1 Introduction

In order to understand traffic congestion, many observations have been made and several types of theoretical models have been proposed (see for example [8]). Among the theoretical models, the three-phase traffic theory is widely accepted to explain the mechanism of congestion emergence [4]. For low vehicle density, traffic flow is free with no congestion at all. As the density increases, a free flow state becomes metastable and finally turns to a congested state, which is considered as a kind of phase transition from free-flow phase to congested phase. It is observed that a synchronized flow appears around the phase transition.

Cellular automata (CA’s) models [11] have been employed to explain the behavior of traffic flow [7]. The rule 184 CA (and its continuous analogue, the Burgers equation) is a well-known basic model among them. A distinguishing feature of the model is that we can use analytical solutions to reproduce free flow and propagation of traffic congestion on highways. However, the model cannot capture the existence of a metastable state and dynamical behavior of the phase transition. The important factor in forming jams is the effect of delay, namely the effect that once a vehicle stops, it takes time to start moving again. An empirical evidence of this effect has been provided, for example, in [10]. The CA model which incorporates the effect of delay is known as the CA with slow-to-start property [9]. It can describe the phase transition from free-flow phase to congested phase.

In a previous paper [3], we have proposed the rule 184 fuzzy CA. It is a nonlinear difference equation which includes the rule 184 CA and the Burgers equation as limiting cases. The fundamental diagram (flux-density relation) of this fuzzy CA consists of three parts; a free-flow part, a congested part and a two-periodic part which may correspond to the synchronized flow. All stationary solutions have been analytically obtained and it is also shown that the congested and the two-periodic solutions are stable, while the free-flow solution is unstable. However, the phase transition from free-flow phase to congested phase can not be described by this model.

In this paper, we generalize the rule 184 fuzzy CA to possess slow-to-start property. Since empirical data indicate that the phase transition takes place when the density of vehicles exceeds about 20% of the full density [2, 7]. This value is different from the theoretical result 1/3 observed by Takayasu and Takayasu for the CA model with slow-to-start property [9]. It is our main concern whether the new fuzzy CA model can analytically predict a closer value of the critical density.

The rest of this paper consists of four sections. In Sect. 2, a new fuzzy CA with slow-to-start property is presented and its basic properties are discussed. In Sect. 3, we give exact stationary solutions which correspond to the free-flow and the congested states with periodic boundary condition for the new model. The statistical quantities such as average densities and average fluxes of both flows are also given. In Sect. 4, we study both numerically and analytically the stabilitiy of the congested solutions and some typical solutions of the free-flow states. We show that the congested state becomes unstable when the mean density falls below a certain critical value about 20%, which is a main result of the present work. To our best knowledge, this is the first analytical result on the stability of congested state of the models with slow-to-start property. Section 5 is devoted to concluding remarks.

2 Fuzzy CA with slow-to-start property

We consider a multi-lane road with many vehicles and divide it into a one-dimensional array of sections by appropriate distance \(\delta \). We assume that the maximum number of vehicles is constant at all sections, and examine the ratio of vehicles in each section. The vehicles are classified into two groups according to their running condition. One is the group of vehicles which have just come into the section or can move to the next section in one-time step. The other is the group of vehicles which are stopped and wait for at least one-time step before they can move. Let the ratio of the former in the interval \([n\delta ,(n+1)\delta ]\) at time t be \(u^t_n\) and the ratio of the latter be \(v^t_n\). They satisfy the inequalities; \(0 \le u^t_n, v^t_n, u^t_n+v^t_n \le 1\) for all n and t.

First, we consider the dynamics among three adjacent sections. From the equation of continuity, we have

$$\begin{aligned} u^{t+1}_{n} + v^{t+1}_{n} - (u^{t}_{n} + v^{t}_{n}) = Q^t_n - Q^t_{n+1}, \end{aligned}$$
(2.1)

where \(Q^t_n\) is the flux of vehicles from the section \([(n-1)\delta , n\delta ]\) to \([n \delta , (n+1) \delta ]\) at time t. As observed empirically, the flux \(Q^t_n\) is approximately a linearly decreasing function of \(u^t_{n}+v^t_{n}\) and proportional to the ratio \(u^t_{n-1}\) [1, 5]. Hence we may assume

$$\begin{aligned} Q^t_n := (1-u^t_{n}-v^t_{n})u^t_{n-1}. \end{aligned}$$
(2.2)

From (2.1) and (2.2), we get

$$\begin{aligned} u^{t+1}_{n} + v^{t+1}_{n} - (u^{t}_{n} + v^{t}_{n}) = (1-u^t_{n}-v^t_{n})u^t_{n-1} - (1-u^t_{n+1}-v^t_{n+1})u^t_{n}. \end{aligned}$$
(2.3)

If all vehicles can move at any time, that is \(v^t_n \equiv 0\) for all n and t, then (2.3) becomes

$$\begin{aligned} u^{t+1}_n = (1-u^{t}_n)u^{t}_{n-1} + u^{t}_{n}u^{t}_{n+1}, \end{aligned}$$
(2.4)

which is the rule 184 fuzzy CA, and all its stationary solutions as well as their stability have been investigated in [3].

We now consider the change of v and u inside the section \([n\delta , (n+1)\delta ]\). Since \(v^t_n\) denotes the ratio of stopped vehicles, its temporal change is given by

$$\begin{aligned} v^{t+1}_{n} - v^{t}_{n}= & {} +\, \text {(the amount of change from } u^t_n\text { to }v^{t+1}_n)\nonumber \\{} & {} -\, \text {(the amount of change from }v^t_n\text { to }u^{t+1}_n). \end{aligned}$$
(2.5)

A vehicle belonging to \(u_n^t\) will stop if there is a vehicle in front of it. Hence the first term of the right-hand side of (2.5) may be assumed to be the product of \(u_n^t\) and the ratio of occupied space in front of the section \([n\delta , (n+1)\delta ]\) as

$$\begin{aligned} (u^{t}_{n+1} + v^{t}_{n+1})u^{t}_{n}. \end{aligned}$$
(2.6)

Similarly, for the second term of the right-hand side of (2.5), we may assume that the term is equal to the product of \(v^t_n\) and the ratio of vacant space, that is,

$$\begin{aligned} (1 - u^{t}_{n+1} - v^{t}_{n+1}) v^{t}_{n}. \end{aligned}$$
(2.7)

From (2.5), (2.6) and (2.7), we obtain

$$\begin{aligned} v^{t+1}_{n} = (u^{t}_{n+1} + v^{t}_{n+1})(u^{t}_{n} + v^{t}_{n}). \end{aligned}$$
(2.8)

Thus, from (2.3) and (2.8), we finally obtain a discrete dynamical system

figure a

This system of nonlinear difference equations (2.9), (2.10) is called the fuzzy CA with slow-to-start property, the analytical properties of which we study in the following sections. Note that the system retains fuzzy property, i.e., \(0\le u_n^t, v_n^t\le 1\) for any n and t when it satisfies the initial condition \(0\le u_n^0, v_n^0, (1-u_n^0-v_n^0) \le 1\) for any n. This fact is easily proved by mathematical induction; for a time step t, if \(0\le u_n^t, v_n^t, (1-u_n^t-v_n^t) \le 1\) for any n, then clearly \(u_n^{t+1}, v_n^{t+1} \ge 0\), and

$$\begin{aligned} u^{t+1}_n + v^{t+1}_n&= v^t_n + (u^t_{n+1} + v^t_{n+1})u^t_{n} + (1 - u^t_n -v^t_n)u^t_{n-1} \\&\le v_n^t+u_n^t+ (1 - u^t_n -v^t_n)=1. \end{aligned}$$

Furthermore, if we impose the initial condition as \(u^{t=0}_n, v^{t=0}_n, u^{t=0}_n + v^{t=0}_n \in \{ 0,1 \}\) for any n, then the system (2.9), (2.10) turns into the CA with slow-to-start property given in [9]. To see this coincidence, we note that the reduced system is a one-dimensional three-states CA, and a state of a cell is updated with those of its adjacent two cells and itself at the previous time step. There are 27 possible updating maps from \(\{0,1,1'\}^3\) to \(\{0,1,1'\}\), which are listed below.

$$\begin{aligned} \begin{array}{ccccccccc} \frac{000}{0}&{}\frac{001'}{0}&{}\frac{001}{0}&{}\frac{01'0}{1}&{}\frac{01'1'}{1'}&{}\frac{01'1}{1'}&{}\frac{010}{0}&{}\frac{011'}{1'}&{}\frac{011}{1'} \\[5pt] \frac{1'00}{\times 0}&{}\frac{1'01'}{0}&{}\frac{1'01}{0}&{}\frac{1'1'0}{1}&{}\frac{1'1'1'}{1'}&{}\frac{1'1'1}{1'}&{}\frac{1'10}{0}&{}\frac{1'11'}{\times 1'}&{}\frac{1'11}{\times 1'}\\[5pt] \frac{100}{1}&{}\frac{101'}{1}&{}\frac{101}{1}&{}\frac{11'0}{1}&{}\frac{11'1'}{1'}&{}\frac{11'1}{1'}&{}\frac{110}{0}&{}\frac{111'}{\times 1'}&{}\frac{111}{\times 1'} \end{array} \end{aligned}$$

Among these, the configurations marked with a symbol \(\times \) do not occur in time evolution of the CA. If we make the following correspondence

$$\begin{aligned}&1 \leftrightarrow u=1,v=0, \nonumber \\&1' \leftrightarrow u=0,v=1, \nonumber \\&0 \leftrightarrow u=v=0 \nonumber , \end{aligned}$$

then we find that (2.9), (2.10) give the same updating rule as the CA with slow-to-start property.

3 Stationary solutions, average density and flux

Statistical properties of traffic phenomena are mostly investigated by the fundamental diagram which gives the relation between density and flux of vehicles. In order to establish the fundamental diagram of the system (2.9) and (2.10), we adopt a periodic boundary condition in which the total number of vehicles does not change. That is,

$$\begin{aligned} \forall n \;\; \forall t \;\; u^t_n = u^t_{n+N},\;\;v^t_n = v^t_{n+N} \end{aligned}$$
(3.1)

for a fixed N. The average density s is defined by

$$\begin{aligned} s := \frac{1}{N}\sum _{n=1}^N (u^t_n + v^t_n), \end{aligned}$$
(3.2)

which is constant in time due to the periodic boundary condition. Moreover, the average flux at time t, \(Q^t\), is defined by

$$\begin{aligned} Q^t := \frac{1}{N} \sum _{n=1}^N (1 - u^t_n - v^t_n) u^t_{n-1}. \end{aligned}$$
(3.3)

We now study stationary solutions corresponding to traffic phases. There are at least two of them; a free-flow solution corresponding to a free-flow state and a congested solution corresponding to a congested state.

Let us put \(u^t_n = \sigma _{n-t}, v^t_n = \tau _{n-t}\) to find a free-flow solution.

Since

figure b

\(\{\sigma _n\},\{\tau _n\}\) give a stationary solution if the following condition is satisfied;

$$\begin{aligned} \forall n \;\; \sigma _{n}\sigma _{n+1}=0, \tau _{n} =0. \end{aligned}$$
(3.6)

This stationary solution represents the situation that each vehicle moves keeping sufficient distance from others. Hence, this is the free-flow solution. Its average flux is given by

$$\begin{aligned} Q(s) = \frac{1}{N}\sum _{n=1}^N u^t_n = s. \end{aligned}$$
(3.7)

Since there is no free-flow solution for \(s>\frac{1}{2}\) from the definition of s and (3.6), the average flux Q(s) is only defined for \(0 \le s \le \frac{1}{2}\).

Next let us put \(u^t_n \equiv q, v^t_n \equiv r\), where \(q,r \in [0,1]\) are constants to find a congested solution. From (3.4) and (3.5), we have

figure c

Substituting \(s=q+r\) into (3.8) and (3.9), we find

$$\begin{aligned} q = (1-s)s, \quad r = s^2. \end{aligned}$$
(3.10)

This solution corresponds to congested state, since inflow and outflow are balanced in any section. The average flow is given by

$$\begin{aligned} Q(s)&= \frac{1}{N} \sum _{n=1}^N \{1 - s(1-s) - s^2\}(1-s)s \nonumber \\&= s(1-s)^2 \;\;\;\; 0 \le s \le 1. \end{aligned}$$
(3.11)

Since \(s(1-s)^2<s \), the average flow given by (3.11) is smaller than that of the free-flow state.

4 Linear stability of the congested, the free-flow state and phase transition

We now investigate the density region where the congested phase does not exist. For this purpose, we first define a critical density \(s_1\) at which a congested state becomes unstable if the density is less than \(s_1\). Since \(s_1\) is the density at which the uniform solution representing the congested state becomes unstable, let us examine the linear stability of the uniform solution.

Applying small perturbation \(\varepsilon ^t_n,\nu ^t_n\) to the uniform solution, we obtain

$$\begin{aligned} u^t_n = (1-s)s + \varepsilon ^t_n, v^t_n = s^2 + \nu ^t_n, \end{aligned}$$
(4.1)

where \(\sum \nolimits _{n=1}^N (\varepsilon ^t_n + \nu ^t_n)=0\) due to the conservation of the total density of vehicles. Equations (2.9) and (2.10) are linearized by neglecting terms more than the quadratic order. We have

figure d

Taking

$$\begin{aligned} \varepsilon ^t_n = C_1\lambda ^t \text{ e}^{\text{ i }k_mn},\;\; \nu ^t_n = C_2\lambda ^t \text{ e}^{\text{ i }k_mn}, \end{aligned}$$
(4.4)

where \(\lambda ,C_1,C_2 \in \mathbb {C}\), and \(k_m:= \frac{2\,m\pi }{N}\;\;(m \in \mathbb {Z})\), we obtain

(4.5)

Since (4.5) holds for any n, it is sufficient to evaluate eigenvalues only for the \(2 \times 2\) matrix. The condition that the characteristic equation

$$\begin{aligned} \lambda ^2 - \{(1-s^2)\cos k_m + s^2 - \text{ i }(1-s)^2\sin k_m\}\lambda - 2\text{ i }s(1-s)\sin k_m = 0 \end{aligned}$$
(4.6)

has a root whose norm is greater than unity is

$$\begin{aligned} f(s,k_m):= & {} 4 - 11 s + 15 s^2 - 10 s^3 + 6 s^5 - 2 s^6\nonumber \\{} & {} - \,2 (-3 + 8 s - 6 s^2 + 2 s^3 - 2 s^5 + s^6) \cos k_m \nonumber \\{} & {} +\, 2(1 - 2 s - 2 s^2 + 5 s^3 - 3 s^5 + s^6) \cos 2k_m \nonumber \\{} & {} - \,2 s^2 (1 - 2 s + 2 s^3 - s^4)\cos 3 k_m < 0. \end{aligned}$$
(4.7)

In the limit of \(N \rightarrow \infty \), the wave number \(k_m\) takes continuous value in \([0,2\pi ]\). Setting \(t:=\cos k\) (\(k \in [0,2\pi ]\)) and \(F(s,t):=f(s,k)\), the implicit equation for the boundary curve which satisfies \(f(s,k)=0\) in st-plane is given by

$$\begin{aligned} F(s,t)= & {} 2 - 7 s + 19 s^2 - 20 s^3 + 12 s^5 - 4 s^6 \nonumber \\{} & {} +\, (6 - 16 s + 18 s^2 - 16 s^3 + 16 s^5 - 8 s^6) t \nonumber \\{} & {} +\, (4 - 8 s - 8 s^2 + 20 s^3 - 12 s^5 + 4 s^6) t^2 \nonumber \\{} & {} +\, (-8 s^2 + 16 s^3 - 16 s^5 + 8 s^6) t^3=0. \end{aligned}$$
(4.8)

From implicit function theorem, a maximal point \(t^*\) satisfies \(\frac{ds}{dt}=-\frac{F_t}{F_s}=0\), namely

$$\begin{aligned}&3(4s^6 - 8s^5 + 8s^3 - 4s^2){t^*}^2 \nonumber \\&+(4s^6 -12s^5 + 20s^3 -8s^2 -8s +4){t^*} \nonumber \\&-8s^6 + 16s^5 -16s^3 + 18s^2 -16s +6 = 0. \end{aligned}$$
(4.9)

If the wave number k and the density s take values inside the curve determined by (4.8), the norm of eigenvalue is greater than unity. Solving the quadratic equation, we can evaluate the magnitude of the critical point \(s_1\) which changes the stability (see Fig. 2). The value \(s_1\) is evaluated as 0.21 which is close to the value reported in [2]. Figure 1 is the fundamental diagram which represents flow-density relation for the free-flow and the congested flow. The straight line and the curved line represent the free flow \(Q=s\) and the congested flow \(Q=s(1-s)^2\), respectively. The dotted line represents the unstable region of the congested flow.

Next, we discuss the stability of free-flow solutions. In Eqs. (2.9) and (2.10), converting to a system moving at velocity 1 yields the following equations:

figure e

Applying small perturbation \(\varepsilon ^t_n,\nu ^t_n\) to free-flow solution:

figure f

where \(\forall n ~~ \alpha _n \alpha _{n+1} = 0 \), we have the linearized equations:

figure g

Since there are infinitely many free-flow solutions for a given average density and it is difficult to treat them all, we consider the two typical cases where the states have a similar forms to those of CA models.

  • Case 1    \(N=2M\), \(\ (\alpha _n)_{n=1}^{2\,M}=(\alpha ,0,\alpha ,0,\ldots ,\alpha ,0)\) (\(0<\alpha \le 1\))

  • Case 2    \(N=3M\), \(\ (\alpha _n)_{n=1}^{3\,M}=(0,0,\beta ,0,0,\beta ,\ldots , 0,0,\beta )\) (\(0<\beta \le 1\))

Case 1 corresponds to a free-flow state of the Rule 184 CA, while case 2 corresponds to that of the slow-to-start CA.

For case 1, where \(\alpha _{2m}=0\) and \(\alpha _{2m-1}=\alpha \) (\(m=1,2,\ldots \)), the equations derived from (4.14) and (4.15) are as follows:

figure h

If we put

figure i

where \(k=\frac{2m \pi }{M}\) (\(m=0,1,2,\ldots ,M\)), then, we have

figure j

The characteristic equation for the \(4 \times 4\) matrix

$$\begin{aligned} \begin{pmatrix} 1&{}-\alpha &{}0&{}1-2\alpha \\ 0&{}1-\alpha &{}\text{ e}^{-\text{ i }k}&{}0\\ 0&{}\alpha &{}0&{}\alpha \\ 0&{}\alpha \text{ e}^{-\text{ i }k} &{} 0&{}\alpha \text{ e}^{-\text{ i }k}\\ \end{pmatrix} \end{aligned}$$
(4.20)

is calculated as

$$\begin{aligned} \lambda (\lambda -1)\left( \lambda ^2-(\alpha \text{ e}^{-\text{ i }k}-\alpha +1)\lambda -\alpha ^2 \text{ e}^{-\text{ i }k}\right) =0. \end{aligned}$$
(4.21)

For \(k=0\),

$$\begin{aligned} \lambda ^2-(\alpha \text{ e}^{-\text{ i }k}-\alpha +1)\lambda -\alpha ^2 \text{ e}^{-\text{ i }k}= \lambda ^2-\lambda -\alpha ^2=0 \end{aligned}$$

which has a root \(\lambda =\frac{1+\sqrt{1+4\alpha ^2}}{2}>1\). Hence, the state is linearly unstable.

Similarly, in case 2, by defining

figure k

for \(j=0,1,2\), we obtain the linearized equations:

figure l

The characteristic equation is given as

$$\begin{aligned} \lambda (\lambda -1)(\lambda -1+\beta )(\lambda ^3-\lambda ^2-\text{ e}^{-\text{ i }k}\beta \lambda +\text{ e}^{-\text{ i }k}\beta ^2)=0. \end{aligned}$$
(4.28)

For \(k=0\),

$$\begin{aligned} \lambda ^3-\lambda ^2-\text{ e}^{-\text{ i }k}\beta \lambda +\text{ e}^{-\text{ i }k}\beta ^2 =\lambda ^3-\lambda ^2-\beta \lambda +\beta ^2=0. \end{aligned}$$
(4.29)

Putting

$$\begin{aligned} f(\lambda ):=\lambda ^3-\lambda ^2-\beta \lambda +\beta ^2, \end{aligned}$$

we have

$$\begin{aligned} f(1)=-\beta +\beta ^2\le 0,\qquad f(+\infty )=+\infty , \end{aligned}$$

and there is a root \(\lambda > 1\) except for \(\beta =1\). In case \(\beta =1\), the roots of the characteristic equation is \(\lambda =0,\,1,\,\text{ e}^{\pm \text{ i }k/2}\). Hence we find that a state of case 2 is linearly unstable for \(0<\beta <1\). Since \(0 < \beta \le 1\), a small perturbation reduces the value of \(\beta \), therefore we consider that a state is unstable for \(\beta =1\) too.

Now let us discuss numerical results of the present model. Figure 3a–c show temporal change of density \(u^t_n\), where (a) free-flow, (b) intermediate, and (c) congested, respectively. Each initial state is chosen so that it is a free-flow state under small perturbation. Precisely speaking, for a given density s, we consider a free-flow state with sN pairs of 0 and 1. The initial state is the state where 0 is increased to 0.1 and 1 is reduced to 0.9 for all the sN pairs in this free-flow state. In the free-flow state (a), regions of high and low density appear alternately and they move at almost constant velocity. On the other hand, in the congested state (c), the density is uniform and the flow is rather small. The state (b) exhibits intermediate behavior between these two states and shows temporal and spatial periodicity which is observed in synchronized flow.

The temporal changes of average flows of two perturbed free-flow states with average densities (i) \(s < s_1~(s = 0.10)\) and (ii) \(s_1 < s~(s=0.22)\) are displayed in Fig. 4. The perturbations are given in the same way as described in Fig. 3. In case (i), after a long time steps, the flow convergences to an almost nearly free-flow state. It implies the existence of other stable, nearly free-flow states for \(s<s_1\). On the other hand, in case (ii), the flow converges to a congested state via oscillations which may be considered a synchronized flow.

Fig. 1
figure 1

Fundamental diagram with flow Q on the vertical axis and density s on the horizontal axis. Straight line and curvy lines represent the free flow \(Q=s\) and the congested flow \(Q=s(1-s)^2\), respectively. Dotted line represents the unstable region of the congested flow

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Fig. 2
figure 2

The congested solution becomes unstable in the hatched region. The horizontal axis is average density s and the vertical axis is wave number k

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Fig. 3
figure 3

Examples of time evolution of \(u^t_n\) under perturbations

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Fig. 4
figure 4

Examples of temporal changes of free-flows under perturbation. In i, \(t_0 =50,\, t_1=200,\, t_2=300,\, t_3=1000, \,t_4=2000\), and in ii \(t_0 =2, \, t_1=3,\, t_2=5,\, t_3=7, \, t_4=300\)

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5 Concluding remarks

In this paper, we have proposed a model which incorporates the effects of slow-to-start property into rule 184 fuzzy CA. The exact solutions corresponding to the free-flow and the congested states are given for the new model. By studying the stability of the uniform solution corresponding to the congested state, the critical point at which congestion does not occur is obtained analytically. This may be the first analytical result on the stability of congested state of the models with slow-to-start property.

We study the stability of some typical free-flow states. As discussed in Chapter 4, they are unstable, but numerical simulations suggest the existence of other stable states which are similar to free-flow states. We have numerically found several exact periodic solutions which exhibit oscillation in fundamental diagrams, though we have not reported them in this paper. The analysis of these solutions as well as the stability of general free-flow solutions are the problems we wish to address in the future.

After this work has completed, we noticed that Nishida et al. had made an equivalent fuzzification [6]. Our results imply that their method of fuzzification is also useful to examine analytical study of CA models. We would like to thank the reviewer for notifying us of the existence of this paper.