Abstract
In this Chapter, we consider an application of the minimization principle (BM) principle for dynamic traffic assignment and other types of network optimization in a network called the network throughput maximization approach. With the use of the network throughput maximization approach, general physical conditions for the maximization of the network throughput at which free flow conditions are ensured in the whole traffic or transportation network can be found. It turns out that there is a physical measure of the network called a network capacity that characterizes general features of the network with respect to the maximization of the network throughput.
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- 1.
This increase in q m on other alternative routes maintaining conditions (11.6) can be obtained with the use of different routing methodologies including those that are based on the Wardrop’s UE or SO.
- 2.
In accordance with results of Sects. 5.7 and 9.3.4, the minimum capacity C min (k) of a network bottleneck is a stochastic value. Therefore, the network capacity C net (11.8) that depends on the values of the minimum capacities of the network bottlenecks is also a stochastic value. A consideration of the effect of stochastic features of minimum capacities C min (k), k = 1, 2, …, N of the network bottlenecks on the application of the network throughput maximization approach and on the network capacity C net is out of scope of this book. This can be an interesting task for future investigations.
- 3.
In city traffic, there is a hypothetical case of “red wave”, when all vehicles approach a traffic signal during the red signal phase only (Sect. 9.8). In this case, the definition of the network capacity C net through conditions (11.8) remains. However, when at one of the network bottlenecks due to the signal “red wave” is realized, then the minimum capacity of the signal C min is equal to the classical signal capacity C cl (Sect. 9.8). Therefore, in (11.8) the value of C min for this signal should be replaced by C cl. To explain this, we note that in the case of “red wave” the classical theory of traffic at the signal is a special case of the three-phase theory (Sect. 9.8): When the average arrival flow rate (flow rate at a bottleneck due to the signal) exceeds C min = C cl, then traffic breakdown, i.e., the transition from under-saturated traffic to over-saturated (congested) traffic occurs at the signal without time delay. In a non-realistic case, when all network bottlenecks are due to the signals at which red wave conditions are realized, then the network capacity C net is found from conditions (11.8) as follows:
$$\displaystyle\begin{array}{rcl} q_{\mathrm{sum}}^{(k)} = C_{\mathrm{ cl}}^{(k)}\ \mathrm{for}\ k = k_{z}^{(1)},& & \\ q_{\mathrm{sum}}^{(k)} <C_{\mathrm{ cl}}^{(k)}\ \mathrm{for}\ k\neq k_{z}^{(1)},& & \\ z = 1, 2,\ldots,Z;\ Z \geq 1,\ Z \leq N;\ k = 1, 2,\ldots,N,& & {}\end{array}$$(11.9)where bottlenecks k = k z (1) and value Z are found in accordance with the constrain “alternative routes” as described in Sect. 10.5.
As explained in Sec. 12.4 of review [11], formula (11.9) is also applicable in the framework of the classical traffic flow theories in which it is assumed that there is a particular value of capacity for any network bottleneck. In particular, the measure “network capacity” introduced in [10], which is discussed in this section, has the same physical sense in the classical theory as that in the three-phase traffic theory. However, as emphasized in Sect. 4.10, the classical understanding of stochastic highway capacity at a bottleneck is inconsistent with the empirical nucleation nature of traffic breakdown at the bottleneck. For this reason, in the book we will not consider the network capacity under classical understanding of stochastic highway capacity at a network bottleneck (see such a consideration in Sec. 12.4 of [11]).
- 4.
- 5.
Under non-steady state conditions in the network, even when free flow is realized in the whole network, nevertheless, at a given time instant the total network inflow rate should not be necessarily equal to the the total network outflow rate: The total network inflow rate is equal to the total network outflow rate on average only.
- 6.
It should be noted that when the network throughput maximization approach is applied under non-steady state conditions in the network, then the set of the network bottlenecks k = k z (1) (where z = 1, 2, …, Z 1) in conditions (11.6) can be different from the set of the network bottlenecks k = k z (1) (where z = 1, 2, …, Z) in conditions (11.8). Respectively, the limit number of network bottlenecks Z 1 = Z in (11.6) should not necessarily coincide with the number of network bottlenecks Z in (11.8).
- 7.
In numerical calculations of the breakdown probability P (B)(q sum) [12] only a finite number N r of simulation realizations (runs) can be made for each given flow rate q sum. As shown in Sect. 5.4.5, for the smallest value of the breakdown probability at the flow rate q sum = q th (B) we get \(P^{\mathrm{(B)}}\mid _{q_{\mathrm{ sum}}=q_{\mathrm{th}}^{\mathrm{(B)}}} = \frac{1} {N_{\mathrm{r}}}.\) This formula determines the accuracy of the calculation of the threshold flow rate q th (B) related to the flow rate ranges (11.15) and (11.17). In other words, the larger the number N r, the more exactly the threshold flow rate q sum = q th (B) can be calculated.
- 8.
- 9.
As the total network inflow rate Q is subsequently increased, the number Y of bottlenecks for which conditions (11.36) are satisfied can change.
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Kerner, B.S. (2017). Maximization of Network Throughput Ensuring Free Flow Conditions in Network. In: Breakdown in Traffic Networks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54473-0_11
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