# Traffic management

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### 7.4.4 Recurrence Model

Using data collected by others (Hillier and Rothery 1967), Robertson (1969) developed an empirical platoon dispersion model using a discrete iterative technique. This recurrence model has received wide application in the various versions of the TRANSYT network optimisation package, which takes account of platoon dispersion in selecting signal offsets to minimise total delay over a road network.

The recurrence model considers traffic flows over a series of equal, small time intervals at two locations on the road, x1 (say, the stop line at a signalised intersection) and x2 (a point some distance downstream from x1 where the level of dispersion is of interest), relating flows at the two locations in different time intervals. Specifically, the recurrence relationship can be written as:

 $q_{2} (i+t) = F . q_{1} (i) + (1-F) . q_{2} (i+t-1)$ 7.23 where q1(n) = the flow at location x1 in time interval n q2(n) = the flow at location x2 in time interval n i = a time interval counter t = $$\beta \cdot T$$ where $$\beta =$$an empirical parameter < 1and T = average undelayed travel time (cruise time) x1 to x2 (in time intervals)with $$\beta \cdot T$$ representing the travel time of the head of a platoon F = smoothing factor = $$\frac{1}{1 + \alpha t}$$where $$\alpha =$$ an empirical parameter

Robertson (1969) fitted this model to field data finding that values of $$\alpha = 0.5$$ and $$\beta = 0.8$$ gave the best predictions of flows at downstream points.

Seddon (1972b) shows that Equation 7.23 can be written equivalently as:

 $q_{2} (j) = \sum _{i=1}^{j-t} q_{1} (i) F (1-F)^{j-t-i}$ 7.24

where j is a count of time intervals at the second point and j=i+t ; all other variables are as previously defined.

Seddon (1972b) points out the similarity between Equation 7.24 and Pacey’s diffusion model as expressed in Equation 7.22, the only difference being the replacement of the travel time function $$g(j-i)$$ with the probability function $$F (1-F)^{j-t-i}$$, which will be recognised as a geometric distribution commonly used to represent the number of failures in a two-outcome trial before the first success. Here it is the probability that a vehicle passing the first point in th ith interval will pass the second point in the jth interval. Seddon (1972b) found good fits of the recurrence model to field data, but with values of the calibration parameters a little different from those suggested by Robertson (1969).