Module 4 : Macroscopic And Mesoscopic Traffic Flow Modeling
Lecture 18 : Cell Transmission Models
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CTM: Network Traffic

General

As sequel to his first paper on CTM, Daganzo (1995) published first paper on CTM applied to network traffic. In this section application of CTM to network traffic considering merging and diverging is discussed. Some basic notations: (The notations used from here on, are adopted from Ziliaskopoulos (2000)) $ \Gamma^{-1}$ = Set of predecessor cells. $ \Gamma$ = Set of successor cells.

Network topologies

The notations introduced in previous section are applied to different types of cells, as shown in Figures 1, 2 & 3. Some valid and invalid representations in a network are shown in Fig 7 & Fig 6.
Figure 1: Source Cell
\begin{figure} \centerline{\epsfig{file=qfSourceCell.eps,width=8 cm}} \end{figure}
Figure 2: Sink Cell
\begin{figure} \centerline{\epsfig{file=qfSinkCell.eps,width=8 cm}} \end{figure}
Figure 3: Ordinary Cell
\begin{figure} \centerline{\epsfig{file=qfOrdinaryCell.eps,width=8 cm}} \end{figure}
Figure 4: Merging Cell
\begin{figure} \centerline{\epsfig{file=qfMergingCell.eps,width=8 cm}} \end{figure}
Figure 5: Diverging Cell
\begin{figure} \centerline{\epsfig{file=qfDivergingCell.eps,width=8 cm}} \end{figure}
Figure 6: Invalid representations
\begin{figure} \centerline{\epsfig{file=qfInvalid.eps,width=8 cm}} \end{figure}
Figure 7: Valid representations
\begin{figure} \centerline{\epsfig{file=qfValid.eps,width=8 cm}} \end{figure}

Ordinary link

Consider an ordinary link with a beginning cell and ending cell, which gives the flow between two cells is simplified as explained below.
Figure 8: Ordinary Link
\begin{figure} \centerline{\epsfig{file=qfOrdinaryLink.eps,width=8 cm}} \end{figure}

$\displaystyle y_k(t) = min(n_{Bk}(t), min[Q_{Bk}(t), Q_{Ek}(t)], \delta_{Ek}[ NEk(t) - nEk(t)])$ (1)

where, $ \delta = w/v$. $ y_k(t)$ is the inflow to cell Ek in the time interval ($ t$,$ t+1$). Defining the maximum flows that can be sent and received by the cell i in the interval between $ t$ to $ t+1$ as $ S_I(t) = min (QI,nI)$, and $ R_I(t) = min (Q_I,\delta_I,[N_I- n_I])$. Therefore, $ y_k(t)$ can be written in a more compact form as: $ y_k(t) = min(S_{Bk},R_{Ek})$. This means that the flow on link k should be the maximum that can be sent by its upstream cell unless prevented to do so by its end cell. If blocked in this manner, the flow is the maximum allowed by the end cell. From equations one can see that a simplification is done by splitting $ y_k(t)$ in to $ S_{Bk}$ and $ R_{Ek}$ terms. 'S' represents sending capacity and 'R' represents receiving capacity. During time periods when $ S_{Bk} < R_{Ek}$ the flow on link k is dictated by upstream traffic conditions-as would be predicted from the forward moving characteristics of the Hydrodynamic model. Conversely, when $ S_{Bk} > R_{Ek}$, flow is dictated by downstream conditions and backward moving characteristics.