Traffic flow

Let ${\displaystyle x(t)}$be the vehicle trajectory. Then,

${\displaystyle {\begin{aligned}[rlr]v(t)&=x'(t)&{\text{(speed)}}\\a(t)&=v'(t)=x''(t)&{\text{(acceleration)}}\\j(t)&=a'(t)=v''(t)=x'''(t)&{\text{(jerk)}}\end{aligned}}}$

Or, equivalently,

${\displaystyle {\begin{aligned}x(t)&=x_{0}+\int _{t_{0}}^{t}v(s)ds\\v(t)&=v_{0}+\int _{t_{0}}^{t}a(s)ds\\a(t)&=a_{0}+\int _{t_{0}}^{t}j(s)ds\end{aligned}}}$

where all the variables with subscript "0" are given initial conditions at time ${\displaystyle t_{0}}$.

In some applications it is convenient to take distance as the independent variable. A vehicle trajectory is represented by ${\textstyle t(x)}$, the inverse function of ${\displaystyle x(t)}$.

• If ${\displaystyle v(x)}$is given, ${\displaystyle t(x)}$can be derived as: ${\textstyle t(x)=t_{0}+\int _{x_{0}}^{x}{\frac {du}{v(u)}}}$.
• If ${\displaystyle a(x)}$is given, ${\displaystyle v(x)}$can be derived using the chain rule: ${\textstyle a(x)={\frac {dv}{dt}}={\frac {dv}{dx}}{\frac {dx}{dt}}={\frac {dv}{dx}}v}$, or ${\displaystyle a(x)=v'(x)v(x)}$. This can also be written as ${\textstyle a(x)={\frac {d[{\frac {1}{2}}v^{2}]}{dx}}}$, or better as ${\textstyle d[{\frac {1}{2}}v(x)^{2}]=a(x)dx}$, which can be integrated to give ${\textstyle v(x)^{2}=v_{0}^{2}+2\int _{x_{0}}^{x}a(x)dx}$. Hence, ${\textstyle v(x)={\sqrt {v_{0}^{2}+2\int _{x_{0}}^{x}a(x)dx}}}$

Vehicle kinematics models give the "desired acceleration" ${\textstyle a=a(v)}$that the driver imposes to the vehicle when traveling at a speed ${\displaystyle v(t)}$at time ${\displaystyle t}$ under free-flow conditions. A desired acceleration model captures both driver behavior and the physical limitations imposed by roadway geometry on the engine.

Noting that ${\textstyle a(v)={\frac {dv}{dt}}}$ we have ${\textstyle dt={\frac {dv}{a(v)}}}$, which by integration gives ${\textstyle t(v)=t_{0}+\int _{v_{0}}^{v}{\frac {dv}{a(v)}}}$. The position ${\textstyle x(v)}$can be derived using the chain rule:

${\displaystyle a(v)={\frac {dv}{dt}}={\frac {dv}{dx}}{\frac {dx}{dt}}={\frac {dv}{dx}}v}$

This gives ${\textstyle dx={\frac {v}{a(v)}}}$and hence

${\displaystyle x(v)=x_{0}+\int _{v_{0}}^{v}{\frac {v}{a(v)}}dv}$

For light vehicles, a good approximation is a linearly decreasing function of the speed:

${\displaystyle {\frac {dv}{dt}}=(v_{c}-v)\beta }$

where ${\textstyle \beta }$has units of ${\displaystyle {\text{time}}^{-1}}$and ${\displaystyle v_{c}}$can be interpreted as a desired speed. The reasonable typical value[4] of ${\displaystyle \beta }$is 0.06${\displaystyle {\text{s}}^{-1}}$.

Dimensionless formulations are convenient because they reduce the number of parameters involved in a problem. Define${\displaystyle {\tilde {t}}=\beta t,{\text{and }}{\tilde {v}}=v/v_{c}}$,which means that we measure time in units of ${\displaystyle \beta ^{-1}}$, and the speed in units of ${\displaystyle v_{c}}$. The quantity

${\displaystyle \tau =\beta ^{-1}}$

is the time scale of the problem. This means that the time for the system to reach equilibrium from a perturbation is comparable to ${\displaystyle \tau }$.

The corresponding transformation for the space variable

${\displaystyle {\tilde {x}}=\int _{{\tilde {t}}_{0}}^{\tilde {t}}{\tilde {v}}({\tilde {s}})d{\tilde {s}}={\frac {1}{\tau v_{c}}}\int _{t_{0}}^{t}v(s)ds=x(t)/(\tau v_{c})}$

is obtained by a change of variable noting that ${\displaystyle d{\tilde {s}}=\beta ds.}$

The linear acceleration model is now

${\displaystyle {\tilde {v}}'=1-{\tilde {v}}}$

with the initial condition ${\textstyle {\tilde {v}}(0)={\tilde {v}}_{0}}$. Setting ${\displaystyle x_{0}=0}$the equations of motion become

${\displaystyle {\begin{aligned}{\tilde {x}}({\tilde {t}})&={\tilde {t}}-(1-{\tilde {v}}_{0})(1-e^{-{\tilde {t}}}),\\{\tilde {v}}({\tilde {t}})&=1-e^{-{\tilde {t}}}(1-{\tilde {v}}_{0}),\\{\tilde {a}}({\tilde {t}})&=e^{-{\tilde {t}}}(1-{\tilde {v}}_{0}),\\{\tilde {j}}({\tilde {t}})&=-e^{-{\tilde {t}}}(1-{\tilde {v}}_{0}),\end{aligned}}}$

and the only parameter is the initial condition ${\displaystyle {\tilde {v}}_{0}}$.

A completely parameter-free formulation is given by the transformation

${\displaystyle {\tilde {t}}=\beta t,{\text{and }}{\tilde {v}}={\frac {v_{c}-v}{v_{c}-v_{0}}}.}$

The acceleration model becomes ${\displaystyle {\tilde {v}}'=-{\tilde {v}}}$with initial condition ${\displaystyle {\tilde {v}}(0)=1}$; this gives

${\displaystyle {\begin{aligned}{\tilde {x}}({\tilde {t}})&=1-e^{-{\tilde {t}}},\\{\tilde {v}}({\tilde {t}})&=e^{-{\tilde {t}}},\\{\tilde {a}}({\tilde {t}})&=-e^{-{\tilde {t}}},\\{\tilde {j}}({\tilde {t}})&=e^{-{\tilde {t}}}.\end{aligned}}}$

Speed is the distance covered per unit time. One cannot track the speed of every vehicle; so, in practice, average speed is measured by sampling vehicles in a given area over a period of time. Two definitions of average speed are identified: "time mean speed" and "space mean speed".

• "Time mean speed" is measured at a reference point on the roadway over a period of time. In practice, it is measured by the use of loop detectors. Loop detectors, when spread over a reference area, can identify each vehicle and can track its speed. However, average speed measurements obtained from this method are not accurate because instantaneous speeds averaged over several vehicles do not account for the difference in travel time for the vehicles that are traveling at different speeds over the same distance.

          ${\displaystyle v_{t}=(1/m)\sum _{i=1}^{m}v_{i}}$

where m represents the number of vehicles passing the fixed point and v_i is the speed of the ith vehicle.

• "Space mean speed" is measured over the whole roadway segment. Consecutive pictures or video of a roadway segment track the speed of individual vehicles, and then the average speed is calculated. It is considered more accurate than the time mean speed. The data for space calculating space mean speed may be taken from satellite pictures, a camera, or both.

          ${\displaystyle v_{s}=\left((1/n)\sum _{i=1}^{n}(1/v_{i})\right)^{-1}}$

where n represents the number of vehicles passing the roadway segment.

The "space mean speed" is thus the harmonic mean of the speeds.

The time mean speed is never less than space mean speed:

          ${\displaystyle v_{t}=v_{s}+{\frac {\sigma _{s}^{2}}{v_{s}}}}$

where ${\displaystyle \sigma _{s}^{2}}$ is the variance of the space mean speed[5]

Figure 2. Space Mean- and Time Mean speeds

In a time-space diagram, the instantaneous velocity, v = dx/dt, of a vehicle is equal to the slope along the vehicle’s trajectory. The average velocity of a vehicle is equal to the slope of the line connecting the trajectory endpoints where a vehicle enters and leaves the roadway segment. The vertical separation (distance) between parallel trajectories is the vehicle spacing (s) between a leading and following vehicle. Similarly, the horizontal separation (time) represents the vehicle headway (h). A time-space diagram is useful for relating headway and spacing to traffic flow and density, respectively.

Density (k) is defined as the number of vehicles per unit length of the roadway. In traffic flow, the two most important densities are the critical density (k_c) and jam density (k_j). The maximum density achievable under free flow is k_c, while k_j is the maximum density achieved under congestion. In general, jam density is seven times the critical density. Inverse of density is spacing (s), which is the center-to-center distance between two vehicles.

     ${\displaystyle k={\frac {1}{s}}}$

Figure 3. Flow Density relationship

Figure 4. Relationship between flow (q), density (k), and speed (v)

The density (k) within a length of roadway (L) at a given time (t₁) is equal to the inverse of the average spacing of the n vehicles.

     ${\displaystyle K(L,t_{1})={\frac {n}{L}}={\frac {1}{{\bar {s}}(t_{1})}}}$

In a time-space diagram, the density may be evaluated in the region A.

     ${\displaystyle k(A)={\frac {n}{L}}={\frac {n\,dt}{L\,dt}}={\frac {tt}{\left|A\right\vert }}}$

where tt is the total travel time in A.

Figure 5.

Flow (q) is the number of vehicles passing a reference point per unit of time, vehicles per hour. The inverse of flow is headway (h), which is the time that elapses between the ith vehicle passing a reference point in space and the (i + 1)th vehicle. In congestion, h remains constant. As a traffic jam forms, h approaches infinity.

     ${\displaystyle q=kv\,}$

     ${\displaystyle q=1/h\,}$

The flow (q) passing a fixed point (x₁) during an interval (T) is equal to the inverse of the average headway of the m vehicles.

     ${\displaystyle q(T,x_{1})={\frac {m}{T}}={\frac {1}{{\bar {h}}(x_{1})}}}$

In a time-space diagram, the flow may be evaluated in the region B.

     ${\displaystyle q(B)={\frac {m}{T}}={\frac {m\,dx}{T\,dx}}={\frac {td}{\left|B\right\vert }}}$

where td is the total distance traveled in B.

Figure 6.

A more general definition of the flow and density in a time-space diagram is illustrated by region C:

     ${\displaystyle q(C)={\frac {td}{\left|C\right\vert }}}$

     ${\displaystyle k(C)={\frac {tt}{\left|C\right\vert }}}$

where:

     ${\displaystyle td=\sum _{i=1}^{m}\,dx_{i}}$

     ${\displaystyle tt=\sum _{i=1}^{n}\,dt_{i}}$

In addition to providing information on the speed, flow, and density of traffic streams, time-space diagrams may illustrate the propagation of congestion upstream from a traffic bottleneck (shockwave). Congestion shockwaves will vary in propagation length, depending upon the upstream traffic flow and density. However, shockwaves will generally travel upstream at a rate of approximately 20 km/h.

Figure 7.

Traffic on a stretch of road is said to be stationary if an observer does not detect movement in an arbitrary area of the time-space diagram. Traffic is stationary if all the vehicle trajectories are parallel and equidistant. It is also stationary if it is a superposition of families of trajectories with these properties (e.g. fast and slow drivers). By using a very small hole in the template one could sometimes view an empty region of the diagram and other times not, so that even in these cases, one could say that traffic was not stationary. Clearly, for such fine level of observation, stationary traffic does not exist. A microscopic level of observation must be excluded from the definition if traffic appears to be similar through larger windows. In fact, we relax the definition even further by only requiring that the quantities t(A) and d(A) be approximately the same, regardless of where the "large" window (A) is placed.

Figure 10. Step function

The traffic light example depicts N-curves as smooth functions. Theoretically, however, plotting N-curves from collected data should result in a step-function (figure 10). Each step represents the arrival or departure of one vehicle at that point in time.[11] When the N-curve is drawn on larger scale reflecting a period of time that covers several cycles, then the steps for individual vehicles can be ignored, and the curve will then look like a smooth function (figure 8).

The N-curve can be used in a number of different traffic analyses, including freeway bottlenecks and dynamic traffic assignment. This is due to the fact that a number of traffic flow characteristics can be derived from the plot of cumulative vehicle count curves. Illustrated in figure 11 are the different traffic flow characteristics that can be derived from the N-curves.

Figure 11. Traffic flow characteristics from two N-curves

These are the different traffic flow characteristics from figure 11:

From these variables, the average delay experienced by each vehicle and the average queue length at any time t can be calculated, using the following formulas:

     ${\displaystyle {\text{average delay (}}w_{\text{avg}}{\text{)}}={\frac {{\text{total delay experienced by }}m{\text{ vehicles}}}{\text{total number of delayed vehicles}}}={\frac {TD}{m}}}$

     ${\displaystyle {\text{average queue (}}Q_{\text{avg}}{\text{)}}={\frac {{\text{total delay experienced by }}m{\text{ vehicles}}}{\text{duration of congestion}}}={\frac {TD}{(t_{2}-t_{1})}}}$

In traffic flow area, an alternative way to solve the kinematic wave model is to treat it like a Hamilton–Jacobi equation, which is particularly useful in identifying conserved quantities for mechanical systems.

Suppose we are interested in finding the cumulative curve as a function of time and space, N(t,x). Based the definition of cumulative curve,${\displaystyle {\frac {\partial N(t,x)}{\partial t}}=\rho (x,t)}$ refers to the flow and ${\displaystyle {\frac {\partial N(t,x)}{\partial x}}=-k(x,t)}$ refers to the density. Note that the sign convention should be consistent. Then the fundamental flow-density (${\displaystyle q-k}$) equation: ${\displaystyle q=F(k)}$ can be expressed in cumulative count form as:
${\displaystyle {\frac {\partial N(t,x)}{\partial t}}-F(-{\frac {\partial N(t,x)}{\partial x}})=0}$
${\displaystyle s.t.N(B)=G(B)}$, where ${\displaystyle B}$ is a known boundary.

Now, for a generic random point ${\displaystyle P(x,t)}$ in the time-space diagram, the solution to the above partial derivative equation is equivalent to solve the following optimization problem, which minimizes the vehicles pass-by:${\displaystyle N(P)=\min {G(x_{b})+(t-t_{b})R({\frac {x-x_{b}}{t-t_{b}}})}}$, where ${\displaystyle b}$ is a random point on the boundary ${\displaystyle B}$.

The function ${\displaystyle R}$ is defined as the maximum passing rate along the observers. In the case of triangular fundamental diagram, we have ${\displaystyle R(v)=Q-K_{c}*v}$. The observer speed ${\displaystyle v\in [-w,u]}$.Here, notation ${\displaystyle Q}$ corresponds to the capacity, ${\displaystyle k_{c}}$ corresponds to the critical density, ${\displaystyle u}$ and ${\displaystyle w}$ are free flow speed and wave speed respectively.

With that being said, the minimization function above simplifies into: ${\displaystyle N(P)=\min {G(x_{b})+(t-t_{b})Q-(x-x_{b})K_{c}}}$, where ${\displaystyle b}$ is a random point on the boundary ${\displaystyle B}$. Here, we limit the solution discussion over initial value problems (IVP) and boundary value problems (BVP).

Initial value problem occurs when boundary condition is given at a fixed time, e.g. at ${\displaystyle t=0}$ and boundary ${\displaystyle B:=N(0,x)}$. As the observer speed is bounded by ${\displaystyle v\in [-w,u]}$, the potential solution is delimited by two lines ${\displaystyle x=x_{U}+ut}$ and ${\displaystyle x=x_{D}-wt}$.

Thus, the IVP is defined as follow:
${\displaystyle N(P)=\min {f(x_{b})\equiv G(x_{b})+(t-t_{b})Q-(x-x_{b})K_{c}}}$
${\displaystyle s.t.N(0,x)=G(x)}$
${\displaystyle x_{U}<x_{b}<x_{D}}$

The local minimum point occurs when the first order derivative is 0 and the second order derivative is greater than 0. Or, the minimum happens at boundaries. So, the set of potential solutions goes as follow:
1.${\displaystyle \forall x_{b}}$ that ${\displaystyle f'(x_{b})=G'(x_{b})+K_{c}=-k(0,x_{b})+K_{c}=0}$ and ${\displaystyle f''(x_{b})=G''(x_{b})=-k_{x}(0,x_{b})>0}$
2.${\displaystyle x_{U}}$ and ${\displaystyle x_{D}}$.
The solution will be the minimum corresponding ${\displaystyle N(P)}$ of all candidate points. ${\displaystyle N(P)=\min(f(x_{U}),f(x_{D}),}$ and all ${\displaystyle f(x_{b})}$ from condition 1).

Specifically, if the initial condition ${\displaystyle G(x)}$is a linear function, ${\displaystyle N(P)=\min(f(x_{U}),f(x_{D}))}$

Similarly, the boundary value problem indicates the boundary condition is given at a fix location, e.g. ${\displaystyle B:=N(x_{0},0)}$. Still, the observer speed is bounded by ${\displaystyle v\in [-w,u]}$. For a random point ${\displaystyle P(x,t)}$, the upper bound for solution candidates: if ${\displaystyle x>x_{0}}$, ${\displaystyle t_{U}=t-{\frac {x-x_{0}}{u}}}$; else, ${\displaystyle t_{D}=t-{\frac {x_{0}-x}{w}}}$.

The BVP is defined as follow:
${\displaystyle N(P)=\min {f(t_{b})\equiv G(t_{b})+(t-t_{b})Q-(x-x_{b})K_{c}}}$
${\displaystyle s.t.N(t,x_{0})=G(t)}$
${\displaystyle {\begin{cases}t<t_{U},&{\text{if }}x>x_{0}\\t<t_{D},&{\text{if }}x<x_{0}\end{cases}}}$

The first order derivative:${\displaystyle f'(t_{b})=G'(t_{b})-Q=q(t_{b},0)-Q}$ is always smaller than 0 because flows won't exceed the capacity. Thus, the minimum happens at the upper bound of the time axis.
${\displaystyle N(P)={\begin{cases}f(t_{U}),&{\text{if }}x>x_{0}\\f(t_{D}),&{\text{if }}x<x_{0}\end{cases}}}$

In practice, people use this method to estimate the traffic states at ${\displaystyle P(t,x)}$ between two loop detectors, which can be viewed as a combination of two boundary value problems (one at upstream and one at down stream). Denote the upstream loop detector location as ${\displaystyle x_{U}}$ and the downstream loop detector location as ${\displaystyle x_{D}}$. Based on the conclusion above, the minimum value occurs at the upper bound along the time axis.
${\displaystyle N(P)=\min(f(t_{U}),f(t_{D}))}$, with ${\displaystyle {\begin{cases}t_{U}=t-{\frac {x-x_{U}}{u}}\\t_{D}=t-{\frac {x_{D}-x}{w}}\end{cases}}}$

Figure 12. Roadway section experiencing a bottleneck

Figure 13. Maximum queue length and delay

One application of the N-curve is the bottleneck model, where the cumulative vehicle count is known at a point before the bottleneck (i.e. this is location X₁). However, the cumulative vehicle count is not known at a point after the bottleneck (i.e. this is location X₂), but rather only the capacity of the bottleneck, or the discharge rate, μ, is known. The bottleneck model can be applied to real-world bottleneck situations such as those resulting from a roadway design problem or a traffic incident.

Take a roadway section where a bottleneck exists such as in figure 12. At some location X₁ before the bottleneck, the arrivals of vehicles follow a regular N-curve. If the bottleneck is absent, the departure rate of vehicles at location X₂ is essentially the same as the arrival rate at X₁ at some later time (i.e. at time TT_FF – free-flow travel time). However, due to the bottleneck, the system at location X₂ is now only able to have a departure rate of μ. When graphing this scenario, essentially we have the same situation as in figure 9, where the arrival curve of vehicles is N₁, the departure curve of vehicles absent the bottleneck is N₂, and the limited departure curve of vehicles given the bottleneck is N′₂. The discharge rate μ is the slope of curve N′₂, and all the same traffic flow characteristics as in figure 11 can be determined from this diagram. The maximum delay and maximum queue length can be found at a point M in figure 13 where the slope of N₂ is the same as the slope of N′₂; i.e. when the virtual arrival rate is equal to the discharge / departure rate μ.

The N-curve in the bottleneck model may also be used to calculate the benefits in removing the bottleneck, whether in terms of a capacity improvement or removing an incident to the side of the roadway.

Figure 14. Tandem Queues

Figure 15. N-Curve of Tandem Queues with Two BNs

Figure 16. N-Curve of Tandem Queues with n BNs

As introduced in the section above, the N-curve is an applicable model to estimate traffic delay during time by setting arrival and departure cumulative counting curve. Since the curve can represent various traffic characteristics and roadway conditions, the delay and queue situations under these conditions will be able to be recognized and modeled using N-curves. Tandem queues occur when multiple bottlenecks exist between the arrival and departure locations. Figure 14 shows a qualitative layout of a tandem-queue roadway segment with a certain initial arrival. The bottlenecks along the stream have their own capacity, 'μ_i [veh/time], and the departure is defined at the downstream end of the entire segment.

To determine the ultimate departure, D(t), it can be an available method to research on the individual departures, D_i(t). As shown in the Figure 15, if the free-flow travel-time is neglected, the departure of BN_i−1 will be the virtual arrival of BN_i, which can also be presented as D_i−1(t) = A_i(t). Thus, the N-curve of a roadway with two bottlenecks (minimum number of BNs along a tandem-queue roadway) can be developed as Figure 15 with μ₁ < μ₂. In this case, D₂(t) will be the ultimate departure of this 2-BN tandem-queue roadway.

Regarding of a tandem-queue roadway having 3 BNs with μ₁ < μ₂, if μ₁ < μ₂ < μ₃, similarly as the 2-BN case, D₃(t) will be the ultimate departure of this 3-BN tandem-queue roadway. If, however, μ₁ < μ₃ < μ₂, D₂(t) will then still be the ultimate departure of the 3-BN tandem-queue roadway. Thus, it can be summarized that, the departure of the bottleneck with the minimum capacity will be the ultimate departure of the entire system, regardless of the other capacities and the number of bottlenecks. Figure 16 shows a general case with n BNs.

The N-curve model describing above represents a significant characteristic of the tandem-queue systems, which is that the ultimate departure only depends on the bottleneck with the minimum capacity. In a practical perspective, when the resources (economy, effort, etc.) of the investment on tandem-queue systems are limited, the investment can mainly focus on the bottleneck with the worst condition.

Figure 17. Departure curve for a signal with a releasing capacity

Figure 18. Saturated case at a traffic light

Figure 19. Unsaturated case at a traffic light with a downstream bottleneck

A signalized intersection will have special departure behaviors. With simplified speaking, a constant releasing free-flow capacity, μ_s, exists during the green phases. On the contrary, the releasing capacity during the red phases should be zero. Thus, the departure N-curve regardless of arrival will look like as Figure 17 below: counts increase with the slope of μ_s during green, and remain the same during red..

Saturated case of a traffic light occurs when the releasing capacity is fully used. This case usually exists when the arriving demand is relatively large. The N-curve representation of the saturated case is shown in the Figure 18.

Unsaturated case of a traffic light occurs when releasing capacity is not fully used. This case usually exists when the arriving demand is relatively small. The N-curve representation of the unsaturated case is shown in the Figure 19. If there is a bottleneck with a capacity of μ_b(<μ_s) downstream of the light, the ultimate departure of the light-bottleneck system will be that of the downstream bottleneck.

Dynamic traffic assignment can also be solved using the N-curve. There are two main approaches to tackle this problem: system optimum, and user equilibrium. This application will be discussed further in the following section.

In short, a network is in system optimum (SO) when the total system cost is the minimum among all possible assignments.
System Optimum is based on the assumption that routes of all vehicles would be controlled by the system, and that rerouting would be based on maximum utilization of resources and minimum total system cost. (Cost can be interpreted as travel time.) Hence, in a System Optimum routing algorithm, all routes between a given OD pair have the same marginal cost. In traditional transportation economics, System Optimum is determined by equilibrium of demand function and marginal cost function. In this approach, marginal cost is roughly depicted as increasing function in traffic congestion. In traffic flow approach, the marginal cost of the trip can be expressed as sum of the cost(delay time, w) experienced by the driver and the externality(e) that a driver imposes on the rest of the users.[12] Suppose there is a freeway(0) and an alternative route(1), which users can be diverted onto off-ramp. Operator knows total arrival rate(A(t)), the capacity of the freeway(μ_0), and the capacity of the alternative route(μ_1). From the time 't_0', when freeway is congested, some of the users start moving to alternative route. However, when 't_1', alternative route is also full of capacity. Now operator decides the number of vehicles(N), which use alternative route. The optimal number of vehicles(N) can be obtained by calculus of variation, to make marginal cost of each route equal. Thus, optimal condition is T_0=T_1+∆_1. In this graph, we can see that the queue on the alternative route should clear ∆_1 time units before it clears from the freeway. This solution does not define how we should allocates vehicles arriving between t_1 and T_1, we just can conclude that the optimal solution is not unique. If operator wants freeway not to be congested, operator can impose the congestion toll, e_0-e_1, which is the difference between the externality of freeway and alternative route. In this situation, freeway will maintain free flow speed, however alternative route will be extremely congested.

In brief, A network is in user equilibrium (UE) when every driver chooses the routes in its lowest cost between origin and destination regardless whether total system cost is minimized.
The user optimum equilibrium assumes that all users choose their own route towards their destination based on the travel time that will be consumed in different route options. The users will choose the route which requires the least travel time. The user optimum model is often used in simulating the impact on traffic assignment by highway bottlenecks. When the congestion occurs on highway, it will extend the delay time in travelling through the highway and create a longer travel time. Under the user optimum assumption, the users would choose to wait until the travel time using a certain freeway is equal to the travel time using city streets, and hence equilibrium is reached. This equilibrium is called User Equilibrium, Wardrop Equilibrium or Nash Equilibrium.

Figure 15. User equilibrium traffic model

The core principle of User Equilibrium is that all used routes between a given OD pair have the same travel time. An alternative route option is enabled to use when the actual travel time in the system has reached the free-flow travel time on that route.

For a highway user optimum model considering one alternative route, a typical process of traffic assignment is shown in figure 15. When the traffic demand stays below the highway capacity, the delay time on highway stays zero. When the traffic demand exceeds the capacity, the queue of vehicle will appear on the highway and the delay time will increase. Some of users will turn to the city streets when the delay time reaches the difference between the free-flow travel time on highway and the free-flow travel time on city streets. It indicates that the users staying on the highway will spend as much travel time as the ones who turn to the city streets. At this stage, the travel time on both the highway and the alternative route stays the same. This situation may be ended when the demand falls below the road capacity, that is the travel time on highway begins to decrease and all the users will stay on the highway. The total of part area 1 and 3 represents the benefits by providing an alternative route. The total of area 4 and area 2 shows the total delay cost in the system, in which area 4 is the total delay occurs on the highway and area 2 is the extra delay by shifting traffic to city streets.

Navigation function in Google Maps can be referred as a typical industrial application of dynamic traffic assignment based on User Equilibrium since it provides every user the routing option in lowest cost (travel time).

Both User Optimum and System Optimum can be subdivided into two categories on the basis of the approach of time delay taken for their solution:

Predictive time delay assumes that the user of the system knows exactly how long the delay is going to be right ahead. Predictive delay knows when a certain congestion level will be reached and when the delay of that system would be more than taking the other system, so the decision for reroute can be made in time. In the vehicle counts-time diagram, predictive delay at time t is horizontal line segment on the right side of time t, between the arrival and departure curve, shown in Figure 16. the corresponding y coordinate is the number nth vehicle that leaves the system at time t.

Reactive time delay is when the user has no knowledge of the traffic conditions ahead. The user waits to experience the point where the delay is observed and the decision to reroute is in reaction to that experience at the moment. Predictive delay gives significantly better results than the reactive delay method. In the vehicle counts-time diagram, predictive delay at time t is horizontal line segment on the left side of time t, between the arrival and departure curve, shown in Figure 16. the corresponding y coordinate is the number nth vehicle that enters the system at time t.

Figure 16. Predictive and Reactive Time Delay

The kinematic wave model of traffic flow theory is the simplest dynamic traffic flow model that reproduces the propagation of traffic waves. It is made up of three components: the fundamental diagram, the conservation equation, and initial conditions. The law of conservation is the fundamental law governing the kinematic wave model:

     ${\displaystyle {\frac {\partial k}{\partial t}}+{\frac {\partial q}{\partial x}}=0,}$

The fundamental diagram of the kinematic wave model relates traffic flow with density, as seen in figure 3 above. It can be written as:

     ${\displaystyle {q}={F(k)}}$

Finally, initial conditions must be defined to solve a problem using the model. A boundary is defined to be ${\displaystyle {k(t,x)}}$, representing density as a function of time and position. These boundaries typically take two different forms, resulting in initial value problems (IVPs) and boundary value problems (BVPs). Initial value problems give the traffic density at time ${\displaystyle {t}={0}}$, such that ${\displaystyle {k(0,x)}={g(x)}}$, where ${\displaystyle {g(x)}}$ is the given density function. Boundary value problems give some function ${\displaystyle {g(t)}}$ that represents the density at the ${\displaystyle {x=0}}$ position, such that ${\displaystyle {k(t,0)}={g(t)}}$. The model has many uses in traffic flow. One of the primary uses is in modeling traffic bottlenecks, as described in the following section.

Firstly, consider the initial value problem (IVP), that is, ${\displaystyle {t}={0}}$, for the transport equation:

${\displaystyle {\begin{cases}k_{t}+wk_{x}=0&{\text{(Transport equation)}}\\k(0,x)=g(x),(t,x)\in \beta &{\text{(Initial values)}}\end{cases}}}$

k can thus be solved as ${\displaystyle {k(t,x)}={g(x-wt)}}$. This is referred to as the IVP solution. This implies that along the lines with the same slope w on the space-time diagram, density k is constant. These lines are called characteristics. More specifically:

${\displaystyle {x-wt}={x_{0}}}$

Consider the Boundary Value Problem (BVP), that is, ${\displaystyle {x}={0}}$, for the transport equation:

${\displaystyle {\begin{cases}k_{t}+wk_{x}=0&{\text{(Transport equation)}}\\k(t,0)=g(t),(t,x)\in \beta &{\text{(Boundary values)}}\end{cases}}}$

k can thus be solved as ${\displaystyle {k(t,x)}={g(x-wt)}}$. This is referred to as the BVP solution. Similar to the IVP solution, this means that along the lines with the same slope w on the space-time diagram, or so called characteristics, density k remains constant.

It is assumed that when the initial conditions are piece wise constant, the wave speed of each piece is also constant thus the transport equation holds.

The Riemann problem provides the foundations to develop numeric solutions for the kinematic wave model. Consider the initial values:

${\displaystyle k(0,x)=g(x)=}$ ${\displaystyle {\begin{cases}k_{U}&{\text{if }}x<x_{0}\\k_{D}&{\text{otherwise}}\end{cases}}}$

Case 1: ${\displaystyle k_{U}<k_{D}}$

This is a deceleration process, with traffic going from wave speed ${\displaystyle w_{V}}$ to ${\displaystyle w_{D}}$, and density from ${\displaystyle k_{V}}$ to ${\displaystyle k_{V}}$. The deceleration creates a discontinuity in traffic condition and results in a "shockwave":

${\displaystyle s={\frac {q(k_{U})-q(k_{D})}{k_{U}-k_{D}}}}$

Figure 17

The shockwave effect is illustrated in Figure 17. Traffic state moves from U (free flow) to D (congested). The slope s of this shockwave in the space time diagram is represented by the straight line that links point U and D.

Case 2: ${\displaystyle k_{U}>k_{D}}$

This is an acceleration process, with traffic going from wave speed ${\displaystyle w_{D}}$ to ${\displaystyle w_{V}}$, and density from ${\displaystyle k_{D}}$ to ${\displaystyle k_{V}}$. The slope s of this shockwave can be same as to case 1, but that solution is not unique and the traffic state does not go back via a straight line from point D to U. Traffic recovers along the fundamental diagram curve, rather than returning to the free flow speed at once. This results in multiple different "solution shockwaves" that radiates from a given x0. These mechanism are shown in Figure 18.

Figure 18

In this case, oftentimes the entropy condition (EC) is used to pick a single solution. EC founds the solution that maximizes the flow at each location using the vanishing viscosity method.

Graphical solution of Newell–Daganzo merge model.

The flows that pass through the merging process, q₁ and q₂, are determined by split priority or, merge ratio. The state of each branch of roadways is determined by graphically with the input of the demands for each branch of roadways, q₁^D and q₂^D. There are four possible states for the merge system, both inlets in free flow, one of the inlets in congestion, and both inlets in congestion.

A common approach to calculate the merge ratio p is called "zipper rule" which p is calculated based on the number of lanes of the single roadway when both inlets are in congestion. If there are n lanes of the single roadway, then under the zipper rule, p=1/(2n-1). This merge ratio is also the ratio of minimum capacities of the inlets μ₁^* and μ₂^*. μ₁^* + μ₂^* = μ. As a result, q₁=(μ₁^*/μ)*μ and q₂=(μ₂^*/μ)*μ.

The state of each branch of roadways is determined by the graphical solution which is shown in the right. The x-axis is the possible value of q₁ and the y-axis is the possible value of q₂.The feasible region of demands is the defined by the maximum possible values for q₁^D and q₂^D which are μ₁ and μ₂. The feasible region for q₁ and q₂ is defined as the intersection between the line of q₁ + q₂ = μ and the feasible region of demands. The merge ration, p, is plotted from the origin to the line of q₁ + q₂ = μ.

The four possible states of the merge system are shown in the graph by the regions noted by A1, A2, A3, and A4. A specific states of a merge system is determined by the region where the input data fall into. The region A1 represents the state when both inlet 1 and inlet 2 are in free flow. The region A2 represents the state when inlet 1 is in free flow and inlet 2 is in congestion. The region A3 represents the state when inlet 1 is in congestion and inlet 2 is in free flow. The region A4 represents the state when both inlet 1 and inlet 2 are in congestion.

The general cause of stationary bottlenecks are lane drops which occurs when the a multilane roadway loses one or more its lane. This causes the vehicular traffic in the ending lanes to merge onto the other lanes.

Figure 18.

Consider a stretch of highway with two lanes in one direction. Suppose that the fundamental diagram is modeled as shown here. The highway has a peak capacity of Q vehicles per hour, corresponding to a density of k_c vehicles per mile. The highway normally becomes jammed at k_j vehicles per mile.

Before capacity is reached, traffic may flow at A vehicles per hour, or a higher B vehicles per hour. In either case, the speed of vehicles is v_f, or "free flow," because the roadway is under capacity.

Now, suppose that at a certain location x₀, the highway narrows to one lane. The maximum capacity is now limited to D', or half of Q, since only one lane of the two is available. D shares the same flowrate as state D', but its vehicular density is higher.

Figure 19.

Using a time-space diagram, we may model the bottleneck event. Suppose that at time 0, traffic begins to flow at rate B and speed v_f. After time t1, vehicles arrive at the lower flowrate A.

Before the first vehicles reach location x₀, the traffic flow is unimpeded. However, downstream of x₀, the roadway narrows, reducing the capacity by half – and to below that of state B. Due to this, vehicles will begin queuing upstream of x₀. This is represented by high-density state D. The vehicle speed in this state is the slower v_d, as taken from the fundamental diagram. Downstream of the bottleneck, vehicles transition to state D', where they again travel at free-flow speed v_f.

Once vehicles arrive at rate A starting at t1, the queue will begin to clear and eventually dissipate. State A has a flowrate below the one-lane capacity of states D and D'.

On the time-space diagram, a sample vehicle trajectory is represented with a dotted arrow line. The diagram can readily represent vehicular delay and queue length. It is a simple matter of taking horizontal and vertical measurements within the region of state D.

As explained above, moving bottlenecks are caused due to slow moving vehicles that cause disruption in traffic. Moving bottlenecks can be active or inactive bottlenecks. If the reduced capacity(q_u) caused due to a moving bottleneck is greater than the actual capacity(μ) downstream of the vehicle, then this bottleneck is said to be an active bottleneck. Figure 20 shows the case of a truck moving with velocity 'v' approaching a downstream location with capacity 'μ'. If the reduced capacity of the truck (q_u) is less than the downstream capacity, then the truck becomes an inactive bottleneck.

Figure 20.

Laval 2009, presents a framework for estimating analytical expressions for the capacity reductions caused by a subset of vehicles forced to slow down at horizontal/vertical curves on multilane freeway. In each of the lane the underperforming stream is described in terms of its desired speed distribution and is modeled as per Newell’s kinematic wave theory for moving bottlenecks. Lane changing in the presence of trucks can lead to a positive or negative impact on capacity. If the target lane is empty then the lane-changing increases capacity

Figure 21. A slow tractor creates a moving bottleneck.

For this example, consider three lanes of traffic in one direction. Assume that a truck starts traveling at speed v, slower than the free flow speed v_f. As shown on the fundamental diagram below, q_u represents the reduced capacity (2/3 of Q, or 2 of 3 lanes available) around the truck.

State A represents normal approaching traffic flow, again at speed v_f. State U, with flowrate q_u, corresponds to the queuing upstream of the truck. On the fundamental diagram, vehicle speed v_u is slower than v_f. But once drivers have navigated around the truck, they can again speed up and transition to downstream state D. While this state travels at free flow, the vehicle density is less because fewer vehicles get around the bottleneck.

Figure 22.

Suppose that, at time t, the truck slows from free-flow to v. A queue builds behind the truck, represented by state U. Within the region of state U, vehicles drive slower as indicated by the sample trajectory. Because state U limits to a smaller flow than state A, the queue will back up behind the truck and eventually crowd out the entire highway (slope s is negative). If state U had the higher flow, there would still be a growing queue. However, it would not back up because the slope s would be positive.

Figure 23.

Imagine a scenario in which a two lane road is reduced to one lane at point x_o from here on the road’s capacity is reduced to half its original (½µ), Case I. Later along the road at point x₁ the 2nd lane is opened and the capacity is restored to its original (µ), Case II.

Case I

There is a bottleneck limiting the flow of traffic which causes an increase in the density of cars (k) at location (x_o). This causes a deceleration for all oncoming cars traveling at speed u to slow to speed v_d. This shockwave will travel at the speed of the slope of line U-D on the fundamental diagram. The wave speed can be calculated as v_shock = (q_D − q_U)/(k_D−k_U). This line delineates the congestion traffic from oncoming free-flow traffic. If the slope of U-D on the fundamental diagram is positive congestion will continue downstream of the highway. If it has a negative slope the congestion will continue upstream (see figure a[22]). This deceleration is the case I of Riemann’s problem (see figure b and c).

Case II

In case II of Riemann’s problem traffic goes from congestion to free-flow and the cars accelerate as the density drops. Again the slope of these shock waves can be calculated using the same formula v_shock = (q_D − q_U)/(k_D−k_U). The difference this time is that traffic flow travels along the fundamental diagram not in a straight line across but many slopes between various points on the curved fundamental diagram (see figure d). This causes many lines emanating from point x₁ all in a fan shape, called rarefaction (see figure e). This model implies that the users later on in time will take longer to accelerate as they meet each of the lines. Instead a better approximation is a triangular diagram where the traffic increases abruptly as it would when a driver sees an opening in front of them (see figures f and g).

Figure 24.

Figure 25.

Figure 26.

Figure 27.

The set of fundamental empirical features of traffic breakdown at a highway bottleneck is as follows:

1. Traffic breakdown at a highway bottleneck is a local phase transition from free flow (F) to congested traffic whose downstream front is usually fixed at the bottleneck location. Such congested traffic is called synchronized flow (S). Within the downstream front of synchronized flow, vehicles accelerate from synchronized flow upstream of the bottleneck to free flow downstream of the bottleneck.

1. At the same bottleneck, traffic breakdown can be either spontaneous or induced.
2. The probability of traffic breakdown is an increasing flow rate function.
3. There is a well-known hysteresis phenomenon associated with traffic breakdown: When the breakdown has occurred at some flow rates with resulting congested pattern formation upstream of the bottleneck, then a return transition to free flow at the bottleneck is usually observed at considerably smaller flow rates.

A spontaneous traffic breakdown occurs, where there are free flows both upstream and downstream of the bottleneck before the breakdown has occurred. In contrast, an induced traffic breakdown is caused by a propagation of a congested pattern that has earlier emerged for example at another downstream bottleneck.

Empirical data that illustrates the set of fundamental empirical features of traffic breakdown at highway bottlenecks as well as explanations of the empirical data can be found in Wikipedia article Kerner’s breakdown minimization principle and in review.[23]

The generally accepted classical fundamentals and methodologies of traffic and transportation theory are as follows:

(i) The Lighthill-Whitham-Richards (LWR) model introduced in 1955–56.[18][24] Daganzo introduced a cell-transmission model (CTM) that is consistent with the LWR model.[25]

(ii) A traffic flow instability that causes a growing wave of a local reduction of the vehicle speed. This classical traffic flow instability was introduced in 1959–61 in the General Motors (GM) car-following model by Herman, Gazis, Montroll, Potts, and Rothery.[26][27] The classical traffic flow instability of the GM model has been incorporated in a huge number of traffic flow models like Gipps's model, Payne's model, Newell's optimal velocity (OV) model, Wiedemann's model, Whitham's model, the Nagel-Schreckenberg (NaSch) cellular automaton (CA) model, Bando et al. OV model, Treiber's IDM, Krauß model, the Aw-Rascle model and many other well-known microscopic and macroscopic traffic-flow models, which are the basis of traffic simulation tools widely used by traffic engineers and researchers (see, e.g., references in review[23]).

(iii) The understanding of highway capacity as a particular value. This understanding of road capacity was probably introduced in 1920–35 (see [28]). Currently, it is assumed that highway capacity of free flow at a highway bottleneck is a stochastic value. However, in accordance with the classical understanding of highway capacity, it is assumed that at a given time instant there can be only one particular value of this stochastic highway capacity (see references in the book[29]).

(iv) Wardrop's user equilibrium (UE) and system optimum (SO) principles for traffic and transportation network optimization and control.[30] –

Kerner explains the failure of the generally accepted classical traffic flow theories as follows:[23]

1. The LWR-theory fails because this theory cannot show empirical induced traffic breakdown observed in real traffic. Correspondingly, all applications of LWR-theory to the description of traffic breakdown at highway bottlenecks (like related applications of Daganzo’s cell-transmission model, cumulative vehicle count curves (N-curves), bottleneck model, highway capacity models as well as associated applications of kinematic wave theory) are also inconsistent with the set of fundamental empirical features of traffic breakdown.

2. Two-phase traffic flow models of the GM model class (see references in[23]) fail because traffic breakdown in the models of the GM class is a phase transition from free flow (F) to a moving jam (J) (called F → J transition): In a traffic flow model belonging to the GM model class due to traffic breakdown, a moving jam(s) appears spontaneously in an initially free flow at a highway bottleneck. In contrast with this model result, real traffic breakdown is a phase transition from free flow (F) to synchronized flow (S) (called F → S transition): Rather than a moving jam(s), due to traffic breakdown in real traffic, synchronized flow occurs whose downstream front is fixed at the bottleneck.

3. The understanding of highway capacity as a particular value (see references in the book [29]) fails because this assumption about the nature of highway capacity contradicts the empirical evidence that traffic breakdown can be induced at a highway bottleneck.

4. Dynamic traffic assignment or/and any kind of traffic optimization and control based on Wardrop's SO or UE principles fail because of possible random transitions between the free flow and synchronized flow at highway bottlenecks. Due to such random transitions, the minimization of travel cost in a traffic network is not possible.

According to Kerner,[23] the inconsistence the generally accepted classical fundamentals and methodologies of traffic and transportation theory with the set of fundamental empirical features of traffic breakdown at a highway bottleneck can explain why network optimization and control approaches based on these fundamentals and methodologies have failed by their applications in the real world. Even several decades of a very intensive effort to improve and validate network optimization models have no success. Indeed, there can be found no examples where on-line implementations of the network optimization models based on these fundamentals and methodologies could reduce congestion in real traffic and transportation networks.

This is due to the fact that the fundamental empirical features of traffic breakdown at highway bottlenecks have been understood only during last 20 years. In contrast, the generally accepted fundamentals and methodologies of traffic and transportation theory have been introduced in the 50s-60s. Thus the scientists whose ideas led to these classical fundamentals and methodologies of traffic and transportation theory could not know the set of empirical features of real traffic breakdown.

The explanation of traffic breakdown at a highway bottleneck by a F → S transition in a metastable free flow at the bottleneck is the basic assumption of Kerner’s three-phase traffic theory.[23] The three-phase traffic theory is consistent with the set of fundamental empirical features of traffic breakdown. None of earlier traffic-flow theories incorporates a F→S transition in a metastable free flow at the bottleneck. Therefore, as above mentioned none of the classical traffic flow theories is consistent with the set of empirical features of real traffic breakdown at a highway bottleneck. The F→S phase transition in metastable free flow at highway bottleneck does explain the empirical evidence of the induced transition from free flow to synchronized flow together with the flow-rate dependence of the breakdown probability. In accordance with the classical book by Kuhn,[31] this shows the incommensurability of three-phase theory and the classical traffic-flow theories (for more details, see [32]):

The minimum highway capacity ${\displaystyle C_{min}}$, at which the F→S phase transition can still be induced at a highway bottleneck as stated in Kerner’s theory, has no sense for other traffic flow theories and models.

The term "incommensurability" has been introduced by Kuhn in his classical book[31] to explain a paradigm shift in a scientific field. The existence of these two traffic phases, free flow (F) and synchronized flow (S) at the same flow rate does not result from the stochastic nature of traffic: Even if there were no stochastic processes in vehicular traffic, the states F and S do exist at the same flow rate. However, classical stochastic approaches to traffic control do not assume a possibility of an F→S phase transition in metastable free flow. For this reason, these stochastic approaches cannot resolve the problem of the inconsistence of classical theories with the set of empirical features of real traffic breakdown.

There're three representations of traffic flow, all these three representations are corresponding to the same surface in the three-dimensional space of vehicle number, position, and time:

• N(t,x): the number of vehicles that have crossed location x by the time t, represented in Eulerian coordinates (t,x).
• X(t,n): the position of vehicle n at the time t, represented in Lagrangian coordinates (t,n).
• T(n,x): the time of vehicle n crosses position x, represented in Lagrangian coordinates (n,x).

Based on the theory of Hamilton-Jacobi equation aforementioned, the solutions (Hopf-Lax formula) of the three models can be represented as:

${\displaystyle N(t,x)=\min _{B\in \beta _{P}}\{G(t_{B},x_{B})+(t-t_{B})R({\frac {x-x_{B}}{t-t_{B}}})\}}$

${\displaystyle X(t,n)=\min _{B\in \beta _{P}}\{G(t_{B},n_{B})+(t-t_{B})R({\frac {n-n_{B}}{t-t_{B}}})\}}$

${\displaystyle T(n,x)=\min _{B\in \beta _{P}}\{G(n_{B},x_{B})+(n-n_{B})R(-{\frac {x-x_{B}}{n-n_{B}}})\}}$

For the N(t,x) model, the Hamilton-Jacobi PDE ${\displaystyle q=F(k)}$ is based on the density-flow fundamental diagram, the Lagrangian function can be represented as ${\displaystyle R({\tilde {v}})=\sup _{k}\{F(k)-k{\tilde {v}}\}}$, in the case of trangle fundamental diagram, ${\displaystyle R({\tilde {v}})=Q-K_{c}{\tilde {v}}}$, ${\displaystyle {\tilde {v}}}$ is the wave speed, ${\displaystyle K_{c}}$ is the critical density, ${\displaystyle Q}$ is the capacity. For the X(t,n) model, the Hamilton-Jacobi PDE ${\displaystyle v=V(s)}$ is based on the spacing-speed fundamental diagram, the Lagrangian function can be represented as ${\displaystyle R({\tilde {q}})=\sup _{s}\{V(s)-s{\tilde {q}}\}}$, in the case of trangle fundamental diagram, ${\displaystyle R({\tilde {q}})=u-S_{c}{\tilde {q}}}$, ${\displaystyle {\tilde {q}}}$ is the wave flow, ${\displaystyle S_{c}}$ is the critical spacing, ${\displaystyle u}$ is the free flow speed. For the T(n,x) model, the Hamilton-Jacobi PDE ${\displaystyle h=H(r)}$ is based on the pace-headway fundamental diagram, the Lagrangian function can be represented as ${\displaystyle R({\tilde {s}})=\inf _{r}\{H(r)-r{\tilde {s}}\}}$, in the case of trangle fundamental diagram, ${\displaystyle R({\tilde {s}})={\frac {1}{Q}}-{\frac {\tilde {s}}{u}}}$, ${\displaystyle {\tilde {s}}}$ is the wave spacing, ${\displaystyle u}$ is the free flow speed, ${\displaystyle Q}$ is the capacity.

Note that ${\displaystyle G(\cdot )}$ of each model is described in the table below:

	Initial Value Problem	Boundary Value Problem
${\displaystyle N(t,x)}$	${\displaystyle N(0,x)}$: cumulative vehicle profile at ${\displaystyle t=0}$	${\displaystyle N(t,0)}$: cumulative cout curve at ${\displaystyle x=0}$
${\displaystyle X(t,n)}$	${\displaystyle X(0,n)}$: position of vehicle n at ${\displaystyle t=0}$	${\displaystyle X(t,0)}$: trajectory of leading vehicle
${\displaystyle T(n,x)}$	${\displaystyle T(0,x)}$: trajectory of leading vehicle	${\displaystyle T(n,0)}$: time for vehicle n enter the road segment

Figure 28.

Figure 29.

Figure 30.

Considering the triangular flow-density fundamental diagram, we can get ${\displaystyle V(s)=min\{u,{\frac {s}{\tau }}-w\}}$ and ${\displaystyle R({\tilde {q}})=u-S_{c}{\tilde {q}}}$, and correspondingly the car-following model can be described via ${\displaystyle X(t,n)}$ model:
${\displaystyle X(t,n)=min\{\min _{y\in \beta _{P}}\{X(0,y)+ut-S_{c}(n-y)\},X(t-n\tau ,0)-n\delta \}}$

where ${\displaystyle \tau }$ is the headway between vehicles, and the bumper-to-bumper spacing at which vehicles are jamed can be derived as ${\displaystyle \delta =-(u\tau -S_{c})}$. The solution of ${\displaystyle X(t,n)}$ can be graphically shown in Figure 28 and Figure 29.

When the constant spacing is applied, the initial data is linear, and the car-following model can be simplified into:
${\displaystyle X(t,n)=min\{X(0,n)+ut,X(t-n\tau ,0)-n\delta \}}$

Then, if we partition the time-space plane into grids of ${\displaystyle (\Delta {t},\Delta {n})}$, and we interpret the origin (0,0) as ${\displaystyle (t-\Delta {t},n-\Delta {n})}$, the general car-following model becomes:
${\displaystyle X(t,n)=\min\{X(t-\Delta {t},n)+u\Delta {t},X(t-\delta {n}\tau ,n-\Delta {n})-\Delta {n}\delta \}}$

The solution can be interpreted intuitively in the time-space diagram of Figure 30: the trajectory vehicle n is the lower envelope between (i) the trajectory of leading vehicle shifting along the characteristics of slope ${\displaystyle w=-{\frac {\Delta {n}\delta }{\Delta {n}\tau }}}$, and (ii) its own trajectory under free-flow conditions.

However, the general car-following model assumes an infinite vehicle acceleration, which is impractical. To compensate for this drawback, we can incorporate a vehicle kinematics model into the car-following model. The vehicle kinematics can be expressed as a linear acceleration model:
${\displaystyle a(v)=\beta (v_{c}-v)}$

in which ${\displaystyle \beta }$ is the acceleration coefficient, ${\displaystyle v_{c}}$ is the desirable speed.

Define ${\displaystyle \xi _{n}(t,v)}$ as the resulting displacement at time ${\displaystyle t}$ for vehicle ${\displaystyle n}$ starting at a speed of ${\displaystyle v}$ at time 0, the general-car following model with acceleration bounds would be:
${\displaystyle X(t,n)=\min\{X(t-\Delta {t},n)+\min\{u\Delta {t},\xi _{n}(\Delta {t},v_{n}(t-\Delta {t}))\},X(t-\delta {n}\tau ,n-\Delta {n})-\Delta {n}\delta \}}$

Recall the general car-following model we obtain from X-model above, Newell's car-following model can be derived via setting ${\displaystyle \Delta {n}=1}$ and ${\displaystyle \Delta {t}=\tau }$:
${\displaystyle X(t,n)=min\{X(t-\tau ,n)+u\tau ,X(t-\tau ,n-1)-\delta \}}$

in which ${\displaystyle X(t-\tau ,n)+u\tau }$ represents vehicle trajectory in the free-flow conditions, and ${\displaystyle X(t-\tau ,n-1)-\delta }$ is the vehicle trajectory in congested conditions.

Some additional explanations and examples can be found in Wikipedia webpage Newell's car-following model.

Louis A. Pipes started researching and gaining acknowledgment from the public in the early 1950s. Pipes car-following model [33] is based on a safe driving rule in the California Motor Vehicle Code, and this model utilized an assumption of safe distance: a good rule for following another vehicle is to allocate an inter-vehicle distance of at least the length of a car for every ten miles per hour of vehicle speed. Mathematically, the safety spacing in Pipes car-following model can be derived as:
${\displaystyle \{x_{i-1}(t)-x_{i}(t)-l_{i-1}\}_{min}=\{s_{i}(t)-l_{i-1}\}_{min}={\frac {{\dot {x}}_{i}}{0.447*10}}l_{i}}$

in which ${\displaystyle s_{i}(t)}$ is the inter-vehicle distance between vehicle ${\displaystyle i}$ and preceding vehicle ${\displaystyle i-1}$, ${\displaystyle x_{i}}$ and ${\displaystyle x_{i-1}}$ is the absolute position of vehicle ${\displaystyle i}$ and vehicle ${\displaystyle i-1}$ respectively, ${\displaystyle {\dot {x}}_{i}}$ is the speed of vehicle ${\displaystyle i}$, ${\displaystyle l_{i}}$ and ${\displaystyle l_{i-1}}$ is the length of vehicle ${\displaystyle i}$ and vehicle ${\displaystyle i-1}$ respectively, ${\displaystyle 0.447}$ is the coefficient of unit conversion from mph to m/s.

More specifically, the safe spacing ${\displaystyle s_{i}(t)_{min}}$ and safe time headway ${\displaystyle h_{i}(t)_{min}}$ in Pipes car-following model can be expressed as:
${\displaystyle s_{i}(t)_{min}={\frac {{\dot {x}}_{i}}{0.447*10}}l_{i}+l_{i-1}}$
${\displaystyle h_{i}(t)_{min}={\frac {l_{i}}{0.447*10}}+{\frac {l_{i-1}}{{\dot {x}}_{i}}}}$

To capture the potential nonlinear effects in the dynamics of car following, G. F. Newell proposed a nonlinear car-following model[34] based on empirical data. Unlike Pipes model which is solely relying on rules of safe driving, Newell nonlinear model aims at capturing the correct shape of fundamental diagrams (e.g., density-speed, flow-speed, density-flow, spacing-speed, pace-headway, etc.). The Newell nonlinear model can be described as:
${\displaystyle {\dot {x}}_{i}(t+\tau _{i})=v_{i}(1-e^{-{\frac {\lambda _{i}}{v_{i}}}(s_{i}(t)-l_{i})})}$

in which ${\displaystyle {\dot {x}}_{i}}$ is the speed of vehicle ${\displaystyle i}$, ${\displaystyle \tau _{i}}$ is the perception-reaction time of driver ${\displaystyle i}$, ${\displaystyle v_{i}}$ is the desirable speed, ${\displaystyle \lambda _{i}}$ is the parameter associated with driver ${\displaystyle i}$, ${\displaystyle s_{i}}$ is the spacing between vehicle ${\displaystyle i}$ and preceding vehicle ${\displaystyle i-1}$, ${\displaystyle l_{i}}$ is the length of vehicle ${\displaystyle i}$.

The Optimal Velocity Model (OVM) is introduced by Bando et al. in 1995 [35] based on the assumption that each driver tries to reach to the optimal velocity according to the inter-vehicle difference and velocity difference between preceding vehicle. In OVM, the acceleration/deceleration of vehicle n is a function of inter-vehicle distance ${\displaystyle \Delta {x_{n}}}$, speed of preceding vehicle ${\displaystyle n-1}$ ${\displaystyle {\dot {x}}_{n-1}}$, and sensitivity coefficient ${\displaystyle \alpha }$ (which represents driver's sensitivity towards acceleration, large value indicates an aggressive driver while small value means a cautious driver):

${\displaystyle {\ddot {x}}_{n}=\alpha \{V(\Delta {x_{n}})-{\dot {x}}_{n-1}\}}$

in which ${\displaystyle V(\cdot )}$ is the Optimal Velocity function (OV-function), it can be expressed as:

${\displaystyle V(\Delta {x_{n}})=tanh(\Delta {x_{n}}-2)+tanh2}$

The OV-function has two following properties:
1. The OV-function is a monotone increasing function.
2. ${\displaystyle \left\vert V(\Delta {x_{n}})\right\vert }$ has an upper-bound: ${\displaystyle V_{max}=V(\Delta {x_{n}}\rightarrow \infty )}$

Intelligent driver model is widely adopted in the research of Connected Vehicle (CV) and Connected and Autonomous Vehicle (CAV). For details about this car-following model, see Wikipedia webpage Intelligent driver model.