Hazard based models for freeway traffic incident duration

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Hazard based models for freeway traffic incident duration

Ahmad Tavassoli Hojati a,∗, Luis Ferreiraa, Simon Washingtonb, Phil Charlesa

a Faculty of Engineering, Architecture and Information Technology, The University of Queensland, Australiab TMR Chair, Faculty of Science and Engineering, Queensland University of Technology, Australia

a r t i c l e i n f o

Article history:Received 1 August 2012Received in revised form13 December 2012Accepted 13 December 2012

Keywords:Traffic incident managementIncident durationSurvival modellingMotor vehicle crashesCongestion management

a b s t r a c t

Assessing and prioritising cost-effective strategies to mitigate the impacts of traffic incidents and acci-dents on non-recurrent congestion on major roads represents a significant challenge for road networkmanagers. This research examines the influence of numerous factors associated with incidents of var-ious types on their duration. It presents a comprehensive traffic incident data mining and analysis bydeveloping an incident duration model based on twelve months of incident data obtained from theAustralian freeway network. Parametric accelerated failure time (AFT) survival models of incident dura-tion were developed, including log-logistic, lognormal, and Weibul—considering both fixed and randomparameters, as well as a Weibull model with gamma heterogeneity. The Weibull AFT models with ran-dom parameters were appropriate for modelling incident duration arising from crashes and hazards.A Weibull model with gamma heterogeneity was most suitable for modelling incident duration of sta-tionary vehicles. Significant variables affecting incident duration include characteristics of the incidents(severity, type, towing requirements, etc.), and location, time of day, and traffic characteristics of theincident. Moreover, the findings reveal no significant effects of infrastructure and weather on incidentduration. A significant and unique contribution of this paper is that the durations of each type of inci-dent are uniquely different and respond to different factors. The results of this study are useful for trafficincident management agencies to implement strategies to reduce incident duration, leading to reducedcongestion, secondary incidents, and the associated human and economic losses.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Traffic congestion in urban areas has steadily increased as aresult of population growth and increased motorisation. This grow-ing congestion has reduced transport mobility and consequentlyhas resulted in significant increases in vehicle delays, travel timevariability, air pollution, fuel consumption, and negative social, eco-nomic and environmental impacts.

Traffic congestion is typically categorised into two types: recur-rent and non-recurrent. Recurrent congestion is predictable andis caused by chronically exceeded road capacity. Non-recurrentcongestion, in contrast, is triggered by random events where thecapacity of a road is temporarily reduced, by traffic incidents, workzones, adverse weather, and where peak demands are higher thannormal as a result of special events (Al-Deek and Emam, 2006).

In a US study, non-recurrent congestion was estimated to bebetween 40% and 60% of total delay observed on US highways

∗ Corresponding author at: School of Civil Engineering, Faculty of Engineering,Architecture and Information Technology, The University of Queensland, Brisbane,St Lucia, QLD 4072, Australia. Tel.: +61 733654156; fax: +61 733654599.

E-mail address: [email protected] (A. Tavassoli Hojati).

(Ikhrata and Michell, 1997; Skabardonis et al., 2003). In a laterstudy, the results from an investigation of congestion levels in 85large US metropolitan areas from 1982 to 2003 showed that non-recurrent congestion contributed up to 60% of all congestion, whiletraffic incidents accounted for 25% of all congestion (CamSys/TTI2005). Thus, traffic incidents appear to be a major contributor tonon-recurrent delay. However, the importance and impacts varyfrom place to place as a function of local conditions.

Acknowledging the effects of incidents on congestion, incidentmanagement programs are administered in order to minimise inci-dent delay by quickly reinstating the capacity of a road networkin the case of an incident. Systematic understanding of incidentcharacteristics and patterns is essential to restore a road networkto its full capacity. Therefore, the collection and analysis of trafficincident data and their components are crucial. In addition, under-standing factors that influence traffic incident duration is vitallyimportant for improving the management of traffic incidents, asit allows appropriate strategies to be implemented to alleviatethe traffic impacts of incidents through an efficient allocation ofequipment and personnel (Konduri et al., 2003). In addition, under-standing and being able to predict incident duration is vital forproviding reliable traffic information and improving travel timereliability (Lyman and Bertini, 2008).

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The purpose of this paper is to provide an effective and practi-cal methodology for predicting incident duration on freeways andfor identifying critical variables to facilitate the improvement andoptimisation of incident management strategies. State of the artduration models are developed, and a unique set of model specifi-cations are presented.

The paper begins with a review of previous research onincident duration. This review is followed by details on modeldevelopment. Attention is then directed towards describing theAustralian incident data examined in this study. The subsequentsection describes the modelling results including estimation ofmodel parameters, identifying significant model variables, and theselection of the ‘best’ incident duration model from among var-ious specifications. The final section provides conclusions basedon the results of the analysis and identifies areas for futureresearch.

2. Literature review

Incident duration is defined as the elapsed time from themoment an incident is detected until the cause is removed fromthe scene (Garib et al., 1997; Nam and Mannering, 2000). Researchover the past few decades has demonstrated that various method-ologies and techniques have been employed to analyse and modelincident duration, mainly on freeways. These models have deter-mined the relationships between incident duration and influencingvariables. Sets of variables affecting incident duration have beenreported, such as incident characteristics (e.g. incident type andseverity; the number and type of vehicles involved; environmen-tal effects; temporal characteristics; geometric characteristics; andoperational factors).

The most representative approaches for incident duration mod-els can be categorised into: (1) linear regression analyses (Garibet al., 1997; Valenti et al., 2010); (2) non-parametric regressionmethods and tree-style classification models (Smith and Smith,2001); (3) support vector regression (Wu et al., 2011); (4) con-ditional probability analyses (Nam and Mannering, 2000; Chung,2010); (5) probabilistic distribution analyses (Golob et al., 1987;Giuliano, 1989); (6) time sequential methods (Khattak et al., 1995);(7) discrete choice models (Lin et al., 2004); (8) Bayesian classifier(Boyles et al., 2007; Kim and Chang, 2012); (9) fuzzy logic models(Kim and Choi, 2001); and (10) artificial neural networks (Wanget al., 2005).

Incident duration has been found to follow a log-normal dis-tribution in numerous studies (Golob et al., 1987; Giuliano, 1989;Skabardonis et al., 1999), while other studies have shown that theduration of incidents is characterised by a log-logistic distribution(Jones et al., 1991; Nam and Mannering, 2000; Chung, 2010).

A hazard-based duration modelling approach considers not onlythe length of time of an incident, but also the relationship betweenthe duration and the probability that an incident will end soon.Therefore, the likelihood of an end-of-duration event depends onthe length of elapsed time since the event began, or durationdependence. This method is suitable for dealing with duration datathat are positive and can be censored and time-varying (Bhat andPinjar, 2008). This hazard-based approach is common in manydisciplines including the biomedical, social sciences, and engineer-ing (Hensher and Mannering, 1994). In the transport field, thismethod has been applied in modelling of many time-related eventsincluding safety, traffic studies, vehicle ownership, and activitybased models over the last two decades. Examples include timebetween planning and execution of an activity (Bhat and Pinjar,2008), duration of shopping activity (Bhat, 1996), length of trafficdelay (Mannering et al., 1994), and the analysis of urban travel time(Anastasopoulos et al., 2012c).

In addition, the method has been applied to model the dura-tion of traffic incidents. In an early study by Jones et al. (1991),an accident duration model was estimated based on the hazardfunction assuming an accelerated lifetime approach. The authorsanalysed approximately two years of accident data including 2156records from six study zones in the Seattle metropolitan areaand found that the log-logistic distribution better replicates acci-dent duration data compared to other distributions including thelog-normal, Weibull and Gamma. In addition, the findings of thisstudy indicated that duration modelling could be used to specifykey relationships between site characteristics and the duration ofaccidents. In a later study, Nam and Mannering (2000) used acceler-ated hazard models with alternative Weibull and log-logistic basedhazards to analyse incident duration components, including inci-dent detection, response times, and clearance times using two yearsof highway incident data in Washington state. The major contribu-tion of this study was the application of the gamma distribution tocapture unobserved heterogeneity. The authors concluded that thisapproach enabled the exploration of additional significant factorsaffecting incident duration.

According to Chung (2010), accident duration was modelledusing a hazard based duration model and a log-logistic acceleratedfailure time metric model based on a two-year accident durationdataset on the Korean freeway system. The author claimed that theduration model could be used for accident prediction purposes;however, the transferability and stability of the model in subse-quent years required validation. In a recent study (Chung and Yoon,2012), accelerated failure time survival models were used to esti-mate accident duration of 6200 accidents on freeways on OrangeCounty, California, in 2001. The log-normal model was the best fit-ting model compared to the log-logistic, Weibull and exponentialmodels. As a result of the study, causal factors influencing accidentdurations were identified.

Many of the research studies on incident duration analyses can-not be generalised to other cases because: (1) the research wasbased on small sample sizes of up to several hundred incidentrecords; (2) the duration data were incomplete or of poor qual-ity; and (3) the results cannot be generalised to other locales as thecharacteristics of the modelled factors were inconsistent with oneanother, or the factor(s) were not available in other locales.

In view of these limitations, an analysis of traffic incidents onAustralian urban roads was undertaken to validate the factors aris-ing in the literature, to uncover other potential factors that mightinfluence the duration of traffic incidents, and to better understandincident duration effects in the Australian context.

3. Model development

In light of the large variance in incident duration, the use ofprobabilistic methods to understand the data generation process iswarranted. Specifically, hazard-based duration concepts in proba-bilistic methods, which are well suited for analysing time relateddata, are utilised to model the incident duration (Hensher andMannering, 1994).

Survival analysis is a collection of statistical procedures for dataanalysis for which the outcome variable of interest is time untilan event occurs (or ends). In the case of an incident occurrence,one of the key variables is incident duration—the length of the timebetween its detection and clearance. This length of time is a contin-uous random variable T with a cumulative distribution function F(t)and probability density function f(t). F(t) is also known as the fail-ure function and gives the probability of having an incident beforesome specified time t. Conversely, the survival function, S(t), is theprobability of the duration being greater than some specific time t.

F(t) = Pr(T ≤ t) = 1 − Pr(T > t) = 1 − S(t) (1)

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The hazard function h(t) gives the instantaneous potential perunit time for the event to occur, given that the individual has sur-vived up to time t (Washington et al., 2011).

h(t) = f (t)1 − F(t)

= f (t)S(t)

= lim�t→0

Pr(t + �t ≥ T ≥ t|T ≥ t)�t

(2)

The proportional hazard (PH) and the accelerated failure time(AFT) models are two alternative parametric approaches that incor-porate the effect of external covariates on hazard function (Greene,2012). With fully parametric models, a variety of distributionalalternatives for the hazard function are possible including theexponential, Weibull, log-logistic and lognormal. In contrast, onlythe exponential and Weibull distributions can accommodate PHassumptions (Jenkins, 2005).

This characteristic allows for simpler interpretation of theresults because the parameters measure the effect of the corre-sponding covariate on the mean survival time.

The key assumption for an AFT model is that survival timeaccelerates (or decelerates) by a constant factor when comparingdifferent levels of covariates. Whereas, the key assumption of a PHmodel is that hazard ratios are constant over time. Nevertheless, theAFT assumption allows for the estimation of an acceleration factorwhich can capture the direct effect of an exposure on survival time(Kleinbaum and Klein, 2012). Considering these model attributes,the AFT model is utilised to model incident duration in this study.

The AFT model assumes a linear relationship between the logof survival time T and a vector of explanatory variables X. In thismodel the effect of external covariates on survival time is directand accelerates or decelerates the time to “failure”:

ln(T) = ˇX + ε (3)

where ̌ is a vector of the estimated coefficient and ε is an errorterm.

A general formulation for AFT can be written:

h(t, X) = h0( t) (4)

where h0(.) is the baseline hazard function, = exp(−(ˇ′0 + ˇ′

1x1 +· · · + ˇ′

nxn)) = exp(−(ˇ′X)), and n is the number of explanatory vari-ables.

In estimating Eq. (4) with fully parametric models, three distri-butional alternatives were considered, namely: Weibull, lognormaland log-logistic for the hazard function are tested to find the bestfit to the incident duration data. The functional forms of the hazardfunction for each model can be derived by using the general func-tion and each distribution model with the positive location andscale parameters. These models are fitted using maximum likeli-hood method, where the distributions and corresponding modelsare given by (Greene, 2007; Washington et al., 2011):

Weibull distribution : h(t) = �p(�t)p−1 (5)

Lognormal distribution : h(t) = �(−pLog(�t))˚[−pLog(�t)]

(6)

Log-logistic distribution : h(t) = �p(�t)p−1

[1 + (�t)p](7)

where � and p are two parameters, known as the location parameterand scale parameter, respectively. �(.) is the standard probabil-ity density function and �(.) is the standard normal cumulativedistribution function. In this study, Log is the neperian logarithm.The results of the modelling can be interpreted according to themeasure of p for each model.

This approach is based on the assumption that the effect of anyindividual explanatory variable is the same for each observation,therefore parameters are treated as constant across observations.However, if incident duration is not homogeneous across obser-vations, erroneous inferences based on the improperly specified

model may result. To examine the homogeneity assumption, bothmodelling approaches are examined. The first approach applies thegamma distribution over the population with mean 1 and variance� to incorporate heterogeneity into the Weibull model as describedby Washington et al. (2011). This specification assumes that thesource of the heterogeneity is unobserved and not associated withincluded explanatory variables:

Weibull with gamma heterogeneity distribution :

h(t) = �p(�t)p−1

1 + �(�t)p(8)

A second approach incorporates unobserved heterogeneity in away that allows parameters to vary across observations based onsome pre-specified distribution (Greene, 2007; Washington et al.,2011). In this regard, random parameters are introduced into dura-tion models by adding a randomly distributed term in Eq. (3) byletting:

ˇn = ̌ + ωn (9)

where ˇn is a vector of parameters and varies across n observationsand ωn is a randomly distributed term (e.g. normally distributedterm with mean zero and variance 2). Estimation of randomparameter incident duration models is achieved by simulationbased maximum likelihood using Halton draws, an efficient alter-native to random draws (Washington et al., 2011). This approachassumes that heterogeneity is associated with one of the observedexplanatory variables. This random parameter specification allowsfor correlation across random parameters (discussed in the mod-elling results section), the examination of which may yield insightsinto the data generating process the underlying heterogeneity.

In this study, likelihood ratio statistics as described inWashington et al. (2011) are calculated and compared to selectthe best fitting model. Higher levels of significance for this statis-tic indicate superior goodness of fit. This statistic has been usedin numerous previous studies to assess model fit and thus is usedhere for direct comparison (see for instance Nam and Mannering,2000; Anastasopoulos et al., 2012a,c; Ghosh, 2012). All statisticalanalyses were performed using LIMDEP (Version 9, EconometricSoftware Inc., NY, USA).

4. Incident data description

Traffic incidents are caused by complex interactions betweenfactors. Generally, the difficulty in recording incident data and theirrelated factors at the desirable level of quality is a critical issue inanalysing the characteristics of traffic incidents. Often, due to thelack of comprehensive incident data, accessing the local historicaldata has been a major challenge for researchers (Konduri et al.,2003; Tavassoli Hojati et al., 2011).

To overcome this limitation, a logical framework analysis(Logframe) was used to establish an analytical process for struc-turing and systematising the data collection and the analysis ofthis research. The Logframe has four main stages: inputs, process,outputs, and outcomes. Logframe contains numerous characteris-tics associated with each incidents, such as environmental effects,geographic information, operational factors, and traffic data. Usingthis framework, all data from different sources were processed andlinked together. More details of Logframe and the process of inci-dent data preparation can be found in the study by Tavassoli Hojatiet al. (2012).

Incident data were obtained from the Queensland Departmentof Transport and Main Roads’ STREAMS Incident Management Sys-tem (SIMS) for South East Queensland (SEQ) urban road networksfor a one-year period to November 2010. SIMS is an incident

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Table 1Freeway incident durations by incident type on weekdays (min).

Incident type Number of incidents Mean Median SD Min Max COVa Skewness Kurtosis

Crash 854 43.4 32.1 99.4 5.1 391.2 99.4 4.1 22.5Hazard 1254 73.7 48.4 110.6 5.6 486.2 110.6 2.5 6.9Stationary vehicle 1143 41.2 25.9 124.9 5.0 430.7 124.9 4.2 22.7

a Coefficient of variation.

management system which is used throughout Queensland to cap-ture incident traffic events which cause an impact on traffic flowon the road network. Weather data were received from 10 Bureauof Meteorology stations around SEQ. The modelled traffic dataand road specifications were captured from the Brisbane StrategicTransport Model (BSTM). Chung (2010) has stated that temporaleffects such as day and type of day – weekday/weekends, publicholiday and school holiday – need to be incorporated into any inci-dent analysis. The analysis was conducted using twelve months ofincident data starting November 2009 on SEQ urban freeways. Thedatabase contains 4926 incidents. Since 99.6% of the incidents arethe result of either a crash, hazard, or stationary vehicle, other typesof incidents (alerts and floods) were excluded from the analysis.

All incident events cause temporary capacity reductions. Someevents occur unexpectedly such as vehicle-based incidents (e.g.crashes, stationary vehicles), other objects or obstructions on theroad (e.g. debris), or extreme weather events (e.g. flood). Thereare also events that might not be expected by all road users, butwhich are planned and are publicly notified (e.g. roadworks andsports/cultural activities). The scope of this study is limited to unex-pected non-recurrent congestion and only unplanned incidentshave been considered in the analysis.

The results of the analysis indicate that incident durations ofcrashes, hazards and stationary vehicles varied significantly moreon freeways compared to freeway ramps. Therefore, incident occur-ring on freeway ramps need to be analysed separately, and thisgroup of incidents was excluded from the analysis and are to beexamined in future research. In addition, no statistically signifi-cant differences in durations were found by month of the year,week of the month, and day of the week (i.e. normal day vs. pub-lic/school holiday, etc.) Conversely, incident durations on weekdaysvs. weekend days revealed significant differences. As a result, free-way incident durations models were estimated for three majorincident types on weekdays. Table 1 shows statistics of freewayincident durations for the three major types of incidents on week-days.

Each incident type has a different classification according tothe definitions in SIMS. There are 7, 12 and 4 categories within‘crashes’, ‘hazards’ and ‘stationary vehicles’, respectively. Statis-tical analyses using the Kruskal–Wallis test for non-parametricdata when comparing more than two unmatched groups fromnon-Gaussian population were performed to measure and test thestatistical significance between classifications within each inci-dent type. Considering the limited numbers of sample sizes withinclassifications, new classifications were defined by logically com-bining classifications as well as by testing significant differences

between incident types. Based on the results, there was a sta-tistically significant difference between the incident duration in‘crashes’ (H(2) = 18.603, P < 0.001) with a mean rank of 833 for C1,413.23 for C2 and 496.59 for C3, in ‘hazards’ (H(3) = 128.9, P < 0.001)with a mean rank of 912.3 for H1, 739.7 for H2, 559.6 for H3 and1067 for H4, and in ‘stationary vehicles’ (H(1) = 14.78, P < 0.001)with a mean rank of 560.4 for S1 and 693.8 for S2. As a result ofthis data reduction process, 3, 4 and 2 categories were redefinedfor ‘crashes’, ‘hazards’ and ‘stationary vehicles’; respectively, forfurther analysis as shown in Table 2.

Potential independent variables were identified from the inci-dent database and are shown in Table 3. In the first step, statisticalsignificance tests were conducted for each independent variable oneach incident types’ duration in order to identify potentially signif-icant variables. Then, a set of potential explanatory variables andvarious combinations were tested to identify potential interactionsbetween variables. Akaike’s Information Criterion (AIC) was usedto compare models including the null model with a constant termonly. A decrease in the AIC value reveals the importance of a set ofvariables in the model in explaining variation across incident dura-tions. A stepwise procedure was employed to select the significantvariables for each incident type as described by Collett (2003).

5. Modelling results

As described in Section 3, seven parametric survival modelswere fitted to the incident data for each of the three types of inci-dent. Tables 4–6 show the estimation results for the crash, hazard,and stationary vehicles models respectively, and show the param-eter estimates and t-statistics for model variables as well as theoverall goodness of fit statistics including model log-likelihood andlikelihood ratio statistics. The best fitting model was selected foreach incident type based on the results of the likelihood ratio tests.

Fixed parameter models were estimated using standard max-imum likelihood methods, whereas random parameters modelswere estimated using simulated maximum likelihood using200 Halton draws, since prior studies have shown that 200 Haltondraws is usually adequate to achieve stable parameter estimates(Anastasopoulos et al., 2012b).

Tables 4–6 list the parameter estimates for models estimated forincident duration for each incident type. A positive sign of a param-eter estimate suggests an increase in the incident duration anda decrease in hazard function associated with an increase in thatvariable. Importantly, unobserved heterogeneity is present in theincident data, whereas the normal distribution best replicated the

Table 2New classifications in each incident type.

Crashes Hazards Stationary vehicles

C1 C2 C3 H1 H2 H3 H4 S1 S2

Heavy vehicle MotorcycleMultiplevehiclePedestrianRoad furniture

Single vehicleOther

Pavement failureWater over road

AnimalSpill/chemicalVegetation other

Debris fire fogPedestrian/cyclistSmoke

Signal fault Breakdown HazardousMegaTow MRTow to Safety

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Table 3Potential independent variables and their descriptions.

Variable Value Variable Value

Incident details Traffic variablesIncident classification (In.Cl.) integer codea Daily traffic volume Vehicles/dayDistance from CBDb (Dist CBD) km Daily CV%c

Injury 1 = Yes; 0 = No Traffic volume Vehicles/hFatality 1 = Yes; 0 = No CV%Major incident 1 = Yes; 0 = No Volume over CapacityTraffic disrupted 1 = Yes; 0 = No Temporal characteristicsMedical required 1 = Yes; 0 = No Day of week (DOW) 1–5d

Towing required 1 = Yes; 0 = No Week of month 1–4Chemical spill 1 = Yes; 0 = No Month of year 1–12Multiple vehicle involved 1 = Yes; 0 = No Hour of day 1–24Diversion required 1 = Yes; 0 = No Time period of day 1–4e

Assistance requested 1 = Yes; 0 = No Weather characteristicsInfrastructure characteristics Air temperature ◦CPosted speed km/h Wind speed km/hNumber of lanes 2–6 Rain precipitation mmLink capacity Vehicles/h

a 3, 4 and 2 categories for ‘crash’, ‘hazard’ and ‘stationary vehicles’, respectively.b CBD: central business district.c CV: Commercial Vehicle.d Monday to Friday.e 1 = evening (18-7), 2 = AM peak (7-9), 3 = off peak (9-16), 4 = PM peak (16-18).

unobserved heterogeneity. All variables are statistically significantat a 95% confidence level.

The results indicate that although a Weibull AFT model with ran-dom parameters is the best fitting model for crashes and hazards,a Weibull AFT model with gamma heterogeneity and fixed param-eters provides the best fit for stationary vehicles. Moreover, thecollection of statistically significant variables is different for eachcrash type. There is no evidence of the use of random parametersduration models for incident data in the literature. However, the

result for third type of incident is consistent with the previousresearch by Nam and Mannering (2000).

In general, none of the infrastructure variables have significanteffects on the durations of different types of incidents, includ-ing posted speeds, number of lanes, and link speeds. Similarly,weather conditions including air temperature, wind speed, andprecipitation did not have significant effects on incident durations.Moreover, the durations of crashes and hazards are not significantlyaffected by traffic variables.

Table 4Summary of survival AFT model estimation results for crashes. **Parameter estimation followed by t-statistics in parentheses. ***Dependent variable is log of incident durationin minutes.

Variable Fixed parameters Random parameters

Log-logistic Lognormal Weibull Weibull withGammaheterogeneity

Log-logistic Lognormal Weibull

Constant 3.21 (55.42) 3.25 (56.65) 3.78 (62.59) 3.21 (47.87) 3.22 (68.53) 3.24 (65.67) 3.57 (113.77)Dist CBD .012 (7.08) .0116 (6.42) .0111 (6.50) 1.2 (7.09) .0135 (9.32) .013 (8.02) .0124 (11.98)Std. Dev.c .0125 (9.42) .013 (11.06) .0204 (21.05)Major incident .489 (8.51) .481 (7.91) .427 (8.73) 0.49 (8.49) .483 (10.72) .479 (9.58) .479 (15.44)Std. Dev.c .163 (3.7) .107 (2.19) .122 (4.04)Diversion .601 (5.25) .571 (5.55) .848 (8.63) 0.6 (5.19) .539 (4.71) .53 (4.22) .472 (5.83)Std. Dev.c .748 (6.33) .779 (5.65) .977 (10.47)Towing .186 (4.30) .198 (4.42) .162 (3.89) 0.19 (4.3) .193 (5.48) .187 (4.85) .174 (6.96)Std. Dev.c .456 (14.43) .399 (11.72) .596 (27.19)Medical .253 (4.26) .260 (4.13) .202 (4.30) 0.25 (4.26) .247 (5.38) .262 (5.07) .223 (7.12)Chemical spill .574 (3.31) .658 (3.90) .661 (4.15) 0.57 (3.3) .481 (3.17) .557 (3.43) .433 (3.96)PM peak −.105 (−2.01) −.116 (−1.96) −.238 (−4.40) −0.1 (−1.99) −.091 (−2.25) −.107 (−2.28) −.125 (−4.51)Std. Dev.c .262 (6.46) .223 (4.84) .287 (10.18)DOW (Tuesday) .179 (3.64) .152 (3.03) .146 (3.03) 0.18 (3.63) .157 (3.9) .147 (3.4) .143 (5.11)In.Cl. C2 −.238 (−4.70) −.273 (−5.59) −.466 (−11.07) −0.24 (−4.64) −.273 (−6.6) −.279 (−6.43) −.396 (−14.03)Std. Dev.c .456 (16.77) .426 (14.23) .569 (29.03)Sigma () .333 (32.23) .588 (42.00) .583 (39.07) 0.33 (15.89) .266 (34.01) .5 (41.77) .324 (36.86)Teta – – – 1.01 (6.29) – – –

P 3.005 1.701 1.717 3.011 3.754 2.0 3.08LL (0)a −936 −939 −995 −995 −936 −939 −995LL (ˇ)b −758 −762 −826 −762 −744 −751 −749Sample size 854 854 854 854 854 854 854Number of covariates 11 11 11 12 11 11 11

Likelihood ratio statistics 356 354 338 466 384 376 594

a Initial log-likelihood.b Log-likelihood at convergence.c Standard deviation of normally distributed parameter.

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Table 5Summary of survival AFT model estimation results for hazards. **Parameter estimation followed by t-statistics in parentheses. ***Dependent variable is log of incident duration in minutes.

Variable Fixed parameters Random parameters

Log-logistic Lognormal Weibull Weibull withGammaheterogeneity

Log-logistic Lognormal Weibull

Constant 4.07 (66.93) 4.08 (66.47) 4.63 (75.61) 4.08 (57.48) 4.04 (85.99) 3.97 (172.96) 4.09 (389.07)Std. Dev.c .78 (20.97) 1.19 (49.47) 1.21 (112.88)Dist CBD .015 (6.81) .014 (6.09) .011 (5.03) .015 (6.78) .017 (9.17) .018 (21.09) .02 (49.43)Std. Dev.c .035 (20.02) .029 (34.93) .035 (85.55)Major incident 1.05 (3.26) 1.03 (2.46) .782 (1.78) 1.04 (3.25) 1.03 (4.78) 1.002 (9.93) 1.05 (24.22)Std. Dev.c .584 (5.46)Assistance −.231 (−4.86) −.22 (−4.45) −.256 (−6.32) −.231 (−4.84) −.2 (−5.86) −.198 (−12.22) −.2 (−27.12)Std. Dev.c .438 (14.2) .235 (15.05) .135 (26.5)AM peak .141 (2.13) .146 (2.21) .230 (4.05) .141 (2.14) .17 (3.49) .118 (5.0) .157 (14.59)Std. Dev.c .512 (10.53) .833 (35.96) .436 (40.10)DOW (Wednesday) −.188 (−3.33) −.160 (−2.75) −.104 (−2.14) −.188 (−3.32) −.173 (−4.17) −.177 (−8.96) −.214 (−23.89)

.64 (16.21) .417 (21.42) .511 (57.55)In.Cl. H1 .386 (3.73) .372 (3.34) .251 (2.11) .385 (3.71) .383 (4.9) .434 (11.39) .36 (20.61)Std. Dev.c 1.2 (31.85) .46 (28.61)In.Cl. H3 −.480 (−9.14) −.485 (−9.13) −.593 (−11.60) −.482 (−9.16) −.482 (−12.43) −.44 (−23.13) −.456 (−52.82)Std. Dev.c 1.03 (62.59) 1.38 (185.64)Sigma (�) .484 (38.52) .851 (49.08) .848 (43.71) .487 (18.71) .354 (41.67) .281 (49.71) .328 (45.31)Teta – – – .986 (7.31) – – –

P 2.064 1.175 1.180 2.053 2.896 3.56 3.05LL (0)a −1690 −1679 −1789 −1789 −1690 −1679 −1789LL (�)b −1577 −1587 −1679 −1587 −1538 −1567 −1532Sample size 1254 1254 1254 1254 1254 1254 1254Number of covariates 9 9 9 10 9 9 9

Likelihood ratio statistics 226 184 220 404 304 224 514

a Initial log-likelihood.b Log-likelihood at convergence.c Standard deviation of normally distributed parameter.

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Table 6Summary of survival AFT model estimation results for stationary vehicles. **Parameter estimation followed by t-statistics in parentheses. ***Dependent variable is log of incident duration in minutes.

Variable Fixed parameters Random parameters

Log-Logistic Lognormal Weibull Weibull with Gamma heterogeneity Log-Logistic Lognormal Weibull

Constant 3.425 (44.67) 3.453 (46.87) 4.040 (54.95) 3.25 (38.66) 3.42 (64.04) 3.43 (83.13) 3.67 (132.9).189 (8.61) .892 (21.40) .822 (30.76)

Major incident .981 (4.86) .939 (3.36) .660 (1.77) 1.026 (5.59) .977 (5.0) .957 (10.71) .937 (16.76)Std. Dev.c .429 (4.77) .492 (8.62)CV% 028 (6.30) .025 (5.53) .023 (6.30) .028 (6.64) .028 (6.45) .027 (12.46) .025 (16.96)Std. Dev.c .014 (7.14) .041 (29.8)In.Cl. H1 −.409 (−5.64) −.377 (−5.49) −.535 (−7.79) −.354 (−5.02) −.406 (−5.76) −.364 (−9.19) −.401 (−15.03)Std. Dev.c .796 (26.92) 1.7 (65.89)Sigma () .442 (39.18) .797 (51.94) .903 (47.46) .358 (16.74) .428 (39.65) .375 (47.86) .253 (43.22)Teta – – – 1.64 (8.97) – – –

P 2.261 1.254 1.108 2.794 2.33 2.67 3.95LL (0)a −1399 −1400 −1585 −1585 −1399 −1400 −1585LL (ˇ)b −1353 −1363 −1554 −1341 −1352 −1349 −1376Sample size 1143 1143 1143 1143 1143 1143 1143Number of covariates 6 6 6 7 6 6 6

Likelihood ratio statistics 92 74 62 488 96 102 418

a Initial log-likelihood.b Log-likelihood at convergence.c Standard deviation of normally distributed parameter.

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(a)Crashe s 1 (b)Hazar ds 1

1 Weibull model with rando m para meters *

2 Weibull model with g amma heter ogenit y and f ixed p aramete rs

* based on parameter mean s

(c) Stat ion ary ve hicles 2

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Fig. 1. Hazard function for different incident types.

There are six and seven random parameters in the crash and haz-ard models, respectively. These random parameters were selectedaccording to the significant variation of estimated parametersacross observations, i.e. the confidence interval of the standarderror does not include zero. The standard deviation of modelparameters and the related t-statistics are shown in the table.The results indicate that parameter P in Weibull model is positiveand significantly different from zero in both crashes and hazardstype, is a monotonically increasing function, and suggests thatthe longer an incident lasts, the more likely it is to end. In thecase of the Weibull model with gamma heterogeneity (stationaryvehicle model), the parameter P is positive and significantly differ-ent from zero, while the parameter � is also statistically differentfrom zero (t-statistic of 9.01), suggesting that the hazard functionincreases from zero until it reaches a maximum at an inflectionpoints (28 min), then decreases over time to approach zero. In otherwords, incident durations are likely to end soon only after they havelasted longer than 28 min. Fig. 1 shows the hazard functions for thedifferent incident types.

The results from the implied correlation matrix of randomparameters of crashes indicate that there are high positive cor-relations (correlations greater than 0.6) between some randomvariables, namely: dist CBD and diversion (0.899), major incident andtowing (0.888), and In.Cl. C2 and PM peak (0.966). The greater thedistance an incident or accident occurs from the CBD, the morelikely a diversion will be required. In addition, major crashes weremore likely to require towing. Finally, more crashes on classifi-cation C2 occurred during the PM peak. In contrast, crashes withclassification C2 occurring during the PM peak were less likely to bediverted.

Similarly; a high positive correlation between random parame-ters for distance from CBD and assistance required (0.889) suggeststhat hazards that occur more distance from the CBD are more likelyto require assistance.

To justify the estimation of separate incident duration modelsacross incident types, a likelihood ratio test was conducted:

2 = −2[LL(ˇT ) − LL(ˇC ) − LL(ˇH) − LL(ˇS)] (10)

where LL(ˇT) is the log-likelihood at convergence of the modelestimated with the data from all incidents combined, and LL(ˇC),LL(ˇH)and LL(ˇS)are the log-likelihoods at convergence of the mod-els estimated with the data from incident types ‘crashes’, ‘hazards’and ‘stationary vehicles’, respectively. The statistic is chi-squaredistributed with degrees of freedom equal to the summation ofthe number of estimated parameters in each type of incidentmodel minus the number of estimated parameters in the com-prehensive model. The likelihood ratio is 492 with 6 degreesof freedom. Therefore, the null hypothesis indicating that themodels are statistically indistinguishable is rejected with 99%confidence.

To gain further insight into the effects of explanatory vari-ables on AFT models for each incident type, the exponents of thecoefficients provide an intuitive way to interpret the results. Theexponents of the coefficients, with all coefficients typically evalu-ated at their mean values, translates to a percent change in incidentdurations resulting from a unit increase for continuous explana-tory variables and a change from zero to one for indicator variables(Jenkins, 2005). For instance, the exponent of the estimated coeffi-cient of ‘Chemical spill’ (random parameters Wiebull for crashes) isexp(.433) = 1.54, and indicates that the duration of crashes involv-ing spilled chemicals will last 54% longer on average (it varies fromcrash to crash due to the random parameter) compared to ‘baseline’crashes. Moreover, for each mile further from the CBD, the dura-tion of an event will last 1.2% longer (this effect is likely to reflectthe length of time required to get services dispatched to clear theincident).

Table 7 shows the impact of each significant variable on durationin each model of incident type. In the fixed parameter models, theestimated effect is assumed to be the same for all incidents, whereasin the random parameters models the effect is expected to varyacross the sample of incidents according to the random parameterdistribution.

For each 1% increase in the percentage of the commercialvehicles, incident durations resulting from stationary vehiclesincrease by 2.5%. While stationary vehicles are not associatedwith any of the temporal effect variables, crashes during the PMpeak resulted in incident durations 11% shorter than those of

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Table 7Percent changes in incident duration WRT unit changesa in each variable. **Elasticity values are estimated at the sample means of all other variables.

Crashes Hazards Stationary vehicles

Variable Changes (%) Variable Changes (%) Variable Changes (%)

Distance from CBD 1.2 Distance from CBD 2 CV% 2.5Major incident 62 Major incident 185 Major incident 155Diversion required 60 Assistance requested −18 Incident classification S2 −33Towing required 19 AM peak 17Medical required 25 Day of week (Wednesday) −19Chemical spill 54 Incident classification H1 43PM peak −11 Incident classification H3 −36Day of week (Tuesday) 15Incident classification C2 −32

a One unit for continuous variables and zero to one for binary variables.

Fig. 2. The typical survival function of Weibull AFT models.

other times of day—perhaps reflecting a rapid response to clearingcrashes during the am peak. This might suggest also an omittedvariable issue regarding a “priority incident” as deemed by a localauthority. In contrast, the duration of hazards increases by 15%during the AM peak. This may reflect a more uncertain nature ofhazards, the caution that must be exercised for the nearby morningpeak.

As anticipated, major incidents are associated with increaseddurations of 61% for crashes, 185% for hazards, and 155% for sta-tionary vehicles. Crashes involving motorcycles, multiple vehicles,pedestrians, and road furniture (Classification C2) are likely to have32% shorter durations.

While the incident durations of hazards including pavementfailure and water over the road last 43% longer than baselinedurations on average, there is a 36% reduction in the case ofhazard classification H3, including debris, fire, fog and smoke.Results also reveal that the incident durations of stationary vehi-cles involving ‘breakdown’ were 33% shorter than those of othercauses.

Although the specific location of an incident was not a significantfactor for stationary vehicle incident durations, durations increaseby 1.2% and 2% per kilometre from the CBD for incident durationsof crashes and hazards, respectively.

Crashes involving chemical spills result in an increase in dura-tion by 54%. Similarly, crashes where diversion was required,medical attention was required, or where towing was requiredincrease incident durations by 60%, 25% and 19% respec-tively.

It should be noted that some of the variables were expected tohave significant influences on incident durations, namely when acrash involved injuries or fatalities. However, these variables didnot reveal significant effects on durations (only four fatalities wererecorded in the 12 months analysed data). These effects might becaptured by other included variables such as ‘major incident’.

As can be seen from the analysis results, different variableshave varying effects on incident durations according to the incidenttype. For example, while distance from CBD is significant factor onincident durations for crashes and hazards, it is not important forstationary vehicles. Conversely, CV% is consequential on the dura-tions of stationary vehicles, whereas, it has no effect on the incidentdurations of crashes and hazards. Despite that temporal charac-teristics have varying effects on the durations of the crashes andhazards, they have no impact on stationary vehicle incident dura-tions. These findings emphasise the importance of modelling eachtype of incident based on its characteristics and contributory fac-tors. They also indicate that types of incidents differ in their abilityto be effectively cleared from the roadway.

6. Summary and conclusions

This paper describes the modelling of incident duration datafrom a South East Queensland network of freeways during theperiod November 2009 to November 2010. Parametric acceleratedfailure time (AFT) survival models including the log-logistic,lognormal and Weibull were estimated. To consider heterogeneityin incident durations, both random parameter specifications as

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well as a Weibull with gamma heterogeneity were examined. Thisis the first known traffic incidents duration model using randomparameters. The study facilitated not only an investigation of thefactors affecting incident durations, but also the exploration ofvarious survival distributions. Based on the some preliminarydata reduction, three major types of incidents including crashes,hazards, and stationary vehicles on weekdays were examined. Atotal of 3251 incidents were recorded in the study dataset, yielding26% of incidents to crashes, 39% to hazards, and 35% to stationaryvehicles. Twenty-eight variables were examined including incidentdetails, infrastructure characteristics, traffic variables, temporalcharacteristics and infrastructure characteristics.

Weibull models with random parameters were most suitable fortwo types of incidents on freeways involving crashes and hazards.In addition, a Weibull model with gamma heterogeneity providedthe best fit for stationary vehicle incidents. The estimated param-eters and collection of significant variables of the models weredifferent across different types of incidents.

The analyses described in this paper indicate that a total ofnine variables significantly affect the duration of crash types.Seven variables reflect incident details, while two were relatedto temporal characteristics. For understanding incident durationsresulting from hazards, seven significant variables were identified,five of them associated with incident details and two related totemporal characteristics. Lastly, three variables were found to sig-nificantly affect the durations of the stationary vehicle incidents,two representing incident details and one representing traffic char-acteristics.

The results clearly indicate that the durations of each typeof incident are uniquely different, and require different types ofresponses to clear them from a road and have differential impact oncumulative clearance times and hence delay. Understanding howdifferent factors influence modelling incident durations allows anetwork manager to identify policy responses to different incidenttypes and to better understand what tools and unique emergencyresponses might be required to improve response times.

Importantly, there were no significant effects associated withinfrastructure and weather characteristics on the durations of dif-ferent types of incidents, although all incidents examined occurredon freeways.

Fig. 2 depicts the typical survival function of a Weibull AFTmodel for a confidence level of 90% (using average incidentresponse). An objective, of course, is to apply the learnings of themodels and attempt to shift the curve to the left, thus shorteningincident durations. The survival functions based on explanatoryvariables can be utilised to make incident duration predictions,real time. Consequently, different traffic incident managementresponse initiatives can be tailed for different incident types. Theimplementation of incident duration predictions within congestionmanagement and mitigation program has significant potential ben-efits, and can assist to achieve more efficient use of limited incidentmanagement resources associated with congestion management.Ultimately, the results are of considerable utility for evaluatingthe potential impacts of different incident management scenarios,program, and systems.

Further research needs to be conducted to investigate the tem-poral stability of incident durations as well as to verify the resultsof this study using a few years of incident data. In addition, it isrecommended that the effects of directly measured (as opposedto modelled) traffic flow parameters on traffic incident durationshould be examined. This direct measurement can more accu-rately capture the potential impacts of investigated variety oftraffic parameters on durations, including flow, speed, and den-sity. Finally, potential omitted variables should be examined, suchas “priority response” of local agencies as potentially captured bytime of day, among other possible omitted variables.

Acknowledgments

The authors would like to express their appreciation to AnnaWebster from the Queensland Department of Transport and MainRoads (DTMR) and also to the Australian Bureau of Meteorology fortheir support and assistance with data collection.

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