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The Reason for Incommensurability of Three-Phase Theory with Classical Traffic Flow Theories

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Breakdown in Traffic Networks

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Abstract

In this chapter, we explain why the classical traffic flow instability is inconsistent with the empirical nucleation nature of traffic breakdown at a highway bottleneck. To reach this goal, we use a concept for the distinguishing of a new paradigm in a scientific field introduced by Kuhn. This concept is mainly based on an analysis whether a new theory is incommensurable with the old one or not. A critical comparison of threephase traffic flow models with two-phase traffic flow models as well as with the classical understanding of highway capacity made in this chapter discloses the incommensurability of the three-phase theory with any other classical traffic and transportation theories.

The original version of this chapter was revised. An erratum to the chapter can be found at DOI 10.1007/978-3-662-54473-0_16.

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Notes

  1. 1.

    Because a detailed consideration of the moving jam emergence is out of scope of the book, we do not discuss the effect of the bottleneck on the value of the critical flow rate q cr (B).

  2. 2.

    A more detailed explanation of the stable and metastable synchronized flow states with respect to the S → J transition can be found in Chap. 6 of the book [52] and Chap. 5 of the book [62].

  3. 3.

    In reality, a wide moving jam exhibits a complex microscopic spatiotemporal structure that consists of alternations of moving blanks with flow interruption intervals [74–76]. For simplicity, moving blanks within the wide moving jam have been neglected in Fig. 8.8. Through this simplification of the real complex microscopic spatiotemporal structure within wide moving jams, the wide moving jam phase is related to the speed that is equal to zero (labeled by J in Fig. 8.8). A detailed consideration of moving blanks within wide moving jams is out of scope of this book (see [74–76] and Sec. 2.6 of the book [62]).

  4. 4.

    The mean flow rate q cong is measured within the pinch region of synchronized flow of the GP. In a theory of GPs, this flow rate has been denoted by q (pinch) (see Sec. 18.3 of the book [52]). Here we use a general designation q cong used in [60] for the mean flow rate in congested traffic.

  5. 5.

    The chosen on-ramp inflow rate q on in Figs. 8.15–8.17 is equal to a characteristic on-ramp inflow rate q on (strong) at which so-called “weak” congestion transforms into “strong” congestion. A theory of “weak” and “strong” congestion in GPs has been presented in Sec. 18.3 of the book [52]: When the on-ramp inflow rate q on exceeds the characteristic on-ramp inflow rate q on (strong), the mean flow rate within the pinch region of the GP q cong does not depend on the on-ramp inflow rate q on any more reaching a limit value denoted by q cong = q lim (pinch) in Sec. 18.3 of the book [52]. In accordance with results of the theory “weak” and “strong” congestion in GPs, simulations show that if we choose the flow rate q on that is larger than q on (strong) = 770 vehicles/h, then neither the discharge flow rate q out (bottle) in Fig. 8.15 nor the flow rate q cong in Fig. 8.17d have on average changed. This is because at q on > q on (strong) congested traffic occurs in the on-ramp lane that limits on average the on-ramp inflow onto the main road to the value q on (strong) = 770 vehicles/h.

  6. 6.

    It should be noted that when the local speed disturbance is close to the critical nucleus for a phase transition, the phase transition exhibits a variety of diverse probabilistic features (Chap. 5 and Sect. 8.3.2). However, qualitative conclusions about critical nuclei made in this chapter below with the use of induced phase transitions are independent of probabilistic features of the phase transitions. For this reason, probabilistic features of induced phase transitions are not considered in the book.

  7. 7.

    It should be noted that when qualitative features of an Z-characteristic for traffic breakdown as a function of the flow rate q sum (Fig. 5.5) has been discussed (Sect. 5.3), we have considered the metastability of the free flow phase (F) with respect to an F → S transition only, rather than the metastability of the synchronized flow phase (S) with respect to an S → F transition. The reason for this limitation of the consideration of the Z-characteristic for traffic breakdown made in Sect. 5.3 is as follows. In the phase F, the sum of the flow rates q in + q on is equal to the flow rate q sum = q in + q on in free flow downstream of the on-ramp bottleneck. For this reason, we can consider the Z-characteristic for traffic breakdown as a function of the flow rate q sum (Fig. 5.5). One of the reasons for such a Z-characteristic is that it shows the minimum highway capacity C min and maximum highway capacity C max of free flow at the bottleneck. However, it must be stressed that strictly speaking the presentation of the Z-characteristic for phase transitions in Fig. 5.5 is valid for the F → S transition only. Contrarily, when the synchronized flow phase (S) is at a highway bottleneck, then in a general case the flow rate within the synchronized flow is not equal to the flow rate q in in free flow upstream of the bottleneck. In this case, the flow rate in free flow downstream of the bottleneck is not equal to the sum q in + q on (see Sects. 8.4.1 and 8.5). For this reason, we present the Z-characteristic for S → J and J → S phase transitions (Fig. 8.8) as well as the 2Z-characteristic (Fig. 8.25) as a function of the on-ramp inflow rate q on at a given flow rate q in.

  8. 8.

    An 2Z-characteristic can also be shown by three-phase traffic flow models that do not satisfy hypothesis of the three-phase theory about 2D-steady states of synchronized flow (Sect. 5.9.1). In other words, 2D-steady states of synchronized flow is not a necessarily condition for a three-phase traffic flow model [72].

  9. 9.

    At model parameters chosen in the three-phase model that simulations presented in Fig. 8.26, condition

    $$\displaystyle{ C_{\mathrm{min}} <q_{\mathrm{out}} }$$
    (8.17)

    is satisfied. However, under conditions

    $$\displaystyle{ C_{\mathrm{min}} \leq q_{\mathrm{sum}} <q_{\mathrm{out}} }$$
    (8.18)

    no F → J transition is possible. Therefore, under condition (8.18) only the F → S transition can occur at the bottleneck in the three-phase model. In contrast, in the two-phase model at any flow rate q sum < q out no phase transition can be induced in free flow at the bottleneck. We have found that the minimum highway capacity C min can depend considerably on the value q on. In particular, at other model parameters than those used in Fig. 8.26 we can also find that C min > q out.

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    Boris S. Kerner

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Kerner, B.S. (2017). The Reason for Incommensurability of Three-Phase Theory with Classical Traffic Flow Theories. In: Breakdown in Traffic Networks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54473-0_8

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