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Transportmetrica A: Transport Science
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Microscopic traffic behaviour modelling
and simulation for lane-blocking
arterial incidents
Jiuh-Biing Sheu aa
Institute of Traffic and Transportation, National Chiao Tung University , 4F, 114 Chung Hsiao W. Rd., Sec. 1, Taipei , Taiwan Published online: 23 May 2011.
To cite this article: Jiuh-Biing Sheu (2013) Microscopic traffic behaviour modelling and simulation
for lane-blocking arterial incidents, Transportmetrica A: Transport Science, 9:4, 335-357, DOI: 10.1080/18128602.2011.577042
To link to this article: http://dx.doi.org/10.1080/18128602.2011.577042
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Microscopic traffic behaviour modelling and simulation for lane-blocking
arterial incidents
Jiuh-Biing Sheu*
Institute of Traffic and Transportation, National Chiao Tung University, 4F, 114 Chung Hsiao W. Rd., Sec. 1, Taipei, Taiwan
(Received 10 March 2010; final version received 29 March 2011) Incident-induced driver behaviour modelling is essential to analyse non-recurrent traffic congestion problems. However, such research is inadequate in such areas as traffic flow prediction, traffic simulation and incident management. This article presents microscopic lane traffic models to characterise incident-induced driver behaviour including car following and lane changing conducted under conditions of lane-blocking arterial incidents. To demonstrate the validity of the proposed models, a specific microscopic traffic simulation program embedded with the proposed incident-induced lane traffic behaviour models is tested by comparing simulation data with video-based incident data collected from five incident events. Preliminary test results indicate that the proposed microscopic traffic behaviour models permit not only reproducing incident-induced traffic behaviour but also characterising incident effects on lane traffic phenomena.
Keywords: incident-induced traffic modelling; lane changing; microscopic traffic simulation
1. Introduction
Modelling incident-induced lane traffic manoeuvres remains challenging in the field of traffic flow theory and related areas such as traffic simulation and incident management (Hawas 2007). In reality, numerous related issues including queue overflows and gridlocks caused by lane-blocking incidents have resulted in difficulty in the operations of traffic control and management (Sheu 2003, Qi et al. 2009). Compared with incident-free cases, lane-blocking incidents may result in anomalous increases in lane changes upstream from incident sites, thus creating unusual queue lengths and delays (Sheu and Ritchie 2001). From a psychological point of view, drivers’ rubbernecking and driving with pressure while passing by an incident site also add complexity and uncertainty to incident traffic characterisation and prediction (Holland 1998, Hamdar and Mahmassani 2008). Second, lane changes can generate other traffic flow phenomena, such as secondary accidents and spillback events, during an incident as lane changes from blocked lanes can cause irregular traffic interference among adjacent lanes that adversely affect road safety. Thus, modelling incident-induced lane traffic manoeuvres is vital to develop sophisticated traffic flow theories, as well as advanced traffic control and management methodologies, that address non-recurrent traffic congestion problems.
*Email: [email protected]
ß 2013 Hong Kong Society for Transportation Studies Limited
Vol. 9, No. 4, 335–357, http://dx.doi.org/10.1080/18128602.2011.577042
Microscopic traffic behaviour modelling has garnered increased recognition for its ability to accurately reproduce incident-induced lane traffic dynamics, particularly lane-changing behaviour during incident-induced lane traffic jams (Chowdhury et al. 2000). In contrast with the aforementioned macroscopic methodology, which assumes that traffic has fluid-like dynamics, microscopic methodology treats incident-induced traffic flow as a nonequilibrium system of particles with transitional jamming phases caused by unusual speed and lane-changing variations. Janowsky and Lebowitz (1992) applied the asymmetrical simple exclusion process that uses a unique transition probability to characterise incident-induced traffic jams on a one-dimensional lattice. Nagatani (1994) extended this asymmetrical simple exclusion model for two-lane highway incident cases. Kurata and Nagatani (2003) employed an optimal velocity model to generate a two-lane traffic model in which symmetrical lane-changing rules were adopted to investigate incident-induced traffic congestion. Kerner and Klenov (2003) and Kerner et al. (2006) proposed parameter-rich microscopic models to investigate spatial-temporal traffic congestion patterns of highway bottlenecks, and discussed the effects of lane changes on the structure of highway bottlenecks and moving traffic jams. This was followed by the development of a cellular automation (CA)-based traffic model by Zhu et al. (2009), in which the symmetrical and asymmetrical lane-changing rules proposed by Chowdhury et al. (1997) were utilised to improve the model’s ability to characterise incident-affected lane changing. Additionally, some studies which did not focus on incident cases have proposed sophisticated microscopic models that rationalise the entire driver decision process associated with lane changing (Choudhury 2005, Ramanujam 2007, Toledo et al. 2007, 2009, Webster et al. 2008).
Despite the advances made by these pioneering scholars in microscopic traffic modelling, incident-induced driver behaviour modelling on urban streets remains challenging. The intricacies in modelling are based on the discontinuity of traffic flows caused by traffic signal control, accompanied by the network effect of traffic flows moving across blocks characterised by various geometric characteristics. For instance, vehicles in blocked lanes must change lanes. However, changing lanes at will in this instance is often difficult due to various traffic conditions such as arriving and discharging vehicles in adjacent lanes. When an incident is located on a short block or at an intersection, a bottleneck and serious network-wide gridlock likely occur, increasing the complexity in reproducing traffic flows on urban streets. Moreover, unlike freeway incidents with a one-dimensional impact on traffic flows, the impact of urban street incidents can be two-dimensional (network-wide) and irregularly change with time when neighbouring traffic signal controls are not coordinated (Sheu 2002).
Accordingly, this work applies a microscopic traffic behaviour modelling approach that characterises incident-induced intra-lane and inter-lane traffic manoeuvres. This study primarily focuses on cases of arterial lane blocking (i.e. incidents on a roadway between two adjacent intersections). The proposed models are applicable to multi-lane blocking as well as single-lane blocking.
The remainder of this article is organised as follows. Section 2 introduces the proposed lane traffic behaviour models to elucidate the rationale of using the proposed models to characterise incident-induced intra-lane and inter-lane traffic manoeuvres. Section 3 presents the primary procedures and analytical results for model tests to verify whether the proposed models permit accurately characterising incident-induced lane traffic phenomena on arterials. Finally, Section 4 gives concluding remarks and directions for future research.
2. Traffic behaviour modelling
The proposed microscopic lane traffic behaviour models contain two components: intra-lane and inter-intra-lane traffic behaviour models to characterise incident-induced intra-lane traffic manoeuvres present in the area upstream of an incident site on an arterial, namely, subsystem 1 of a detection zone (Figure 1). According to analytical results and simple identification logic, the microscopic intra-lane and inter-lane traffic manoeuvres were classified into several types to model the movement of an individual vehicle within subsystem 1 (Figure 2). This work aims at modelling incident-induced car-following and lane-changing manoeuvres conducted in blocked lanes upstream from an incident site, which are described in the following subsections.
2.1. Intra-lane traffic behaviour modelling
In this subsection, we first introduce three types of inter-vehicle dynamic spacing to facilitate intra-lane traffic behaviour characterisation, followed by the modelling of specific intra-lane traffic behaviour present in these three dynamic spacing scenarios.
2.1.1. Inter-vehicle dynamic spacing
In adopting the concept of dynamic thresholds of inter-vehicle spacing proposed by Sheu (2007), this work specified three inter-vehicle dynamic spacing types: (1) free-flow movement; (2) incident-induced car following and (3) mandatory braking. These three
Subsystem 1 Subsystem 2 Detection zone
(for arterial incident cases)
Point detector Incident
Traffic flow direction Figure 1. Definition of a detection zone.
spacing types are defined to characterise three types of driver behaviour of a given vehicle moving in a blocked lane of subsystem 1. Figure 3 shows the three types of inter-vehicle
spacing, and the corresponding dynamic boundaries, i.e. XMðkÞand XFðkÞ, which indicate
the required spacing for mandatory braking (M) and free-flow movement (F) at a given time step k. Vehicles upstream from an incident site must brake when the dynamic
inter-vehicle spacing is 5XMðkÞ. Conversely, when the spacing exceeds XFðkÞ, free-flow lane
manoeuvres are allowed; otherwise, vehicles may change lanes to pass the incident site. The following gives the derivational procedures for these dynamic thresholds, followed by the description of the proposed models.
Individual vehicle Incident is perceived Incident is not perceived
Blocked lane Adjacent lane
In ci dent-i n d u ce d la n e chan gin g In ci dent-indu ced ca r fo llo w in g Brakin g Discretionary ca r fo llo w in g Discretionary car fo llo wing / la n e chan gin g Yes No Yes No Dis cretionary la n e chan gin g Front most vehicle Lane changing needed Lane changing possible Front most vehicle In ci dent-indu ced ca r fo llo w in g In ci dent-indu ced ca r fo llo w in g In ci dent-indu ced la n e chan gin g Front most vehicle Lane changing possible Yes No Yes No Yes No No Yes
Figure 2. Classification of lane traffic manoeuvres in subsystem 1.
XM(k) XF(k)
Mandatory
Incident-induced
Free-flow
braking area
car-following area
moving area
0
Spacing relative to the front vehicle
Figure 3. Illustration of the dynamic spacing for intra-lane traffic modelling.
The derivation of XMðkÞ is based primarily on dynamic safety spacing that must be
maintained between a target and the vehicle in front when both drivers brake (Figure 4). Such braking may exist under the condition that both the target and front vehicles are approaching vehicles queued upstream from the incident site. In considering the mandatory braking that may be conducted by a front vehicle at any given time step k, the target vehicle must maintain the minimum safety spacing to ensure safe stopping.
Accordingly, this work derives XMðkÞas follows:
XMðkÞ ¼ XsþZnðk þ nÞ Zn1ðk þ n1Þ ð1Þ )XMðkÞ ¼ Xsþ VnðkÞ n VnðkÞ ½ 2 2Dn þ½Vn1ðkÞ 2 2Dn1 ð2Þ
where Xs (unit: m) is the minimum safety spacing between the target and front vehicles,
and is to be calibrated in the section of model tests; Zn(k þ n) and Zn1(k þ n1) are the
braking distances (unit: m) associated with target vehicle n and front vehicle n 1 that are Time step n k+π −1 + n k π Zn−1(k+πn−1) Zn(k+πn) Xs k ) (k XM Relative spacing The moving target vehicle (coded with n)
The stopped target vehicle (coded with n)
The moving front vehicle (coded with n–1)
The stopped front vehicle (coded with n–1)
n n–1
n n−1
Figure 4. Illustration of the critical spacing for mandatory braking.
fully stopped at time steps k þ n and k þ n1, respectively; VnðkÞ and Vn1ðkÞare the
speeds (unit: m=s) of target vehicle n and front vehicle n 1 at time step k, respectively; Dn
and Dn1represent maximum deceleration rates (m=s2) of target vehicle n and front vehicle
n 1, respectively, and both depend on the type of vehicle and nis the reaction time (unit: s)
associated with target vehicle n, which is calibrated in the section of model tests.
This work now considers the following interesting cases to determine the validity of the
aforementioned derivation. For simplification, we assume parameters Dn and Dn1
depending only on vehicle type. Thus, if the target vehicle and front vehicle are the same
vehicle type (Dn¼Dn1), then Equation (2) can be rewritten as
XMðkÞ ¼ Xsþ VnðkÞ n VnðkÞ ½ 2½Vn1ðkÞ2 2Dn ð3Þ
Equation (3) characterises the relative speed relationship between the target and front
vehicles, and the corresponding effect on the determination of XMðkÞ. Given the
instantaneous speed of the front vehicle (i.e. Vn1ðkÞ), as the speed of the target vehicle
increases, the value of XMðkÞ increases, implying that the target vehicle needs a longer
braking distance when it runs at a higher speed relative to low speed conditions. Conversely, when the driver of the target vehicle perceives that the speed of the front
vehicle has increased (i.e. VnðkÞ5 Vn1ðkÞ), the value of XMðkÞmay decline. Notably, in
applications such as microscopic simulation, Equation (3) can be integrated with specific stochastic processes that represent various driver characteristics to determine the specific
value of XMðkÞfor any given vehicle.
In contrast with XMðkÞ, XFðkÞis derived based on the assumption that target vehicle
movement is independent of any behaviour of the front vehicle in a free-flow area. This work now considers a specific case, in which the target vehicle can accelerate without considering the front vehicle braking at time step k. Therefore, the generalised form of
XFðkÞshould theoretically satisfy the following condition:
XFðkÞ XMðkÞ þ Vnðk þ1Þ ½ 2½VnðkÞ2 2anðkÞ ½Vn1ðkÞ 2 2dn1ðkÞ ð4Þ
where Vnðk þ1Þ (unit: m=s) is the speed of target vehicle n at time step kþ1; anðkÞis the
acceleration (unit: m=s2) of target vehicle n at time step k and d
n1ðkÞis the instantaneous
deceleration (unit: m=s2) by front vehicle n 1 at time step k.
Similarly, the following critical condition is used to determine the lower bound of XFðkÞ
( ~XFðkÞ). According to the aforementioned assumption for XFðkÞ, any given target vehicle n
moving in a free-flow area should be allowed to undertake maximum acceleration even
when the front vehicle undertakes maximum deceleration. Thus, the lower bound of XFðkÞ
( ~XFðkÞ) is ~ XFðkÞ ¼ XMðkÞ þ VnðkÞ þ An 2 ½Vn1ðkÞ 2 2Dn1 ð5Þ
where An is maximum acceleration (unit: m=s2) of target vehicle n.
Based on the specifications of the three dynamic spacing types bounded by XMðkÞand
XFðkÞ, intra-lane traffic manoeuvres that potentially occur in these three dynamic spacing
scenarios are modelled, and described as follows.
2.1.2. Scenario 1 – mandatory braking
This scenario models the mandatory braking behaviour of any vehicle under the condition
that it cannot change lanes at a given time step. According to the boundaries of XMðkÞin
Equation (2), any target vehicle may be forced to brake at the current time step k once the
relative spacing (Xn,n1ðkÞ) between target vehicle n and front vehicle (n 1) is XMðkÞ.
Thus, the following condition (Equation (6)) must be satisfied:
Xn,n1ðkÞ Xsþ VnðkÞ n VnðkÞ ½ 2 2Dn þ½Vn1ðkÞ 2 2Dn1 ð6Þ
The instantaneous deceleration (dnðk þ nÞ) by target vehicle n at time step k þ n can
then be derived as dnðk þ nÞ ¼ VnðkÞ ½ 2 2 XsþVnðkÞ nþ½Vn1ðkÞ 2 2Dn1 Xn,n1ðkÞ n o ð7Þ
Notably, the aforementioned mandatory braking occurs most frequently under the condition that the target vehicle in a blocked lane is approaching either an incident site or
a stopped vehicle (i.e. Vn1ðkÞ ¼0). In such a case, Equation (7) can be further
simplified as dnðk þ nÞ ¼ VnðkÞ ½ 2 2 XsþVnðkÞ nXn,n1ðkÞ ð8Þ
Furthermore, consider a special case in which reaction time n is too small. Under this
condition, the formulation of deceleration in response to stalled vehicles in the mandatory braking area can be expressed as
dnðkÞ ¼
VnðkÞ
½ 2
2 XsXn,n1ðkÞ
ð9Þ
Notably, the aforementioned case may be acceptable for automatic highway systems (AHSs) because the effect of such human factors as reaction time on traffic manoeuvres may be insignificant.
2.1.3. Scenario 2 – incident-induced car-following
This scenario describes procedure for deriving the proposed incident-induced car-following model, which characterises the speed adjustment of any target vehicle moving
within the incident-induced car-following area bounded by XMðkÞ and XFðkÞ. The
proposed incident-induced car-following model is constructed based on the fundamentals
of collision avoidance (CA) models1(Gipps 1981, Benekohal and Treiterer 1989, Brokua
et al. 1991, McDonald et al. 1994, Kumamoto et al. 1995, Brackstone and McDonald 1999). However, compared with existing CA models, the target vehicle in the proposed model adjusts its speed based on the front vehicle and front platoon, which are further ahead and approaching the incident site. Arguments supporting this idea can also be found in the literature of multi-anticipative car-following which hinges on the manoeuvres of perceived more than one vehicles moving ahead (Bexelius 1968, Lenz et al. 1999,
Hoogendoorn and Ossen 2006, Hoogendoorn et al. 2009). Note that the proposed model considers the phenomenon in that, under incident conditions, any target vehicle can adjust to incident effects on traffic flows, and thus, tailor its speed based on a safe distance from the vehicle in front and the perceived aggregate speed of the traffic platoon upstream of the incident site in a blocked lane. Particularly, the corresponding effect of the front platoon may increase as the target vehicle approaches the incident site. The incident-induced car-following model is proposed as Equation (10).
Enðk þ nÞ ¼ 1 ! Xn,pðkÞ, k 1 Xn,IðkÞ Xn1,IðkÞ Sn,n1ðkÞ þ! Xn,pðkÞ, k 2 PNpðkÞ i¼1 ViðkÞ NpðkÞ VnðkÞ " # ð10Þ
where Enðk þ nÞ(unit: m=s2) is the incident-induced speed adjustment by target vehicle n
at time step k þ n; NpðkÞ is the number of vehicles in the platoon upstream from the
incident site in the blocked lane at time step k; Xn,IðkÞ(unit: m) is the geographical spacing
between target vehicle n and incident I observed at time step k; Xn1,IðkÞ(unit: m) is the
spacing between front vehicle n 1 and incident I; Sn,n1ðkÞ(unit: m) is the dynamic safety
space between target vehicle n and front vehicle n 1 at time step k that is acceptable to
target vehicle n; 1and 2are two pre-set parameters (units: s2and s1); and !ðXn,pðkÞ, kÞ
is a weighting value indicating the relative magnitude of the effect of platoon p on the speed adjustment of target vehicle n compared with the effect caused by front vehicle n 1.
In this work, we assume ! Xn,pðkÞ, k
is a negative exponential function with respect to
Xn,IðkÞ, and is given by
! Xn,pðkÞ, k
¼ebXn,IðkÞ ð11Þ
where b is a positive parameter (unit, m1). Equation (11) represents a weighting value
indicating the relative influence of the perceived platoon p on the speed adjustment of target vehicle n compared with the influence from the front vehicle n 1. Herein,
!ðXn,pðkÞ, kÞ is formulated based on the following postulation from a psychological point
of view. When approaching incident site I, the closer target vehicle n is to I, its speed adjustment relies on platoon p in the blocked lane more than on front vehicle n 1 due to the target driver’s curiosity about incident situations, termed incident-induced
rubber-necking effect in Sheu (2008). At the current model developmental stage, !ðXn,pðkÞ, kÞ is
assumed to be a negative exponential function for simplicity. It is noted that !ðXn,pðkÞ, kÞ
should not be limited to a negative exponential function. Instead, it can also be formulated in other forms (e.g. linear functions) to appropriately characterise the aforementioned relationship between the target vehicle’s speed adjustment and perceived platoon relative
to the front vehicle. Additionally, Sn,n1ðkÞis regarded as the dynamic length (unit: m) of
the front vehicle n 1 perceived by target vehicle n (Chou and Sheu 1992), and thus Sn,n1ðkÞis denoted by
Sn,n1ðkÞ ¼ Ln1þVnðkÞ n ð12Þ
where Ln1 represents the physical length (unit: m) of front vehicle n 1.
The following case discusses two interesting cases to which the proposed incident-induced car-following model is applied.
. Case 1: Perception of platoon queuing at an incident site
This case illustrates the situation in which the target driver perceives the queuing platoon near an incident site, which may lead to
PNpðkÞ i¼1 ViðkÞ
NpðkÞ ¼0. Under this condition, the
speed adjustment estimation in Eqation (10) becomes
Enðk þ nÞ ¼ 1 ! Xn,pðkÞ, k 1 Xn,IðkÞ Xn1,IðkÞ Ln1 1n 1 ! Xn,pðkÞ, k þ2! Xn,IðkÞ, k VnðkÞ ð13Þ
where Enðk þ nÞ can be either positive or negative, depending on the difference (i.e.
Xn,IðkÞ Xn1,IðkÞ Ln1) in the geographic spacing between the target and front vehicles,
as well as the instantaneous speed (VnðkÞ) of target vehicle n.
. Case 2: No front vehicle perceived
Further, consider a special case in which the target vehicle perceives no front vehicle as
it approaches the incident site. Correspondingly, Xn,IðkÞ Xn1,IðkÞ Ln1, which is on the
right-hand side of Equation (13), is zero. Thus, Enðk þ nÞ is negative, and can be
denoted by Enðk þ nÞ ¼ 1n 1 ! Xn,pðkÞ, k þ2! Xn,pðkÞ, k VnðkÞ ð14Þ
That is, target vehicle n typically decelerates, the absolute value of which is proportional to the running speed when approaching an incident site. Notably, the generalisations in this scenario also apply to the case in which no platoon exists at the incident site.
Additionally, the meaning of the upper and lower bounds of the weight value
!ðXn,pðkÞ, kÞ is also worth discussing. When !ðXn,pðkÞ, kÞ¼0, the target vehicle’s speed
adjustment depends on front vehicle n 1, and thus, may lead to an estimation result
similar to that estimated by existing CA models. Conversely, when !ðXn,pðkÞ, kÞ¼1, target
vehicle n is closest to the study site, and the same result as that in Equation (14) can be derived. This condition also applies to the case in which the target vehicle and front vehicle are considered members of the same platoon when approaching the incident site. A related application example worth noting is that in which the target vehicle is linked with other vehicles, including the front vehicle in the AHS operating environment, in response to a lane-blocking incident occurring ahead. In other cases, various human factors can be
considered to rationalise the !ðXn,pðkÞ, kÞ function.
2.1.4. Scenario 3 – free-flow moving
The proposed traffic behaviour model in this scenario is mainly used to estimate the speed adjustment of target vehicle n, which is in free-flow area under the condition
Xn,IðkÞ Xn1,IðkÞ ~XFðkÞ. Under this condition, the speed adjustment by the target
vehicle at time step k þ (Fnðk þ Þ) is estimated by
Fnðk þ nÞ ¼ V^nVnðkÞ
h i
ð15Þ
where ^Vn is the free-flow speed (unit: m=s) of target vehicle n under the current traffic
condition and is a pre-determined parameter (unit: s1).
2.2. Inter-lane traffic behaviour modelling
This subsection models mandatory lane-changing manoeuvres of vehicles present upstream from an incident site using a two-stage modelling approach. According to Sheu and Ritchie (2001), lane-changing manoeuvres within a defined detection zone (Figure 1) can be classified as mandatory lane changing in subsystem 1, and discretionary lane changing in subsystem 2. Compared with conventional lane-changing models, the primary feature of the proposed model is that mandatory lane-changing behaviour of any vehicle in subsystem 1 is completed via two stages – pre-action judgement and in-action lane changing. These two stages are described as follows.
2.2.1. Stage 1: pre-action judgement
The decision-making model in this stage determines whether mandatory lane changing is needed during the subsequent time steps. The logic rules developed here are based on concepts of driving psychology and psychophysical theory. This work adopts the ideas from quantum mechanics-based optical flow models (Baker 1999, Sheu 2008) that utilise psychophysical momentum to quantify the stimulus-response behaviour of a driver. Therefore, the proposed model considers the following two psychological factors: (1) perceived lane-changing status in the target lane (i.e. the lane the target vehicle is in) and (2) the relative platoon speeds in adjacent lanes compared with that in the target lane. The determination of perceived lane-changing status relies on the comparison of frequencies of lane changing from the target lane to adjacent lanes (termed lane-changing-out) and from adjacent lanes to the target lane (termed lane-changing-in) within the visual scope of a target driver. Figure 5 shows the visual scope q specified for target vehicle n in
recognising the lane-changing status in the target lane, where Xq
nðkÞ is the longitudinal
boundary of visual scope (Figure 5), and is denoted by
XqnðkÞ ¼2w Xn,IðkÞ Xn1,IðkÞ Ln1
Wn1
ð16Þ
where w is lane width (unit: m), and Wn1 is the physical width (unit: m) of front vehicle
n 1. This study infers that the driver of the target vehicle (termed target driver in the
following text) is prone to change lanes when the perceived lane-changing-out frequency is greater than lane-changing-in frequency within the visual scope over a given period.
Notably, mandatory lane changing can be conducted anywhere in subsystem 1, and
Xq
nðkÞis time-dependent. Therefore, the corresponding values of XqnðkÞin different dynamic
spacing cases are worth discussing.
. Case 1: Moving in the mandatory braking area
Although changing lanes in the mandatory braking area is difficult, aggressive drivers
exist. In this case, Equation (6) is applied to remove the safety spacing, Xs, from
Equation (6), yielding XqnðkÞ ¼ 2w VnðkÞ n½VnðkþnÞ 2 2Dn þ Vn1ðkÞ ½ 2 2Dn1 Ln1 n o Wn1 ð17Þ
Similarly, one noteworthy case is that in which the target vehicle is the only vehicle in front of an incident in the blocked lane in subsystem 1. Under this condition, the inter-lane behaviour of target vehicle n is directly processed by the model constructed in the
second-stage once incident I is within visual boundary Xqn,IðkÞwhich is given by
Xqn,IðkÞ ¼ 2w VnðkÞ n½VnðkÞ 2 2Dn LI n o Wn1 ð18Þ
where LI is longitudinal distance (unit: m) of incident site I.
. Case 2: Movement in the incident-induced car-following area
In this case, the dynamic spacing between target vehicle n and front vehicle n 1 is estimated using Equation (5); this value is then input into Equation (16) to obtain the
corresponding value of Xq
nðkÞ (Equation (19)) for the case of vehicles moving in the
incident-induced car-following area
XqnðkÞ ¼ 2w VnðkÞ þA2n ½Vn1ðkÞ2 2Dn1 Ln1 n o Wn1 ð19Þ
. Case 3: Movement in the free-flow area
Note that in Case 2, the lower bound of XFðkÞ(i.e. ~XFðkÞ in Equation (5)), is used.
Thus, Equation (19) can be rewritten as Equation (20) to determine the corresponding
value of Xq
nðkÞfor the case of a target vehicle moving in the free-flow area.
XqnðkÞ4 2w VnðkÞ þA2n ½Vn1ðkÞ2 2Dn1 Ln1 n o Wn1 ð20Þ Moving vehicle ) (k Xnq
Visual scope q in the target lane
n n–1
Figure 5. Visual scope for judging the lane-changing status in the target lane.
In reality, Equation (20) implies that any target vehicle in the free-flow area may remain in the same lane for the following two reasons. First, as mentioned, any target vehicle in the free-flow area can move at will in any lane. Therefore, the target vehicle does not need to make a lane change decision before entering the incident-induced car-following
area. Second, the estimate of Xq
nðkÞin Equation (20) may be outside the normal range of a
driver’s visibility, particularly when a vehicle is travelling at high speeds. Thus, we assume that the lane-changing status in front of the front vehicle is difficult to observe when the target vehicle is moving in the free-flow area.
In addition to perceived lane-changing status in the target lane, the relative platoon speed in an adjacent lane is also a significant factor when determining lane-changing
behaviour. By utilising XqnðkÞ, the target vehicle may compare the relative aggregate
platoon speed in the target lane with that in any lane directly adjacent to the target lane. Once platoon speed in the adjacent lane is perceived as higher than that in the target lane, the target vehicle may change lanes in subsequent time steps. Conversely, given the lane-changing-out frequency is greater than the lane-changing-in frequency in the target lane for a prolonged period, this phenomenon may reduce the speeds of vehicles moving in adjacent lanes as lane densities are increasing. Under this situation, the willingness of the target driver to change lanes may decrease, despite the perceived high lane-changing-out frequency, which may, according to the proposed model, cause driver to change lanes. Furthermore, the lane-changing-out frequency may not remain high continuously as adjacent lane densities increase. Briefly, a trade-off exists between the effects of perceived lane-changing status in the target lane and the relative platoon speeds in adjacent lanes. Accordingly, the corresponding logic rules are developed to determine whether or not the target driver considers change lanes in the next time step.
Clearly, given that an incident is known to all drivers, the first stage of pre-action decision-making can be omitted, and the model constructed in the second stage can be applied directly to process the lane changing action of any given target vehicle in subsystem 1.
2.2.2. Stage 2: in-action lane-changing operation
This scenario investigates the factors influencing lane-changing manoeuvres, given that the target driver feels the necessity of conducting lane changing which is determined in the previous stage. Compared to the previous decision, the decision made at this stage helps the driver change lanes safely and smoothly in response to varied traffic conditions in adjacent lanes and blocked lanes. For this purpose, this work proposes four required constraints – two constraints for intra-lane traffic factors, and the other two constraints for inter-lane traffic factors. They are then used to determine whether incident-induced mandatory lane changing can be completed successfully at the next time step by the target vehicle. That is, mandatory lane changing can be implemented only when these four conditions are satisfied during the current time step. The details of the derivations of these constraints are as follows.
. Constraint 1: Restricted target vehicle turning angle n (nðkÞ)
This constraint accounts for potential conflict points that may exist between the target and front vehicles in the target lane when the target vehicle changes lanes (Figure 6).
To change lanes, the condition of nðkÞ n1ðkÞ must be satisfied, where n1ðkÞ is given by n1ðkÞ ¼tan1 Wn1ðkÞ Xn,IðkÞ Xn1,IðkÞ Ln1Xs ð21Þ
. Constraint 2: Restricted latitudinal spacing between the target and front vehicles in the original target lane
Given nðkÞ, the following condition must hold to avoid a collision between the target
vehicle and front vehicle in the original target lane during lane changing.
sinðnÞ
n ðk þ tÞ WDn1ðk þ tÞ ð22Þ
where sinðnÞ
n ðk þ tÞ is the moving distance (unit: m) of target vehicle n in the latitudinal
direction in while changing lanes at time step kþt; WDn1ðk þ tÞ is the dynamic width
(unit: m) of front vehicle n 1 in the original target lane at time step kþt and t is to the amount of time spent by target vehicle n moving at a maximum safe distance in the longitudinal direction while changing lanes to avoid a collision with front vehicle n 1 in
the original target lane. Notably, the concept of WDn1ðk þ tÞ, which is similar to the
dynamic vehicular length described previously, is utilised to determine the latitudinal safe space between target vehicle n and front vehicle n 1 in the original target lane during lane
changing. Theoretically, sinðnÞ
n ðk þ tÞ, WDn1ðk þ tÞ and t are functions of microscopic
Target vehicle (conducting lane-changing behaviour)
Front vehicle (in the original target lane)
Rear vehicle (in the adjacent lane)
Front vehicle (in the adjacent lane)
) (k n θ ) (k n−1 θ n n – 1 n n–1 n n + 1 n + 1 n – 2 n–2
Figure 6. Illustration of the turning angle restriction.
characteristics of the target and front vehicles in the original target lane, and can be derived as: sin n n ðk þ tÞ ¼ 3 ~Vcosn ðkÞ h i2 þ8alnnðkÞsðkÞ 2 ~Vcosn ðkÞ h i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ~Vcos n ðkÞ h i2 þ8aln nðkÞsðkÞ r þ2 ~Vsin n ðkÞ h i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ~Vcos n ðkÞ h i2 þ8aln nðkÞsðkÞ r V~cos n ðkÞ h i ( ) 8 > > > < > > > : 9 > > > = > > > ; 2aln nðkÞ ð23Þ WDn1ðk þ tÞ ¼ Wn1½1 þ 3Vn1ðkÞ ð24Þ t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ~Vcos n ðkÞ h i2 þ8aln nðkÞsðkÞ r 2 ~Vcosn ðkÞ 2aln nðkÞ ð25Þ where aln
nðkÞis the acceleration/deceleration (unit: m=s2) of the target vehicle at time step k
in while changing lanes; 3 is a pre-set parameter (unit: s=m); ~Vcosn ðkÞis the relative speed
(unit: m=s) of target vehicle n compared with that of front vehicle n 1 in the longitudinal
direction at time step k, and, similarly, ~Vsin
n ðkÞis that in the latitudinal direction; sðkÞ is the
relative moving distance (unit: m) allowed for target vehicle n in the longitudinal direction while changing lanes to avoid a collision with the front vehicle in the original target lane at
time step k. Herein, ~Vcos
n ðkÞ, ~Vsinn ðkÞand sðkÞ can be further expressed as Equations (26),
(27) and (28), respectively. ~ Vcosn ðkÞ ¼ VnðkÞ cos ½ nðkÞ Vn1ðkÞ ð26Þ ~ Vsinn ðkÞ ¼ VnðkÞ sin ½ nðkÞ ð27Þ sðkÞ ¼ Xn,IðkÞ Xn1,IðkÞ Ln1Xs ð28Þ
Notably, we assume the speed of front vehicle n 1 is constant when target vehicle n changes lanes. Thus, the lane-changing manoeuvres of the target vehicle in response to the changing speed of the front vehicle warrant further investigation.
. Constraint 3: Restriction of the dynamic safety space between target vehicle n and rear vehicle n þ 1 in the adjacent lane into which the target vehicle will move This constraint applies a minimum safety spacing between target vehicle n and rear vehicle nþ1 in the adjacent lane when the target vehicle moves into the adjacent lane.
In this scenario, this work applies a novel derived conflict point (Xcðk þ tÞ) to formulate
the constraint rather than employing the concept of XFðkÞdirectly, which was constructed
previously in addressing issues associated with intra-lane traffic manoeuvres. That is, the proposed intra-lane models may not be applicable at the moment of lane changing. Under such a transitional condition where the front target vehicle is moving from the original target lane to an adjacent lane, the rear vehicle in the adjacent lane cannot respond safely
to the traffic manoeuvre of the target vehicle within an extremely short time interval.
Thus, Xcðk þ tÞis given by
Xcðk þ tÞ ¼ Xn,IðkÞ sðkÞ ð29Þ
which corresponds to the instantaneous distance between target vehicle n and rear vehicle nþ1 in the longitudinal direction in the adjacent lane.
Utilising the notation of Xcðk þ tÞdefined previously, the following condition must be
satisfied to consider the dynamic safety spacing (Xnþ1,XCðk þ tÞ) between the conflict point
Xcðk þ tÞand rear vehicle nþ1 in the adjacent lane when target vehicle n changes lanes at
time step k. Xnþ1,XCðk þ tÞ XsVnþ1ðk þ tÞ n Vnþ1ðk þ t þ nÞ ½ 2 2Dnþ1 þ½Vnðk þ tÞ 2 2Dn ð30Þ
where Vnðk þ tÞ and Vnþ1ðk þ tÞ are the speeds (unit: m=s) of target vehicle n and rear
vehicle n þ 1 at time step k þ t, respectively; similarly, Vnþ1ðk þ tÞis the speed (unit: m=s) of
rear vehicle nþ1 at time step k þ t þ n; Dnþ1is the predetermined maximum deceleration
(unit: m=s2) of rear vehicle nþ1.
. Constraint 4: Restricted dynamic safety spacing between target vehicle n and the new front vehicle n 2 in the adjacent lane, into which the target vehicle will move
This constraint deals with the constraint of the dynamic safety spacing (XXC,n2ðk þ tÞ)
between target vehicle n and front vehicle n 2 in the adjacent lane when the target vehicle changes lanes at time step k. Similar to Constraint 3, the following condition must be satisfied to ensure that the mandatory lane-changing manoeuvre of the target vehicle is allowed at time step k:
XXC,n2ðk þ tÞ XsVnðk þ tÞ n Vnðk þ t þ nÞ ½ 2 2Dn þ½Vn2ðk þ tÞ 2 2Dn2 ð31Þ
In Equation (31), Vnðk þ t þ nÞis the speed (unit: m=s) of target vehicle n at time step
k þ t þ ; and similarly, Vn2ðk þ tÞ and Dn2 are the speed (unit: m=s) and preset
maximum deceleration (unit: m=s2) of the front vehicle in the adjacent lane at time step
kþt, respectively.
3. Model tests
This section describes the procedures for testing the ability of the proposed microscopic traffic behaviour models to accurately characterise incident-induced lane traffic phenom-ena on arterials. The technique utilised for performance evaluation is a microscopic traffic simulation program developed for this work. The proposed incident-induced lane traffic behaviour models are integrated into the developed traffic simulation program. Evaluation measurements were based mainly on a comparison of simulation data generated by the traffic simulation program and video-based real incident data collected with the assistance of the Traffic Engineering Office (TEO) of Taipei City Government, Taiwan.
The database used in model tests was primarily generated by processing the video-based data collected from five respective traffic incidents occurring at five arterials located at the city of Taipei, as indicated in Figure 7. Therein, one camera installed at an
intersection near each studied location was used to monitor real-time local traffic situations. The studied incident cases were videotaped by these cameras. Among these five sampled incident events, one is a low-volume incident case (Case 1), two are medium-volume incidents (Cases 2 and 3) and two are high-medium-volume incidents (Case 4 and 5). The geometric and traffic characteristics observed in the study cases are summarised in Table 1. The data acquisition procedure in this study involves two scenarios: (1) input data processing for simulation and (2) output data generated for calibration and validation. We used videotapes provided by TEO of Taipei City Government to generate the incident database. Observed from the videotapes, we generated 30 s lane-based data including the arrival rates, arrival speeds and traffic composition associated with those links where incidents were located (termed incident links) for the use as input data of simulation. Additionally, we collected several traffic measurements, including vehicle-based incident-induced link travel time, lane usage and lane-changing fractions (for each 30 seconds), for model evaluation.
To test the proposed incident-induced lane traffic behaviour models, a microscopic traffic simulation program was developed in Turbo C computer language. Model calibration was conducted in advance of model tests in this study. The data used in mode calibration was a part of the collected video-based data. Therein, one half of the collected video-based data was used for model calibration and the rest for model tests. Table 2 lists the primary parameters set for simulation. Based on Abdulhai et al. (1999, 2002), this work
primarily calibrated the parameters of minimum safety spacing (Xs) and mean reaction
time () associated with each type of vehicle as these parameters determine driver decisions during car-following and lane-changing behaviour in microscopic traffic simulation. To
seek for appropriate values for Xs and , this work adopted a two-dimensional search
process (Abdulhai et al. 1999) which permits minimising the discrepancy between the
Figure 7. Locations of studied incident cases.
simulation output and field observations, but requires numerous simulation runs for every point in the search space. As the search domain was unknown, we used our own empirical
iterative search process that searches in one direction only at a time. For example, given Xs
is fixed at its best known value obtained from the previous search iterations, one-directional search is performed for the other parameter, say , within reasonable upper and lower bounds. Once the optimal value is found, it gets fixed and its range is empirically
reduced around this value, followed by the next-round search for Xs. Such an iterative
search process continues until improvements diminish. The primary steps of the calibration procedure can also be found in Abdulhai et al. (1999), and omitted in this
work for our space concern. In terms of maximum acceleration (A) and deceleration (D),
this work referred to Pline (1999), and then determined these two parameters while considering emergency braking, which is common in incident cases. Such vehicular characteristics as physical length and width of different vehicles were pre-determined.
The model tests in this work aim to verify the performance of the proposed lane traffic behaviour models for arterial incident cases and to determine the advantages of these models for use in simulating incident-induced traffic manoeuvres. Thus, this work compared simulation data generated by the proposed incident-induced traffic simulation program with video-based real incident data.
Table 2. Preset vehicle characteristics and calibrated traffic parameters.
Vehicle characteristics Calibrated traffic parameters
Vehicle type Length L(m) Width W(m) Maximum acceleration A(m/s2) Maximum deceleration D(m/s2) Minimum safety spacing Xs(m) Mean reaction time (s)
Light vehicle Car 4.0 3.56 3.56 7.30 5.2 0.88
Light goods vehicle
6.0 2.22 2.22 7.30 7.2 0.97
Heavy vehicle Truck 11.0 1.4 1.4 5.63 13.6 1.25
Bus 10.0 1.4 1.4 5.63 12.6 1.25
Other key parameter 1(s2) 2(s1) (s1) b(m1)
Calibrated value 0.22 0.13 0.31 0.06
Table 1. Geometric and traffic characteristics of studied incident sites. Characteristics Incident case Number of lanes in arterial (ln) Block length (m) Hourly volume
(veh/h/ln) Incident location
Incident duration (min)
Case 1 (low-volume) 3 101 453 Upstream, outside lane 27
Case 2 (medium-volume) 4 208 980 Midstream, inside lane 42 Case 3 (medium-volume) 3 592 864 Upstream, central lane 28 Case 4 (high-volume) 4 290 1124 Downstream, central lane 34 Case 5 (high-volume) 4 187 1277 Downstream, outside lane 39
The following five traffic measurements were utilised: (1) traffic arrivals, (2) average arrival speeds, (3) incident-induced lane-changing fractions, (4) link travel time and (5) lane usage on the incident-affected link. Therein, simulation data used to test traffic arrivals, average arrival speed, and lane-changing fractions were collected at 30 s intervals, whereas link travel time was measured on a vehicle-by-vehicle basis during incidents. Herein, lane usage was measured by a lane-specific traffic density divided by the associate link traffic density observed during an incident. Accordingly, it seems agreeable that lane usage has a close relationship with either lane-specific flows or time-based lane occupancies, and thus is also suitable for indicating the distribution of lane-specific traffic loads revealed in an incident link. Traffic arrivals and arrival speeds were used to test the acceptability of simulated traffic flow conditions upstream of the incident site. The measurements of lane-changing fractions were used to assess the validity of the proposed lane-changing logic under lane-blocking incident conditions. Conversely, measurements of link travel time and lane usage were utilised to assess the performance of the proposed models in characterising incident effects on macroscopic traffic states.
This study examined simulated traffic arrivals, average arrival speed, and lane-changing fractions using mean absolute percentage errors (MAPEs). In this test scenario, the 30 s simulation data samples were generated using the proposed simulation program, and then compared with video-based data via the MAPE statistics. All simulated values were aggregated measurements from 10 simulations for each incident case. To demonstrate the relative advantage of the proposed model used in characterising incident-induced traffic behaviour, the MAPE statistics yielded from the simulation program without incident-induced lane traffic behaviour models were also generated for comparison. Table 3 summarises test results for traffic arrivals, average arrival speeds and lane-changing fractions.
The test results of Table 3 indicate that, overall, the simulated values of lane traffic variables such as traffic arrivals, arrival speeds and lane-changing fractions reproduced by the proposed models fit video-based data better than those without incident-induced behaviour models. Although there are no general guidelines for when an MAPE value is sufficiently high, it is commonly agreed that forecast output should not be accepted when a value of MAPE 450%. Moreover, several classes are suggested in Lewis (1982) for the assessment using MAPE values, where 0.2–0.5 is regarded as a reasonable range for accepting MAPE statistics, and relatively 10% is a demanding threshold for MAPE values. Conversely, some MAPE values yielded in the case without incident-induced traffic behaviour models exceed the loose threshold 0.5, indicating the necessity of incorporating the proposed models into the reproduction of incident-induced lane traffic manoeuvres.
Furthermore, the MAPE values obtained under medium-volume incident conditions (i.e. Cases 2 and 3) are, overall, higher than those obtained in low-volume and high-volume incident cases. Therein, we found that unlike low- and high-high-volume incident cases in which the lane traffic flow conditions change stably, the lane traffic variables (particularly traffic arrivals and lane-changing fractions) observed in Cases 2 and 3 are likely to vary irregularly with time. Such a feature of significant traffic randomness existing in the medium-volume incident cases can be the major reason for resulting in higher MAPE values in the study.
Conversely, test results for lane usage and averaged link travel time and lane usage (Table 4) indicate the performance of the proposed models in dealing with lane traffic manoeuvres as vehicles move through the link on which the incident occurred. Overall, the
absolute values of the resulting relative errors are lower than 0.2, and 90% of them are lower than 0.1, i.e. satisfactory. That is, the proposed models can estimate average link travel time for arterial incident cases by appropriately reproducing incident-induced driver behaviour. Such a generalisation, on the other hand, implies that the effects of incidents on intra-lane and inter-lane traffic manoeuvres can be addressed utilising the proposed microscopic lane traffic behaviour models. Furthermore, the ability of the proposed models to reproduce lane usage of the incident site is also satisfactory. According to test results, the proposed models permit reproducing incident-induced mandatory lane changes from blocked lanes to adjacent lanes upstream of the incident site, as well as discretionary lane changes once vehicles pass the incident. Thus, simulated measurements of lane usage obtained by the proposed models generally match real values.
Table 3. Test results (I). MAPE measurement Incident case Traffic arrivals Arrival speed Lane-changing fractions Case 1 (low-volume) 0.15 (0.18) 0.19 (0.37) 0.23 (0.44) Case 2 (medium-volume) 0.43 (0.53) 0.38 (0.66) 0.47 (0.83) Case 3 (medium-volume) 0.32 (0.44) 0.25 (0.59) 0.39 (0.72) Case 4 (high-volume) 0.26 (0.41) 0.21 (0.44) 0.31 (0.63) Case 5 (high-volume) 0.22 (0.38) 0.28 (0.48) 0.30 (0.58) Overall performance 0.28 (0.39) 0.26 (0.51) 0.34 (0.64)
Test result (compared with threshold: 0.5) Accepted Accepted Accepted Note: MAPE values yielded without incident-induced traffic behaviour models in parentheses. Table 4. Test results (II).
Lane usage (%) Incident case Measurement data source Inside lane Central lane(s) Outside lane Averaged link travel time (s)
Case 1 (low-volume) Videotape 33.7 42.7 23.6 43.4
Model 36.3 38.6 25.1 41.7
Relative error (%) 7.7 9.6 6.4 3.9
Case 2 (medium-volume) Videotape 14.2 46.9 38.9 65.5
Model 11.7 47.5 40.8 70.2
Relative error (%) 17.6 1.3 4.9 7.2
Case 3 (medium-volume) Videotape 36.1 25.2 38.7 69.8
Model 39.6 23.5 36.9 75.4
Relative error (%) 9.7 6.7 4.7 8.0
Case 4 (high-volume) Videotape 41.6 11.9 46.5 78.3
Model 42.8 12.4 43.8 83.2
Relative error (%) 2.8 4.2 5.8 6.3
Case 5 (high-volume) Videotape 38.9 51.4 9.7 86.6
Model 42.3 49.3 8.4 82.7
Relative error (%) 8.7 4.1 13.4 4.5
4. Conclusions and recommendations
This study modelled inter-lane and intra-lane traffic manoeuvres under arterial lane-blocking incidents and simulated incident-induced lane traffic behaviour using a microscopic traffic modelling approach. This work also analysed the effects of lane-blocking incidents on lane traffic behaviour, including lane-changing and car-following behaviours. Novel microscopic traffic behaviour models were formulated to characterise incident-induced lane traffic manoeuvres, followed by the development of a microscopic simulation program to demonstrate the applications of the proposed models. In analysing microscopic incident-induced lane manoeuvres, this work specified three types of dynamic spacing for free-flow moving traffic, incident-induced car following and mandatory braking, respectively. The intra-lane traffic manoeuvres and related factors were then characterised and formulated. Additionally, a two-stage lane-changing mechanism, which has two levels of functionality – (1) pre-action decision-making and (2) in-action lane-changing operation, was proposed to reproduce mandatory lane-lane-changing behaviour of any given vehicle moving in blocked lanes upstream of an incident site.
Preliminary test results indicate that the proposed microscopic lane traffic behaviour models can characterise incident-induced lane traffic behaviour and incident effects on inter-lane and intra-lane traffic manoeuvres. Analytical results of model tests under different traffic flow conditions have also demonstrated the potential effect of traffic randomness on the proposed models’ overall performance particularly under medium-volume traffic flow conditions. Therein, we found that lane traffic states such as traffic arrivals and lane-changing fractions under medium-volume incident conditions are likely vary irregularly with time, not as stably as those observed in low- and high-volume incident cases. This finding may also imply that specific coordinated traffic signal controls are needed at two successive intersections that cover an incident site to regulate traffic arrivals upstream of the incident site, particularly for arterial incidents with medium volumes. Furthermore, there is still room for model improvement and tests with more demanding thresholds (e.g. MAPE values 10%) although the MAPE test results are overall acceptable in this study. The heterogeneity of driver reaction time can be another issue. As the proposed models are microscopic, and formulated in a personal decision domain, a more appropriate measure is to treat driver reaction time as a random variable following a constrained distribution bounded by lower and upper bounds.
The proposed models are currently being extended to intersection incident cases in which incident-induced lane traffic manoeuvres may be relatively more complicated due to the effects of signal controls. Further testing the proposed models using relatively more real data for incident cases is needed. Moreover, applications of the proposed lane traffic models also warrant further research. The applications suggested include uses of the proposed models in developing a network-based microscopic traffic simulator, and exploring the effects of human factors (e.g. awareness and driving pressure) on microscopic traffic manoeuvres, such as queuing and changing lanes, for various lane-blocking incidents. Comparison or integration of related macroscopic and microscopic models with the proposed models also warrants additional research. For example, some early studies (e.g. Hidas 2002, 2005, Chevallier and Leclercq 2009) are outstanding in dealing with the response of a following vehicle to, and interaction with, the subject vehicle changing lanes based on the concept of gap acceptance. Further research comparing existing models with the proposed models may advance incident-induced
lane-changing modelling. Additionally, the effect of the perceived vehicular platoons in adjacent lanes on the target vehicle’s speed adjustment also needs further investigation to improve the proposed incident-induced car-following model. Most importantly, we hope that this study can stimulate research devoted to the development of microscopic incident-related traffic simulators.
Acknowledgements
This research was supported by grant NSC 99-2410-H-009-031-MY3 from the National Science Council of Taiwan. The author also wishes to thank the referees for their helpful comments. Any errors or omissions remain the sole responsibility of the authors.
Note
1. The idea common to existing CA models is that by manipulating the basic Newtonian equations, a safe car-following spacing is determined to avoid a collision, given that the front vehicle acts unpredictably.
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