### 7.4.3 Diffusion Theory

Pacey (1956) presented a simple, purely kinematic model of platoon dispersion, which considered the situation of normally distributed speeds in the traffic stream and argued that, in this case, the dispersion of a traffic platoon could be described by the dispersion in speeds. In developing the theory, Pacey also made the following simplifying (and somewhat unrealistic) assumptions:

- each vehicle travels at a constant speed
- vehicle speed is independent of position in the platoon
- there are no impediments to any one vehicle overtaking another.

In spite of these questionable assumptions, Pacey (1956) showed that the theory was quite successful in predicting flow profiles.

A presentation and extension of the theory by Grace and Potts (1964) starts with Pacey’s assumptions, including that of normally distributed speeds, with mean m and standard deviation σ. It is further assumed that, at the start of the signal green phase (time t = 0), the traffic density k(x,0) is known, where x represents the position along the road from the signal stop line. The authors then extend the theory by noting that, with a change of variables, the traffic density at location x and time t can be obtained as the solution to the standard one-dimensional diffusion equation:

\[\frac{\partial k}{\partial t} + m \frac{\partial k}{\partial X} = \alpha ^{2} m^{2} t \frac{\partial ^{2} k}{\partial \text{ x }^{2} } \] | 7.20 |

with the diffusion constant \(\alpha \) being equal to \(\sigma /m\), the coefficient of variation of the distribution of vehicle speeds in the platoon.

Seddon (1972a) presents the diffusion theory in terms of the traffic flows at the stop line and at a downstream location (where the level of dispersion is of interest) and of the vehicle travel times between those two points. Seddon notes that, if the distribution of vehicle speeds is assumed to be normal, it is possible to derive the distribution of travel times \(g(\tau ) d\tau \) between the two points. If the flow past the first point in the time interval t to t+dt is q_{1}(t)dt then, of these vehicles, there will be \(q_{1} (t) g(\tau ) dt d\tau \) that will pass the downstream point at time \(T = t+\tau \). Overall, therefore, the flow past the downstream point in the time interval T to T+dT will be:

\[q_{2} (T) dT = \int q_{1} (t) g(T-t) dt dT \] | 7.21 |

the integration being over all values of t for which q_{1}(t) exceeds zero.

In practical applications a more convenient expression of this dispersion relationship may be the discrete form:

\[q_{2} (j) = \sum _{i}q_{1} (i) g(j-i) \] | 7.22 |

Where i and j are counts of discrete intervals of time at the first and second points respectively. In words, this expression says that the flow in the j^{th} interval at the second point is the sum, over all values of i, of the flow in the i^{th} interval at the first point multiplied by the probability of a travel time between the two points of j–i intervals.

Seddon (1972a) fitted the model to field data by selecting values of m and \(\sigma \), the mean and standard deviation of platoon speeds to derive the travel time function \(g(\tau )\) giving the best fit between observed and predicted arrival patter ns at the downstream point. He found the model then produced good predictions of arrival patterns at other downstream points. Denney (1989) and the references already noted in this section provide further detail for the interested reader.