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# PDF (DX094490.pdf) - White Rose Etheses Online

PDF (DX094490.pdf) - White Rose Etheses Online

## 98 counted, while

98 counted, while interval ones describe the distribution of the time intervals between events. In this section one counting distribution has been included, the Poisson distribution - and the following interval distributions: the negative exponential, the shifted negative exponential, the Pearson Type III and Schuhl t s composite headway model. 4.8.2.1 The Poisson Counting Distribution The use of this distribution in trafffic studies was introduced by Kinzer (1934), Adams (1936) and Greenshields et al (1947). This distribution gives the probability of any number of vehicles to arrive during a period of given length. If this probability is P(x), its mathematical formulation is where -m x P(x) = e m (eq. 4.17) m: the mean number of arrivals expected in the given time e: the base of natural logarithms = 2.71828 Figure 4.11 shows the distribution for m = 5. The Poisson distribution is appropriate for describ- ing discrete random events. Gerlough and Hucer (1975) note that it will provide satisfactory results when the traffic flow is light and it is not affected by any disturbing control systems. However, at high flow or when there is some cyclic disturbance the Poisson distribution does not describe the conditions adequately. The Poisson distribution has equal mean and variance. Therefore, if the observed data have markedly different mean and variance the Poisson distribution is not suitable.

99 4.8.2.2 The Negative Exponential Distribution The Poisson distribution Ls discrete. However, another traffic characteristic of interest is the interval between the occurrence of events, for example the gap size between successive vehicles along a road. Adams showed that P(0), i.e. the probability of zero arrivals using the Poisson counting distribution, is also the probability for a headway equal or greater than t, the time interval used in the Poisson distribution. If h is the headway then where P(h ^ t) shown in Fig. 4.12. Furthermore _qt q: the average flow (veh/sec). (eq. 4.18) The probability of a headway being less than t is -qt P(h < t) 1 - e (eq.4.19) The distributions of equatiors 4.18 and 4.19 are -qt1 -qt2 P(t 1 < h < t 2 ) = e - e (eq. 4.20) The negative exponential distribution predicts the greatest number of headways in the smallest time interval between t = 0 sec and t = t 1 sec, where t 1 is the time interval considered. This coincides with observations only when traffic flows are light and there are several lanes available to the traffic. The agreement becomes poor as soon as the traffic increases in intensity when interaction between vehicles

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