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PDF (DX094490.pdf) - White Rose Etheses Online

PDF (DX094490.pdf) - White Rose Etheses Online

66 q: the major road

66 q: the major road (circulating) flow (veh/sec). Figure 4.1 shows a typical example of a cumulative gap- acceptance distribution. Miller (1971) compared the last three methods using simulated data, and determined that the methods proposed by Bluriden et al (1962) and Drew (1967) gave very biased results, while the correction given by Ashworth (1968, 1970) did remove the bias and gave satisfact- ory results. Ashworth and Bottom (1977) carried out repeated observations on a number of drivers entering into major roads from a T-junction. That enabled them to build acceptance probability distributions for each driver. To obtain each driver's mean critical gap they fitted cumulative normal distributions on each driver's data. Blumenfeld and Weiss (1978, 1979), analysing the same data, used a shifted negative exponential distribution to describe each driver's behaviour. The mean value and the variance of the distribution can be expressed in terms of the two parameters which define each driver's distribution. 4.2.2 Raff's Critical Lag One of the first definitions of a gap-acceptance parameter was by Raff and Hart (1950). They only considered lags presented to the minor road flow. They defined as critical lag, L, the size lag for whichthe nurrber of accepted lags shorter than L is the same as the number of rejected lags longer than L. The value of L was determined graphically as shown in Fig. 4.2. They noted that if lags and gaps are considered together, the percentage of intervals accepted for a particular size is not a true measure of the proportion

67 of drivers who accept such gaps, since several rejected intervals, but only one acceptable, may be counted for each driver. Similar definitions were used by other researchers, specifically Drew (1967), Armitage and McDonald (1974), and Bendtsen (1972). Ashworth (1970) compared Raff's critical lag to the mean a of the critical gap distribution. When this distribution has variance s 2 , and the circulating (major road) flow is q veh/sec, the relationship is L = a - s 2q/2. Thus it is incorrect to equate the two parameters apart from the case of a constant critical gap associated with a step function. Miller (1971) arrived at the same relationship. He compared this method with other estimators of critical gaps to conclude that it is biased. 4.2.3 Other Methods to Determine the Critical Gap When the distribution of the acceptance probability is known Maximum Likelihood Estimates (LE) equations can be derived to give the maximum likelihood values of the gap acceptance parameters. Moran (1966) and Miller (1971) derived MLE equations assuming normal distributions. Miller compared his method to eight other estimators to conclude that the maximum likelihood method and Ashworth's method both gave satisfactory results, the NLE method being slightly more precise but, also, more laborious. Since then Maher and Dowse (1982) have used MLE methods (see section 4.4.2). Ramsey and Routledge (1973) evaluated the critical gap using a histogram of all offered gaps and a histogram of the accepted gaps. They assume that all drivers are consistent

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