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# Delay-Volume Relations for Travel Forecasting: Based on the 1985 Highway Capacity Manual

## Appendix A: Sample Specifications for Intersection Delay

The following specification of intersection delay models assumes prior knowledge of the HCM. References are made to equations, tables, and figures from Chapters 9 and 10 of the HCM.

### Signalized Intersections

When a signalized intersection is included in a network, the model should only require information about:

1. the cycle length;
2. the saturation flow rate for the through lanes of each approach;
3. the existence of exclusive lanes at each approach;
4. the link's arrival type; and

The model should be able to calculate all other intersection information that normally would be part of a capacity/delay analysis.

The signalized intersection specification follows the HCM, except as noted here.

Adjustment Factors. The model not does not necessarily have to make adjustments for lane width, grade, parking, buses, heavy vehicles, and/or area type. For example, deviations from ideal conditions can be incorporated by the user into the saturation flow rate for the through lanes at the approach.

Green Times. The model should determine whether protected left phases are required and should determine the amount of green time to be allocated to each phase. When a protected phase is warranted the model should always adopt the phase sequence [(L + L),(LTR + LTR)], sometimes referred to as dual leading lefts with overlap. The model should not determine optimal green times. Rather, the model adheres to standard traffic engineering practice by allocating time to a phase in proportion to the maximum flow ratio (ratio of volume to saturation flow rate) during that phase.

Protected Lefts. The model should introduce a protected left phase, if there is insufficient capacity to process all left-turning vehicles without one. In ascertaining this capacity, the model should consider the number of gaps available during the unblocked green time and the number of sneakers. The protected left phase is given only sufficient time to process vehicles that cannot be handled during the LTR phase of the worst approach. The model then divides left turning traffic between the L and LTR phases for all approaches, nearly filling the protected left phase with traffic. The saturation flow rate for the LTR lane group includes the left lane capacity, if the left lane can be shared.

Left Lane Saturation Flow Rate. The left turn factor for exclusive lanes should be calculated according to Cases 1 or 2 from Table 9-12. The model should be able to modify the saturation flow rate for left turn lanes by using the implied reduction from the ideal saturation flow rate for the through lanes (e.g., for heavy vehicles and grades).

Shared Left Lanes Acting as Exclusive Lanes. To avoid discontinuities in delay, the model should create an exclusive left lane from a shared LT lane, only if a protected phase is warranted. The HCM's procedure for determining defacto left lanes should not be used.

Exclusive Right Lanes. The model need not create a separate lane group for an exclusive right turn lane. Rather, the saturation flow rate for the LTR or TR lane group can be adjusted upward to reflect the additional lane. The model should add sufficient capacity to just accommodate the right turning vehicles, with a maximum adjustment equal to a single lane's saturation flow rate.

Right Turns from Shared Lanes. The model need not provide for pedestrians. Consequently, the right turn adjustment factor would be calculated according to Case 4 on Table 9-1 1.

Period of Analysis. Because the model forecasts travel during whole hours, the peak-hour-factor is unnecessary. For multihour assignments, the model should take a volume-weighted average of the delay in each hour.

Delay Function. The model should calculate stopped delay from the HCM delay function (i.e., total delay divided by 1.3). The HCM delay function can become undefined for volume-to-capacity ratios only slightly greater than 1.0. Consequently, the model can use the HCM delay function only up to a volume-to-capacity of 1.0. Beyond 1.0, delay should be calculated as a linear extrapolation of the delay at a volume-to-capacity ratio of 1.0.

Acceleration Delay. The model should estimate the fraction of stopping vehicles and add acceleration delays for those vehicles. The fraction of stopping vehicles depends upon the arrival type and the volume-to-capacity ratio. The acceleration delay depends upon the link speed. For stopping vehicles,

Acceleration Delay

(Speed/2)(l/Acceleration Rate + 1/Deceleration Rate)

As a convenience, the speed can be taken from the link constituting the approach. For the simulations of this report, acceleration rate was set at 3.5 mph/second and deceleration rate was set at 5.0 mph/second.

Fraction of Stopped Vehicles. The model can determine the number of stopped vehicles by interpolating between 1.0 (at the value of the volume-to-capacity ratio, X, where all vehicles are assumed to have stopped, e.g., 1.2) and the fraction assumed to stop when the volume-to-capacity ratio is zero. This latter value will be referred to as the lowerbounds, L. There are separate lowerbounds for each possible arrival type. For an arrival type of 1 (least favorable progression), all vehicles must stop. So,

The lowerbound for arrival type 2 is found from averaging the lowerbound for arrival types 1 and 3. Similarly, the lowerbound for arrival type 4 is found from averaging the lowerbound for arrival types 3 and 5.

Regardless of the arrival type, all vehicles are assumed to stop when the volume-to-capacity ratio exceeds the user-specified value of the volume-to-capacity ratio, X.

It should be noted that the fraction of vehicles stopping at a signalized intersection under arrival type 3 can be easily derived from elementary traffic flow theory. The resulting nonlinear relationship is closely approximated by application of Equation A.3, above. A linear relation was chosen for consistency with the other arrival types.

Lane Utilization. Because the model calculates average delay across all lanes, a lane utilization factor is not needed.

Progression Adjustment. Like the HCM, the model should adjust delay as a function of the arrival type and the volume-to-capacity ratio. To avoid discontinuities, the model should use a set of linear equations to estimate the adjustment factor - one equation for each arrival type. The linear equations range from a volume-to-capacity ratio of 0.0 to a volume-to-capacity ratio of 1.2 (or another user-supplied parameter value), where the progression adjustment factor always becomes 1.0 (equivalent to no adjustment). Beyond a volume-to-capacity ratio of 1.2, no adjustment to delay is made. No adjustment is made to delay for vehicles in exclusive left-turn lanes.

Define F as the lowerbound value of the progression factor, i.e., when X is zero. For an arrival type of 1 (least favorable progression) the value of delay must be increased. Consequently,

For values of the volume-to-capacity ratio less than the user-specified maximum, the model interpolates between the lowerbound, F, and 1.0. The progression factor when the arrival type is 2 is found by averaging those for 1 and 3. The progression factor for a arrival type of 3 is found by averaging those for 3 and 5.

Overflow Time Period. Unlike the HCM, the model must allow the user to vary the overflow delay time period, T, fixed at 0.25 hours in the HCM. In addition, it should be possible to vary the ratio of total to stopped delay, η, fixed at 1.3 in the HCM. These changes affect the three constants in Equation 9-18. (See Akcelik, 1988, for a technical analysis of the HCM delay function.) The constant leading the first term (seen as 0.38) is found from:

First Constant = 0.5/r7

The constant leading the second term (seen as 173) is found from:

Second Constant = 90OT/η

The last constant appears within the radical (seen as 16), and is calculated from:

Third Constant = 4/T

### Some-Way Stop Intersections

In order to calculate delay at some-way stop intersections, the specification requires information about the locations of stop signs and the lane geometry at approaches with signs. Three types of lane configurations can be readily handled: one LTR lane; one LT and one R lane; and one LT and one TR lane. The model also needs the speeds of traffic on all links at the intersection.

The some-way stop model is consistent with the unsignalized model in the HCM, except as follows.

Potential Capacity Curves. The curves for potential capacity as a function of conflicting volume, Figure 10-3 in HCM, must be extended to handle any amount of conflicting volume (Baass, 1987). Figure 10-3 suggests that there should be a minimum capacity of 33 vehicles per hour, regardless of the amount of conflicting volume. The user should be able to change this minimum for all intersections or for any given intersection.

Treatment of Left Turns. The model need not make a distinction between left and through vehicles at signed approaches. Consequently, a left-turning vehicle would not impact the capacity of its opposing approach. However, the model should be consistent with the HCM in its treatment of left turns from unsigned approaches.

Acceleration Delay. The specification provides for acceleration delay for all vehicles at signed approaches and for left-turning vehicle at unsigned approaches. The acceleration delay depends upon the link speed.

Right-turn Lane Geometry. The model can consider right-turn lane geometry. For example, the user should be able to make adjustments to the acceptable right-turn gap at signed approaches.

Number of Lanes for the Major Street. The number of lanes for the major street can be determined by observing the capacity (or saturation flow rate) of the unsigned approaches. The number of lanes may be found by dividing the capacity by the ideal saturation flow rate and rounding to a whole number. The number of lanes is taken to be the maximum over all unsigned approaches.

Capacity. Capacity of a movement is computed by the German method as summarized by Baass (1987). This method produces almost exactly the same results as the HCM, but permits any value for the critical gap and any value for conflicting traffic.

Stopped Delay. The HCM provides relationships for estimating the capacity of some-way stops, but does not provide relationships for estimating delay. The specification includes queuing delay for all vehicles at signed approaches and for left-turning vehicles at unsigned approaches. Following the Swedish Highway Capacity Manual (Hansson, 1978), the model estimates delay, D, for any lane assuming Poisson arrivals and exponential service times:

D = 1 /(VI - c)

where D is measured in seconds, VI is the lane volume (in vehicles per second), and c is the lane capacity (in vehicle per second). Equation A.11 is used for volume-to-capacity ratios less than or equal to 0.9. For greater volume-to-capacity ratios the model should compute delay from the tangent to Equation A.11 at a volume-to-capacity ratio of 0.9. Thus, delay can still be calculated even when volume exceeds capacity.

Distribution of Through Vehicles Across Lanes. At signed approaches with two shared lanes, the model must divide the through traffic between the LT and TR lanes. An attempt should be made to equalize the volume-to-capacity ratios of the two lanes. To do this, the model calculates the proportion of through to be allocated to the right lane, PR

If PR is greater than 1 or less than 0, all through vehicles are allocated to either the right or left lanes, respectively.

### All-Way Stop Intersections

The HCM does not contain methods for estimating capacity or delay at all-way stop intersections. Consequently, the model must adopt other procedures for delay at all-way stop intersections. An enhanced version of Richardson's M/G/1 queuing model is chosen. Unlike Richardson's original formulation, the specification considers delays due to turning and delays caused by the need for coordination between drivers on the same and opposing approaches.

Definition of Processing Time and Service Time. The M/G/1 model estimates delay at an approach from the rate of arriving vehicles and from the mean and variance of the amount of time it takes for vehicles to pass through the intersection, referred to as the service time. The service time for an approach is equal to the sum of the time necessary to process a vehicle through the subject approach and the time necessary to process a vehicle through a conflicting approach, provided there is a vehicle at the conflicting approach. Both of these processing times (subject and conflicting) are computed by the same method, although they will have different values because of differing traffic characteristics. A typical processing time is about 4 seconds, so a service time is either about 4 seconds or about 8 seconds, depending upon the absence or presence of a conflicting vehicle.

Capacity in Relation to Service Time. The capacity of an intersection is inversely related to service time. For example, a single-lane approach at an intersection with heavy traffic in all directions would have a uniform service time of about 8 seconds, because there will always be conflicting vehicles. The capacity of such an approach would be 1/8 vehicle per second or 450 vehicles per hour.

Factors in Processing Time. For single lane approaches, the processing time depends upon (1) the presence or absence of right and left turning vehicles on the subject or opposing approaches and (2) the presence or absence of any vehicle on the opposing approach. This is handled by adding and subtracting constants for each effect. In general, left turns increase processing time, while right turns decrease processing time. For two lane approaches, the processing time also depends upon the presence or absence of a second vehicle on either the subject or opposing approaches. These additional vehicles introduce a need for coordination among drivers and, therefore, tend to increase processing time.

Lane Distribution. Each vehicle arriving at an approach has a different service time, but the average service time is assumed to be the same for all vehicles, regardless of their turning behavior. Consequently, traffic is distributed across lanes, at multilane approaches, as evenly as possible (taking into consideration the required lane assignments for left and right turning vehicles).

Lane Configurations. Possible lane configurations for approaches at all-way stops are the same as for some-way stops.

Acceleration Delay. Since all the vehicles stop, the model must add an acceleration delay to the queuing delay found from the M/G/1 model.

Stopping Delay. One of two delay relations could be used, depending upon user preference. First, delay can calculated from the following relation for each lane (Kyte, 1989),

Equation A.15 differs from Richardson's (1987) by including terms for coordination of vehicles on the subject and opposing approaches. This expression for variance is an approximation because it only includes variation due to the presence or absence of conflicting traffic, ignoring variation due to turning and due to the presence or absence of other vehicles on the subject approach or opposing approach.

Delay for a lane is computed by the following equation for values of less than or equal to 0.9:

For values of X greater than 0.9, the model should take the delay from the tangent to Equation A.16 at a value of X of 0.9. This second method was used for the simulations in this report.

Parameters. The parameters of the all-way stop model consist of "waits" in units of seconds. The following "waits" affects processing time.

a. Subject Unit Wait = 3.6

(Processing with no other vehicle present.)

b. One Left Wait = 1.

(Additional processing time if there is exactly one left turning vehicle on the subject or opposing approaches.)

c. Two Lefts Wait = 1.

(Additional processing time if there is exactly two left turning vehicles on the subject and opposing approaches.)

d. One Right Wait = -0.5

(Additional processing time if there is exactly one right turning vehicle on the subject or opposing approaches.)

e. Two Rights Wait = -1.

(Additional processing time if there is exactly two right turning vehicles on the subject and opposing approaches.)

f. Another Lane Wait = 1.

(Additional processing time if there is a second vehicle at the subject approach.)

g. One Opposing Lane Wait = 0.25

(Additional processing time if there is exactly one vehicle on the opposing approach.)

h. Two Opposing Lanes Wait = 1.

(Additional processing time if there is exactly two vehicles on the opposing approach.)

The remain "waits" affect service time, only if there is a vehicle at an conflicting approach.

i. One-Lane Added Wait = -0.5

(Additional service time when the subject approach has one lane.)

j. One+ Right Added Wait = 0.

(Additional service time when the subject approach has one left/through lane and one right lane.)

1. Two-Lane Added Wait = 0.5

(Additional service time when the subject approach has two lanes.)

These parameters were selected to match data collected by Kyte (1989).

Updated: 3/25/2014
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