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Chapter 35
Evaluation of a Traffic Signal: Delay
Models
35.1 Introduction
Signalized intersections are the important points or nodes within a system of highways and streets. To describe the MOE or to describe the quality of operations is a difficult task to perform than defining uninterrupted flow facilities. There are a number of measures have been used in capacity analysis and simulation, all of which quantity some aspect of experience of traversing a signalized intersection in terms the driver comprehends, the most common of these include:
Delay is the measure that most directly relates the driver’s experience, in that it describes the amount of time consumed in traversing the intersection. Length of queue at any time is a useful measure, and is critical in determining when a given intersection will begin to impede the discharge from an adjacent upstream intersection. Number of stops made is an important input parameter in air quality models. Among these three, delay is the most frequently used measure of effectiveness for signalized intersections.
35.2 Delay
Delay is the amount of time consumed in traversing the intersection. Delay can be quantified in many different ways. The most frequently used forms of delay are defined below:
Figure 35:1: Illustration of delay measures
- Stopped time delay
- Approach delay
- Travel time delay
- Time-in-queue delay
- Control delay
These delay measures can be quite different, depending on conditions at the signalized inter- section. Fig 35:1 shows the differences among stopped time, approach and travel time delay for single vehicle traversing a signalized intersection. The desired path of the vehicle is shown, as well as the actual progress of the vehicle, which includes a stop at a red signal.
35.2.1 Stopped Time Delay
Stopped-time delay is defined as the time a vehicle is stopped in queue while waiting to pass through the intersection. It begins when the vehicle is fully stopped and ends when the vehicle begins to accelerate. Average stopped-time delay is the average for all vehicles during a specified time period.
35.2.2 Approach Delay
Approach delay includes stopped-time delay but adds the time loss due to deceleration from the approach speed to a stop and the time loss due to reacceleration back to the desired speed.
G R G
W(i)
Cummulative Vehicles
Slope =v Q(t)
Vehi
(veh−secs)
Aggregate delay
Slope =s
Time t
Figure 35:2: Illustration of delay, waiting time and queue length
curve. Uniform arrivals assume that the inter-vehicle arrival time between vehicles is a constant.
Assuming no preexisting queue, arriving vehicles depart instantaneously when the signal is green (ie., the departure curve is same as the arrival curve). When the red phase begins, vehicles begin to queue, as none are being discharged. Thus, the departure curve is parallel to the x-axis during the red interval. When the next effective green begins, vehicles queued during the red interval depart from the intersection, at a rate called saturation flow rate, s, in veh/s. For stable operations, depicted here, the departure curve catches up with the arrival curve before the next red interval begins (i.e., there is no residual queue left at the end of the effective green). In this simple model:
- The total time that any vehicle i spends waiting in the queue, W(i), is given by the horizontal time-scale difference between the time of arrival and the time of departure
- The total number of vehicles queued at any time t, Q(t), is the vertical vehicle-scale difference between the number of vehicles that have arrived and the number of vehicles that have departed
- The aggregate delay for all vehicles passing through the signal is the area between the arrival and departure curves (vehicles X time)
In analytic models for predicting delay, there are three distinct components of delay. They are:
Cummulative vehicles, i
arrival function, a(t)
departure function,d(t)
Time ,t
Figure 35:3: Illustration of uniform delay
- Random Delay
- Overflow Delay
35.3.2 Uniform Delay
Uniform delay is the delay based on an assumption of uniform arrivals and stable flow with no individual cycle failures. Fig. 35:3, shows stable flow throughout the period depicted. No signal cycle fails here, i.e., no vehicles are forced to wait for more than one green phase to be discharged. During every green phase, the departure function catches up with the arrival function. Total aggregate delay during this period is the total of all the triangular areas between the arrival and departure curves. This type of delay is known as Uniform delay.
35.3.3 Random Delay
Random delay is the additional delay, above and beyond uniform delay, because flow is randomly distributed rather than uniform at isolated intersections. In Fig 35:4some of the signal phases fail. At the end of the second and third green intervals, some vehicles are not served (i.e., they must wait for a second green interval to depart the intersection). By the time the entire period ends, however, the departure function has caught up with the arrival function and there is no residual queue left unserved. This case represents a situation in which the overall period of analysis is stable (ie.,total demand does not exceed total capacity). Individual cycle failures within the period, however, have occurred. For these periods, there is a second component of delay in addition to uniform delay. It consists of the area between the arrival function and the dashed line, which represents the capacity of the intersection to discharge vehicles, and has the slope c. This type of delay is referred to as Random delay.
Slope =s
G R G
V Cummulative Vehicles
Slope =v Aggregate delay (veh−secs)
R=C[1−g/C] tc
Time t
Figure 35:6: Illustration of Webster’s uniform delay model
35.4 Various Delay Models
35.4.1 Webster’s Uniform Delay Model
Model is explained based on the assumptions of stable flow and a simple uniform arrival func- tion. As explained in the section 3, aggregate delay can be estimated as the area between the arrival and departure curves. Thus, Webster’s model for uniform delay is the area of the tri- angle formed by the arrival and departure functions. From Fig. 35:6 the area of the aggregate delay triangle is simply one-half the base times the height, or:
Uniform Delay(UD) = RV 2 (35.1)
Where, UD = Uniform Delay R = Duration of Red V = Number of vehicles arriving during the time R+t or no. of vehicles departing at time t
Length of red phase is given as the proportion of the cycle length which is not green, or:
R = C(1 − g C
The height of the triangle is found by setting the number of vehicles arriving during the time (R+tc) equal to the number of vehicles departing in time tc, or:
V = v(R + tC ) = stC (35.3)
Substituting for R in equation and solving for tc and then for V gives,
v(C(1 − (^) Cg ) + tC ) = stC tc(s − v) = vC(1 − (^) Cg ) V s (s^ −^ v) =^ vC(1^ −^
g C ) V = C(1 − (^) Cg )( (^) s vs− v ) Therefore UD = Area of triangle(in veh − sec) =
2 RV
2 C
(^2) (1 − g C )(^
vs s − v )
Where UD is the aggregate delay, in vehicle seconds. To obtain the average delay per vehicle, this divided by the number of vehicles processed during the cycle, which is the arrival rate, v, times the full cycle length, C (divide by vC).
UD = 12 C^2 (1 − (^) Cg )( (^) s vs− v ) vC^1
= C 2
(1 − (^) Cg )^2 (1 − v s )
Another form of the equation uses the capacity, c, rather than the saturation flow rate, s. We know, s =
c g/C (35.4)
So, the relation for uniform delay changes to,
UD = C 2
(1 − (^) Cg )^2 (1 − (^) Ccgv )
= C 2
(1 − (^) Cg )^2 (1 − (^) Cg X)
Where, UD = Uniform Delay R = Duration of Red V = Number of vehicles arriving during the time (R+t) or no. of vehicles departing at time t C = Cycle time v = Vehicle arrival rate
slope = v slope = c slope = s
Time ,t
Cummulative vehicles, i
Overflow Delay
Uniform Delay
Figure 35:7: Illustration of overflow delay model
models generally assume that arrivals are Poisson distributed, with an underlying average rate of v vehicles per unit time. The models account for random arrivals and the fact that some individual cycles within a demand period with v/c ¡ 1.00 could fail due to this randomness. This additional delay is often referred to as Random delay. The most frequently used model for random delay is Webster’s formulation:
RD = (X)
2 2 v(1 − X) (35.5)
Where, RD = average random delay per vehicle, s/veh X = v/c ratio The formula was found to somewhat overestimate delay. Webster proposed that total delay is the sum of uniform delay and random delay, and it can be estimated as,
D = 0.90(UD + RD) (35.6)
35.4.3 Webster’s Overflow Delay Model
Model is explained based on the assumption that arrival function is uniform. In this model a new term called over saturation is used to describe the extended time periods during which arrival vehicles exceeds the capacity of the intersection approach to the discharged vehicles. In such cases queue grows and there will be overflow delay in addition to the uniform delay. Fig. 35:7 illustrates a time period for which v/c¿1.0. During the period of oversaturation delay consists of both uniform delay (in the triangles between the capacity and departure curves) and overflow delay (in the triangle between arrival and capacity curves).
UD = C 2
(1 − (^) Cg )^2 (1 − (^) Cg X) (35.7)
Time
Cummulative vehicles
Slope = v
Slope = c
T
cT
vT
Figure 35:8: Derivation of the overflow delay model
As the maximum value of X is 1.0 for uniform delay, it can be simplified as,
UD = C 2 (1^ −^
g C )
2 (1 − (^) Cg X) = C 2 (1 − (^) Cg )
From Fig. 35:8 overflow delay can be estimated as,
OD =^12 T (vT − cT ) = T^
2 2 (v^ −^ c)^ (35.8)
Where, OD= aggregate overflow delay (in veh-secs) Average delay is obtained by dividing the aggregate delay by the number of vehicles discharged with in the time T, cT.
OD = T 2 (v c − 1) (35.9)
The delay includes only the delay accrued by vehicles through time T, and excludes additional delay that vehicles still stuck in the queue will experience after time T. The above said delay equation is time dependent i.e., the longer the period of oversaturation exists, the larger delay becomes. A model for average overflow delay during a time period T1 through T2 may be developed, as illustrated in Fig. 35:9 Note that the delay area formed is a trapezoid, not a triangle. The resulting model for average delay per vehicle during the time period T1 through T2 is: OD =
T 1 + T 2
v c −^ 1)^ (35.10) Formulation predicts the average delay per vehicle that occurs during the specified interval, T 1 through T 2. Thus, delays to vehicles arriving before time T 1 but discharging after T 1 are
Since v/c is greater than 1.15 for which the overflow delay model is good so it can be used to find the delay.
Delay = UD + OD UD = C 2 (1 − (^) Cg ) = 0. 5 ∗ 90[1 − 0 .55] = 20. 3 sec/veh OD = T 2 (v c − 1)
= 36002 (1. 23 − 1) = 414 sec/veh D = 20 .3 + 414 = 434. 3 sec/veh
This is a very large value but represents an average over the full hour Period of over saturation
35.4.5 Inconsistencies between Random and Overflow delay
As explained earlier random and overflow delay is given as, Random delay,
RD = (X)
2 2 v(1 − X) (35.11) Overflow delay,
OD = T 2 (X − 1) (35.12) The inconsistency occurs when the X is in the vicinity of 1.0. When X¡1.0 random delay model is used. As the Webster’s random delay contains 1-X term in the denominator, when X approaches to 1.0 random delay increases asymptotically to infinite value. When X¿1.0 overflow delay model is used. Overflow delay contains 1-X term in the nominator, when X approaches to 1.0 overflow becomes zero and increases uniformly with increasing value of X. Both models are not accurate in the vicinity of X=1.0. Delay does not become infinite at X=1.0. There is no true overflow at X=1.0, although individual cycle failures due to random arrivals do occur. Similarly, the overflow model, with overflow delay = 0 s/veh at X= 1.00 is also unrealistic. The additional delay of individual cycle failures due to the randomness of arrivals is not reflected in this model. Most studies show that uniform delay model holds well in the range X 0.85. In this range true value of random delay is minimum and there is no overflow delay. Also overflow delay model holds well in the range X 1.15. The inconsistency occurs in the range 0.85 X
Average Overflow Delay
Theoretical overflow delay model
Webster’s random delay model
Ratio
0.80 0.90 1.00 1.
Figure 35:10: Comparison of overflow & random delay model
1.15; here both the models are not accurate. Much of the more recent work in delay modeling involves attempts to bridge this gap, creating a model that closely follows the uniform delay model at low X ratios, and approaches the theoretical overflow delay model at high X ratios, producing ”reasonable” delay estimates in between. Fig. 35:10 illustrates this as the dashed line. The most commonly used model for bridging this gap was developed by Akcelik for the Australian Road Research Board’s signalized intersection.
35.4.6 Akcelik Delay Model
To address the above said problem Akcelik proposed a delay model. In his delay model, overflow component is given by,
OD = cT 4 [(X − 1) +
√ (X − 1)^2 + 12(X cT^ − X^0 )] F orX > X 0 OD = 0; f or X ≤ X 0
Where X 0 is given by the equation,
X 0 = 0.67 + 600 sg (35.13)
Where, T= analysis period, h X=v/c ratio c= capacity, veh/hour s= saturation flow rate, veh/sg (vehicles per second of green) g=effective green time, sec
As v/c¿1.0 in the same problem, what will happen if we use overflow delay model. Uniform delay will be the same, but we have to find the overflow delay.
OD = T 2
(v c
= 70. 2 sec/veh
As per Akcelik model, Overflow delay obtained is 39.1 sec/veh which is very much lesser com- pared to overflow delay obtained by OD delay model. This is because of the inconsistency of overflow delay model in the range 0.85-1.
35.4.8 Delay Models in the HCM 2000
The delay model incorporated into the HCM 2000 includes the uniform delay model, a version of Akcelik’s overflow delay model, and a term covering delay from an existing or residual queue at the beginning of the analysis period. The delay is given as,
d = d 1 P F + d2 + d 3 d 1 = 2 c (1^ −^
g c )
2 1 − [min(1, X)(g c )]
d 2 = 900 T [(X − 1) +
√ (X − 1)^2 +^8 klX cT ]
Where, d = control delay, s/veh d1 = uniform delay component, s/veh PF = progression adjustment factor d2 = overflow delay component, s/veh d3 = delay due to pre-existing queue, s/veh T = analysis period, h X = v/c ratio C = cycle length, s k = incremental delay factor for actuatedcontroller settings; 0.50 for all pre-timed controllers l = upstream filtering/metering adjustment factor; 1.0 for all individual intersection analyses c = capacity, veh/h
35.5 Conclusion
In this section measure of effectiveness at signalized intersection are explained in terms of delay. Different forms of delay like stopped time delay, approach delay, travel delay, time- in-queue delay and control delay are explained. Types of delay like uniform delay, random delay and overflow delay are explained and corresponding delay models are also explained above. Inconsistency between delay models at v/c=1.0 is explained in the above section and the solution to the inconsistency, delay model proposed by Akcelik is explained. At last HCM 2000 delay model is also explained in this section. From the study, various form of delay occurring at the intersection is explained through different models, but the delay calculated using such models may not be accurate as the models are explained on the theoretical basis only.
35.6 Acknowledgments
I wish to thank my student Mr. NagRaj R for his assistance in developing the lecture note, and my staff Mr. Rayan in typesetting the materials. I also wish to thank several of my students and staff of NPTEL for their contribution in this lecture.
