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Macroscopic Fundamental Diagram

A macroscopic fundamental diagram (MFD) is type of traffic flow fundamental diagram that relates space-mean flow, density and speed of an entire network with n number of links as shown in Figure 1. The MFD thus represents the capacity, $\mu (n)$, of the network in terms of vehicle density with $\mu _{1}$ being the maximum capacity of the network and $\eta$ being the jam density of the network. The maximum capacity or “sweet spot” of the network is the region at the peak of the MFD function.

Figure 1: Sample Traffic Flow Macroscopic Fundamental Diagram

Figure 2: Space-time Diagram for the ith link in a traffic flow network

Flow

The space-mean flow, ${\bar {q}}$ , across all the links of a given network can be expressed by:

${\bar {q}}={\frac {\sum _{k=1}^{n}d_{i}(B)}{nTL}}$, where B is the area in the time-space diagram shown in Figure 2.

Density

The space-mean density, ${\bar {k}}$ , across all the links of a given network can be expressed by:

${\bar {k}}={\frac {\sum _{k=1}^{n}t_{i}(B)}{nTL}}$, where B is the area in the time-space diagram shown in Figure 2.

Speed

The space-mean speed, ${\bar {v}}$ , across all the links of a given network can be expressed by:

${\bar {v}}={\frac {\bar {q}}{\bar {k}}}$, where B is the area in the space-time diagram shown in Figure 2.

Average Travel Time

The MFD function can expressed in terms of the number of vehicles in the network such that:

$n={\bar {k}}\sum _{k=1}^{n}l_{i}={\bar {k}}L$ where $L$ represents the total lane miles of the network.

Let $d$ be the average distance driven by a user in the network. The average travel time ($\tau$) is:

$\tau ={\frac {d}{\bar {v}}}={\frac {nd}{MFD(n)L}}$

Application of the MFD

In 2008, the traffic flow data of the city street network of Yokohama, Japan was collected using 500 fixed sensors and 140 mobile sensors. The study revealed that city sectors with approximate area of 10km^2 are expected to have well-defined MFD functions. However, the observed MFD does not produce the full MFD function in the congested region of higher densities. Most beneficially though, the MFD function of a city network was shown to be independent of the traffic demand. Thus, through the continuous collection of traffic flow data the MFD for urban neighborhoods and cities can be obtained and used to for analysis and traffic engineering purposes. These MFD functions can aid agencies in improving network accessibility and help to reduce congestion by monitoring the number of vehicles in the network. In turn, using congestion pricing, perimeter control an other various traffic control methods, agencies can maintain optimum network performance at the "sweet spot" peak capacity. Agencies can also use the MFD to estimate average trip times for public information and engineering purposes.