Abstract
Based on the stochastic concept for freeway capacity, the procedure of capacity estimation is developed. Due to the fact that it is impossible to observe the value of the capacity and to obtain the probability distribution of the capacity, the product-limit method is used in this paper to estimate the capacity. In order to implement estimation of capacity using this technology, the lifetime table based on statistical methods for lifetime data analysis is introduced and the corresponding procedure is developed. Simulated data based on freeway sections in Beijing, China, were analyzed and the results indicate that the methodology and procedure are applicable and validated.
1. Introduction
In the last decades, the capacity of a freeway section is most taken as a constant value. For example, in the currently published Highway Capacity Manual (HCM2000) [1], it was still defined as the maximum hourly rate at which vehicles reasonably can be expected to traverse a point or a uniform section of a lane or roadway during a given time period under prevailing roadway, traffic, and control conditions. However, more and more investigations [2–11] showed that the traditional definition of capacity cannot explain the traffic phenomena such as traffic breakdown occurring at various flow rate levels. And it is shown that the freeway capacity is not a deterministic but a stochastic phenomenon.
Despite the fact that traditional definition of capacity has played an important role in traffic planning, design, and operational analysis, it was challenged due to the inadequacy and impracticality. As a result, a stochastic concept for freeway capacity was defined and new methods were suggested [6–8]. But how to estimate capacity based on the new concept and methods was not discussed in detail.
The aim of this paper is to develop a procedure for estimating the capacity of a freeway section based on the stochastic concept for freeway capacity. To fulfill this purpose, the traffic flow rate is divided into several intervals according to the surveyed data and requirement of capacity estimation precision. Based on the data, the distribution of traffic breakdown flow was estimated using maximum likelihood estimation. Then, the definition of capacity based on the probability of breakdown was used and the capacity compute model was developed. In order to illustrate how to use the method and traffic breakdown data to estimate the capacity of a freeway section, the simulation result was provided.
2. Literature Review
In order to understand the freeway capacity, researchers have made great efforts to model and describe stochastic nature of freeway capacity and breakdown phenomenon. Polus and Pollatschek [2] studied the stochastic nature of freeway capacity and used Gamma distribution to describe the capacity. But the definition of capacity is still based on the maximum flow rates. Lorenz and Elefteriadou [6, 7] examined the process of breakdown in detail and noted that the current HCM freeway capacity definition does not accurately reflect the relationship between breakdown and flow rate. They also suggested that freeway capacity should be described by incorporating a probability of breakdown component in the definition. Brilon et al. [8] introduced a new understanding of freeway capacity as the traffic flow below which traffic still flows and above which the traffic breaks down. In their study, a traffic breakdown is regarded as a failure event and the Product Limit Method for lifetime data analysis was used to estimate the capacity. Elefteriadou et al. [9] studied the breakdown flow rates. It was found that breakdown may occur at flows lower than the capacity and that breakdown is a probabilistic event. A linear model for describing the process of breakdown at ramp-freeway junction was developed. Elefteriadou and Ponlathep [10] examined the breakdown flow, prebreakdown flow, and maximum discharge flow to determine which one would be more appropriate for use in defining the “capacity” of a freeway. It was recommended that the breakdown flow be used. But they also pointed out that the numerical value of each of these three parameters varies and their range is relatively large, in the order of several hundred veh/h/lanes. Ozbay and Ozguven [11] provided the parameter estimating method for capacity probability distribution. It was concluded that including random component in the model results in better representation of observed data and thus improves understanding of real-life situation.
From the efforts of these researches, it is concluded that there is variability in the maximum flow rates and that the breakdown flows vary greatly. The breakdown phenomenon and breakdown flow rate should be considered in the model. So, the traditional definition of freeway capacity is not proper, and the definition based on probability distribution should be the better one.
3. Methodology
According to the definition of traffic breakdown, when the traffic flow exceeds the capacity, the traffic will break down. Hence, the capacity cannot be observed directly. The breakdown flow rate which is defined as the flow rate [7] (expressed as a per-lane, equivalent hourly rate) observed immediately prior to breakdown is used to estimate the capacity. Due to the fact that the breakdown flow rate is not the real observed value of capacity, the lifetime analysis method was used to model the capacity distribution and based on it a procedure is suggested to estimate the capacity.
3.1. The Theory of Lifetime Data Analysis and Product Limit Method
In order to capture the real-life situation of the freeway capacity, a lifetime analysis application was proposed by Brilon et al. [8]. The capacity was regarded as a random variable having its distribution as where is the probability function of capacity , is the freeway capacity, and is the flow rate.
According to the definition of breakdown, when the traffic flow rate reaches the capacity , breakdown will occur. Therefore, the is also the probability that the breakdown occurs at the flow rate .
Analogically, the capacity survival function was defined to describe the probability that when traffic flow is above capacity, the traffic breakdown does not occur: where is the survival function.
In addition, the “product-limit method” [8] can be used to estimate the capacity survival function and capacity distribution function as where is the estimate of survival function, is the flow rate, is the traffic flow rate in interval , is the number of intervals with traffic flow rate and breakdown does not occur, , is the observed number of breakdown at a flow rate of , , and is the estimated probability function of capacity .
Equation (3.3) is the basis of estimating freeway capacity distribution, and it is directly transplanted from the lifetime data analysis. However, there is some difference between traffic data and lifetime data. First, to the lifetime data, the real individual death time can be observed in theory, but to the traffic flow, breakdown often lags traffic flow rate (traffic breakdown does not occur immediately after traffic flow overruns the capacity), and the observed time interval has a great effect on the analysis result. Second, there are “censored” observations which are used to explain that some individuals’ explicit lifetimes are not observed in the lifetime data. In this sense, all traffic flow rates are “censored” data due to the reason that the capacity is not deterministic and it cannot be observed directly. If all the flow rates are taken into account, the capacity will be underestimated because that breakdown only occurred at few time intervals and it even can be taken as a small probability event. For example, for some days, the breakdown only occurred during morning and afternoon peak hours. If the time interval is 5 minutes, the probability of breakdown is only 1/144. Thereby, it is not appropriate to use observed traffic flow rate for all time intervals to estimate the capacity. In other words, only breakdown flow rates should be used.
3.2. Estimation of the Freeway Capacity Distribution
Based on the above analysis, only the breakdown flow rates are used to estimate the capacity. A procedure is developed as follows by using lifetime table method and theory in survival analysis [12].
Step 1. The breakdown traffic flow rates are divided into intervals and the th interval is noted as , (), where can be determined by , where is the observed traffic breakdown flow rate at which the th traffic breakdown occurs, is the number of observed traffic breakdowns, is a constant greater than 0; , where is the interval length ( can be determined according to the requirement of estimation precision), .
Step 2. Calculate the statistics. Let be the number of traffic breakdowns with a traffic flow rate , the number of traffic breakdowns with a traffic flow at the interval of , the probability that traffic breakdown does not occur with traffic flow rate , the probability that traffic breakdown does not occur with flow rate of while on the condition that traffic breakdown does not occur with flow rate of , and the probability that traffic breakdown occurs with a traffic volume at the interval of while on the condition that traffic breakdown does not occur with flow rate of .
Then the following results can be obtained:
Step 3. Estimate the distribution of freeway capacity.
According to the knowledge of the probability theory [12], the number of traffic breakdowns which are observed on the volume interval has a multinomial distribution
Inputing (3.7) and (3.8) into (3.10), it is derived that where . Using the maximum likelihood estimation (MLE) method, the MLE of and can be obtained: where . And freeway capacity can be estimated by (3.13).
In order to use the procedure simply, the “lifetime table” for capacity can be made referring to lifetime table methods used in the survival analysis [12]. Table 1 provides the lifetime table for capacity analysis, and the distribution of capacity can be estimated using it.
3.3. The Capacity Definition and Calculation Based on Traffic Breakdown Probability
According to the definition of freeway capacity based on the breakdown probability, “the rate of flow along a uniform freeway segment corresponding to the expected probability of breakdown was deemed acceptable under prevailing traffic and roadway conditions in a specified direction” [6]. The principle and method of estimating the freeway capacity can be illustrated as follows for the given acceptable breakdown probability , there exists a value of “” and flow rate “” which satisfies , and ( can be derived from formula (3.13) or Table 1. Then the value of “” is one of the estimates of capacity and it can be computed as follows.(1)If , the capacity can be estimated by (2)If , the capacity can be estimated as
Applying (3.14) or (3.15) to estimate capacity, the accuracy is closely related to the division of interval (). In order to improve the accuracy, one way is to divide more intervals.
4. Empirical Analysis
4.1. Field Data Collection and Data Analysis
The field data were collected on the 3rd ring in Beijing City, China, by using microwave detectors. The posted speed limit on the 3rd ring is 80 km/h. Traffic volume is generally heavy and breakdowns are typical during morning and afternoon peak periods. One-month-period data are used in this paper for both traffic directions.
Figure 1 illustrates volume and speed changes on time-series plot. It is found from the figure that when traffic breakdown occurs, the speed of traffic flow will drop suddenly and the average speed will be less than a given threshold. Tables 2 and 3 represent the statistical results of the speed before and after breakdown in 10-minute analysis intervals. According to the analysis results, it is found that the speed value is lower than 35 km/h after breakdown and that the average speed is higher than 40 km/h.
Tables 4 and 5 are the statistical results of the flow rates before and after breakdown in 10-minute analysis intervals. It can be found that the average of flow rates after breakdown is higher than the one before breakdown and that the flow rates after breakdown are more stable then breakdown flow rates.
4.2. Simulation Analysis Based on Field Data
Based on data collection and data analysis, the traffic flow is simulated by Monte-Carlo method and simulated data were used to validate the method and to demonstrate how to estimate the freeway capacity using the procedure. Tables 6, 7, and 8 list the estimated distributions of the freeway capacity for the inside lane, median lane, and shoulder lane, respectively.
If the acceptable breakdown probability is taken as , then the estimated capacity is 1950 veh/h/lane, 1870 veh/h/lane, and 1850 veh/h/lane, respectively, for the three lanes. Also, the capacities for the three lanes are calculated using speed-volume relationship and the results are 2060 veh/h/lane, 2040 veh/h/lane, and 1990 veh/h/lane (The probability that breakdown occurs with the flow rates 2060 veh/h/lane, 2040 veh/h/lane, and 1990 veh/h/lane is also obtained as 0.63, 0.77, and 0.64), which is more than the expectation values of the breakdown flow rates. Therefore, the capacity based on the breakdown probability is more moderate than the traditional defined capacity comparatively.
5. Conclusion and Future Research
Based on the exiting research, the breakdown flow rate is determined to be used to study the capacity of the freeway sections in this paper. Due to the fact that, when the traffic flow approaches the level of capacity, breakdown will occur, it is impossible to observe the value of the capacity. The product-limit method is suggested to estimate the capacity. In order to complete this method, the lifetime table method and the estimate procedure are suggested. The results indicate that the lifetime table method and the suggested procedure are feasible and reasonable.
However, there are still some problems to be solved in the future.(1)The effect of geometric characteristics and traffic condition on the breakdown probability should be studied. Only by this, the research result may be applied generally.(2)An acceptable breakdown probability value should be determined. This value reflects the balance between risk of breakdown and higher traffic volume. So, it is determined by the operating agency and questionnaires should be conducted.(3)The definition of breakdown flow rate should be studied. Although, in this paper, the breakdown flow rate is defined as the flow rate observed immediately prior to breakdown, there are still some questions need to be answered, such as should the traffic flow be obtained for 5-minute, 10-minute, or 15-minute time interval prior to breakdown?