Module 7 : Traffic Signal Design

Lecture 37 : Capacity and Los Analysis of a Signalized I/S

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Determining delay

The values derived from the delay calculations represent the average control delay experienced by all vehicles that arrive in the analysis period, including delays incurred beyond the analysis period when the lane group is oversaturated. The average control delay per vehicle for a given lane group is given by Equation,

$$d = d_1(PF) + d_2 + d_3$$

where, d = control delay per vehicle (s/veh); $d_1$ = uniform control delay assuming uniform arrivals (s/veh); PF = uniform delay progression adjustment factor, $d_2$ = incremental delay to account for effect of random arrivals and $d_3$ = initial queue delay, which accounts for delay to all vehicles in analysis period

Progression adjustment factor

Good signal progression will result in a high proportion of vehicles arriving on the uniform delay Green and vice-versa. Progression primarily affects uniform delay, and for this reason, the adjustment is applied only to d1. The value of PF may be determined using Equation,

$$PF = \frac{(1-P) f_{PA}}{1-(\frac{g}{C})}$$ (1)

where, PF = progression adjustment factor, P = proportion of vehicles arriving on green, g/C = proportion of green time available, $f_PA$ = supplemental adjustment factor for platoon arriving during green. The approximate ranges of RP are related to arrival type as shown below.

Table 1: Relation between arrival type (AT) and platoon ratio

AT	Ration	Default $R_p$	Progression quality
1	$\leq 0.50$	0.333	very poor
2	0.50-0.85	0.667	Unfavorable
3	0.85-1.15	1.000	Random arrivals
4	1.15-1.50	1.333	Favorable
5	1.50-2.00	1.667	Highly favorable
6	2.00	2.000	Exceptional

PF may be calculated from measured values of P using the given values of $f_{PA}$ or the following table can be used to determine PF as a function of the arrival type.

Table 2: Progression adjustment factor for uniform delay calculation

Green Ratio	Arrival Type (AT)
(g/C)	AT1	AT2	AT3	AT4	AT5	AT6
0.2	1.167	1.007	1	1	0.833	0.75
0.3	1.286	1.063	1	0.986	0.714	0.571
0.4	1.445	1.136	1	0.895	0.555	0.333
0.5	1.667	1.24	1	0.767	0.333	0
0.6	2.001	1.395	1	0.576	0	0
0.7	2.556	1.653	1	0.256	0	0
$f_{PA}$	1	0.93	1	1.15	1	1
Default, $R_p$	0.333	0.667	1	1.333	1.667	2

Uniform delay

It is based on assuming uniform arrival, uniform flow rate & no initial queue. The formula for uniform delay is,

$$d_1 = \frac{0.5C(1-\frac{g}{C})^2}{1-[min(1,X)\frac{g}{C}]}$$ (2)

where, $d_1$ = uniform control delay assuming uniform arrivals (s/veh), C = cycle length (s); cycle length used in pre-timed signal control, g = effective green time for lane group, X = v/c ratio or degree of saturation for lane group.

Incremental delay

The equation below is used to estimate the incremental delay due to nonuniform arrivals and temporary cycle failures (random delay. The equation assumes that there is no unmet demand that causes initial queues at the start of the analysis period (T).

$$d_2=900~T\left[(X-1)+\sqrt{(X-1)^2+\frac{8klX}{cT}}\right]$$ (3)

where, $d_2$ = incremental delay queues, T = duration of analysis period (h); k = incremental delay factor that is dependent on controller settings, I = upstream filtering/metering adjustment factor; c = lane group capacity (veh/h), X = lane group v/c ratio or degree of saturation, and K can be found out from the following table.

Table 3: k-values to account for controller type

Unit	Degree of Saturation (X)
Extension (s)	$\leq 0.50$	0.6	0.7	0.8	0.9	$\geq 1.0$
$\leq 2.0$	0.04	0.13	0.22	0.32	0.41	0.5
2.5	0.08	0.16	0.25	0.33	0.42	0.5
3	0.11	0.19	0.27	0.34	0.42	0.5
3.5	0.13	0.2	0.28	0.35	0.43	0.5
4	0.15	0.22	0.29	0.36	0.43	0.5
4.5	0.19	0.25	0.31	0.38	0.44	0.5
$5.0^a$	0.23	0.28	0.34	0.39	0.45	0.5
Pre-timed	0.5	0.5	0.5	0.5	0.5	0.5

Aggregate delay estimates

The delay obtained has to be aggregated, first for each approach and then for the intersection The weighted average of control delay is given as:

$$d_A = \sum d_iv_i/\sum v_i$$

where, $d_i$ = delay per vehicle for each movement (s/veh), $d_A$ = delay for Approach A (s/veh), and $v_A$ = adjusted flow for Approach A (veh/h).

$$d_1 = \sum d_A\times v_A/\sum v_A$$

Determination of LOS

Intersection LOS is directly related to the average control delay per vehicle. Any v/c ratio greater than 1.0 is an indication of actual or potential breakdown. In such cases, multi-period analyses are advised. These analyses encompass all periods in which queue carryover due to oversaturation occurs. A critical v/c ratio greater than 1.0 indicates that the overall signal and geometric design provides inadequate capacity for the given flows. In some cases, delay will be high even when v/c ratios are low.

Table 4: LOS criteria for signalized intersection in term of control delay per vehicle (s/veh)

LOS	Delay
A	$\leq 10$
B	10-20
C	20-35
D	35-55
E	55-80
F	$>$80

Sensitivity of results to input variables

The predicted delay is highly sensitive to signal control characteristics and the quality of progression. The predicted delay is sensitive to the estimated saturation flow only when demand approaches or exceeds 90 percent of the capacity for a lane group or an intersection approach. The following graph shows the sensitivity of the predicted control delay per vehicle to demand to capacity ratio, g/c, cycle length and length of analysis period.

Figure 1: sensitivity of delay to demand to capacity ratio

Assumptions are : Cycle length = 100s, g/c = 0.5, T =1h, k = 0.5, l= 1, s = 1800 veh/hr

Figure 2: sensitivity of delay Vs g/c ratio

Figure 3: sensitivity of delay Vs cycle length