RELATED RESOUCES:
Many theoretical analyses of intersections assume that vehicles arrive at an intersection in a random fashion. The ability to verify that traffic arrives in a statistically random pattern is of great value to the traffic engineer. For example, procedures from the Highway Capacity Manual for the analysis of signalized intersections require the analyst to state whether the traffic stream follows a random pattern or is affected by some other source, such as a signal upstream from the approach. The Poisson distribution is a simple, random, statistical distribution. It is referred to as a counting distribution used to predict the number of occurrences of an event within a certain interval. Often, when used in traffic engineering, the occurrences are vehicle arriving, and the time interval is cycle length. The Poisson equation can be used to estimate the number of vehicles arriving at an intersection within a given time span so that the amount of green needed to process a predicted number of vehicles can be estimated. In conjunction with the saturation flow rate, the Poisson distribution is used to estimate the number of signal cycles that will successfully process the approach traffic and the number that can be expected to fail. Also, when left-turning traffic is of concern, the length of a left-turn lane can be estimated by using this random distribution. This lab session has two objectives. First, it provides a procedure for estimating the randomness of the traffic stream by comparing field measurements to computed theoretical values. Secondly, it demonstrates how a statistical distribution can be used in conjunction with field-measured saturation flow rates, startup delays, and volumes to calculate the minimum green time for an approach to an intersection.
The equipment list for data collection procedure is a stopwatch (cell phone) and note-taking materials.
Select a signalized intersection that has sufficient traffic to accomplish the session
The student group must first decide which lanes or lane groups should be measured. A lane group can be thought of as one lane or group of lanes from which drivers can complete the same move during the same phase. For instance, an exclusive left-turn and a combined right-turn and through-lane are examined as two separate lane groups. The Poisson distribution would be applied separately for each of the two lane groups. However, a combined left-turn and through-lane and a combined right-turn and through-lane approach controlled by a single-signal phase would be one lane group, and the lab session would look at the distribution for the entire lane group. Each of the lanes should have sufficient traffic for 0 to 6 or more vehicles to arrive per twenty-second interval. This will provide a sufficient number of data points for the analysis.
For 100 consecutive twenty-second intervals, record the number of vehicles arriving at the intersection during each interval
The interval length will affect the quality of the results and should be chosen carefully. The twenty-second interval offers both enough observations and a reasonable time in the field. An interval shorter than 15 seconds would yield very few points for the probability curve and not allow a significant comparison to the theoretical Poisson distribution. On the other hand, longer interval lengths necessitate longer and impractical field-data collection periods. The data collection will be done for each of the lanes separately but can be accomplished concurrently. The student group can accomplish this task by designating a point on the approach to the intersection for determining whether a vehicle has arrived. This point is best chosen upstream of the intersection but before the point representing the maximum queue for that lane. This will allow vehicles arriving at the intersection to be counted without interference from the queue of vehicles waiting at the intersection when the signal is red. In many instances, the best location for this point will be the beginning of the separate left-turn lane. A data collection form is attached to this handout.
Check your work before leaving the field
Before leaving the field, the data collection form should be checked to see that all field data have been collected correctly. Any unusual observations should be listed as comments.
Summarize field data on form provided
A Poisson Distribution Data Summary Form is attached to this handout. Key to the understanding of the Poisson distribution is to take great care in the treatment of the collected materials; small errors can have a serious effect on results. For the 100 observations recorded, sum those having the same number of vehicles observed, and enter it in the corresponding row. When complete, sum the values in Column 2 and check to see that the total equals the number of intervals observed. Next, calculate the number of vehicles observed. Multiply the values in Column 1 by the values in Column 2, and enter the product in Column 3. The sum of the values in Column 3 is the total number of vehicles arriving during the observation period. Divide this total by the total number of intervals observed to yield the average number of vehicles arriving during the 20-seconds interval. This is the average arrival rate (m or AAR) and will be used in the Poisson equation to calculate theoretical probabilities.
Calculate observed probabilities
An assessment of traffic randomness requires a comparison of the observed distribution of arrivals to the theoretical distribution of arrivals. To do this, it is necessary to compute the observed probabilities. The observed probability is a measure of the likelihood that an observer randomly choosing a 20-second interval would witness a particular number of vehicles arriving. The values are a result of dividing the observed frequency for each level by the total number of observed intervals, or Column 2 divided by the sum of Column 2. The cumulative probability represents the likelihood a group will count a particular number of vehicles, or less, during a randomly selected interval. It is computed by adding all of the observed probabilities for that value and all lower values.
Compute theoretical Poisson probabilities
Once the actual cumulative probability has been computed, the theoretical values are calculated using the Poisson equation. If the distribution of arrival is a random one, then the theoretical and observed values should be close. The form of the Poisson equation to be used is as follows:
\(Pr(X)= \frac{m^xe^{-m}}{x!}\)
where m is the average arrival rate, x is the number of vehicles expected to arrive and the constant e is the base for natural logarithms. For instance, the computed probability of exactly two vehicles arriving during an interval is \(0.2225\), or \(22.25%\), and is calculated using \(m = 1.24\) vehicles per interval, and x = 2 vehicles. The cumulative Poisson probabilities in the last column are calculated, similarly to the cumulative observed probabilities, by adding all values up to, and indicating, the row of interest. The probability that two or fewer vehicles will arrive is theoretically \(0.2894\) plus \(0.3588\) plus \(0.2225\) or \(0.8707\).
Compare observed probabilities to Poisson probabilities The statistically valid method to compare field observations to a theoretical distribution is to apply a Chi-squared test, the application of which is demonstrated here. The question to be answered is whether the collected data indicate that the Poisson equation is a good indicator of how vehicles arrive at the intersection. Also, recognizing the limited number of data gathered, would it be unreasonable to use a Poisson distribution to forecast future events, such as the change in the capacity of an approach because of a change in green time or to find the length of a turning lane? Comparing cumulative-observed and cumulative-theoretical probabilities in the example data summary yield some interesting results. For this example, the values compare favorably, especially at the higher levels. The predicted values are within three percent of the observed values. So, it appears reasonable to use the Poisson equation to predict arrivals on this approach. However, the Goodness-of-Fit must be applied before proceeding. In a nutshell, the Goodness-of-Fit test identifies significant differences between the expected values and the observed values by computing the statistic Chi-squared, \(\chi^2\), and comparing the computed value to the standard \(\chi^2\) value. The following equation is used for the computation: \[\chi^2 = \sum_{i=1}^k \frac{(o_i-e_i)^2}{e_i}\]
where the symbols o_i and e_i represent the observed and expected frequencies, respectively, for the i-th cell. The decision criteria should not be used if the expected frequency of a cell is less than five. The observed and expected frequencies computed using the Poisson equation are shown below. The Goodness-of-Fit test will be applied to this data to determine if the Poisson distribution properly reflects driver behavior on this approach. If the Poisson distribution is a good estimator of the arrivals, then the tabular or theoretical Chi-squared will be greater than the computed Chi-squared. In order to identify the proper Chi-squared, the number of degrees of freedom must be found. By definition, the number of degrees of freedom in a Chi-square goodness-of-fit test is equal to the number of cells minus 1. Computing the \(\chi^2\) using the stated equation yields the following:
Data Summary Number of Vehicles Observed Frequency Computed Frequency:
| # veh | Observed Freq. | Computed Freq. |
|---|---|---|
| 0 | 34 | 28.9 |
| 1 | 34 | 35.9 |
| 2 | 16 | 22.2 |
| 3 | 9 | 9.2 |
| 4 | 5 | 2.9 |
| 5 | 1 | 0.7 |
| 6 | 1 | 0.2 |
The student will notice that the values for 3, 4, 5, and 6 vehicles were combined because the number of observations (computed frequency) in the cells was less than 5.
\(\chi^2 =\frac{(34-28.9)^2}{28.9}+\frac{(34-35.9)^2}{35.9}+\frac{(16-22.2)^2}{22.2}+\frac{(16-13)^2}{13}\)
\(\chi^2=3.424\)
For this example, the average arrival rate was the only quantity obtained from the observed data used to compute the Poisson frequencies. Therefore, the number of degrees of freedom for this example is \(4 - 1 = 3\). From the table, \(\chi^2_{0.05}=7.81\) and \(3.424\ \le\ 7.81\), so it can be concluded that the Poisson distribution provides a good fit for the distribution of arrival rates.
Calculate maximum arrivals The next step in the analysis focuses on the application of the Poisson equation to the identification of the capacity of a lane group. The values presented will be hypothetical, but the reader may wish to refer to the saturation flow rate. Should a quick determination of an existing approach be desired, then the length of the cycle and the length of green time would be useful. Note that many signal timing procedures exist, and the application of a Poisson distribution is but one of them. Determining the capacity of an existing lane group, or the minimum green time required, can be accomplished using the Poisson equation. It is used to determine the highest number of vehicles that may be expected to arrive during a signal cycle. Knowing this number, the saturation flow rate and startup delay can be used to compute the minimum green time needed to process the traffic. First, however, the analyst must determine how often the signal should be able to process all of the demand. Since it is usually impossible to time a signal to process all levels of traffic all the time, a level of acceptance is chosen. For this example, the signal should exhibit sufficient green time to process the traffic for 9 out 10 cycles. The Poisson equation can be used to calculate the maximum number of vehicles expected to arrive during 90% of the cycles. So, timing the green-signal time to process this traffic would result in the green time being insufficient only 10% of the time. Calculating the maximum number of vehicles expected to arrive in 9 of 10 cycles is computed using the cumulative Poisson distribution. If the cycle length is 60 seconds and the demand volume is 180 vehicles per hour, then the average arrival rate is (180)(60)/3600 = 3 vehicles per cycle. Now, let us consider the class example. Calculating the probabilities of exactly 0, 1, 2, … vehicles arriving during the cycle and summing the values, 3 is the maximum number of vehicles expected. The computed probability of having 3, or less, vehicles arriving during the cycle is 0.963 or 96.3%. Therefore, allocating sufficient green time to process three vehicles should allow the approach to process all of the traffic 96.3% of the time.
Calculate minimum green time The minimum green time can be calculated using the saturation flow rate and startup delay for the approach. For this example, it may be assumed that the saturation flow rate is 1800 vehicles per hour of green time and the startup delay is 3 seconds. Using these values, the minimum green time is calculated by the equation 3 + (3600/1800) X = G, where X equals the number of vehicles which must be processed and G is the minimum green time needed to move the traffic. Substituting X = 3 vehicles, the equation yields a green time of 9 seconds.
Summarize and draw conclusions The student group should summarize the findings and draw conclusions from the effort. Questions to be addressed are:
Does the Poisson equation provide a good indication of driver behavior at this intersection?
Can it be used to predict the minimum green time needed?
Should observed frequencies be used for analysis at this location rather than the Poisson theoretical frequencies?
Does this effort indicate that statistical functions are useful in traffic engineering?