Failure of classical traffic flow theories: a critical review

Abstract

We explain that the fundamental empirical basis for automatic driving, reliable control and optimization of traffic and transportation networks is the set of empirical features of traffic breakdown at a road bottleneck. We show why generally accepted traffic and transportation theories and models are not consistent with this empirical fundament of traffic science. In particular, these classical traffic theories are as follows:

• (i) the Lighthill-Whitham-Richards (LWR) theory and traffic flow models in the framework of the LWR theory (for example, Daganzo’s cell transmission model) that explain traffic breakdown through a fundamental diagram of traffic flow,

• (ii) General Motors (GM) class of traffic-flow models that explain traffic breakdown through traffic flow instability due to a driver reaction time (for example, the following well-known models belong to the GM model class: Gipps’s model, Payne’s model, Newell’s optimal velocity (OV) model, Wiedemann’s model (VISSIM traffic simulation tool), Bando et al. OV model, Treiber’s Intelligent Driver Model, Krauß model (SUMO tool), the Aw-Rascle model),

• (iii) the classical understanding of stochastic highway capacity, and

• (iv) Wardrop’s principles for dynamic control, assignment, and optimization of traffic and transportation networks.

In turn, this can explain why dynamics network optimization and control approaches based on these classical traffic flow theories failed by their applications in the real world. We discuss why rather that the assumption about the existence of stochastic highway capacity, at any time instant there should be the infinite number of highway capacities within a range of the flow rate between a minimum capacity and a maximum capacity as assumed in three-phase theory introduced by the author. Because the assumption about the infinite number of highway capacities is consistent with the set of the fundamental empirical features of traffic breakdown at highway bottlenecks, this can be considered a theoretical fundament for the development of reliable automatic driving, control and optimization of vehicular traffic and transportation networks. We discuss briefly some features of the three-phase theory explaining the empirical fundament of transportation science.

Zusammenfassung

Wir zeigen, dass das Set der empirischen Eigenschaften eines Verkehrszusammenbruchs an einer Engstelle der Straße die fundamentale empirische Basis für automatisiertes Fahren, für zuverlässige Kontrolle und Optimierung von Verkehrs- und Transportnetzen darstellt. Wir zeigen, warum allgemein akzeptierte Theorien und Modelle des Straßenverkehrs und des Transports nicht mit diesem empirischen Fundament der Verkehrswissenschaften übereinstimmen. Im Einzelnen sind dies die folgenden klassischen Verkehrstheorien:

• (i) die Lighthill-Witham-Richards(LWR)-Theorie und Verkehrsfluss-Modelle im Rahmen der LWR-Theorie (zum Beispiel Daganzos Cell Transmission-Modell), die Verkehrszusammenbrüche durch ein Fundamentaldiagramm des Verkehrsflusses erklären,

• (ii) die Klasse der General-Motors(GM)-Verkehrsflussmodelle, die Verkehrszusammenbrüche durch eine Instabilität des Verkehrsflusses aufgrund der Reaktionszeit des Fahrers erklären (zum Beispiel gehören die folgenden bekannten Modelle zur Klasse der GM-Modelle: Gipps-Modell, Payne-Modell, Newells Optimal Velocity(OV)-Modell, Wiedemann-Modell (Verkehrssimulations-Programm VISSIM), das OV-Modell von Bando et al., Treibers Intelligent Driver-Modell, Krauß-Modell (Verkehrssimulationsprogramm SUMO), das Aw-Rascle-Modell),

• (iii) das klassische Verständnis der stochastischen Kapazität einer Straße,

• (iv) das Wardrop-Prinzip für die dynamische Kontrolle, dynamische Verkehrsumlegung und Optimierung von Verkehrs- und Transportnetzwerken.

Aus dem Scheitern der klassischen Verkehrstheorien folgt, warum auch die Ansätze zur Optimierung und Kontrolle von dynamischen Netzwerken, die auf diesen klassischen Verkehrsflusstheorien basieren, in den Anwendungen der realen Welt scheitern. Wir diskutieren, warum es anstelle der Annahme der Existenz eines bestimmten Wertes der stochastischen Kapazität einer Straße zu jedem Zeitpunkt stattdessen zu jedem Zeitpunkt eine unendliche Anzahl von Kapazitäten gibt, die in einem Bereich des Flusses zwischen einer minimalen und einer maximalen Kapazität liegen, so wie in der Drei-Phasen-Theorie angenommen wird. Weil die Annahme der Existenz einer unendlichen Anzahl von Kapazitäten einer Straße zu jedem Zeitpunkt in Übereinstimmung mit dem Set der fundamentalen empirischen Eigenschaften eines Verkehrszusammenbruchs an einer Engstelle der Straße steht, kann dies als theoretisches Fundament angesehen werden für die Entwicklung von zuverlässigem automatisierten Fahren, von Kontrolle und Optimierung des Fahrzeugverkehrs und von Transportnetzwerken. Wir diskutieren kurz einige Eigenschaften der Drei-Phasen-Verkehrstheorie, die das empirische Fundament der Verkehrswissenschaften erklären.

Appendices

Appendix A: The classical understanding of the nature of stochastic highway capacity

To illustrate the above critical conclusion about the generally accepted fundamentals and methodologies of traffic and transportation theory (Sects. 2 and 5), we discuss here the criticism of the generally accepted understanding of stochastic highway capacity of free flow at a highway bottleneck [1, 32].

The classical highway capacity is defined through the occurrence of traffic breakdown at a bottleneck: The highway capacity is equal to the flow rate in an initially free flow at the bottleneck at which traffic breakdown is observed at the bottleneck (see e.g., [12, 15, 16, 20–29]).

During last 20 years it was found that empirical traffic breakdown exhibits a probabilistic character and the probability of the spontaneous breakdown is an increasing flow rate function $P^{(B)}(q_{\mathrm{sum}})$, where $q_{\mathrm{sum}}$ is the flow rate in an initially free flow at the bottleneck (see references in the book [20]). Respectively, Brilon has introduced the following concept for stochastic highway capacity [27–29].

In accordance with the classical capacity definition, Brilon’s stochastic highway capacity $C$ is equal to the flow rate $q_{\mathrm{sum}}$ at the bottleneck. At any time instant, there is a particular value of stochastic capacity of free flow at the bottleneck.

However, as long as free flow is observed at the bottleneck, this particular value of stochastic capacity cannot be measured. Therefore, stochastic capacity is defined through a capacity distribution function $F_{C}^{(B)}(q_{\mathrm{sum}})$ [27–29]:

$$ F_{C}^{(B)}(q_{\mathrm{sum}}) = p(C \le q_{\mathrm{sum}}), $$

(8)

where $p(C \le q_{\mathrm{sum}})$ is the probability that stochastic highway capacity $C$ is equal to or smaller than the flow rate $q_{\mathrm{sum}}$ in free flow at the bottleneck.

Thus the basic theoretical assumption of the classical understanding of stochastic highway capacity is that traffic breakdown is observed at a time instant t at which the flow rate $q_{\mathrm{sum}}(t)$ reaches the capacity $C(t)$. This means that the flow rate function of the probability of traffic breakdown $P^{(B)}(q_{\mathrm{sum}})$ should be determined by the capacity distribution function $F_{C}^{(B)}(q_{\mathrm{sum}})$ [27–29]:

$$ P^{(B)}(q_{\mathrm{sum}}) = F_{C}^{(B)}(q_{\mathrm{sum}}). $$

(9)

It must be noted that the breakdown probability function $P^{(B)}(q_{\mathrm{sum}})$ found in empirical observations [20, 25–29] is the empirical evidence. However, condition (9) is a theoretical hypothesis only. This is because in contrast with the breakdown probability function $P^{(B)}(q_{\mathrm{sum}})$, the capacity distribution function $F_{C}^{(B)}(q_{\mathrm{sum}})$ cannot be measured. Below we explain why the hypothesis (9) and, therefore, Brilon’s stochastic capacity contradicts the set of fundamental empirical features of traffic breakdown.

This understanding of stochastic capacity of free flow at a bottleneck, which is currently well accepted in the community of traffic and transportation researchers [20], is illustrated in Fig. 8.

Fig. 8.

Qualitative explanation of Brilon’s stochastic highway capacity of free flow at a highway bottleneck. The flow rate function of the probability of the spontaneous breakdown $P^{(B)}(q_{\mathrm{sum}})$ (left figure) is the same as that shown in Fig. 4(a)

In Fig. 8, we show a qualitative hypothetical fragment of the time-dependence of stochastic capacity $C(t)$ over time $t$. Left in Fig. 8, a qualitative flow rate dependence of the probability of spontaneous traffic breakdown $P^{(B)}(q_{\mathrm{sum}})$ is shown that is the same as that in Fig. 4(a). In accordance with Eq. (9), capacity $C(t)$ can stochastically change over time (Fig. 8).

It is often assumed that a stochastic behavior of highway capacity is associated with a stochastic change in traffic parameters over time [20, 27–29]. Examples of the traffic parameters, which can indeed be stochastic time-functions in real traffic, are weather, mean driver’s characteristics (e.g., mean driver reaction time), share of long vehicles, etc.

In accordance with the definition of stochastic capacity (8), (9), no traffic breakdown can occur, when the time dependence of the flow rate is given by a hypothetical time dependence $q_{\mathrm{sum}}(t) = q_{\mathrm{sum}}^{(1)}(t)$. This is because at all-time instants $q_{\mathrm{sum}}^{(1)}(t) < C(t)$ (Fig. 9(a)).

Fig. 9.

Qualitative explanation of traffic breakdown with the use of Brilon’s stochastic highway capacity of free flow at a highway bottleneck. The fragment of the hypothetical time-function of Brilon’s stochastic highway capacity $C(t)$ is taken from Fig. 8

In contrast, for another hypothetical time dependence $q_{\mathrm{sum}}(t) = q_{\mathrm{sum}}^{(2)}(t)$ traffic breakdown should occur at time instant $t_{1}$ at which $q_{\mathrm{sum}}^{(2)}(t_{1}) = C(t_{1})$, i.e., this flow rate is equal to the capacity value (Fig. 9(b)).

In other words, the classical understanding of a particular value of stochastic capacity can be explained as follows: At a given time instant no traffic breakdown can occur at a highway bottleneck if the flow rate in free flow at the bottleneck at the time instant is smaller than the value of the capacity at this time instant.

The basic importance of the words “at a given time instant” in the capacity definition is as follows: Brilon’s stochastic capacity $C(t)$ changes stochastically over time (Fig. 8). Thus at a given time instant traffic breakdown can occur at the flow rate that is smaller than the value of the stochastic capacity was at another time instant.

In the classical understanding of stochastic capacity, free flow is stable under condition $q_{\mathrm{sum}}(t) < C(t)$. This means that $\mathit{no}$ traffic breakdown can occur or be induced at the bottleneck at long as the flow rate in free flow at the bottleneck is smaller than the stochastic capacity. This contradicts to the empirical fact that traffic breakdown can be induced at the bottleneck due to the upstream propagation of a localized congested pattern (Fig. 2(b)).

This is because stochastic highway capacity cannot depend on whether there is a congested pattern, which has occurred outside of the bottleneck and independent of the bottleneck existence, or not. Indeed, the empirical evidence of induced traffic breakdown is the empirical proof that at a given flow rate at a bottleneck there can be one of two different traffic states at the bottleneck: (i) A traffic state related to free flow and (ii) a congested traffic state labeled as synchronized flow in Fig. 2(b). Due to the upstream propagation of a localized congested pattern, a transition from the state of free flow to the state of synchronized flow, i.e., traffic breakdown is induced (see Appendix B).

The induced traffic breakdown is impossible to occur under the classical understanding of the nature of highway capacity [12, 20, 27–29]. This is because in this classical understanding of highway capacity, free flow is stable under condition $q_{\mathrm{sum}}(t) < C(t)$, i.e., no traffic breakdown can occur (Fig. 9(a)).

In contrast with this classical understanding of the nature of highway capacity, the evidence of the empirical induced breakdown means that free flow is in a metastable state with respect to the breakdown. The metastability of free flow at the bottleneck should exist for all flow rates at which traffic breakdown can be induced at the bottleneck. This empirical evidence of the metastability of free flow at the bottleneck contradicts fundamentally the concept of Brilon’s stochastic capacity, in which free flow is stable under condition $q_{\mathrm{sum}}(t) < C(t)$.

Thus the currently accepted understanding of stochastic highway capacity [20, 27–29] failed because this understanding about the nature of highway capacity contradicts the empirical evidence that traffic breakdown can be induced at a highway bottleneck as observed in real traffic (Fig. 2(b)) (see also Appendix B).

Appendix B: Empirical induced breakdown—empirical proof of the metastability of free flow at highway bottlenecks

In this appendix, following empirical studies of traffic breakdown presented in [1, 32–34], we show the special importance of the following two empirical features of traffic breakdown at a highway bottleneck:

• i. The downstream front of congested traffic resulting from the breakdown is usually fixed at the bottleneck location; as above-mentioned this congested traffic is called synchronized flow (S) (Fig. 2).

• ii. Empirical observations of induced traffic breakdown at highway bottlenecks (Fig. 2(b)).

The importance of these empirical features of traffic breakdown is as follows: They prove that traffic breakdown is an $\mathrm{F} \to \mathrm{S}$ transition occurring in a metastable free flow at a highway bottleneck [33–37]. In more details, the empirical prove of the metastability of free flow at a highway bottleneck one can find in [34].

However, before we consider the empirical proof of the metastability of free flow at highway bottlenecks, we should define and explain the term “nucleus” for traffic breakdown (Appendix B.1) as well as define “empirical spontaneous traffic breakdown” and “empirical induced traffic breakdown” (Appendix B.2).

2.1 B.1 Explanation of nucleus for traffic breakdown

The term “metastable free flow with respect to an $\mathrm{F} \to \mathrm{S}$ transition” means that a small enough disturbance for free flow at a bottleneck decays; therefore, in this case free flow persists at the bottleneck over time. However, when a critical disturbance (or a disturbance that is larger than the critical one) appears in free flow in a neighborhood of the bottleneck, traffic breakdown occurs at the bottleneck. In accordance with general theory of metastable systems of natural science [67], such a (speed, density and/or flow rate) disturbance in free traffic flow can be called a nucleus for traffic breakdown ($\mathrm{F} \to \mathrm{S}$ transition) at a bottleneck.

For this reason, the term “the metastability of free flow with respect to the $\mathrm{F} \to \mathrm{S}$ transition” means that traffic breakdown at the bottleneck exhibits the nucleation nature: If the nucleus for traffic breakdown occurs in free flow at the bottleneck, traffic breakdown does occur. In contrast, as long as no nucleus appears, no breakdown occurs in a metastable state of free flow. It must be noted that there are two ways for nucleus occurrence:

• 1. The nucleus for traffic breakdown can occur spontaneously in free flow, for example, through random fluctuations of the free flow speed, the density, or/and the flow rate at the bottleneck. Recently empirical nuclei for spontaneous traffic breakdown have been revealed in studied of real field traffic data [34].

• 2. The nucleus for traffic breakdown can be induced in free flow at the bottleneck. There can be the following scenario for the induced traffic breakdown in real free flow at the bottleneck. Firstly, a local congested pattern occurs at a downstream bottleneck. Then the pattern propagates upstream to the location of the bottleneck under consideration. When this congested pattern reaches the bottleneck, the pattern induces traffic breakdown at the bottleneck. In accordance with above consideration of the nucleation nature of traffic breakdown, this local congested pattern can be considered the nucleus that induces traffic breakdown at the bottleneck.

2.2 B.2 Definitions of empirical spontaneous and induced traffic breakdowns

This consideration can explain why in empirical data (i.e., field data measured in real traffic) we distinguish two typed of traffic breakdowns at highway bottlenecks: (i) Empirical spontaneous traffic breakdown. (ii) Empirical induced traffic breakdown [32–34].

• Empirical spontaneous traffic breakdown is defined as follows. If before traffic breakdown occurs at the bottleneck, there is free flow at the bottleneck as well as upstream and downstream in a neighborhood of the bottleneck, then traffic breakdown at the bottleneck is called spontaneous traffic breakdown (Fig. 2(a)).

• Empirical induced traffic breakdown at the bottleneck is traffic breakdown induced by the propagation of a spatiotemporal congested traffic pattern. This congested pattern has occurred earlier than the time instant of traffic breakdown at the bottleneck and at a different road location (for example at a downstream bottleneck) than the bottleneck location (Fig. 2(b)).

Example of empirical spontaneous traffic breakdown is shown in Figs. 2(a) and 10(a–c). This is a well-known traffic breakdown studied by many researchers (see, e.g., [12, 15, 16, 20, 22–29] and references there). It is also well-known that states of free flow at a bottleneck overlap in the flow rate with states of synchronized flow (congested traffic) measured at the bottleneck location (Fig. 10). However, this well-known empirical fact considered as a solely empirical fact does not prove that real free flow is a metastable state with an $\mathrm{F}\to\mathrm{S}$ transition.

Fig. 10.

Empirical features of synchronized flow occurring through spontaneous traffic breakdown (a–c) and induced traffic breakdown (d, e) related to Figs. 2(a) and 2(b), respectively (1 min averaged data). (a, b, d, e) Average speed (a, d) and flow rate (b, e) as time-function related to locations 6.4 km (a, b) and 16.2 km (d, e) associated with merging regions of on-ramp bottlenecks on freeway A5-South in Germany (see schema of freeway section in Fig. 2.1 of the book [33]). (c) Data related to (a, b) in the speed-flow rate plane; arrow $\mathrm{F} \to \mathrm{S}$ marks spontaneous traffic breakdown whose duration is about 1 min

The proof of the metastability of free flow with respect to traffic breakdown ($\mathrm{F}\to\mathrm{S}$ transition) is the empirical evidence of induced $\mathrm{F}\to\mathrm{S}$ transition caused by the propagation of a local spatiotemporal congested traffic pattern (Fig. 2(b)) [32–34]. Thus the local congested pattern is the nucleus for the transition between these two flow phases F and S.

After the induced breakdown has occurred, the emergent synchronized flow can persist over time at the bottleneck independent on a further behavior of the congested pattern that has initially induced the breakdown. In the case shown in Fig. 2(b), this pattern is a wide moving jam that propagates far away from the bottleneck, while the induced synchronized flow remains to be localized at the bottleneck. Synchronized flow that results from this induced synchronized flow at the bottleneck (Fig. 10(d, e)) exhibits qualitatively the same empirical features as those of synchronized flow occurring through empirical spontaneous breakdown. A more detailed consideration of empirical features of spontaneous and induced traffic breakdowns can be found in [34].

There can be many different scenarios of empirical induced traffic breakdowns (Fig. 11). All these scenarios show qualitatively the same nucleation nature of traffic breakdown at highway bottlenecks discussed in this appendix. Therefore, we can make the conclusion:

• Empirical induced breakdown is the empirical proof of the metastability of free flow at highway bottlenecks.

Fig. 11.

Examples of empirical induced traffic breakdown at highway bottlenecks measured on four different days by road detectors installed on freeway A5-South in Germany (1 min averaged field data): Representation of measured speed data in the time–space plane; explanations of the reconstruction of spatiotemporal traffic dynamics based on road detector measurements shown in these figures are given in Sect. C.2 of [57]. Freeway section schema is shown in Fig. 2.1 of the book [33]

It should be noted that a number of other empirical examples of traffic breakdown at highway bottlenecks that substantiate the above conclusion can be found in Chap. 2 and Part II of the book [33], in Chaps. 2, 3, 5, and 7 of the book [32] as well as in [34].

2.3 B.3 Empirical induced traffic breakdown as one of the consequences of spillover

Most of the traffic researchers (see, e.g., [12, 15, 16, 20, 22–29] and references there) do not consider the empirical evidence of empirical induced traffic breakdown. The upstream propagation of traffic congestion occurring at a downstream bottleneck is usually called by traffic researchers as spillback. If this traffic congestion forces congested traffic at an upstream bottleneck, it is called the spillover effect.

When the wide moving jam shown in Fig. 2(b) reaches the bottleneck, the jam can indeed be considered spillover: The jam forces congested traffic at the bottleneck. However, due to the upstream jam propagation, the jam can be considered as spillover only during a short time interval: When the jam is far away upstream of the bottleneck, the jam does not force congested traffic at the bottleneck any more.

In [34], we have explained the reason why we do not use the term spillover: This is because there can be at least the following qualitatively different empirical effects of spillover:

(i) An empirical induced traffic breakdown occurs due to congested pattern propagation through a bottleneck (Figs. 2(b) and 11).

(ii) The jam propagation through a bottleneck does not lead to induced traffic breakdown (see Fig. 16(b) of [34]).

(iii) An expanded congested pattern (EP) occurs due to spillover at a bottleneck (see Fig. 18(b) of [34]). This spillover cannot be considered as induced traffic breakdown. This is because during the whole time of the existence of traffic congestion at the bottleneck this traffic congestion is forced by downstream traffic congestion.

Therefore, rather than consider all these qualitatively different traffic phenomena as the same effect spillover, to understand real vehicular traffic, one should consider each of these cases of spillover separately each other, i.e., as qualitatively different traffic phenomena.

Appendix C: Infinite number of stochastic highway capacities of three-phase traffic theory

Follow [68], in this Appendix we explain the understanding of stochastic capacity introduced in the three-phase traffic theory [32, 33]. As explained in Appendix A, in the classical understanding of stochastic capacity, condition (9) is assumed. However, the condition (9) contradicts the empirical fact about observations of induced traffic breakdowns (Appendix B).

3.1 C.1 Basic assumption of three-phase traffic theory about the nature of traffic breakdown at highway bottlenecks

In contrast with condition (9), in three-phase traffic theory the following basic assumption about the nature of traffic breakdown ($\mathrm{F} \to \mathrm{S}$ transition) is made [32, 33, 35–37]:

$$ P^{(B)}(q_{\mathrm{sum}}) = P_{\mathrm{nucleus}}^{(B)}(q_{\mathrm{sum}}), $$

(10)

where $P_{\mathrm{nucleus}}^{(B)}(q_{\mathrm{sum}})$ is the flow-rate dependence of the probability that during a given time interval (that is the same as that used in the definition of the breakdown probability $P^{(B)}(q_{\mathrm{sum}})$) a nucleus for traffic breakdown occurs spontaneously in free flow at a bottleneck. A related mathematical nucleation theory of traffic breakdown can be found in [69–71].

For qualitative explanations of condition (10), firstly we assume that traffic parameters (weather, mean driver’s characteristics, share of long vehicles, etc,) remain the same for all flow rates in free flow. Under this condition, we can also assume that the larger is the flow rate $q_{\mathrm{sum}}$ in free flow at the bottleneck, the smaller is the nucleus required for the breakdown at a bottleneck. Obviously, the probability of the occurrence of a small speed disturbance in free flow is considerably larger than the probability of the occurrence of a large disturbance. This means that probability of the spontaneous occurrence of a nucleus for traffic breakdown $P_{\mathrm{nucleus}}^{(B)}(q_{\mathrm{sum}})$ is an increasing function of the flow rate $q_{\mathrm{sum}}$. In accordance with (10), this explains the increasing flow rate function of the breakdown probability $P^{(B)}(q_{\mathrm{sum}})$ (Fig. 4(a)).

As an example of this general discussion of condition (10), we consider the occurrence of a nucleus associated with a time-limited critical local decrease in the speed in an initial free flow at a bottleneck denoted by $\Delta v_{\mathrm{cr}}^{(\mathrm{FS})}$ (Fig. 12(a)). The larger the flow rate $q_{\mathrm{sum}}$ in free flow at the bottleneck, the smaller should be the value $\Delta v_{\mathrm{cr}}^{(\mathrm{FS})}(q_{\mathrm{sum}})$ that initiates traffic breakdown at the bottleneck. The related decreasing function $\Delta v_{\mathrm{cr}}^{(\mathrm{FS})}(q_{\mathrm{sum}})$, which is qualitatively shown in Fig. 12(a), has indeed been found in simulations with Kerner-Klenov stochastic microscopic three-phase traffic flow model [64].

Fig. 12.

Explanation of condition (10): (a) Qualitative flow rate dependence of function $\Delta v_{\mathrm{cr}}^{(\mathrm{FS})}(q_{\mathrm{sum}})$. (b) Breakdown probability $P^{(B)}(q_{\mathrm{sum}})$ taken from Fig. 4(a). Flow rate ranges I, II, III, and IV have the same sense as those shown in Fig. 4(a)

Condition (10) explains flow rate ranges II–IV discussed in Sect. 6 of the main text as follows (Fig. 12). In flow rate range II (condition (4)), a very large value $\Delta v_{\mathrm{cr}}^{(\mathrm{FS})}(q_{\mathrm{sum}})$ (large nucleus) is required for the breakdown, so we can assume that the probability of spontaneous occurrence of such very large speed disturbance in free flow during a given time interval is zero, i.e., $P_{\mathrm{nucleus}}^{(B)}(q_{\mathrm{sum}}) = 0$. In accordance with (10), the probability of spontaneous breakdown $P^{(B)}(q_{\mathrm{sum}}) = 0$. This means that in this case only induced traffic breakdown is possible.

In flow rate range III (condition (5)), the value $\Delta v_{\mathrm{cr}}^{(\mathrm{FS})}(q_{\mathrm{sum}})$ required for the breakdown decreases sharply. Therefore, the probability of the spontaneous occurrence of such a speed disturbance due to fluctuations in free flow during a given time interval can satisfy conditions $0 < P_{\mathrm{nucleus}}^{(B)}(q_{\mathrm{sum}}) < 1$.

In flow rate range IV (condition (6)), the value $\Delta v_{\mathrm{cr}}^{(\mathrm{FS})}(q_{\mathrm{sum}})$ required for the breakdown is as small as zero; therefore, the probability of the spontaneous occurrence of a nucleus for traffic breakdown $P_{\mathrm{nucleus}}^{(B)}(q_{\mathrm{sum}}) = 1$. Therefore, in accordance with (10), the probability of spontaneous traffic breakdown $P^{(B)}(q_{\mathrm{sum}}) = 1$.

3.2 C.2 Maximum and minimum capacities as stochastic functions

It must be noted that the maximum capacity $C_{\max}$, the minimum capacity $C_{\min}$, and the value $q_{\mathrm{th}}^{(B)}$ depend on traffic parameters, like weather, mean driver’s characteristics (e.g., mean driver reaction time), share of long vehicles, etc. In real traffic flow, these traffic parameters change over time. For this reason, the values $C_{\max}$, $C_{\min}$, and $q_{\mathrm{th}}^{(B)}$ change also over time.

Moreover, in real traffic flow, the traffic parameters are stochastic time functions.

Therefore, in real traffic flow we should consider some stochastic maximum capacity $C_{\max}^{(\mathrm{stoch})}(t)$, stochastic minimum capacity $C_{\min}^{(\mathrm{stoch})}(t)$, and a stochastic threshold flow rate $q_{\mathrm{th}}^{(B, \mathrm{stoch})}(t)$ whose time dependence is determined by stochastic characteristics of traffic parameters. Qualitative hypothetical fragment of these time-functions within a time interval is shown in Fig. 13.

Fig. 13.

Qualitative explanation of the infinite number of capacities of free flow at a highway bottleneck in three-phase traffic theory. Probability function for traffic breakdown $P^{(B)}(q_{\mathrm{sum}})$ (figure left) is the same as that shown in Figs. 4(a), 8, and 12(b). Flow rate regions I, II, and III mentioned in labeling are the same as those shown in Fig. 4(a) and explained in Sect. 6 of the main text. Taken from [68]

Stochastic functions $C_{\max}^{(\mathrm{stoch})}(t)$, $C_{\min}^{(\mathrm{stoch})}(t)$, and $q_{\mathrm{th}}^{(B, \mathrm{stoch})}(t)$ shown in Fig. 13 are qualitative hypothetical functions that cannot be measured in empirical observations. Only their mean values (respectively, $C_{\max}$, $C_{\min}$, and $q_{\mathrm{th}}^{(B)}$) can be found in empirical studies of measured traffic data. In particular, the mean values $C_{\max}$ and $q_{\mathrm{th}}^{(B)}$ can be found from an empirical study of the flow rate function of the breakdown probability $P^{(B)}(q_{\mathrm{sum}})$ (Fig. 4(a)).

It must be noted that in empirical observations the mean value of the minimum capacity $C_{\min}$ can be found from a study of a finite number of different days at which induced traffic breakdowns have been observed at a given bottleneck. The value $C_{\min}$ is related to these empirical days of observations only. In other words, it can occur that at another day, which is not within the days used for the calculation of $C_{\min}$, traffic breakdown at this bottleneck can be induced at a smaller flow rate than the minimum capacity $C_{\min}$ found before. A similar comment is related to the physical meaning of the mean value of $q_{\mathrm{th}}^{(B)}$. To explain this, we should note that with a finite number of measurements it is not possible to find some “exact value” of the minimum flow rate at which traffic breakdown can occur.

In other words, strictly speaking, mean values $C_{\min}$, $C_{\max}$, and $q_{\mathrm{th}}^{(B)}$ are valid only for the days of the observing of traffic breakdown that have been used for the calculations of these mean values.

From Fig. 13 we can see that in three-phase traffic theory traffic breakdown cannot occur spontaneously at “any flow rate”. Indeed, at any time when the flow rate in free flow is smaller than the minimum capacity $C_{\min}^{(\mathrm{stoch})}(t)$, no traffic breakdown can occur at the bottleneck. When the flow rate $q_{\mathrm{sum}}(t)$ satisfies conditions (4), specifically, $C_{\min}^{(\mathrm{stoch})}(t) \le q_{\mathrm{sum}}(t) < q_{\mathrm{th}}^{(B, \mathrm{stoch})}(t)$, traffic breakdown can be induced only. Only under conditions $q_{\mathrm{th}}^{(B, \mathrm{stoch})}(t) \le q_{\mathrm{sum}}(t) < C_{\max}^{(\mathrm{stoch})}(t)$ traffic breakdown can occur spontaneously with some probability $0 < P^{(B)}(q_{\mathrm{sum}}) < 1$ during a given observation time.

Thus, we can see in Fig. 13 that in accordance with the highway capacity definition made in three-phase traffic theory, under conditions $C_{\min}^{(\mathrm{stoch})}(t) \le q_{\mathrm{sum}}(t) < C_{\max}^{(\mathrm{stoch})}(t)$ at any time instant there is the infinite number of highway capacities at which traffic breakdown can occur with some probability or can be induced at the bottleneck.

Appendix D: Classical understanding of stochastic highway capacity versus infinite number of stochastic highway capacities of three-phase traffic theory

The objective of this appendix is to make a critical analysis of the classical definition of stochastic highway capacity that is generally accepted by most of the traffic researches (see references in the book [20]). We follow the associated critical analysis of the understanding of stochastic highway capacity made recently in [68].

The classical understanding of the nature of stochastic highway capacity (Appendix A) [20, 27–29] is based on the assumption that the empirical probability of traffic breakdown is determined by the capacity distribution function, i.e., that condition (9) is valid.

In contrast, the assumption of three-phase traffic theory about the metastability of traffic breakdown with respect to traffic breakdown (condition (10) of Appendix C) is based on the empirical evidence that traffic breakdown can be induced at a bottleneck (Appendix B).

As mentioned in Appendix A, the observation of empirical induced breakdowns proves that condition (9) of Brilon’s stochastic capacity [20, 27–29] cannot be valid for real traffic. However, the following question arises:

• What are the consequences of this controversial understanding of the nature of traffic breakdown?

With the use of Fig. 13, we can qualitatively illustrate in Fig. 14 the basic difference between the classical understanding of the nature of stochastic highway capacity (Appendix A) and the understanding of the infinite number of stochastic highway capacities made in three-phase traffic theory (Appendix C).

Fig. 14.

Qualitative explanation of traffic breakdown with the use of the infinite number of capacities of free flow at a highway bottleneck of three-phase traffic theory [68]. Hypothetical time-functions $C_{\max}^{(\mathrm{stoch})}(t)$, $C_{\min}^{(\mathrm{stoch})}(t)$, and $q_{\mathrm{th}}^{(B, \mathrm{stoch})}(t)$ are taken from Fig. 13. Hypothetical time functions of the flow rates $q_{\mathrm{sum}}(t) = q_{\mathrm{sum}}^{(2)}(t)$ in (a) and $q_{\mathrm{sum}}(t) = q_{\mathrm{sum}}^{(1)}(t)$ in (b) as well as time instant $t_{1}$ in (a) are, respectively, the same as those in Fig. 9

In the classical understanding of stochastic capacity (Appendix A), for the hypothetical time dependence of the flow rate $q_{\mathrm{sum}}(t) = q_{\mathrm{sum}}^{(2)}(t)$ shown in Fig. 9(b), traffic breakdown has occurred at time instant $t_{1}$ at which $q_{\mathrm{sum}}^{(2)}(t_{1}) = C(t_{1})$, i.e., when the flow rate is equal to the capacity value. In contrast, in three-phase traffic theory for the same time dependence of the flow rate $q_{\mathrm{sum}}^{(2)}(t)$, for which conditions $C_{\min}^{(\mathrm{stoch})}(t) \le q_{\mathrm{sum}}^{(2)}(t) < C_{\max}^{(\mathrm{stoch})}(t)$ are satisfied, no breakdown should be necessarily occur both at time instant $t_{1}$ and for a later time interval (Fig. 14(a)).

In the classical understanding of stochastic capacity (Appendix A), for the hypothetical time dependence of the flow rate $q_{\mathrm{sum}}(t) = q_{\mathrm{sum}}^{(1)}(t)$ shown in Fig. 9(a), traffic breakdown could not occur because for all time instants $q_{\mathrm{sum}}^{(1)}(t) < C(t)$. In contrast, in three-phase traffic theory for the same time dependence of the flow rate $q_{\mathrm{sum}}^{(1)}(t)$ traffic breakdown can occur spontaneously as this is shown for time instant $t_{2}$ in Fig. 14(b).

Because the classical understanding of stochastic highway capacity (8), (9) contradicts the empirical nucleation nature of real traffic breakdown, the understanding of stochastic highway capacity made in traditional traffic research community [20, 27–29] cannot be used for reliable highway design and highway operations.