing with respect to 8 and approaches unity as 8. increases. However, for a given level of service 8, the reliability value for different demand levels can be very different. As an example,

Note that the lowest value of 8 starts at one.

198 Anthony Chen, Hai Yang, Hong K . Lo and Wilson H. Tang

consider 8 = 2.0 Othe reliability value for the two lower demand cases (0.3 and 0.5) is loo%, while it drops to 86% and 68% as the demand increases to 0.7 and 0.9, respectively. This result is no surprise since traffic congestion grows as a result of higher demand, which directly reduces the travel time reliability for a given level of service 8.

1 .o s .- a 0.8

.- E

d z

- -

i= 5 0.4

0.2 > m

I

r I c ' 0.0 d

1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50

e

Figure 8. Source terminal travel time reliability of OD pair (1,4) for different demand levels

Conclusions And Future Research

A capacity related reliability measure has been introduced using the concept of network reserve capacity. This method of estimating maximum network capacity differs from the conventional maximum flow model in which driver behavior is explicitly considered. To model everyday situations, capacity of a Iink is treated as a random variable to allow for different levels of degradation. This reliability measure provides a probabilistic way to assess the existing network capacity, which could be used to improve the planning, design, and operation of transportation networks. It can be integrated into the traditional planning process to model networks under uncertainty to better quantify the performance of the network. While the concept of network reserve


capacity has provided a feasible approach to determine network capacity with a route choice, it is restricted to a common multiplier for all OD pairs. This assumes every OD pair will have a uniform growth or decline in its OD demand pattern. Relaxing this limitation can yield information regarding the spatial distribution of the demand pattern to reflect non- uniform growth, which can be very useful in zonal land-use development plans. In addition, link dependency should be considered when generating capacity degradation of each network link. It is also necessary to develop practical methods to identify critical links in the network whose degradation would have a large reduction in the network reserve capacitiy.

Acknowledgement

This research is supported by the Hong Kong Research Grants Council through a RGC-CERG grant (HKUST6208/99E).

References

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A capacity related reliability for transportation networks