Modeling freeway incident response time: A mechanism-based approach

Transportation Research Part C 28 (2013) 87–100

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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc

Modeling freeway incident response time: A mechanism-based approach Lin Hou a, Yunteng Lao b, Yinhai Wang b,⇑, Zuo Zhang a, Yi Zhang a, Zhiheng Li a a b

National Laboratory for Information Science and Technology, Department of Automation, Tsinghua University, Beijing 100084, China Department of Civil and Environmental Engineering, University of Washington, Seattle, WA 98195, USA

a r t i c l e

i n f o

Article history: Received 13 September 2011 Received in revised form 3 December 2012 Accepted 7 December 2012

Keywords: Freeway incident Response time Incident response truck Preparation delay Travel time

a b s t r a c t Incident response time is critical for incident management. The sooner an incident is responded to, the lower the negative impact comes from it. There have been some achievements on incident response time modeling. However, most of them were based on empirical observations rather than the mechanism of the system and hence their findings were highly dependent on the proposed hypotheses and study sites. A more general analytical method is needed for response time analysis. To fill up the gap, a mechanism based approach is proposed to model the incident response process and explore the contributing explanatory attributes in this paper. A typical incident response process is mathematically formulated based on the incident response truck (IRT)’s activity. Response time is considered being comprised of both preparation delay and travel time to the incident site. Both components are modeled using probability distributions to take their stochastic features into account. The response time model is calibrated using the Washington State Incident Tracking System (WITS) data and dual-loop detector data collected in 2009. Seven variables were found to significantly increase the response preparation delay (e.g. injury involved, heavy truck involved, and weekends) and eleven variables were found having a decreasing effect on the preparation delay (e.g. peak hour and average annual daily traffic). The model has the potential to be used for incident response resource optimization and identification of measures for incident response time improvement. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Freeway incidents have been regarded as one of the major causes of traffic congestion. Thus, incident management is important for alleviating incident-induced congestion by either increasing the road capacity or decreasing/re-allocating the traffic demand. Some strategies have attracted research interest: temporary utilization of shoulders (Aron et al., 2013), dynamic/responsive congestion pricing (De Palma and Lindsey, 2011; Lou et al., 2011a,b), route guidance using traffic advisory information (Al-Deek and Kanafani, 1993; Levinson, 2003), ramp metering, signal optimization and coordination (Chen et al., 1990; Papageorgiou, 1995). Moreover, incident management resource optimization is also an important research topic (Zografos et al., 2002; Lou et al., 2011a,b). To determine if a particular incident requires the use of some drastic response, it is necessary to estimate its severity and predict its duration during the incident management process. Moreover, a clear understanding of the contributing factors and their effects on incident duration is essential for optimal incident management resource allocations.

⇑ Corresponding author. E-mail address: [email protected] (Y. Wang). 0968-090X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.trc.2012.12.005

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Nomenclature Symbols t q k u V Vl d l Vw

q f() k

c e(t) X b D P() f(|) dtr dit dir F() L() W g() s E() var() C() di d(i)

response time traffic volume of a specific freeway segment traffic density of a specific freeway segment space mean speed for general traffic on a specific freeway segment space mean speed for IRTs on a specific freeway segment speed limit of a specific freeway segment travel distance from the IRT to the incident site incident-induced queue length speed of the shock wave induced by the incident impact factor of the incident-induced congestion distribution density function Weibull distribution’s shape parameter for preparation delays Weibull distribution’s scale parameter for preparation delays unit step function vector of explanatory variables affecting preparation delays vector of estimable coefficients for explanatory variables alignment distance of the IRT’s territory probability conditional distribution density function travel distance from the IRT’s location downstream to the nearest off-ramp travel distance from the incident site to the IRT’s location travel distance from the incident site upstream to the nearest on-ramp cumulative distribution of Weibull function log-likelihood function weight factor of foundational components for the overall density function foundational component for the overall density function parameter for the foundational density component expectation of response times variance of response times gamma function travel distance between the ith IRT and the incident site order statistics of IRT’s travel distances

Subscripts p preparation delay t travel time i ideal travel time without disturbance from the incident o opposite side to the incident site s same side with the incident site q occupied by the incident-induced queue u upstream d downstream r on/off-ramps

According to the Highway Capacity Manual (TRB, 1994), incident duration can be divided into four typical phases: incident detection (the time interval between incident occurrence and incident notification), incident response (the time interval between incident notification and response team arrival), incident clearance (the time interval between response team arrival and incident clearance), and traffic recovery (the time interval between incident clearance and traffic recovery into normal condition). As a critical component of incident duration, the response time, which reflects the ability of the emergency reaction to an incident, is the focus of this study. Over the past several decades, a lot of research efforts have been made in analyzing the incident duration, especially response duration. Previous research work mainly focused on two aspects: (1) exploring the contribution of external factors on the incident duration; (2) capturing the probability distribution for the incident duration. From the first aspect, previous studies can be classified into two categories: (1) Parametric models, which quantify the relationship between incident duration and external contributing factors. Linear Regression Model was a preferred choice considering its concise formulation and implementation (Garib et al., 1997; Ozbay and Kachroo, 1999; Wang, 1991). The truncated regression method (Khattak et al., 1994, 1995) and ridge

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regression method (Yu et al., in press) were also used for incident duration analysis and prediction. Hazard-based models, mainly including proportional hazard models (Lee and Fazio, 2005; Nam and Mannering, 2000) and accelerated failure time models (Alkaabi et al., 2011, 2012; Chung, 2010), were applied to formulating the contribution of external factors on durations’ conditional probability (Hensher and Mannering, 1994). (2) Semi-parametric or non-parametric models. When an explicit relationship between the incident duration and contributing factors cannot be specified, semi or non-parametric models were preferred choices. Popular methods include artificial neural network (Lee and Wei, 2010; Wei and Lee, 2007), kernel and nearest-neighbor method (Smith and Smith, 2001), decision trees (Ozbay and Kachroo, 1999; Smith and Smith, 2001), fuzzy logic (Kim and Choi, 2001; Teodorovic, 1999), Bayesian classifier (Boyles et al., 2007), and Bayesian Network (Demiroluk and Ozbay, 2011; Li and Cheng, 2011; Ozbay and Noyan, 2006). As for the second aspect, several achievements have been made in capturing the probability distribution for the incident duration. Golob et al. (1987) found that the incident duration was log-normally distributed with classified truck-involved highway incident data. This distribution was empirically supported by several following studies (Garib et al., 1997; Giuliano, 1989; Hojati et al., 2012; Sullivan, 1997). Jones et al. (1991) stated that the log–logistic distribution was a better choice considering the ‘‘wider probability tails’’. Among hazard-based studies, Weibull (Alkaabi et al., 2011, 2012; Nam and Mannering, 2000) and log–logistic (Chung, 2010; Ghosh and Savolainen, 2012; Jones et al., 1991) distributions have enjoyed considerable popularity. In general, the majority of previous research chose a better distribution based on the goodness of fit. Although previous achievements have provided valuable insight into the incident response time analysis, most simply statistically analyzed the sample data and were separated from the response mechanism. The typical procedures of previous approaches consisted of three stages: hypothesis (of the factor-duration relationship expression or probability distribution), parameter training (using collected field data) and testing (if the data support the hypothesis). Consequently, the reliability of the results was highly dependent on the validity of the proposed hypothesis and the representativeness of the testing data. One exception is that Hall (2002) modeled the travel time for the response crew to the scene of the incident and compared different dispatching strategies, ‘‘accounting for spacing between interchanges and the time penalty for changing directions’’. However, the probability distribution of response time was not formulated and the effects of incident characteristics on response process were not quantified. To fill up the gap, this paper aims at conducting a new analytical probability model for the incident response time. It takes the incident response process and operating mechanism into account when formulating the model and this separates it from most existing methods. The rest of this paper is organized as follows: Next section describes the response process and mathematically formulates the response time. Then statistic properties of the proposed model are presented. This is followed by the utilization of the proposed model for evaluating the performance of different response strategies. A case study is then presented before the last section that concludes this research and offers future research directions. 2. Methodology Of all the traffic emergency events, only those handled by Incident Response Trucks (IRTs) are taken into consideration. The response process can be divided into two stages: preparation stage (from the incident notification to the IRT’s dispatching) and IRT travel stage (from the IRT’s departure to arrival). Taking the incident response teams from the Washington Department of Transportation (WSDOT) as an example, soon after an incident is detected or reported, the response teams will dispatch the IRT in charge of the corresponding area to deal with the incident. The period between the incident notification and the time point beginning to dispatch IRTs is called preparation delay. The IRT travel stage begins as soon as the IRTs leave for the incident site and ends after the first IRT arrives at the incident site. 2.1. Modeling response time The response duration t can be subdivided into two parts: the response team’s preparation delay tp, which is related to the response resource availability and the incident type, and the IRT’s travel time tt, which depends on the travel distance and the traffic condition. Then one can get

t ¼ tp þ tt

ð1Þ

Based on the IRT deployment records provided by WSDOT, an IRT normally takes charge of a bidirectional freeway segment ranging from 5 miles to 30 miles. When an incident occurs, the IRT can be at any location of the response range. Thus the travel distance depends on the locations of both the IRT and the incident. If the IRT and the incident are on different sides, the IRT has to get off the freeway and get on the other side via the nearest on-ramp upstream of the incident site. Compared with the whole freeway segment, the distance off the freeway is trivial and ignored in this study. Fig. 1 illustrates the IRT’s territory, with the IRT’s trajectory marked by the blue arrows. The whole travel distance for each IRT can be divided into three parts as illustrated in Fig. 1: Part O in the opposite direction to the incident site, Part S in the same direction without incident-induced queue occupied, and Part Q occupied by the incident-induced queue. Four performance measures can be used to characterize the traffic condition for each part: volume,

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Fig. 1. IRT’s territory and activity.

density, space mean speed for general traffic and space mean speed for the IRTs, represented by (qo, ko, uo, vo) for Part O, (qs, ks, us, vs) for Part S and (qq, kq, uq, vq) for Part Q, respectively. Since general vehicles are required to make space for the IRTs, the IRT speed v should exceed the general traffic speed u under the same traffic condition. Thus, the difference D = v  u should be positive and can be expressed by an exponential function of u:

D ¼ v  u ¼ expðauÞ

ð2Þ

where a is the estimable coefficient vector. The IRT’s speed has to be restricted by a speed constraint Vl for the freeway segment. Hence, IRT speed can be calculated as

v ¼ minðV l ; u þ expðauÞÞ

ð3Þ

As shown in Fig. 1, IRT’s travel time can be divided into three parts in response to the three travel phases: the travel time through Part O, Part S, and Part Q. Let the travel distance between the IRT and the incident site be d, consisting of the part in the same direction ds and the part in the opposite direction do, i.e. d = ds + do. Now we have

tt ¼

do

vo

þ

ds  l

vs

þ

l

vq

ð4Þ

where l denotes the queue length immediately upstream of the incident when the response truck arrives at the end of the queue. Since only non-recurrent congestion is considered in this study, the queue length is solely induced by the incident. The kinematic shock wave theory has been considered as a concise yet powerful tool to investigate the spatio-temporal nature of the generation and dissipation of traffic congestion (Lighthill and Whitham, 1955; Wong and Wong, 2002), especially the non-recurrent congestion induced by incidents (Mongeot and Lesort, 2000; Newell, 1993). In this study, the shock wave theory is used to calculate the incident-induced queue length. The queue length is related to the shock wave speed and the time it travels. First, the speed of the shock wave is calculated as

vw ¼ 

qs  qq @q ¼ @k ks  kq

ð5Þ

where qs and ks are the volume and density for Part S, and qq and kq are the volume and density for Part Q. The travel time of the shock wave equals to the sum of the incident detection time, the preparation delay and the IRT’s travel time. The incident detection time is neglected in this study because of the following two reasons: it is unavailable, because the exact instant of occurrence is usually unknown; it may be of low importance, because incidents may be reported and confirmed within a short time interval after occurrence with the wide usage of cell phones. Thus, the queue length l is formulated as

  do ds  l l ¼ v w tp þ þ

vo

vs

ð6Þ

Then, the total response time can be induced from Eqs. (1), (4), and (6):

t ¼ qðt p þ t i Þ

ð7Þ

where q is the impact factor of the incident-induced congestion and ti is the IRT’s ideal travel time under the incident-disturbance-free condition. They can be respectively expressed as:

q¼

v s ðv q þ v w Þ v q ðv s þ v w Þ

ð8Þ

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91

and

ti ¼

do

þ

ds

ð9Þ

vo vs

The response times can be analytically calculated from Eq. (7), given the values of:  Traffic parameters for link under normal conditions, including vo and vs, which can be derived from the roadway detectors.  Traffic parameters for link under incident-induced-congested conditions, including vq and vw, which can be analytically calculated or logically constructed from three types of data: upstream/downstream detector data, road facility data, and incident data (Heydecker, 1994; Mongeot and Lesort, 2000).  Travel distances do and ds, which can be considered as stochastic variables and determined by temporal locations of the IRT and the incident site.  Response preparation delay tp, which can be considered as a stochastic variable and determined by the response resource availability and the incident type. With the detector data, road facility data, and incident characteristics known, the values of vo, vs, vq and vw can be determined. Thus, the randomness of response times only comes from tp, do and ds. 2.2. Probability distributions of response times In this section, the probability model of response time is derived. The probability distribution for each component of the response time is formulated first and then the convolution integral is conducted for the overall model. 2.2.1. Probability distributions of the preparation delay tp The variable of preparation delay is affected by a number of factors, including the describable (such as the incident type) and the random (unobservable) ones. If the variables with the same describable influential factors are considered identically distributed, the normal distribution is the first choice since the variable is affected by a large number of trivial random factors. The expectation of the distribution is jointly decided by the describable influential factors. In this study, Weibull distribution is assumed as an alternative, considering its empirical flexibility and close approximation to the normal distribution. The probability density function of the Weibull distribution is expressed as

fp ðt p Þ ¼

 k1 k tp

c c

"  # k tp

exp 

c

eðtp Þ

ð10Þ

where k and c are the shape and scale parameters, respectively, and e(t) is the unit step function and defined as

eðtÞ ¼



0 ðt < 0Þ

ð11Þ

1 ðt > 0Þ

Previous research pointed out that Weibull distribution with the shape parameter k ¼ 3:25 is a good approximation of normal distribution (Plait, 1962). Scale parameter c is linearly correlated with the expectation and should be decided by the explanatory variables. Since c is positive, it can be expressed by an exponential function

c ¼ expðbXÞ

ð12Þ

where X and b are explanatory variables affecting preparation delays and corresponding coefficients, respectively. 2.2.2. Probability distributions of the ideal travel time ti The relationships among geographical locations of the IRT, the incident site, and the on/off-ramps need to be taken into consideration in this part. As shown in Fig. 2, four scenarios are proposed depending on the IRT’s temporal location when dispatched, since it is the key factor to bring in randomness: S1: S2: S3: S4:

the the the the

IRT IRT IRT IRT

being being being being

in in in in

the the the the

same direction but at the downstream of the incident site. opposite direction but at the upstream of the incident cross section. opposite direction but at the downstream of the incident cross section. same direction but at the upstream of the incident site.

Let D be the alignment distance of the IRT’s territory. Let Du and Dd be the alignment distance from the incident site upstream/downstream to the responsible segment boundary, respectively. Now we have D = Du + Dd. Let Dr be the average alignment distance between two adjacent on/off-ramps in the scope. Thus, the probabilities of different scenarios are

PðS1Þ ¼ PðS2Þ ¼

Dd 2D

ð13Þ

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Fig. 2. Four scenarios of the relationship between the IRT’s location and the incident site.

and

PðS3Þ ¼ PðS4Þ ¼

Du 2D

ð14Þ

One can calculate the conditional probabilities of travel distances and times under different scenarios. Under Scenario S1, with the IRT’s trajectory marked by the green arrows, do and ds can be calculated as:

do ¼ dtr þ dit þ dir

ð15Þ

ds ¼ dtr þ dir

ð16Þ

and

where dtr indicates the travel distance from the IRT to the nearest downstream off-ramp (or the cross segment), dit represents the travel distance from the incident cross section to the IRT cross section, and dir is the travel distance from the nearest upstream on-ramp to the incident site (or the cross segment), as shown in Fig. 2. Since the IRT is equiprobably located along the responsible segment, dit follows the uniform distribution with the conditional probability density function represented as

f ðdit jS1Þ ¼

1 ½eðdit Þ  eðdit  Dd Þ Dd

ð17Þ

Likewise, dtr and dir also follow uniform distributions without consideration of their mutual dependence. However, it is noted that the distance between upstream and downstream loop stations usually ranges from 0.5 to 1.5 miles, contrary to the distance of the whole responsible range which typically ranges from 5 to 30 miles. Correspondingly, the variances of dtr and dir are rather small compared with that of dit. To further simplify the issue, both dtr and dir are assumed to be deterministic and equal to their expected value Dr/2. Thus, the conditional probability density of ti can be calculated from Eqs. (9), (15)–(17) as

fi ðt i jS1Þ ¼

vo Dd

        1 1 1 1 Dd  e t i  Dr  e ti  Dr þ þ

vo vs

vo vs

vo

ð18Þ

Similarly, the conditional density functions under scenarios S2, S3 and S4 can be calculated as

        D 1 1 Dr 1 1 Dd  e ti   e ti  r þ þ Dd 2 vo vs 2 vo vs vo        vs Dr 1 1 Dr 1 1 Du Þ  e ti   eðti  þ þ fi ðt i jS3Þ ¼ Du 2 vo vs 2 vo vs vs

fi ðt i jS2Þ ¼

vo

ð19Þ ð20Þ

and

fi ðt i jS4Þ ¼

vs Du





eðti Þ  e ti 

Du



vs

ð21Þ

Then the density function of ti for the whole scenario can be obtained from

fi ðt i Þ ¼ fi ðt i jS1ÞPðS1Þ þ fi ðt i jS2ÞPðS2Þ þ fi ðt i jS3ÞPðS3Þ þ fi ðt i jS4ÞPðS4Þ

ð22Þ

2.2.3. Overall probability model and estimation approach Now consider the overall response time t = q(tp + ti). For the sake of simplification, the preparation delays and the IRT’ travel times are assumed to be mutually independent. In this case the convolution method can be used to calculate the overall density function from Eqs. (7), (10), and (22).

L. Hou et al. / Transportation Research Part C 28 (2013) 87–100

  tp dt p h       i Dd vo F t  D 1 þ 1 t 1 1 ¼ q1 2D  D þ  F  r r q vo vs q vo vs vo h       i Dd v o F t  Dr 1 þ 1 Dr t 1 1  þ þ q1 2D  F  q 2 vo vs q 2 vo vs vo h       i v s F t  Dr 1 þ 1  F qt  D2r v1o þ v1s  Dv us þ q1 2D q 2 vo vs h    i v s F t  F t  Du þ q1 2D q q vs f ðtÞ ¼

93



R1

1 f ðt Þf t t p ¼1 q p p i q

ð23Þ

where F(t) is the cumulative distribution of Weibull function and can be expressed as

( "   #) k t FðtÞ ¼ eðtÞ 1  exp 

ð24Þ

c

Let {t1, . . . , tn} be the response times for the incident sample set of size n. For the ith incident sample, the overall density function f(ti) can be calculated from Eq. (23). Then the parameters of the overall model can be estimated by standard maximum likelihood method with the log-likelihood function shown as

Lðb; aÞ ¼

X

lnðf ðt i ÞÞ

ð25Þ

i

3. Statistic properties From the probability density function in Eq. (23), some valuable statistic properties can be derived for thoroughly understanding response times. Due to space limitations, only the scenario of D  Dr is considered in the following analysis. This scenario is the most common situation during the incident response process. Other situations can be analyzed in a similar way. For the scenario of D  Dr, the density function of Eq. (23) can be simplified as

    Du Dd þ W d  g t; f ðtÞ ¼ W u  g t;

vs

vo

ð26Þ

where

Du D Dd ¼ 1  Wu Wd ¼ D

Wu ¼

ð27Þ ð28Þ

and

gðt; sÞ ¼

     1 t t F s F qs q q

ð29Þ

As shown in Eq. (26), the overall density function can be considered as the weighted combination of two foundational density components g(t, Du/vs) and g(t, Dd/vo), which indicate the IRT’s travel on the same/opposite side of freeways with/ to the incident site, respectively. 3.1. Density trends To visualize the overall density trends, the distributions of the component g(t, s) need to be analyzed first. According to Eqs. (24) and (29), three parameters are found to have effects on the distribution of g(t, s): q, which is the impact factor of the incident-induced congestion in Eq. (8); c, which is the scale parameter and reflects the expectation of the preparation delay in Eq. (12); and s, which refers to either Du/vs or Dd/vo, the travel time on a certain side of freeways in Eq. (26). Fig. 3a–c displays the curves of g(t, s) for different values of q, c and s, respectively. One can observe that the effect of both q and c is to rescale the coordinates of the density function. The average response time will increase and the density curve of g(t, s) will get more dispersed with the increase of q or c. The average response time increases and the shape of the curve becomes wider when s increases from 5 to 10 and further to 15. Combining Eqs. (24) and (29), one can find that the shape of the distribution curve is determined by s/c, which reflects the ratio of travel time to preparation delay. The variable s/c can be considered as the shape parameter. Meanwhile, since the scale of the curve is co-determined by parameters q and c, the variable qc can be considered as the scale parameter. In general, the trend of function g(t) is monotone increasing followed by monotone decreasing. Fig. 3d shows the curves of the overall density function for different values of weight factors Wu and Wd. The foundational component density functions g(t, Du/vs) and g(t, Dd/vo) in Eq. (26) are represented by dashed lines, and the overall density

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distributions for different weight factors are revealed by solid lines, respectively. The shape of overall density function is determined jointly by the shapes of component density functions and the weight factors. From Eqs. (27) and (28), one can see that the weight factors Wu and Wd are decided by the location of the incident site in the segment. 3.2. Expectation and variance Two important statistical indices, the expectation and variance, are deduced from Eq. (26) and respectively expressed as:

!   1 q D2d D2u þ EðtÞ ¼ qcC 1 þ þ 3:25 2D v o v s

ð30Þ

and

varðtÞ ¼ q

2



8 ! !2 9     < 1 D3 D3 2 1 1 D2d D2u = 2 2 2 d u c C 1þ þq  2 c C 1þ þ þ :3D v 2o v 2s 3:25 3:25 4D v o v s ; 2

ð31Þ

where C(z) represents the gamma function and can be defined as

CðzÞ ¼

Z

1

t z1 et dt

ð32Þ

t¼0

The average value and variance of response times reflect the efficiency and reliability of the response process, respectively. Basically, the less E(t) and var(t) are, the more effective and more reliable the response mechanism will be. 4. Operational issues This section will indicate how to apply the proposed model to evaluating different strategies for IRT deployment with an example. The evaluation will be helpful for the improvement of response operation.

Fig. 3. Curves of g(t, s) and f(t) for different parameters.

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95

For the proposed model in Section 2, one underlying assumption is that there is one and only one IRT in charge of the incident site. However, more than one IRT may be assigned to be responsible to the same segment of the freeway at the same time. Given the fixed density (number per mile) of IRTs, which alternative is more effective? That is a concern of the transportation agencies and also the focus of this section. Since our concern in this part is not a specific case but the general situation, the difference between the traffic conditions of two sides from the freeway segment will not be taken into consideration. The variable v is used to represent the IRT’s speed on both sides. Then the Eq. (7) can be rewritten as

  d t ¼ q tp þ

ð33Þ

v

Suppose that there are N IRTs in charge of the same segment with the alignment distance D and the nearest one from the incident site will be dispatched. Let the travel distance between the ith IRT and the incident site be di (i = 1, . . . , N). Let the corresponding order statistics be d(i). Evidently, the travel distance between the nearest IRT and the incident site is d(1) and given by

dð1Þ ¼ minðdi Þ

ð34Þ

i

The density function of di can be given by

f ðdi Þ ¼

2ðD  di Þ D2

½eðdi Þ  eðD  di Þ

ð35Þ

The probability density function of the first order statistic (David and Nagaraja, 2003) can be calculated as

f ðdÞ ¼ f ðdð1Þ Þ ¼

2NðD  dÞ2N1 D2N

½eðdi Þ  eðD  di Þ

ð36Þ

Thus, the expectation of the response time can be derived from Eqs. (10), (33), and (36) and expressed as

  1 q D þ EðtÞ ¼ qcC 1 þ 3:25 v 2N þ 1

ð37Þ

Similarly, the variance of the response time is expressed as

" #      2 1 q2 N 2  c2 C2 1 þ þ 2 varðtÞ ¼ q2 c2 C 1 þ D 3:25 3:25 v ðN þ 1Þð2N þ 1Þ2

ð38Þ

For a given IRT density N/D (e.g. 0.1, representing average 1 IRT per 10 miles), the performance (response times’ expectations indicated by solid line and variance by dashed line) of different operational strategies are illustrated in Fig. 4, where the x-axis represents the number of IRTs in the same group (e.g. 5, representing 5 IRTs jointly take charge of the area of 50 miles with the density N/D = 0.1). One can find from Fig. 4 that both the expectation and variance of the response time increase when IRTs’ territories overlap, implying that it is better to divide the liability area for individual IRT.

Fig. 4. Expectations and variances of response time for different operational strategies.

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5. Case study This section applies the mechanism-based model to four selected freeway segments in the Puget Sound area as a case study. These selected segments are typical urban freeways: I-5 between milepost 139.00 and 206.00, I-90 between milepost 2.00 and 10.00, SR-167 between milepost 10.00 and 27.00, and SR-520 between milepost 0.00 and 5.00. 5.1. Data sources The study period was from January 1st to December 31st, 2009. All the incidents were disposed by one and only one IRT. The data inputs were divided into two categories: explanatory variables affecting preparation delays, and geographic information and traffic conditions affecting IRT’s travel times. Incident explanatory variables can be extracted from the incident logs, geographic information can be obtained from the IRT deployment logs, and traffic condition can be derived from the roadway facility information and the loop detector data. The incident logs, the IRT deployment logs, and the inductive detector data used in this study were all provided by WSDOT. The incident data were mainly retrieved from the WITS database. The loop detector data were archived in the database of the Smart Transportation Application and Research Laboratory (STAR Lab) at University of Washington (Corey et al., 2011). Besides, the roadway facility information of the studied freeway sections was extracted from the Highway Safety Information System (HSIS) database. 5.2. Model implementation To apply the proposed model to the specific study case, the following four stages were implemented. (1) Raw data cleansing This stage mainly aimed at improving the quality of the raw incident logs and the collected dual-loop detector data. In this study, the cases with missing or inaccurate data were excluded from the sample set. After data selection and cleansing, a total of 828 valid incident cases were gathered during the study period. Table 1 contains the summary statistics of incident response times for total samples and sorted groups by locations and incident types. (2) Attribute selection for preparation delays Recall Eqs. (10) and (12). The scale parameter of the preparation delays’ density function, which denotes the expectation, should be decided by the explanatory variables. Based on the operational experience of the incident response teams from WSDOT, four categories of variables were selected: incident, temporal, geographic, and traffic characteristics. The candidate variables are listed in Table 2. Both the categorical and numerical variables were involved. The categorical variables (e.g. lane closure type) were converted into dummy binary variable sets with the baseline level for reference marked in bold in Table 2. The numerical variables (like annual average daily traffic) were normalized. (3) Speed calculation for travel times Recall Eqs. (7)–(9). The traffic speed vo and vs under normal conditions were directly derived from the detectors along the freeway segment. The speed vq and vw for incident-induced congestion were calculated using the kinematic wave theory (Heydecker, 1994).

Table 1 Statistical summary of response times.

a

Categories

Frequency

Meana

Median

Min

Max

Std.

Total samples I-5 I-90 SR-167 SR-520

828 528 93 118 89

8.99 9.11 8.10 9.97 7.97

8 8 6 10 7

1 1 1 1 1

65 65 29 27 30

6.31 6.82 5.61 5.20 4.76

Disabled vehicles Abandoned vehicles Debris Collision Other incident types

373 22 153 241 39

9.32 7.82 8.99 8.73 8.23

8 7 8 7 7

1 1 1 1 1

65 17 30 47 37

6.80 4.58 5.30 6.22 6.40

The statistics of response times are measured in minutes.

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Table 2 Description of explanatory variables for preparation delays.

a

Categories

Variable

Value set

Incident

Incident type Lane closure type Injury involved Fire involved Work zone involved Heavy truck involved

{Other a, Disabled vehicles, abandoned vehicles, debris, collision} {Single lane blocked, Shoulder/median, multiple lanes, all travel lanes, total closure} {0, 1} {0, 1} {0, 1} {0, 1}

Temporal

Time of day Day of week Month of year

{Off peak, AM peak (6:00–9:00), PM peak (15:00–18:00)} {Weekdays, Weekends (Saturday, Sunday)} {Other, Summer (June, July, August), Winter (December, January, February)}

Geographic

Incident on HOV

{0, 1}

Traffic

Annual average daily traffic (vehicles)

R+

Bold figures are the baseline level of variables for reference.

(4) Model estimation and data analysis After conducting the model structure, the maximum likelihood estimation method was applied to estimating the coefficients for external variables, using the log-likelihood function expressed in Eq. (25). Two coefficient vectors were estimated: the coefficient vector b for explanatory variable vector X affecting preparation delays in Eq. (12) and the vector a used for adjusting general traffic speed u in Eq. (3). The procedures of this step were carried out using the R statistical analysis package. 5.3. Estimation results The estimation results of the proposed model on incident response times are enumerated in Table 3, including the estimated coefficients, standard errors and t-values for all the significant variables at 95% level of significance. The log-likelihood ratio index q2 (Ben-Akiva and Lerman, 1985) was used to reflect the effectiveness of the model compared with the baseline model. In this case, the value of q2 (0.228) was acceptable but not that high. It may be due to the nature of the mechanistic modeling. For the baseline model, the structure and some parameters have already been identified. Only a few parameters were left to be calibrated. At 95% level of significance, a total of 21 variables were found to significantly affect the probability distribution of the response time. The variables affecting the preparation delay and the travel time will be discussed in the following part, respectively. (1) Variables affecting preparation delays According to Eqs. (10) and (12), the positive value of coefficient indicates that the increase of the explanatory variable value shall increase the expected preparation delay. Next, the detailed analysis of the estimated results will be presented based on different characteristic categories. As for incident characteristics, eleven variables were found to significantly affect the response preparation process. As two common incident types, disabled vehicles (coef = 0.364, t = 5.943) and debris (coef = 0.466, t = 7.176) were found to be associated with longer preparation delays, compared with the baseline level (other incident types). The potential reason is that these two categories have few effects on drivers’ safety or transportation and consequently gain little priority to be disposed. Besides, the incident type of abandoned vehicles (coef = 0.777, t = 8.252) was associated with shorter preparation delays. This may be because the disposal strategies of this incident type were quite convenient. The incidents with collision (coef = 0.259, t = 4.002) were found to be associated with shorter preparation delays. This may be because the disposal priority and rapid response were given to them considering the severity. Among all the variables for lane closure type, three had significant effects on preparation delays: shoulder/median blocked (coef = 0.381, t = 10.121), all travel lanes blocked (coef = 1.272, t = 3.466) and total closure (coef = 2.960, t = 3.123). The positive coefficient for shoulder/median blockage indicates that the corresponding incidents were associated with longer response preparation delays, compared with the baseline scenario (single lane blockage). Meanwhile, the incidents with all travel lanes blocked tended to have shorter preparation delays. This may be because incidents with more blocked travel lanes are more severe and consequently deserve the priority of disposals. The only exception is the estimated result for total closure, indicating that the corresponding incidents were associated with longer preparation delays. However, this finding is not convincing, since the samples in the categories were rare (seven samples) and the model on this attribute might not be thoroughly and correctly trained. The variables of injury involved (coef = 1.252, t = 13.762) and heavy truck involved (coef = 0.626, t = 5.846) were found to significantly increase the preparation delays. Although the incidents with injury are more severe than the others and deserve

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Table 3 Estimation results of the proposed model on response times. Variables Variables affecting the preparation delays Constant b0

a

se

ta

1.981

0.063

31.597

Incident characteristics Incident type: Disabled vehicles (Yes: 1; No: 0) Incident type: debris (Yes: 1; No: 0) Incident type: abandoned vehicles (Yes: 1; No: 0) Incident type: collision (Yes: 1; No: 0) Lane closure: shoulder/median blocked (Yes: 1; No: 0) Lane closure: all travel lanes blocked (Yes: 1; No: 0) Lane closure: total closure (Yes: 1; No: 0) Injury involved (Yes: 1; No: 0) Fire involved (Yes: 1; No: 0) Work zone involved (Yes: 1; No: 0) Heavy truck involved (Yes: 1; No: 0)

0.364 0.466 0.777 0.259 0.381 1.272 2.960 1.252 1.944 1.219 0.626

0.061 0.065 0.094 0.065 0.038 0.367 0.948 0.091 0.223 0.207 0.107

5.943 7.176 8.252 4.002 10.121 3.466 3.123 13.762 8.730 5.878 5.846

Temporal characteristics Time of day: AM peak (Yes: 1; No: 0) Time of day: PM peak (Yes: 1; No: 0) Day of week: Weekends (Yes: 1; No: 0) Month of year: Summer (Yes: 1; No: 0) Month of year: Winter (Yes: 1; No: 0)

0.836 0.118 0.150 0.125 0.335

0.058 0.038 0.038 0.041 0.041

14.429 3.131 3.972 3.035 8.109

Geographic characteristics Incident on HOV lane (Yes: 1; No: 0)

0.872

0.079

11.050

Traffic characteristics Average annual daily traffic

0.596

0.004

144.927

3.199 0.529 5022.84 3878.40 0.228

0.140 0.140

22.879 3.785

Variables affecting the travel times IRT speed adjustment factors Constant a0 Adjustment factor for general traffic speed a1 Log likelihood with constants only Log likelihood at convergence Likelihood ratio index q2 b b

Coef

Coef: estimated coefficient; se: standard error; t: t-value. q2: the log-likelihood ratio index.

priority of disposal, the response process is more complex and time-consuming: the response teams have to communicate with other management agencies such as the police and first-aid personnel. Likewise, it was more difficult to dispose the incidents involving heavy trucks than any others. The finding is consistent with the research conducted by Giuliano (1989) and Garib et al. (1997), who pointed out that truck involvement would significantly increase the incident duration. As expected, the estimated results for the variables of fire involved (coef = 1.944, t = 8.730) and work zone involved (coef = 1.219, t = 5.878) indicated that the corresponding incidents attracted more attention from the response team and were disposed rapidly. As for temporal characteristics, five variables were found to be highly significant for the preparation delays. Incidents occurring in either morning peak hours (coef = 0.836, t = 14.429) or evening peak hours (coef = 0.118, t = 3.131) were associated with shorter preparation delays, compared with other situations. The reason was found after investigating related personnel and examining the response procedures: peak hours are important for daily traffic so that the response team has a common perception to alleviate non-recurrent congestion by responding to the incidents as soon as possible. This result is consistent with the research conducted by Jones et al. (1991) that showed a decrease in incident duration during the peak hours. The absolute coefficient for morning peak is bigger than that of the evening peak, implying that the response process in the morning was more efficient. The incidents occurring on weekends (coef = 0.150, t = 3.972) were found to be associated with longer preparation delays. This is expected, because the staffing deployment on weekends is not as concentrated as that on weekdays according to the WSDOT’s work schedule. This finding is consistent with the research conducted by Nam and Mannering (2000) that pointed out incidents occurring on weekends had longer response times. As for month of year, both the variables of summer (coef = 0.125, t = 3.035) and winter (coef = -0.335, t = 8.109) significantly decreased the response preparation delays. The results imply that the work efficiency of the response teams is higher in summer and winter. As for geographic characteristics, the factor of High Occupancy Vehicle (HOV) lanes involved (coef = 0.872, t = 11.050) was found to be associated with shorter preparation delays. This may be because they had the disposal priority compared with the incidents occurring on general purpose lanes. As for traffic characteristics, the factor of average annual daily traffic was found to be associated with shorter preparation delays (coef = 0.596, t = 144.927). As previously mentioned, this variable reflects the traffic demand and the importance

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of the roadway. The estimated result indicates that disposal priority will be given to the freeway segment with heavier average annual daily traffic and implies a potential model classification along this characteristic. (2) Variables affecting travel times Recall Eq. (3). One can see that the coefficient vector has the effect of converting the general traffic speed into the IRT speed. Both the constant a0 (coef = 3.199, t = 22.879) and the linear term factor a1 (coef = 0.529, t = 3.785) were found to be significant, indicating the significant difference between the general traffic speed and the IRTs’ speed. Based on the above analysis, several suggestions can be given to the response teams. For instance, the response agency should reasonably arrange the deployment of personnel and response trucks. Meanwhile, the response efficiency needs to be improved on weekends and in springs and autumns. 6. Conclusion Being an important portion of incident duration, response time reflects the response ability of the incident management agencies. Response time analysis is essential for better understanding the response process and developing effective countermeasures for incident management agencies. However, most of the previous models left the response mechanism out of consideration, which made their findings less reliable. This paper presented a new probability model of response times based on the IRT’s operational mechanism. The response process was divided into two phases: preparation phase from the incident notification to the IRT dispatch and IRT’s travel process from departure to arrival at the incident site. First, the probability density for the preparation delay was formulated, taking into consideration the external explanatory variables. Then the probability density for the IRT’s travel time was explored under different scenarios, after building up the relationship between travel times and the roadway traffic conditions based on the macroscopic traffic flow theory. The overall model was conducted finally with the convolution method. This mechanism-based model enabled new statistical analysis, e.g. exploring the trends of density, expectation, and variance, that is highly valuable for us to understand the probability patterns of incident response time. Besides, the model showed its potential in evaluating the efficiency of different response strategies and improving the operational deployment schedule for IRTs. Based on the proposed model, significant factors affecting response preparation delays and travel times were investigated using the freeway incident data in Washington State. Seven explanatory variables (disabled vehicles, debris, shoulder/median involved, total closure, injury involved, heavy trucks involved and weekends) were identified associated with longer response preparation delays. Meanwhile, eleven factors (abandoned vehicles, collision, all travel lanes blocked, fire involved, work zone involved, AM peak, PM peak, summer, winter, incident on HOV lane, and average annual daily traffic) tended to decrease the preparation delays. The adjustment variable vector for IRT speed was found to significantly affect travel times. The estimation results can help incident management personnel find out the drawbacks of the current response procedures and develop more effective countermeasures. Although the proposed model has been proven effective using the incident data in Washington State, there is still a need to exploit the transferability of this model for other kinds of response process on other geographic locations in the future. Moreover, the accuracy of this model can be improved with additional data. For instance, the probability distributions of the preparation delay and travel time can be statistically tested if both of them are recorded in the incident logs, and it will be helpful if the relationship is known between the general traffic’s speed and the IRT’s speed under different traffic conditions. Besides, the improved formulation of shock wave modeling can be adopted with the support of vehicle trajectory data. Acknowledgements Authors would like to acknowledge the National Natural Science Foundation of China (Project No. 90924002, 51028802, and 60834001) for their partial funding support to this research. Appreciation also goes to the staff of Washington Incident Tracking System (WITS) and Highway Safety Information System (HSIS) for providing the incident data. The authors also would like to give thanks to Sherry Kim from University of Washington for her English editing effort. References Al-Deek, H., Kanafani, A., 1993. Modeling the benefits of advanced traveler information systems in corridors with incidents. Transportation Research Part C: Emerging Technologies 1 (4), 303–324. Alkaabi, A., Dissanayake, D., Bird, R., 2011. Analyzing clearance time of urban traffic accidents in Abu Dhabi using hazard-based duration modeling method. In: Presented at 90th Annual Meeting of the Transportation Research Board, Washington, DC. Alkaabi, A., Dissanayake, D., Bird, R., 2012. Investigating response time of highway traffic accidents in Abu Dhabi using hazard-based duration modeling technique. In: Presented at 91th Annual Meeting of the Transportation Research Board, Washington, DC. Aron, M., Seidowsky, R., Cohen, S., 2013. Safety impact of using the hard shoulder during congested traffic: the case of a managed lane operation on a French urban motorway. Transportation Research Part C: Emerging Technologies 28, 168–180. Ben-Akiva, M.E., Lerman, S.R., 1985. Discrete Choice Analysis: Theory and Application to Travel Demand. MIT Press Series in Transportation Studies, vol. 9. MIT Press, Cambridge, Mass, pp. 167–168.

100

L. Hou et al. / Transportation Research Part C 28 (2013) 87–100

Boyles, S., Fajardo, D., Waller, S.T., 2007. A naive Bayesian classifier for incident duration prediction. In: Presented at 86th Annual Meeting of the Transportation Research Board, Washington, DC. Chen, L.L., May, A.D., Auslander, D.M., 1990. Freeway ramp control using fuzzy set theory for reasoning. Transportation Research Part A: General 24 (1), 15– 25. Chung, Y., 2010. Development of an accident duration prediction model on the Korean Freeway Systems. Accident Analysis and Prevention 42 (1), 282–289. Corey, J., Lao, Y., Wu, Y., Wang, Y., 2011. Detection and correction of inductive loop detector sensitivity errors using Gaussian Mixture Models. Transportation Research Record: Journal of the Transportation Research Board. 2256, 120–129. David, H.A., Nagaraja, H.N., 2003. Order Statistics, third ed. Wiley, New Jersey. De Palma, A., Lindsey, R., 2011. Traffic congestion pricing methodologies and technologies. Transportation Research Part C: Emerging Technologies 19 (6), 1377–1399. Demiroluk, S., Ozbay, K., 2011. Structure learning for estimation of nonparametric incident duration prediction models. In: Presented at 90th Annual Meeting of the Transportation Research Board, Washington, DC. Garib, A., Radwan, A.E., Al-Deek, H., 1997. Estimating magnitude and duration of incident delays. Journal of Transportation Engineering 123 (6), 459–466. Ghosh, I., Savolainen, P.T., 2012. Analysis of factors affecting clearance time of freeway incidents. In: Presented at 91th Annual Meeting of the Transportation Research Board, Washington, DC. Giuliano, G., 1989. Incident characteristics, frequency, and duration on a high volume urban freeway. Transportation Research Part A: General 23 (5), 387– 396. Golob, T.F., Recker, W.W., Leonard, J.D., 1987. An analysis of the severity and incident duration of truck-involved freeway accidents. Accident Analysis and Prevention 19 (5), 375–395. Hall, R.W., 2002. Incident dispatching, clearance and delay. Transportation Research Part A: Policy and Practice 36 (1), 1–16. Hensher, D.A., Mannering, F.L., 1994. Hazard-based duration models and their application to transport analysis. Transport Reviews 14 (1), 63–82. Heydecker, B., 1994. Incidents and Intervention on Freeways. Research Report No. UCB-ITS-PRR-94-5. California PATH Program, University of California, Berkeley. Hojati, A.T., Ferreira, L., Charles, P., bin Kabit, M.R., 2012. Analysing freeway traffic-incident duration using an Australian data set. Road and Transport Research 21 (2), 19–31. Jones, B., Janssen, L., Mannering, F., 1991. Analysis of the frequency and duration of freeway accidents in Seattle. Accident Analysis and Prevention 23 (4), 239–255. Khattak, A.J., Schofer, J.L., Wang, M., 1994. A Simple Time Sequential Procedure For Predicting Freeway Incident Duration. Research Report No. UCB-ITS-PRR94-26. California PATH Program, University of California, Berkeley. Khattak, A.J., Schofer, J.L., Wang, M., 1995. A simple time sequential procedure for predicting freeway incident duration. IVHS Journal 2 (2), 113–138. Kim, H.J., Choi, H.K., 2001. A comparative analysis of incident service time on urban freeways. Journal of International Association of Traffic and Safety Sciences 25 (1), 62–72. Lee, J., Fazio, J., 2005. Influential factors in freeway crash response and clearance times by emergency management services in peak periods. Traffic Injury Prevention 6 (4), 331–339. Lee, Y., Wei, C.H., 2010. A computerized feature selection method using genetic algorithms to forecast freeway accident duration times. Computer-Aided Civil and Infrastructure Engineering 25 (2), 132–148. Levinson, D., 2003. The value of advanced traveler information systems for route choice. Transportation Research Part C: Emerging Technologies 11 (1), 75– 87. Li, D., Cheng, L., 2011. Bayesian network classifiers for incident duration prediction. In: Presented at 90th Annual Meeting of the Transportation Research Board, Washington, DC. Lighthill, M.H., Whitham, G.B., 1955. On kinematic waves II. A theory of traffic flow on long crowded roads. Proceedings of the Royal Society A229, 317–345. Lou, Y., Yin, Y., Lavalc, J.A., 2011a. Optimal dynamic pricing strategies for high-occupancy/toll lanes. Transportation Research Part C: Emerging Technologies 19 (1), 64–74. Lou, Y., Yin, Y., Lawphongpanich, S., 2011b. Freeway service patrol deployment planning for incident management and congestion mitigation. Transportation Research Part C: Emerging Technologies 19 (2), 283–295. Mongeot, H., Lesort, J., 2000. Analytical Expressions of Incident-induced flow Dynamics Perturbations Using Macroscopic Theory and Extension of Lighthill– Whitham Theory. Transportation Research Record: Journal of the Transportation Research Board No. 1710, Transportation Research Board of the National Academies, pp. 58–68. Nam, D., Mannering, F., 2000. An exploratory hazard-based analysis of highway incident duration. Transportation Research Part A: Policy and Practice 34 (2), 85–102. Newell, G.F., 1993. A simplified theory of kinematic waves in highway traffic, Part II: queuing at freeway bottlenecks. Transportation Research Part B: Methodological 27 (4), 289–303. Ozbay, K., Kachroo, P., 1999. Incident Management in Intelligent Transportation Systems. Faculty Publications (ECE). Paper 103. Artech House Intelligent Transportation Systems Library. . Ozbay, K., Noyan, N., 2006. Estimation of incident clearance times using Bayesian Networks Approach. Accident Analysis and Prevention 38 (3), 542–555. Papageorgiou, M., 1995. An integrated control approach for traffic corridors. Transportation Research Part C: Emerging Technologies 3 (1), 19–30. Plait, A., 1962. The Weibull distribution – with the tables. Industrial Quality Control 19 (1), 17–26. Smith, B.L., Smith, K., 2001. Forecasting the Clearance Time of Freeway Accidents. Research Report No. UVACTS-15-0-35. USDOT University Transportation Center. Sullivan, E.C., 1997. New model for predicting freeway incidents and incident delays. Journal of Transportation Engineering 123 (4), 267–275. Teodorovic, D., 1999. Fuzzy logic systems for transportation engineering: the state of the art. Transportation Research Part A: Policy and Practice 33 (5), 337–364. Transportation Research Board, 1994. Special Report 209: Highway Capacity Manual. National Research Council, Washington, DC. Wang, M., 1991. Modeling Freeway Incident Clearance Time. MS Thesis. Civil Engineering Department, Northwestern University. Wei, C.H., Lee, Y., 2007. Sequential forecast of incident duration using Artificial Neural Network models. Accident Analysis and Prevention 39 (5), 944–954. Wong, S.C., Wong, G.C.K., 2002. An analytical shock-fitting algorithm for LWR kinematic wave model embedded with linear speed–density relationship. Transportation Research Part B: Methodological 36 (8), 683–706. Yu, R., Lao, Y., Ma, X., Wang. Y., in press. Short-term traffic flow forecasting for freeway incident induced delay estimation. Journal of Intelligent Transportation Systems. Zografos, K.G., Androutsopoulos, K.N., Vasilakis, G.M., 2002. A real-time decision support system for roadway network incident response logistics. Transportation Research Part C: Emerging Technologies 10 (1), 1–18.