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PDF (DX094490.pdf) - White Rose Etheses Online

PDF (DX094490.pdf) - White Rose Etheses Online

74 (iv) after estimating

74 (iv) after estimating q 5 and L, T was varied to give a least squares fit to the flow data; (v) t was assumed to be the reciprocal of the satur- ation flow of the arm from which the main circulating flow emerges. Method (iv) was described as the most consistent, with the disadvantage that for certain flow conditions it did not give satisfactory results; however Armitage and McDonald preferred to use method (v) as can be seen from equation 4.6 where the denominator of the right-hand side is the expression for the saturation flow. The other two parameters, q 5 and L, were estimated together by the method of least squares. Taking the simplest case of a single lane of traffic entering a roundabout, two straight lines were fitted to a plot of the number of entries (y) during each gap against the length, (x), of the gap. The line for x ^ L was y = 0, while for x ^ L, it was y = q*(x_L). This is illustrated by Fig. 4.6 for a 2-lane entry where the model is fitted to some sample data and compared with the conventional gap-acceptance step function model which uses parameters cL and 3. The model uses both accepted and rejected gaps. However it should be noted that all rejected gaps less than L have a zero contribution to the least squares value. Also all accepted gaps less than L have a constant contrib- ution since the line for x L cannot change slope being defined as y 0. Therefore those points have no influence on the slope of the line for x > L which determines q 5 . As L decreases more rejected gaps are contributing to the sum of the squares of differences, but it is not possible to know in

75 advance which rejected gaps shculd or should not be abstracted from the data. This results in a considerable number of rejected gaps which although abstracted from the data, are not utilized finally. It should be noted that the rejected gaps will be numerous and proportionally the majority of all the gaps, especially at high circulating flows. Therefore this method is very inefficient in the use of data which have to be manually abstracted. It should be noted that Fig. 4.6 refers to a 2-lane entry of a roundabout. The slope indicated by q 5 on the figure is in fact half the value of the actual slope. This is necessary in order to estimate the saturation flow per lane. Also, it should be noted that no rejected gaps less than L were included on the diagram. It is of interest to examine the relationship between the parameters q 5 and L, used by Armitage and McDonald, and the parameters critical gap, a, and move-up time, , as used in the present study. As can be seen from Fig. 4.6, the move-up time, , is the reciprocal of the saturation flow, and the critical gap, a, is related to L and q 5 as is shown in eq. 4.7 a = = 1 1 (eq. 4.7) (eq. 4.8) These two relationships allow the reinterpretation of the data given in Armitage and McDonald (1977) into the conventional parameters. These are included in Table 4.1.

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