TEMPORARY LOSSES OF HIGHWAY CAPACITY AND IMPACTS ON PERFORMANCE: PHASE 2

Title | Contents | Acknowledgements | Exec. Summary
1. Intro | 2. Approach | 3. Crashes | 4. Breakdowns | 5. Work Zones | 6. Weather | 7. Signal Timing
8. RR Crossings | 9. Toll Facilities | 10. PUD | 11. Results Summary | 12. Next Steps | 13. References

2.  Approach and General Methodology

2.1  Approach

The study uses traffic engineering modeling methods, the best available data, and engineering judgment to derive estimates of capacity losses and delays.  Because direct measurements are scarce and available data are generally incomplete, the validity of the estimates is dependent on the reasonableness of a number of critical assumptions.  The study used rigorous procedures and methodologies firmly grounded in traffic theory and practice, but attempted to keep the methods as straightforward and transparent as possible. Existing data from a variety of sources was used, as well as results from evaluation studies sponsored by the Intelligent Transportation System (ITS) Joint Program Office (JPO).  Reliable data was used where possible.  For areas where data was not available, data or findings from literature sources, reasonable assumptions, and Monte Carlo simulations were used.  The philosophy followed throughout the study was to rely on published peer-reviewed studies whenever possible and, when assumptions must be based solely on the researchers' judgment, to err on the side of underestimating losses of capacity and delay.  There is one general exception to this rule:  the assumption that traffic volumes are not changed by the occurrence of a TLC event.  This issue is discussed below.

The study's analytical framework is portrayed in Fig.2.  Events causing temporary capacity losses occur in an environment comprised of roadway characteristics, location, time, and ambient conditions (e.g., weather).  The characteristics of the event and its environment provide the information based on which traffic impact models can predict impacts.  These travel impacts include delay and the four Rs: (1) re-routing, (2) re-scheduling, (3) reduced mobility (foregone travel, cancelled trips), and (4) reduced reliability. In general, delay is probably the more useful, and certainly the more intuitive, measure of loss of functionality.  In general, an event will generate all five types of impacts, with the relative importance of each depending on the nature of the event and its context.  For example, an unexpected event such as a crash is likely to produce relatively less re-routing and re-scheduling than a work zone whose existence can be known in advance and which may persist for days, weeks, or even longer.

A critical distinction is made between the loss of capacity and its impacts.  Capacity is a measure of potential:  it describes the maximum sustainable throughput of a highway.  As such, it is independent of the highway's actual use.  Impacts, however, depend not only on the loss of capacity, but also on the volume of traffic on the highway when the loss occurs.  A crash occurring on an Interstate highway in the middle of the night will cause far less delay than, but possibly the same loss of capacity as, the same crash occurring during rush hour.  Delay is measured in vehicle-hours, which can be converted to person-hours by multiplying by an appropriate vehicle occupancy.  Capacity loss, on the other hand, is a loss of potential throughput (measured in vehicles per lane per hour, or vplph), integrated over time and a length of roadway. 

Another distinction is important to note. While capacity is typically given as a rate of vehicle throughput, the capacity reduction estimates presented in this study are not given as a rate. Since the general methodology used in this study attempts to estimate capacity reductions over a finite period of time and along a finite length of roadway with a given number of lanes affected, the algorithm produces an estimate of vehicles not serviced through those lanes during the given amount of time. Thus, capacity loss is given in units of vehicles rather than vehicles per hour per lane. The more typical unit of capacity, in terms of vphpl, was used in the algorithms to estimate total capacity loss and delay.

In the course of this study, methods were developed for estimating the impacts of temporary events on the loss of capacity and delay, but not for estimating the impacts of TLCs on the four Rs. Thus, the impacts of events on traffic volumes were not predicted.  For example, a heavy snowstorm might reduce traffic volumes drastically due to travelers re-scheduling or canceling planned travel.  Such impacts have not been estimated.  On the other hand, because normal traffic volumes are assumed, delay will be overestimated.  Thus, in general, the delay estimates presented here reflect, to an unknown degree, the other negative impacts of TLCs on re-scheduling, re-routing, reduced mobility, and reliability.  A high priority for future analysis should be to develop methods for analyzing all five types of impacts.

Figure 2. Analytical Framework

2.2  Basic Concepts of Capacity, Capacity Reduction, and Delay

This section describes the general concepts and methods used in this study to calculate the impacts of TLC events on freeways and major highways.  Methodologies to measure other adverse consequences, such as trip re-scheduling, trip re-routing, reduced mobility (i.e., trip cancellation), and to describe the impacts on reliability should be considered candidates for further research.

2.2.1  Determining Roadway Capacity

Roadway capacity is the maximum vehicular flow rate at a point on a segment of the roadway.  The capacity of a roadway segment with relatively homogeneous physical and operational characteristics is the maximum number of vehicles the segment can accommodate within a unit of time.  In cases where the capacity is different at different points on the segment, the point with the least capacity is considered to represent the capacity for the entire segment.  For example, for a roadway segment with a narrow bridge, the capacity of the narrow bridge is assumed to represent the capacity of the roadway segment. Usually, an access-controlled, divided, multi-lane highway can carry approximately 2,000 vehicles per hour per lane (vphpl).  This results in a total of approximately 48-thousand vehicles per day per lane or over 17.5 million vehicles per year per lane.

2.2.2  Estimating Capacity Reductions

Traffic flow theory asserts that, in a steady state, the flow-density curve of a roadway has a shape similar to the one shown in Fig. 3.  For a given demand Q1, the segment of freeway that is represented by the density-flow curve operates at a density D1 and at a speed S1, represented by the slope of the line from the origin to point P1.  The segment can also operate at higher densities and slower speeds (point P2), but with unstable flow.  The maximum of the flow-density curve is the segment's capacity C.

Figure 3.  Flow rate vs. density for a highway segment with capacity C and a highway segment with capacity C' = α C

Assuming a segment of freeway, with a given capacity C, and a section within that segment with a reduced capacity C' = α C, where a < 1, capacity C' was computed using the capacity reduction factor a for incidents obtained from Blumentritt et al. (1981) (Table 3).  For work zones, the entire term C' was obtained from the Highway Capacity Manual, Special Report 209 (TRB 1985) (Table 4).

Table 4 shows the freeway capacity at work zones as a function of the total number of lanes and the number of lanes open. More recently, Dixon et al. (1996) analyzed speed-flow behavior in work zones in North Carolina.  The study also presents a comparison with an earlier research conducted in Texas (see Table 5).  The differences in the results from those two states are less than 10 percent.

Figure 4 shows the flow-density curves for the freeway segment with capacity C (curve fd) and for the section with reduced capacity C' (curve fd').

Figure 4.  Flow rate vs. density for a highway segment

Two cases must be analyzed: one in which the demand on the highway segment is less or equal to the reduced capacity C', and one in which the demand is larger than C'.

Case 1: Demand Less Than or Equal to the Reduced Capacity

Figure 5.  Demand Q1 equal to or less than bottleneck capacity C' = α C

Assuming the bottleneck section is L' miles long, the average delay can be computed as follows.  The travel time (TT) in seconds for the bottleneck section under normal conditions (i.e., no incident) is

Equation 1:

TT = (L' ÷ S1) × 3,600

where S1 is the speed in mph obtained from curve fd for demand Q1, and 3,600 is merely a factor for converting hours to seconds. Under reduced capacity conditions, the travel time (in seconds) within the bottleneck section (TT') becomes

Equation 2:

TT' = (L' ÷ S2) × 3,600

where S2 is the speed in mph obtained from curve fd' for demand Q1.  The average delay each vehicle experiences (AD₁) is calculated as the difference between these average travel times:

Equation 3:

AD₁= TT' - TT = ((S1 - S2) ÷ (S1 × S2)) × L' × 3,600

where AD₁ is given in seconds per vehicle.

This statistic may be consequential for work zones where L' could be significant.  For incidents, it may be irrelevant (i.e., L' is almost 0), but it could be used as a proxy for the rubbernecking delays at the incident location.  To account for this effect, this study assumes a length of L' = 1/2 mile for incidents and computes the travel time delays on this segment.

Case 2: Demand Greater than the Reduced Capacity

The next case examined is where demand Q1 is larger than the capacity C' of the bottleneck section (Fig. 6).  If there were no loss of capacity, the highway would operate at point P1 with a density D1 and an average speed of S1 mph.  However, because of the bottleneck, only a demand equal to C' can be passed through the reduced capacity section, which will operate at density D2 and speed S2 (point P2). Downstream of the bottleneck, the freeway has a normal capacity C, and, with a demand C' (i.e., the demand that the bottleneck section can handle), it operates at point P3 with density D3 and speed S3 > S1.

Figure 6.  Demand Q1 larger than bottleneck capacity C' = α C

Equation 4:

Q1 - C' = SW × (D4 - D1)

SW = (Q1 - C') ÷ (D4 - D1)

2.2.3  Estimating Delays for Localized Events

Fig. 7 shows the effect over time that a loss of capacity can have on a section of freeway.  This can also be used to compute several statistics, such as total duration of the congestion produced by the incident, total number of cars affected, the maximum individual delay, and the average delay per vehicle.

Figure 7.  Cumulative flow vs. time for a highway bottleneck

In Fig. 7, it is assumed that, on a freeway with constant demand Q1, an event occurs at time Ts, reducing the capacity of a section of that freeway from C to C'.  Line P0-P4 represents the cumulative demand over the entire period from Ts to Te in which there is congestion on the freeway due to an event (the slope of this line is Q1).  The event ends at time Tc (i.e., the duration of the event is Tc - Ts), and during this period this section of freeway is operating at reduced capacity C'.  Line P0-P2 represents the cumulative capacity of the bottleneck (the slope of this line is C' < Q1).  Once the event ends (time Tc), the freeway returns to its normal capacity C (line P2-P4 represents the cumulative capacity under normal conditions; the slope of this line is C > Q1).

During the period Ts to Tc, the demand exceeds capacity, causing an accumulation of vehicles upstream of the bottleneck as discussed before.  When the capacity of the freeway is restored (time Tc), there is a queue with a length given by the difference in the ordinates of points P1 and P2 (i.e., CF1 - CF2).  It takes Te - Tc hrs to dissipate this queue once capacity is restored to C vph.  Therefore, the total duration of the event and its effects are calculated as Te - Ts, and the maximum individual delay is given by the difference of the abscissas of points P3 and P1 (i.e., T3 - Tc).  It can also be shown that the area enclosed between the cumulative demand line and the cumulative capacity lines gives the total delay.  Equations 6 and 7 show the areas of triangles P1-P0-P2 and P1-P4-P2, respectively, and equation 8 shows the total delay (TD).

Equation 6:

Equation 7:

Equation 8:

Note: The average delay per vehicle can be computed by dividing TD by the total number of vehicles involved (i.e., Q1 × (Te-Ts)).

If the event is a crash, several actions may be taken to safeguard public safety while the incident is being cleared.  This produces a situation in which the reduced capacity of the roadway changes over time.  Also, the demand may not be constant during the duration of the incident and its effects.  In Fig. 7 and equation 8, it is assumed that both demand Q1 and capacity C' are constant over time.  Fig. 8 shows a more general case where this assumption has been relaxed.

Figure 8. Cumulative flow vs. time for a highway bottleneck with demand and capacity varying over time

For example, given an incident that occurs at time TC₀ and is cleared at time TC_m-1, events occur during the interval (e.g., arrival of personnel from emergency agencies, removal of affected vehicles to the shoulder, etc.) that affect the capacity of the roadway.  In this case, those m - 1 events are represented by a starting time TC_j-1, an ending time TC_j, and a capacity level C_j that is assumed constant during that interval (i.e., the slope of the Cumulative Capacity Flow line in the interval [TC_j-1, TC_j]).  At time TC_m-1, the capacity of the road is restored to normal (i.e., C_m = C).

Thus, during an event and after its clearance, the demand may not be constant.  Figure 8 shows a case where the demand has been divided into intervals [TQ_i-1, TQ_i] within which it has a constant value of Q_i (i.e., the slope of the Cumulative Demand Flow line in the interval [TQ_i-1, TQi]).  As opposed to the capacity side, it is not known beforehand how many intervals need to be considered for the demand.  All of the intervals with constant demand that have starting and ending times within the interval [TC₀ = TQ₀, TC_m-1] must be included.  A simple iterative process must be used to determine which constant demand intervals having initial times later than TC_m-1 are to be considered.[1]  This is because point (TQ_n, CFQ_n) ≡ (TC_m, CFC_m) is not known beforehand.

The cumulative demand and capacity at the end of each respective constant interval can be found from equations 9 and 10.

Equation 9:

Equation 10:

Assuming that there are n intervals with constant demand and m intervals with constant capacity, and considering that C_m = C, then

Equation 11:

and the total delay

Equation 12:

Note:  Another way of computing the delay would be to calculate S from equation 4 as a function of time, given Q(t) and C(t) and using the following equation to determine D(t).

Equation 13:

where Jd is the "jam density" and C is the capacity of the roadway.

S(t) would provide the speed at which the queue is growing upstream of the bottleneck.  Furthermore, since the density in that area is D4(t), it is possible to know the queue length QL(t) by multiplying S(t) × D4(t).  The summation of QL(t) × dt (where dt is a small discrete interval of time) would give the total delay.

2.3  Delay in the Context of Traffic Patterns and Other Factors

The methodology presented in section 2.2 above describes the complex interaction among capacity reduction, demand, and delay factors for a given event. For most of the estimates in this report, this general methodology is applied to each incident on the network, and the capacity loss and delay impacts are summed across area type (including urban area size), highway type, peak period, and (for each peak period) congestion level. These detailed breakdowns of capacity loss and delay present a challenge when trying to (1) compare the relative impacts on different urban area sizes, highway types, and environments and (2) determine the reasonableness of the estimates relative to one another.

A detailed analysis as to why delay estimates vary across city sizes, road types, and time of the day (peak vs. non-peak) is beyond the scope of this study. The amount of vehicle travel (VMT) is certainly a key factor, but other factors such as congestion level and the nature of the events themselves can also affect delay dramatically.

The VMT information in Figure 9 and Table 6 provide a useful context for assessing the order of magnitude of the delay estimates provided for various area, highway, and time-of-day categories presented in this report. However, the reader should also bear in mind that other factors may certainly contribute to differences among various estimates.

Fig. 9. VMT for highways within scope of TLC2 by area type & size, 1999

1. For each interval k with constant demand that starts after time TC_m-1, find the intersection with the line defined by point (TC_m-1, CFC_m-1) and slope C_m. If the intersection is outside the interval k, then go to the next interval with constant demand. If it is inside, then stop.

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