MODELLING OF VEHICLE ARRIVALS AT UNCONTROLLED INTERSECTIONS
Most of the studies conducted on vehicle arrivals have found that fitting or non-fitting of a probability distribution depends upon the traffic volume. Poisson distribution fits in low volume of arrivals and negative binomial in higher ranges of traffic volume. The present study shows that mean rate of arrival and variance of arrivals from mean are the controlling factors for a probability distribution to describe the arrival pattern of the vehicles. It is found that the binomial distribution fits the vehicle arrival patterns when coefficient of variation is less than 1. Poisson distribution best defines the vehicle arrivals when coefficient of variation lies between 1 and 1.3, inclusive of both values. Negative binomial distribution provides a good fit for higher values of coefficient of variation. At coefficient of variation equal to 1.3, both Poisson distribution and negative binomial distributions fit the observed vehicle arrival pattern equally satisfactorily, indicating a transition state.
Intersections are the important part of a highway system governing efficiency, speed, capacity and safety of traffic operations. They are also the critical locations where major accidents, delays and major bottleneck for smooth flow of traffic fake place. The heterogeneous nature of road traffic adds to the complexity involved at uncontrolled intersections. This heterogeneous traffic is mainly dominated by two-wheelers, three-wheelers and bicycles owing to their low initial and operating cost, need of small space for operation and storage, and flexibility to go where other vehicles cannot go. These vehicles are responsible for high accident rates on urban roads, as they lead to congestion and interfere with the movement of pedestrians and other vehicles. Lack of proper control and lane discipline makes traffic behavior further complex and difficult to predict.
The present study was undertaken to model the vehicle arrival patterns using different probability distributions on urban and semi-urban uncontrolled intersections. Data were collected at 18 single lane and two lane approaches of uncontrolled intersections. The effect of coefficient of variation, approach volume and traffic composition were studied on fitting of a specific probability distribution to the observed vehicle arrival pattern.
2.0 LITERATURE REVIEW
Most of the models for analyzing the traffic flow on intersections have been developed in countries having homogeneous traffic conditions. Schul (1955)’ modified the Poisson distribution to establish closer agreement between theoretical and observed spacing in case of rare events. Goode et al. (1956)2 generated the vehicle arrivals on the basis of Poisson distribution for a signalized intersection with two-lanes in each direction. Mahalel and Hakkert (1983)3 developed a model for four-lane inter-urban divided highways describing the sequence of vehicle arrivals in adjacent lanes as a Markov renewal process. Kumar (1986)4 used cumulative probability distribution for vehicle arrivals for developing a simulation model for controlled intersection in mixed traffic flow. Rao and Rengaraju (1995)5 observed that for mixed traffic condition, the Poisson distribution fitted the observed vehicle arrivals for traffic volume up to 500 veh/h/lane and for exceeding flows up to 1000 veh/h/lane negative binomial distribution was appropriate. A multivariate distribution was suggested for higher traffic volumes.
Most of the above studies were conducted on homogeneous traffic with low percentage of slow moving vehicles. In a mixed traffic situation prevailing in India and other developing countries, the behavior of traffic is extremely difficult to predict. Due to poor structure of regulatory system proper lane discipline is not observed by the vehicles and they can move abreast utilizing every. possible gap in lateral and longitudinal direction. Therefore, it will be interesting to study the vehicle arrival characteristics at uncontrolled intersections in mixed traffic condition.
3.0 DATA COLLECTION
The basic consideration in selection of an intersection was the variation in traffic volume, traffic composition and number of lanes (one lane approach and two lane approach). The data were collected at 18 approaches of uncontrolled intersections in Delhi, Lucknow, Roorkee, Trivendrum and Warrangal cities of India. Eight approaches were with one lane in each direction and rest were with two lanes in each direction. The video recording technique was used to collect the data at about 40-50 m upstream of the intersection for about two hours on a typical weekday. A white line was drawn in transverse direction for reference purpose. The vehicle arrival, traffic volume and composition details were extracted from the recorded film by playing it in laboratory. The data covered the traffic volume ranging from 120 veh/h/lane to 2484 veh/h/lane.
4.0 DATA ANALYSIS
For a meaningful analysis of data, all vehicles were classified into five categories namely car, heavy vehicles (bus and truck), 2-wheelers, auto rickshaw and bicycle. Table 1 and 2 provide details of traffic composition and volume at single lane approaches and double lane approaches, respectively. Auto rickshaws were almost absent in all sections of single lane approaches while these were present in good proportions in five locations of two lane approaches. Similarly, heavy vehicles were very few at two lane approaches while these were in appreciable proportions at all sections of single lane approaches. Due to poor lane discipline imposed on road users in India, lane wise count of vehicles was not possible in the field. Therefore, total traffic approaching the intersection in case of a two lane approach was equally divided in two lanes to get traffic volume in each lane.
Therefore, total traffic approaching the intersection in case of a two lane approach was equally divided in two lanes to get traffic volume in each lane.
The selection of a suitable time interval is important since some traffic phenomena may follow random distribution when observed for an interval of one length but non random when observed for an interval of a different length (Drew, 1968)6. A time interval of 10 seconds was used to formulate the frequency tables for all the vehicles at 18 approaches selected for the study. The mean rate of arrival and variance of arrivals from mean were computed for each approach. These are given in Table 3.
Theoretically, Poisson distribution defines the number of random events in a specified time interval (Johnson and Kotz, 1969)7. The probability ofx arrivals P(x) in a time interval t is given by recurrence formula as in equation (1).
respectively. Similar analysis was done at other sections also, where mean and variance were not equal. Negative binomial distribution provided a good fit for the arrivals at approaches where variance of arrivals from mean was greater than mean rate of arrival. Similarly, binomial distribution defined the arrival patterns satisfactorily at approaches where mean was greater than variance of arrivals. Figures 2 and 3 show the difference between the observed arrivals and the arrivals estimated by binomial and negative binomial distribution respectively, at approach XVIII and approach IV. The Chi-square test of goodness of fit was used throughout the study to test the goodness of fit of specific statistical distribution to the observed data. Table 7 shows the Chi-square values for all the 18 approaches selected of the study. It was observed that traffic volume and traffic composition do not provide of both values. For higher value of coefficient of variation, negative binomial distribution provided a good fit. At coefficient of variation equal to 1.3, both Poisson and negative binomial distributions fit the observed vehicle arrival pattern equally satisfactorily, indicating a transition state. Figure 4 shows the fitting of these distributions at approach I where coefficient of variation was 1.3. The transition value for binomial and Poisson distribution could not be obtained as no approach had the coefficient of variation between 0.75 and 1.00. But it must certainly lie between these two values.
The available literature indicates that Poisson distribution fits the vehicle arrival distribution on the approaches of uncontrolled intersections where traffic volumes are less than 500 vehicles/hr/lane. For greater volumes up to 1000 vehicles/hr/lane negative binomial distribution provides a good fit and for still higher volumes no statistical distribution is able to define the vehicle arrival characteristics (Rao and Rengaraju, 1995)5. The present study shows that volume of an approach is not an important factor for defining the vehicle arrival patterns. It is observed that modelling of vehicle arrival patterns at uncontrolled intersections can be done on the basis of coefficient of variation alone. The knowledge of complete pattern in terms of mean rate of arrival and variance of arrivals from mean rate is important to model the vehicle arrivals. It is further observed that when coefficient of variation is less than 1 binomial distribution fits the vehicle arrival patterns, irrespective of traffic volume. Poisson distribution best defines the vehicle arrivals when coefficient of variation lies between 1 and 1.3, inclusive of both values. Negative binomial distribution provides a good fit for higher values of coefficient of variation. At coefficient of variation equal to 1.3, both Poisson and negative binomial distributions fit the observed vehicle arrival pattern equally satisfactorily, indicating a transition state.
1. Schul, Andre. (1955). “The Probability Theory Applied to Distribution of Vehicles on Two-Lane Highways”, The Eno foundation for highway traffic control, Saugatuck.