The travel course is divided into units called gap sessions. A gap session is a time period over which the subject vehicle has the same set of vehicles in the relative positions around it, specifically the {lead, rear, left lead, left rear, right rear, right lead, right rear} positions, and the same gap availability. The concept of gap session is illustrated in Figure 2 which shows how one gap session transitions into another. In Figure 2 the spatial size of this gap session is the length in meters from the back bumper of vehicle D to the front bumper of vehicle C.

Figure 2. Transition between gap session

In this paper a traffic dataset was collected to provide a core set of driver behaviour data and algorithms for verification and validation purposes.

A lane change performance function ULC is proposed and is computed for each vehicle, for each point in the parameter search space covering lane change model parameters. ULC is the proportion of the lane changes for each gap session in the analysed vehicle trajectory for which the model gave the incorrect lane action. The purpose of the values used in the model’s parameters is to minimise the ULC.

This paper seems to introduce a way to evaluate an algorithm made on lane changing according to real traffic data with Gipps decision tree altered in same aspects.

Treiber, M., Kesting, A. Modelling Lane-Changing Decisions with MOBIL

In the model presented in this paper the lane changing is considered as a multi-step process of three stages. First a strategic level, where the driver knows about his or her route on a network which influences the lane choice. Secondly the tactical stage, an intended lane change is prepared and initiated by advance accelerations or decelerations of the driver, and possibly by cooperation of drivers in the target lane. Finally, in the operational stage, one determines if an immediate lane change is both safe and desired.

In this contribution is examined only the operational decision process. The basic idea in this lane-changing model is to formulate the anticipated advantages and disadvantages of a prospective lane change in terms of single-lane accelerations. The criterion is the difference of the accelerations after and before the lane change, at least, if the acceleration of the longitudinal model is sensitive to velocity differences. For a driver considering a lane change, the subjective utility of a change increases with the gap to the new leader on the target lane. However, if the velocity of this leader is lower, it may be favourable to stay on the present lane despite of the smaller gap.

Advantages according to the author: the assessment of the traffic situation is transferred to the acceleration function of the car-following model, which allows for a compact and general model formulation with only a small number of additional parameters. In contrast to the classical gap-acceptance approach, critical gaps are not taken into account explicitly. Second, it is ensured that both longitudinal and lane-changing models are consistent with each other. Third, any complexity of the longitudinal model such as anticipation is transferred automatically to a similarly complex lane-changing model. Finally, the braking deceleration imposed on the new follower on the target lane to avoid accidents is an obvious measure for the (lack of) safety. Thus, safety and motivational criteria can be formulated in a uniﬁed way.

Equations of the model:

Acceleration function of the form:

Where:

ua is the velocity

Sa is the gap to the front vehicle (a-1)

Δua=ua- ua-1 is the relative velocity

Generally a lane change depends on the leader and the follower on the present and target lane. According to Figure 1 we use the following notation: C is the vehicle which is changing lanes and O the old follower and N the new follower of that vehicle. Furthermore refers to the prospective situation on the target lane.

Figure1

Safety criterion:

Compared to conventional gap-acceptance models, this approach depends on gaps only indirectly, via the dependence on the longitudinal acceleration.

In this way, larger gaps between the following vehicle in the target lane and the own position are required to satisfy the safety constraint if the following vehicle is faster than the changing vehicle. In contrast, smaller gaps are acceptable if the following vehicle is slower.

For realistic longitudinal models, bsafe should be well below the maximum possible deceleration b which is about 9 m/s2 on dry road surfaces.

Incentive criterion:

In the presented model is proposed an incentive criterion that includes a consideration of the immediately affected neighbors as well. A politeness factor p determines to which degree these vehicles inﬂuence the lane-changing decision of a driver. For symmetric overtaking rules differences between the lanes is neglected and for a lane-changing decision of the driver of vehicle c:

The ﬁrst two terms denote the advantage (utility) of a possible lane change for the driver himself. The third term with the prefactor p is an innovation of the presented model and it denotes the total advantage (acceleration gain–or loss,if negative) of the two immediately affected neighbors. Finally, the switching threshold Δath models a certain inertia and prevents lane changes if the overall advantage is only marginal compared to a “keep lane” directive.

In summary, the incentive criterion is fulﬁlled if the own advantage (acceleration gain) is greater than the weighted sum of the disadvantages (acceleration losses) of the new and old successors augmented by the threshold.

By means of simulation it was found that realistic lane-changing behaviour results for politeness parameter was in the range of 0.2 to 0.5.

For asymmetric passing rules the restrictions are based in the same concepts and they are slightly changed to address the differencies that exist.

Chang, G.-L., Kao, Y.-M. 1990. An empirical investigation of macroscopic lane-changing characteristics on uncongested multilane freeways

This paper as posted at his title is based on empirical investigation of lane-changing. Data from two diffent locations were gathered, analysed and discrte choise models were constructed according to them.

An exploratory analysis was conducted first to investigate the interrelations between the lane-changing characteristics and key traffic flow variables. Although only limited observations were available in the study, the results clearly indicated that the distribution of headways, and the speed and density ratios between neighboring lanes are principal factors affecting the lane-changing behavior. Such a relation was further confirmed with two generalized linear models. The first model, a binary logistic regression, was to study the effects of key traffic flow variables on the fraction of vehicles changing lanes, because an individual driver’s choice of lanes is either change, or no change. The second model focused on estimating the frequency of lane changes, where occurrences of lane-changing were assumed to follow a Poisson process with a time-varying mean. Both models with the standard maximum likelihood estimation yielded expected signs for all parameters and achieved reasonable levels of fit. It is fully recognized that the reported results are based only on limited observations from two different locations, which may not be sufficient to represent the general lane-changing characteristics.

Hidas, P. 2002. Modelling lane changing and merging in microscopic traffic simulation

The model presented in this paper is called SITRAS. The model of Gipps assumes that a lane changing manoeuvre takes place only when it is safe, i.e. when a gap of suffcient size is available in the target lane. This assumption was found to be a serious limitation in congested

and incident-affected conditions which needed further consideration. Another arguable aspect of Gipps model is that the check of the feasibility of lane changing is performed before actually checking whether the vehicle needs to change lane and thus this check needs to be done for every vehicle during the vehicle update process. This appears to be illogical, however, from a computational efficiency point of view it is beneﬁcial to perform the fastest check ﬁrst. In Gipps_ model the feasibility of a lane change is based on relatively simple conditions, which may justify the selected order. However, in a model where more complex procedures are applied when a lane change is necessary but unfeasible, it is better to establish ﬁrst the need for a lane change before dealing with the feasibility of the manoeuvre.

The overall structure of the SITRAS model is presented in Figure 1. The main modules of the

model are: (i) route building; (ii) vehicle generation; (iii) route selection, based on individual driver characteristics; and (iv) vehicle-progression, based on car following and lane changing theory.

Figure 1. The SITRAS structure

One important element of SITRAS is the driver-vehicle objects (DVOs) as the authors calls them which are divided in two classes i)unguided and ii)guided.

It is important to mention the limitation of the DVOs in order to undertsand how the lane changing procedure is srtuctured:

InSITRAS, DVOs have no memory––they do not remember their past, and they can only plan ahead for the next 1 s. Thus DVOs cannot learn from experience––all the knowledge they possess is procedural knowledge (Bigus and Bigus, 1998) encoded in the program.

DVOs have little direct contact with surrounding DVOs. Vehicles in the same lane of each road section are linked to their immediate leader and follower vehicles, as shown in Figure 2. They do not have any information concerned the vehicles in the adjacent lanes.

Figure 2. Links between DVOs in SITRAS

The lane changing in SITRAS is formed based on the Gipps model with some alterationsand additions.

To begin with : is lane changing necessary?

The result of the evaluation of a reason may be one of ‘Essential’, ‘Desirable’ and ‘Unnecessary’. If lane changing is found to be ‘Essential’ for any reason, the rest of the reasons are not evaluated.

Summary flowchart of the lane changing process in SITRAS

For the following situations the ‘Essential’, ‘Desirable’ and ‘Unnecessary’ are defined in seconds or in meters according to case:

Turning movement

End-of-lane

Incident

Transit lane

Speed advantage

Qeueu advantage

The selection of terget lane is as in Gipps model.

Is lane change to target lane feasible?

The lane change is feasible if:

(a) the deceleration (or acceleration) required for the subject vehicle to move behind the new

leader vehicle is acceptable, and

(b) the deceleration required for the new follower vehicle to allow the subject vehicle to move into the lane is acceptable

The accelaration is calculated with the car following model of this program and then is compared with the ‘acceptable acceleration’ bn calculated using a modefied format of the formula suggested by Gipps:

Where:

bn is the acceptable deceleration of vehicle n

D is the location of the intended turn or lane blockage

xn(t) is the location of vehicle n at time t

Vn is the desired speed of vehicle n(free)

bLC is the average deceleration a vehicle is willing to accept in lane changing

θ is the driver aggressivity parameter

The driver aggressivity parameter is included to represent individual differences among drivers. In SITRAS each driver-vehicle object is assigned a driver type parameter, a number between 0 and 99. Larger numbers represent more aggressive drivers. The driver type is drawn from a normal distribution when the driver-vehicle object is created. In the formula of the acceptable deceleration the driver aggressivity parameter is calculated differently for the two conditions:

For condition (a) it is the ratio of the subject vehicle driver type to the _average_ driver type (i.e. driver type =50). Thus, the leader vehicle has no effect on the acceptable acceleration, it depends only on the aggressivity parameter of the subject driver.

For condition (b) it is the ratio of the subject vehicle driver type to the new follower vehicle

driver type. Thus, the more aggressive the subject vehicle driver compared to the new follower vehicle driver, the higher the acceptable deceleration.

The forced lane changing algorithm developed in SITRAS is based on a ‘driver courtesy’

Concept as called by the author. The vehicle which wants to change lane sends a ‘courtesy’ request to subsequent vehicles in the target lane; the request is evaluated by each vehicle and depending on several factors such as the speed, position and driver type of the responding vehicle, it is either refused or accepted. When a vehicle ‘provides courtesy’ to another vehicle it reduces its acceleration to ensure that a free gap of sufficient length is created during the next few seconds for the lane changing vehicle. The concept is illustrated in the Figure 4.

Figure 4. Schematic represantation of the forced lane changing concept

Starting from the ﬁrst vehicle in the target lane which is located behind the subject vehicle, the deceleration required for each potential new follower vehicle to allow the subject vehicle to move into the target lane is calculated from the car-following model. This deceleration is compared with the ‘acceptable’ deceleration with the only deference that the driver aggressivity parameter is taken as the ratio of the ‘average’ driver type (i.e. driver type=50) to the driver type of the potential new follower vehicle; that is, it depends only on the aggressivity of the potential follower vehicle, and that the more aggressive the driver of the potential follower vehicle, the less deceleration it is prepared to accept. If the deceleration required for the vehicle is less than the maximum acceptable, the vehicle is selected to provide ‘courtesy’ to the subject vehicle, otherwise the evaluation continues with the next vehicle in the platoon. Once the new follower vehicle is found in the target lane, its acceleration is calculated by the car following model with respect to the subject vehicle in the adjacent lane instead of its current leader in the same lane. At the same time, the acceleration of the subject vehicle is calculated with respect to its new leader vehicle in the target lane instead of its current leader in the same lane. Consequently, the new follower vehicle will gradually slow down, while the new leader vehicle will pass the subject vehicle and a gap of suﬃcient size will be created which will ﬁnally allow the subject vehicle to move into the target lane.

In SITRAS the driver-vehicle-units (objects) are stored in their current lane in chains so they only know about their immediate leader and follower. We the abovementioned procedure the interaction between vehicles in different lanes is implemented.

For the merging situations a new concept has been developed to handle merging in a more intelligent and realistic manner. The procedure is described in the following table:

Li, X.-L., Jia, B., Gao, Z.-Y., Jiang, R. 2005. A realistic two-lane cellular automata traffic model considering aggressive lane-changing behavior of fast vehicle.

For the structure of the lane changing rules in this model the NaSch model is used which is a discrete model for traffic flow. The road is divided into L cells, which can be either empty or

occupied by a vehicle with a velocity v=0, 1, … , vmax. The vehicles which are numbered 1, 2, 3, … , N move from the left to the right on a lane with periodic boundary conditions. At each discrete time step t → t+1, the system update is performed in parallel according to the following four rules:

Step 1: acceleration, vn→ min (vmax, vn+1);

Step 2: deceleration, vn→ min (vn, dn);

Step 3: randomization, vn→ max (vn-1,0) with probability p;

Step 4: position update, xn→xn+vn

Here vn and xn denote the velocity and position of the vehicle n respectively

vmax is the maximum velocity

dn = xn+1-xn-1 denotes the number of empty cells in front of the vehicle n

p is the randomization probability.

This set of rules control the forward motion of vehicles. In order to extend the model to multi-lane trafﬁc, one has to introduce lane-changing rules, which control the parallel motion of vehicles. So in multi-lane models the update step is usually divided into two sub-steps: In the ﬁrst sub-step, vehicles may change lanes in parallel according to lane-changing rules and in the second sub-step the lanes are considered as independent single-lane NaSch models.

In this paper the symmetric two-lane model is investigated. Chowdhury et al. [10] have assumed a symmetric rule set where vehicles change lanes if the following criteria are fulﬁlled [hereafter referred to as symmetric two-lane cellular automata (STCA) model]:

dn< min(vn+1, vmax) and dn,other > n and dn,back >5 and rand() < pn,change

Where:

dn,other, dn,back denote the number of free cells between the nth vehicle and its two neighbor vehicles on the other lane at time t, respectively

pn,change is the lane-change probability

rand() is for a random number between 0 and 1.

Next the lane-changing rules of STCA are revised to take into account the aggressive lane-changing behaviour of fast vehicle and the different lane-changing manoeuvres of different types of vehicles. The new conditions are:

Tn=1 and Tn+1=0 and dn < min (vn,+1,vmax) and dn,other > dn and dn,back2 and vn vn,back,other

and rand() < pn,change

If these conditions are fulfilled the vehicle n will change the lane.

Here Tn stands for the type of the nth vehicle. Tn = 1 (0) means that the nth vehicle is fast (slow). Conditions dn,back2 and vn vn,back,other make sure that there are not less than 2 free cells between nth vehicle and its following vehicle on the other lane and the velocity of the former is greater than or equal to the latter. This corresponds to the following facts of lane-changing behaviors: 95% of lane-changing vehicles have a rearward distance beyond 47.1 ft (14.36 m, approximately corresponding to 2 cells in CA model) and the relative velocity beyond 0 ft/s between the lane-change vehicle and the nearest rearward vehicle on the desired lane. In addition, different from that in STCA model, the lane-changing probability pn,change is deﬁned as:

pn,change= p0 if Tn=1 and Tn+1=0

pn,change= p0 otherwise

Here, p0>>p1 is adopted in the new model to embody the fact in real trafﬁc: the fast vehicles hindered by a slow vehicle is more likely to change lane than that in all other cases (i.e., the fast vehicle hindered by a fast one, a slow vehicle hindered by a slow one or a slow vehicle hindered by a fast one).

Laval, J., Daganzo, C. 2005. Lane-changing in traffic streams

This paper considers freeway sections away from diverges, where the main incentive for drivers to change lanes is increasing their speed. The main thesis is that a lane-changing vehicle acts as a moving bottleneck on its destination lane while accelerating to the speed prevailing on the lane, and that the ensuing disruption can trigger other lane changes. The freeway is therefore modelled as a set of interacting streams linked by the lane changes. The proposed model only needs one more parameter than the simplest traﬃc ﬂow models (which require three) and explains several puzzling phenomena without re-calibration.

Each lane is modelled as a separate KW stream interrupted by lane-changing particles that completely block traffic. The incremental-transfer (IT) principle for multilane KW problems (Daganzo et al., 1997), coupled with a one-parameter model for lane-changing demand, is used to predict the ﬂow transfers between neighboring lanes.

Τhe model is presented in two parts: i) the multilane KW module and ii) the lane-changing particles.

The multilane KW model:

For a highway with n=2 lanes the conservation equation for a single lane, l, is

, l=1,2….,n (1)

where kl (t,x) and ql (t, x) give the density and flow on l, at the time-space point (t,x). The inhomogeneous term Φl is the net lane-changing rate onto lane l, in units of veh/time–distance.

Then the vector k(t,x)=[k1(t, x),…., kn(t, x)] is defined and assumed that the one-directional lane-changing rate from lane l to lane l’ (with l≠l’) is a function, Φll’, of k, t and x. The net lane-changing rates are related to the one-directional rates by:

(2)

The proposed model speciﬁes the Φll’ instead of the Φl and does not require linearity. The Φll’

must realistically represent the competition between drivers desires for changing lanes, and the available space capacity in the target lane. To strike a balance between these two factors, we ﬁrst specify three sets of functions of (k, t, x) deﬁning: (i) a desired lane-changing rate from l to l’ (i.e.,a demand for lane-changing in units of veh/time–distance) Lll’, (ii) a desired set of through ﬂows on l, Tl , (in units of veh/time) and (iii) the available capacity on lane l, μl (in units of veh/time). Formally,

Lll’= Lll’ (k, t, x) (3a)

Tl = Tl (k, t, x) (3b)

μl = μl (k, t, x) (3c)

A competition mechanism, F, then determines the actual one-directional lane-changing rates Φll’ and through ﬂows ql from the abovementioned equations, i.e.

(Φl-1,l, ql, Φl+1,l) = F (Ll-1,l, Tl, Ll+1,l, μl) (4)

The demand functions L and T are obtained by disaggregating with a choice model the sending (or demand) function of KW theory. The capacity function μl is the receiving (or supply) function of KW theory. The transformation F should reﬂect sensible priority rules, which depend upon the nature of the lane-changing maneuvers (discretional or mandatory).

It is assumed here that the fundamental diagram (FD) of each lane is triangular with free-ﬂow

speed u, wave speed _w and jam density j. (This accounts for three of the four model parameters). All lanes are partitioned into small cells of length Δx and time is discretised into steps of duration Δt; see Figure 1.

We transform equations (1) and (2) according to the above figure. We iterate and calculate L, T and μ using the current densities as arguments. The lane changing rates are computed through flows q, Φ and the IT principle.

The IT principle transforms the values of L, T and μ for every cell into the actual lane-changing rates and through ﬂows. The IT recipe allocates differentials of ﬂow to the desired target cell (i, l) on a ﬁrst-come-ﬁrst-served basis. When total demand is less than the available capacity μl all the demands are fulﬁlled and able to advance to the target cell; otherwise the IT recipe prorates that available capacity to the different origin lanes according to their demands. It has been shown in Leclercq (2004) that if γl represents the fraction of the demand able to advance the IT result reduces to

(5)

And the transfers to the target lane l are

Φll’ = γlLl’l (6a)

ql = γlTl (6b)

From (5) and (6a,6b) we find:

Discrete lane-changing particles

The basic idea consists in quantizing the lane-changing rates from the above model to generate discrete particles, and then treating them as temporary blockages that move with bounded accelerations. This is possible because the blockages have a known (zero) passing rate and trajectories that can be determined endogenously with the CM model of vehicle dynamics. In the CM model, particles move with maximum acceleration, but are constrained by their own power and the speed of traﬃc ahead. A distinguishing feature of the method is that particles are tracked with very high resolution in continuous space (no jumping).

To quantize the process we can simply evaluate the cumulative number of lane changes from

(i, l) to (i + 1, l’ ) by time tj, , and then use the “floor” function [to generate integer jumps.

The complete hybrid model has good estimation and convergence properties. It is parsimonious since it only requires the relaxation time for lane-changing, τ, and the three usual KW parameters (free-ﬂow speed, capacity and jam density), which are readily observed in the ﬁeld.