Network capacity

Network capacity introduced by Boris Kerner in 2016 is a measure (or “metric") of a traffic or transportation network. The network capacity determines the maximum total network inflow rate that is still possible to assign in the network while keeping free flow conditions in the whole network. The network capacity is denoted by $C_{net}$.

Network capacity allows us to formulate a general condition for the maximization of the network throughput at which free flow does persist in the whole network: Under application of Kerner’s network throughput maximization approach, as long as the total network inflow rate denoted by $Q$ is smaller than the network capacity $C_{net}$ traffic breakdown with resulting traffic congestion cannot occur in the network, i.e., free flow remains in the whole network.

Traffic Breakdown at Network Bottlenecks

Traffic breakdown in a traffic or transportation network occurs usually at some road locations of a traffic network called network bottlenecks. Traffic breakdown at a network bottleneck is a transition from free flow to congested traffic at the bottleneck (see, e.g., reviews and books about an analysis of traffic and transportation networks). Network bottlenecks are caused, for example, by on- and off-ramps, road gradients, road-works, a decrease in the number of road lines (in the flow direction), traffic signals in city traffic, etc.

It is assumed that there is a traffic or transportation network with $N$ network bottlenecks, where $N>1$. It is also assumed that there are $M$ network links (where $M>1$) for which the inflow rates $q_{m}$ can be adjusted; $q_{m}$ is the link inflow rate for a link with index $m$, where $m=1,2,\ldots ,M$. For each of the network bottlenecks $k=1,2,\ldots ,N$, the flow rate at the bottleneck will be denoted by $q_{sum}^{(k)}$ and the minimum capacity will be denoted by $C_{min}^{(k)}$.

At a small enough flow rate in free flow at a network bottleneck no traffic breakdown can occur at the bottleneck. When the flow rate at the bottleneck denoted by $q_{sum}$ reaches some critical value, traffic breakdown can occur at the bottleneck with some finite probability during a given time interval. This critical flow rate is called as a minimum bottleneck capacity denoted by $C_{min}$. Thus, when $q_{sum}<C_{min}$, no traffic breakdown can occur at the bottleneck; therefore, free flow persists at the bottleneck. Only when $q_{sum}\geq C_{min}$, traffic breakdown can occur at the bottleneck.

Kerner’s Network Throughput Maximization Approach

The network throughput maximization approach is an approach to dynamic traffic assignment in a network. Dynamic traffic assignment is traffic assignment in a traffic or transportation network considering the temporal dimension of the assignment problem (see reviews of another approach to dynamic traffic assignment in traffic and transportation networks based on Wardrop’s equilibria, e.g., in ; see also Sect. below)).

To understand the network throughput maximization approach, firstly, constrain “alternative network routes” should be considered. This constrain is applied for dynamic traffic assignment under the use of the network throughput maximization approach.

Constrain “Alternative Network Routes”

It is assumed that in a network there are several different routes from each origin to each destination. To avoid the use of network routes with considerably longer travel times in comparison with the shortest routes, in the network throughput maximization approach discussed below dynamic traffic assignment is applied for some “alternative network routes (paths)" (for short, alternative routes) only. This means that there is constrain of alternative network routes (paths) for any application of dynamic traffic assignment in the network. Constrain for the alternative routes prevents the use of too long routes in dynamic traffic assignment with the use of the network throughput maximization approach.

The set of alternative network routes is related to possible different network routes (paths) from an origin to a destination for which the difference between route travel times under free flow conditions does not exceed a chosen threshold value. After the sets of alternative routes have been determined, dynamic traffic assignment is performed through an application of the network throughput maximization approach with the use of these sets of alternative routes only. This means that dynamic traffic assignment in the network made through the use of the network throughput maximization approach does not depend on current values of the route travel times in the network.

Procedure of Maximization of Network Throughput

In real traffic and transportation networks, the total network inflow rate $Q$ and, therefore, the flow rates $q_{\rm {sum}}^{(k)},\ k=1,2,\ldots ,N$ in the network increase from very small values (at night) to large values during daytime related to rush hours in urban networks. At small initial values rate $Q$ and $q_{\rm {sum}}^{(k)}$ condition

$$q_{\rm {sum}}^{(k)}<C_{\rm {min}}^{(k)}$$

(1)

is valid for each of the network bottlenecks $k=1,2,\ldots ,N$. This mean that free flow is stable with respect to traffic breakdown at each of the network bottlenecks. Therefore, no traffic breakdown can occur in the network.

When the total network inflow rate $Q$ is small enough, no dynamic traffic assignment is needed. When the total network inflow rate $Q$ increases subsequently, the network throughput maximization approach is applied for dynamic traffic assignment in the network that is as follows. Due to the increase in $Q$ over time, the flow rate $q_{\rm {sum}}^{(k)}$ at least for one of the network bottlenecks $k=k_{1}^{(1)}$ becomes very close to $C_{\rm {min}}^{(k)}$. Therefore, in formula

$$q_{\rm {sum}}^{(k)}+\epsilon ^{(k)}=C_{\rm {min}}^{(k)}$$

(2)

for bottleneck $k=k_{1}^{(1)}$ a positive value $\epsilon ^{(k)}=C_{\rm {min}}^{(k)}-q_{\rm {sum}}^{(k)}$ becomes very small:

$$\epsilon ^{(k)}/C_{\rm {min}}^{(k)}\ll 1$$

(3)

for bottleneck $k=k_{1}^{(1)}$.

Conditions (2), (3) are maintained at the expense of the increase in the link inflow rates $q_{m}$ on other alternative routes of the network. As a result, conditions (2), (3) are satisfied for another bottleneck $k=k_{2}^{(1)}$.

When $Q$ increases further, the above procedure for other network bottlenecks is repeated. Consequently, conditions

$$q_{\rm {sum}}^{(k)}+\epsilon ^{(k)}=C_{\rm {min}}^{(k)}$$

(4)

are valid for bottlenecks $k=k_{z}^{(1)}$ and conditions $q_{\rm {sum}}^{(k)}<C_{\rm {min}}^{(k)}$ are valid for bottlenecks $k\neq k_{z}^{(1)}$, where $z=1,2,\ldots ,Z_{1}$, $Z_{1}\geq 1,\ Z_{1}\leq N;\ k=1,2,\ldots ,N$; values $\epsilon ^{(k)}$ satisfy conditions

$$0<\epsilon ^{(k)}/C_{\rm {min}}^{(k)}\ll 1$$

(5)

for bottlenecks $k=k_{z}^{(1)}$, $z=1,2,\ldots ,Z_{1}$. Thus, when the network inflow rate $Q$ increases, conditions (4), (5) are maintained at the expense of the increase in the link inflow rates $q_{m}$ on other alternative routes.

However, due to the constrain “alternative routes", the number $Z_{1}$ of bottlenecks satisfying (4), (5) is limited by some value $Z$, where $Z\leq N$. All values $\epsilon ^{(k)}$ in (4) accordingly to (5) are positive ones. Therefore, under conditions (4), conditions (1) are satisfied. Thus, due to the application of the network throughput maximization approach free flow remains to be stable with respect to traffic breakdown at each of the network bottlenecks. For this reason, no traffic breakdown can occur in the network.

This means that conditions (4), (5) in the limit case $Z_{1}=Z$ and $\epsilon ^{(k)}\rightarrow 0$ are related to ‘’the maximization of network throughput” at which free flow conditions are still ensured in the whole network. This is because at the subsequent increase in the total network inflow rate $Q$ the constrain “alternative routes" does not allow us to maintain conditions (4), (5) at the expense of the increase in the link inflow rates $q_{m}$ on other alternative routes: At least for one of the network bottlenecks condition $q_{\rm {sum}}^{(k)}\geq C_{\rm {min}}^{(k)}$ is satisfied. Therefore, traffic breakdown can occur in the network.

Kerner’s network throughput maximization approach is the approach to dynamic traffic assignment in a traffic or transportation network that maximizes the network throughput while keeping free flow conditions in the whole network. Through the maximization of the network throughput with the network throughput maximization approach no traffic breakdown can occur in the network.

Steady-State Analysis of Traffic and Transportation Networks

In many cases, the total network inflow rate $Q(t)$ changes in a traffic or transportation network over time $t$ very slowly in comparison with any characteristic times of dynamic traffic effects at network bottlenecks under free flow conditions in the network. For this reason, as it is usually assumed in classical theories of traffic and transportation networks, at any given total network inflow rate $Q$ free flow distribution in the network can be considered as a steady state (steady-state analysis of traffic and transportation networks).

Formulation of Network Capacity

Within a steady-state analysis of traffic and transportation networks, the limit case $Z_{1}=Z$ in conditions (4), (5) allows us to define a network measure (or “metric") – “network capacity” $C_{net}$ as follows. The network capacity $C_{net}$ is the maximum total network inflow rate $Q$ at which conditions

$$q_{\rm {sum}}^{(k)}=C_{\rm {min}}^{(k)}$$

(6)

for $k=k_{z}^{(1)}$ and conditions

$$q_{\rm {sum}}^{(k)}<C_{\rm {min}}^{(k)}$$

(7)

for $k\neq k_{z}^{(1)}$, where $z=1,2,\ldots ,Z;\ Z\geq 1,\ Z\leq N;\ k=1,2,\ldots ,N$ are satisfied. From a comparison of conditions (4) and (6) it can be seen that the total network inflow rate $Q$ reaches the network capacity

$$Q=C_{net}$$

(8)

when the limit case $Z_{1}=Z$ in (4) is realized and all values $\epsilon ^{(k)}$ in (5) are set to zero: $\epsilon ^{(k)}=0$. When

$$Q<C_{net}$$

(9)

conditions (4), (5) are satisfied. Therefore, free flow is stable with respect to traffic breakdown in the whole network. Therefore, no traffic breakdown can occur in the network. This explains the physical sense of the network capacity $C_{net}$: Dynamic traffic assignment in the network in accordance with the network throughput maximization approach maximizes the network throughput at which traffic breakdown cannot occur in the whole network. Respectively, when

$$Q\geq C_{net}$$

(10)

due to the constrain “alternative routes" of the network throughput maximization approach at least for one of the network bottlenecks the flow rate in free flow at the bottleneck $q_{sum}^{(k)}$ becomes equal to or larger than the minimum bottleneck capacity $C_{min}^{(k)}$. Therefore, even under application of the network throughput maximization approach under conditions (10) traffic congestion can occur in the network.

The network capacity allows us to formulate a general physical condition for the maximization of the network throughput at which free flow does persist in the whole network: Under application of Kerner’s network throughput maximization approach, as long as the total network inflow rate is smaller than the network capacity traffic breakdown cannot occur in the network. The network capacity is a stochastic value.

Kerner’s Network Throughput Maximization Approach versus Wardrop's Equilibria

Standard Dynamic Traffic Assignment

To find traffic optima and make an effective traffic control in traffic and transportation networks, a huge number of models for dynamic traffic assignment as well as other advanced traffic models have been introduced (see, e.g., reviews and books).

Travel times and/or other travel costs in a network seems to be the most important parameters of traffic and transportation networks. Therefore, most of the traffic researchers and practitioners are strongly believe that the minimization of travel times and/or other travel costs in the network should be the explicit objective of any reasonable principle for dynamic control, optimization, and assignment in traffic and transportation networks. In other words, travel times or/and other travel costs a traffic or transportation network have been generally accepted in classical traffic and transportation theories to be self-evident traffic characteristics for objective functions used for the optimization of transportation networks, like dynamic traffic assignment. The main aim of the classical approaches is to minimize travel times or/and other travel costs in the network (see, e.g., works as well as references in the reviews and books). Dynamic traffic assignment in traffic and transportation networks that objective is the minimization of network travel times (or/and other travel costs) can be considered standard dynamic traffic assignment.

However, it has been shown that already under condition (9), i.e., when the total network inflow rate $Q$ is less than the network capacity $C_{net}$ applications of this standard dynamic traffic assignment can provoke traffic congestion in the network. This critical conclusion will be explained below.

Wardrop's Equilibria

To show that applications of the standard methodology of the minimization of network travel times provokes heavy traffic congestion in a traffic or transportation network when the total network inflow rate $Q$ is less than the network capacity $C_{net}$, we consider below the most famous approach to an analysis of traffic and transportation networks that is based on the Wardrop's user equilibrium (UE) and system optimum (SO) equilibrium introduced by Wardrop in 1952. The Wardrop's UE and SO are also called the Wardrop's principles or the Wardrop's equilibria (see, e.g., reviews and books).

“Wardrop's UE”: Traffic on a network distributes itself in such a way that the travel times on all routes used from any origin to any destination are equal, while all unused routes have equal or greater travel times.

“Wardrop's SO”: The network-wide travel time should be a minimum.

The Wardrop's principles reflect either the wish of drivers to reach their destinations as soon as possible (UE) or the wish of network operators to reach the minimum network-wide travel time (SO).

Network Capacity and Standard Dynamic Traffic Assignment: A General Consideration

Here it is explained why standard dynamic traffic assignment leads to traffic congestion already at the total network inflow rate $Q$ that is less than the network capacity $C_{net}$, i.e., when condition $Q<C_{net}$ (9) is satisfied. In contrast with standard dynamic traffic assignment based on the Wardrop's UE or SO, the application of the network throughput maximization approach under condition $Q<C_{net}$ (9) ensures that traffic breakdown cannot occur in the network.

A real traffic or transportation network consists of alternative routes with very different lengths. Therefore, at small enough link inflow rates $q_{m}$, there are routes with short travel times (“short routes") and routes with longer travel times (“long routes"). When the total network inflow rate $Q$ and, consequently, values of the link inflow rates $q_{m}$ increase, the minimization of travel times in the network with the use of standard dynamic traffic assignment based on the Wardrop's UE or SO leads to larger increases in the flow rates on short routes in comparison with increases in the flow rates on long routes.

Thus, through the application of the Wardrop's UE or SO, the flow rate on a short route should be larger than the flow rate on a long route. This is independent of capacity values $C_{min}^{(k)}$ for bottlenecks on the route. Contrarily, in accordance with conditions (4), (5) resulting from the network throughput maximization approach, the flow rate at any network bottleneck $q_{sum}^{(k)}$ should be smaller than the minimum capacity $C_{min}^{(k)}$of the bottleneck. Through the application of the network throughput maximization approach, the flow rate increases firstly on the short route. Only when this flow rate on the short route achieves the minimum capacity of free flow at the bottleneck on the short route, the flow rates on longer routes begin to increase.

Therefore, at $Q\rightarrow C_{net}$ rather than conditions (4) (5), any application of the Wardrop's UE or SO leads to conditions $q_{sum}^{(k)}<C_{min}^{(k)}$ for some of the bottlenecks on long routes, whereas for some of the bottlenecks on short routes, we get $q_{sum}^{(k)}>C_{min}^{(k)}$. This condition means that traffic breakdown can occur at network bottlenecks on the short routes. One of the consequences of this general conclusion is that already at relatively small total network inflow rates $Q<C_{net}$ the application of the Wardrop's equilibria leads to the occurrence of congestion in urban networks.

From this general analysis, it can be seen that the main benefit of the network throughput maximization approach in comparison with the Wardrop's equilibria is as follows. The wish of humans to use shortest routes of a network contradicts fundamentally another wish of humans to drive under free flow conditions in the network. Therefore, the use of the classical Wardrop's equilibria can provoke the occurrence of congestion in urban networks already at relatively small total network inflow rates $Q<C_{net}$.

Traffic Breakdown in Networks through the Use of Wardrop's UE: Numerical Simulations

Here, the above general conclusion that already at relatively small total network inflow rates $Q<C_{net}$ the application of the Wardrop's equilibria leads to the occurrence of congestion in urban networks is illustrated with the use of numerical simulations made for a simple three-route network model (Fig. 1).

Figure 1: Model of simple network. (a) Sketch a network model with three routes and three on-ramp bottlenecks with on-ramp inflow rates $q_{on\ 1}$, $q_{on\ 2}$, and $q_{on\ 3}$, respectively. (b) Time-dependence of the inflow rate $q_{O}(t)$ used in simulations. The total network inflow rate $Q=q_{O}+q_{on\ 1}+q_{on\ 2}+q_{on\ 3}$.

Figure 2: Simulations of dynamic traffic assignment in three-route network (Fig. 1) based on Wardrop's UE under condition $Q<C_{net}$. (a—c) Speed in space and time on the three routes of the network. (d) Time functions of travel times $T_{s}^{(UE)}(t)$ on different routes ($s=1,2,3$).

It is assumed that at any time instant the total network inflow rate $Q$ is smaller than the network capacity $C_{net}$, i.e., condition $Q<C_{net}$ (9) is satisfied. Nevertheless, as it can be seen in Fig. 2, traffic congestion occurs in the network. It has been found that there is a random process of sequences of the congested pattern emergence and dissolution on different routes with resulting oscillations of route travel times (Fig. 2).

Contrarily, simulations show that at the same total network inflow rate $Q$ as that in Fig. 2 no congestion occurs and free flow persists in the whole network under application of dynamic traffic assignment based on the network throughput maximization approach (Fig. 3).

Figure 3: Simulations of dynamic traffic assignment in three-route network (Fig. 1) based on the network throughput maximization approach at the same set of model parameters as that used in Fig. 2. Time functions of travel times $T_{s}^{(TM)}(t)$ on different routes ($s=1,2,3$).

See also

• Active traffic management
• Fundamental diagram
• Intelligent transportation system
• Microscopic traffic flow model
• Traffic bottleneck
• Traffic flow
• Traffic wave
• Traffic congestion
• Traffic congestion: Reconstruction with Kerner’s three-phase theory
• Three-phase traffic theory
• Kerner’s breakdown minimization principle
• Transportation forecasting

This article "Network capacity" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Network capacity. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

📰 Article(s) of the same category(ies)

• N-Dimensional rotation matrix generation algorithm
• Blancride
• Origin and disparities in technological advancements & infrastructure compared to automobile safety sector
• Phase Retrieval
• List of OECD countries by road network size
• Paul A. Singh
• New Mobility Agenda

---