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1 Introduction and OV Model

Traffic jams appear on a highway in several places, with different numbers of cars and under different conditions of the road. However, the properties of traffic jams, such as the critical car-density of emergence of a jam, the fundamental diagram (the relation between car-density and flow), the velocity of a cluster of jam, etc. These properties are very common independent of such conditions. We naturally wonder why traffic jams act highly universal. The question seems naive, but the answer is not trivial at all. In this paper we provide a physical and mathematical answer using the exact solution of a traffic flow model, the Optimal Velocity model.

The OV model is a minimal 1-dimensional system of particles with nonlinear asymmetric interactions and a dissipative (viscous) term, which was first introduced as a traffic flow model in [1, 2]. The equation of motion is formulated as

$$\displaystyle{ \frac{d^{2}x_{n}} {\mathit{dt}^{2}} = a\Big(V (\varDelta x_{n}) -\frac{\mathit{dx}_{n}} {\mathit{dt}} \Big), }$$
(1)

where x n is the position of the nth car, and \(\varDelta x_{n} = x_{n+1} - x_{n}\) is the headway distance. a is a control parameter, which dimension is the inverse of time. The OV-function V (Δ x n ) determines the interaction with a car moving ahead. V (Δ x n ) should be a sigmoidal function. The model well describes the emergence of a jam in traffic flow and clearly explains its physical mechanism.

As for the simple case choosing the Heaviside step-function as OV-function, such that V (Δ x) = 0 for Δ x < d, and V (Δ x) = v max for Δ x ≥ d, the emergence of a traffic jam is observed as in Fig. 1.

Fig. 1
figure 1

Plot of all cars, N = 100, for the process of jam formation on a circuit in the OV model with Heaviside step function as OV-function. The initial condition is set as homogeneous flow. The orbit for a car is represented as an example

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The jam flow solution in this model can be obtained analytically based on some hypothesis, which is justified for the case of large N [3]. Moreover, the N-body problem for the jam flow solution is exactly solved for \(N = 2,3,4,\cdots \), in the special case of car-density \(L/N = d\) [4]. Recently, we have obtained the most general jam flow solution for arbitrary N and L.

The profile of a jam flow is described by a limit cycle solution in the phase space \((\varDelta x_{n}, \frac{\mathit{dx}_{n}} {\mathit{dt}} )\), which expresses the two regions in a jam cluster and that cars move smoothly [2]. Profiles of jam flow solutions for several N, L, and car-densities are presented in Fig. 2.

We notice that solutions with the same density L∕N but with different number of cars N, are different. However, these differences seem to be reduced for large N.

Fig. 2
figure 2

Profiles of jam flow solutions (limit cycles) for N = 4, 5, 10, 30 with several densities, \(L/N = d\pm \epsilon\). The horizontal axis is the headway distance Δ x, and the vertical axis is the velocity \(\frac{\mathit{dx}} {\mathit{dt}}\)

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In this paper, using the exact solution for an arbitrary number of cars N and length of circuit L, we discuss the universality of traffic jams.

2 Procedure of Solving the OV Model

The equation of motion Eq. (1) consists of two parts according to Δ x < d or Δ x ≥ d. Each case is easily solved as follows. At the initial condition, x n (t 0), \(\dot{x}_{n}(t_{0})\) for t = t 0, in the case Δ x < d,

$$\displaystyle{ x_{n}(t) = x_{n}(t_{0}) + \frac{\dot{x}_{n}(t_{0})} {a} (1 - e^{-a(t-t_{0})}), }$$
(2)

and in the other case Δ x ≥ d,

$$\displaystyle{ x_{n}(t) = x_{n}(t_{0}) + v_{\mathit{max}}(t - t_{0}) -\frac{v_{\mathit{max}} -\dot{ x}_{n}(t_{0})} {a} (1 - e^{-a(t-t_{0})}) }$$
(3)

The motion of each particle is constructed by changing these two solutions depending on its headway distance Δ x. The important point for obtaining a jam flow solution is how to connect the above two solutions in the appropriate condition.

If we obtain five unknown variables, we can determine the connection condition to build up a jam flow solution for N and L. They are the following: The velocity v RB of a particle at the time when Δ x = d where the solution changes from Eqs. (2) to (3), and the velocity v BR when the solution changes from Eqs. (3) to (2); the time delay τ; the period of the same relative position shifting one number of particle, satisfying \(x_{n}(t+\tau ) = x_{n+1} - v_{c}\tau\); the shift of time beyond τ, denoted by Δ t, that is the difference between the period in Δ x < d and that in Δ x > d; and v c , the velocity of a cluster moving in the opposite direction of the car motion. We can write down five independent equations including the above five unknown variables and derive these variables from the solution.

3 Exact Solution of a Jam Flow

A cluster flow solution for a given arbitrary number of cars, N and circuit length, L, with a density \(L/N = d+\epsilon\), \((\epsilon \geq 0)\) is expressed by using five unknown variables; v RB , v BR , v c , τ, Δ t.Footnote 1 They are obtained as

$$\displaystyle\begin{array}{rcl} v_{\mathit{RB}} = \frac{v_{\mathit{max}}(1 - e^{\frac{N} {2} a(\tau -\varDelta t)})} {(1 + e^{\frac{N} {2} a\tau })(1 - e^{\frac{N} {2} a\tau })},& &{}\end{array}$$
(4)
$$\displaystyle\begin{array}{rcl} v_{\mathit{BR}} = \frac{v_{\mathit{max}}(1 - e^{-\frac{N} {2} a(\tau -\varDelta t)})} {(1 + e^{-\frac{N} {2} a\tau })(1 - e^{-\frac{N} {2} a\tau }),}& &{}\end{array}$$
(5)
$$\displaystyle\begin{array}{rcl} v_{c} = \frac{d -\frac{1} {2}v_{\mathit{max}}\tau } {\tau } + f(\varDelta t;N),& &{}\end{array}$$
(6)
$$\displaystyle\begin{array}{rcl} \frac{a\tau (1 + e^{-\frac{N} {2} a\tau })(1 - e^{-\frac{N} {2} a\tau })} {1 -\frac{e^{-\frac{N}{2} a\tau }(e^{-\frac{N}{2} a\varDelta t}+e^{\frac{N}{2} a\varDelta t})} {2} } = 2(1 - e^{-a\tau }),& &{}\end{array}$$
(7)

where f(Δ t; N) is given as

$$\displaystyle\begin{array}{rcl} f(\varDelta t;N) = \frac{v_{\mathit{max}}} {2a\tau } \frac{e^{\frac{N} {2} a\varDelta t} - e^{-\frac{N} {2} a\varDelta t}} {e^{\frac{N} {2} a\tau } - e^{-\frac{N} {2} a\tau }} (1 - e^{-a\tau }).& &{}\end{array}$$
(8)

The difference of density from the self-dual case \(d = L/N\), denoted by \(\epsilon\), is expressed with Δ t as

$$\displaystyle\begin{array}{rcl} \epsilon = \frac{v_{\mathit{max}}\varDelta t} {2} - f(\varDelta t;N)\tau & &{}\end{array}$$
(9)

4 Rapid Convergence to the Universal Solution for Large N

Fig. 3
figure 3

N-dependence of a τ and a Δ t

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The property of convergence for large N in the formula of the obtained solution is determined by the largest of the two values a τ and a Δ t. Figure 3 shows the relation obtained by solving Eq. (7) numerically for given N, which indicates

$$\displaystyle\begin{array}{rcl} a\tau > a\varDelta t,& &{}\end{array}$$
(10)

for N ≥ 3. Thus, \(f(\varDelta t;N) \rightarrow 0\) for \(N \rightarrow \infty \) in Eq. (8), and Eqs. (4)–(7) are reduced to

$$\displaystyle\begin{array}{rcl} v_{\mathit{RB}} = 0,& &{}\end{array}$$
(11)
$$\displaystyle\begin{array}{rcl} v_{\mathit{BR}} = v_{\mathit{max}},& &{}\end{array}$$
(12)
$$\displaystyle\begin{array}{rcl} v_{c} = \frac{d -\frac{1} {2}v_{\mathit{max}}\tau } {\tau },& &{}\end{array}$$
(13)
$$\displaystyle\begin{array}{rcl} a\tau = 2(1 - e^{-a\tau }),& &{}\end{array}$$
(14)

independent of Δ t or \(\epsilon\), which means independent of the density \(L/N\). The jam flow solution for \(N \rightarrow \infty \) is already known 10 years before [3]. In addition, as for Eq. (9),

$$\displaystyle\begin{array}{rcl} \epsilon = \frac{v_{\mathit{max}}\varDelta t} {2} \,.& &{}\end{array}$$
(15)

Jam flow solutions depending on \(N\) and L converge exponentially, as \(e^{-\frac{N} {2} a\tau }\) and \(e^{-\frac{N} {2} a\varDelta t} \rightarrow 0\), for large N to the universal self-dual solution. They are independent of the car density \(L/N = d\pm \epsilon\) and the number of cars N. Based on these results, we can explain analytically that the motion of every car in a jam flow built with more than several 10 cars shows the same universal profile of the limit cycle as in Fig. 2.

5 Answer of the Question

The property of the rapid convergence to the universal profile of the limit cycle solution for jam flow explains that any jam flow shows the same behavior with the common features independent of the condition and situation on real highways.

The characteristic factor of convergence as \(e^{-\frac{N} {2} a\tau }\) is originated in the dissipative (viscous) term in the basic equation of motion in OV model. This is evidence that phenomena of traffic flow, such as jam flow, follow from the dissipative features in physics.