Publish Information: Xu, H., H. Liu, and Z. Tian (2010). “Control Delay at Signalized Interchanges Considering Internal Queue Spillback.” Journal of Transportation Research Record, No.2173, 123‐132. Control Delay at Signalized Diamond Interchanges Considering Internal Queue Spillback
Paper No. 10-1148
Hao Xu
Department of Civil & Environmental Engineering
Texas Tech University
Lubbock, TX 79409-1023
Phone: (806) 786-0934
E-mail: hao.xu@ttu.edu
Hongchao Liu (Corresponding Author)
Department of Civil & Environmental Engineering,
Texas Tech University
Lubbock, TX 79409-1023
Phone: (806) 742-3523 Ext. 229
E-mail: Hongchao.Liu@ttu.edu
Zong Tian
Department of Civil & Environmental Engineering
University of Nevada Reno
Reno, NV 89557
Phone: (775) 784-1232
E-mail: zongt@unr.edu
Total Words: 3852
Total Figures: 13
Total Tables: 1
Total Combined: 7352
Revised manuscript for publication in the pre-print CD of the 89th Annual Meeting of the Transportation
Research Board, and possible publication in the Journal of Transportation Research Record.
November 15, 2009
Xu, Liu, Tian
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 -1-
ABSTRACT
Control delay is the primary measure for determination of the level of service of signalized intersections.
The existing analytical delay models usually work well for isolated intersections, but not as effective
when applied to diamond interchanges. The limited storage space between the two closely spaced
intersections of a diamond interchange may cause queue spillback from the internal link to the outside
roads. This property would give rise to unrealistic delay calculation for diamond interchanges with high
traffic volumes. This technical document describes the development of a new analytical delay model that
takes into account the effects of internal queue spillback at diamond interchanges. Simulation studies are
also conducted to compare the effectiveness of the proposed approach with existing methods. The study
shows that for low overlap time conditions, the proposed model tends to agree with the Synchro and
Vissim simulation and is better than Elefteriadou’s method, which tends to over predict delay. For high
overlaps, the Elefteriadou’s method, Vissim simulation, and the proposed model tend to agree, while
Synchro diverges significantly by over estimating delay. The study contributes to the literature and
practice by providing an open-source analytical model that can either be used as a standalone delay
calculation model or as a supplement to the existing methods.
Keywords: diamond interchange, control delay, internal queue, spillback, traffic signal capacity
Xu, Liu, Tian
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1. INTRODUCTION AND LITERATURE REVIEW
Control delay is the primary measure for determination of the level of service of signalized intersections.
Simulation methods (1) are advantageous in conducting “what if” studies and testing the scenarios and
phenomenon that may not occur or hard to capture in the field. Nonetheless, simulation approaches are
usually less effective in providing generalized results. For that reason, a combined analytical and
simulation approach would be ideal for developing control delay models.
In the past several decades, mathematical models for calculating (signalized) intersection delay
have been studied extensively by numerous researchers. Notable works include, but are not limited to
Beckmann et al. (2) and Webster (3) who developed and tested through simulation their fundamental
delay models, and van Zuylen and Viti (4) who provided comprehensive summaries of analytical delay
models and improved some of them. Currently, the commonly used method is described in Chapter 16 of
the Highway Capacity Manual, 2000 edition (5), which came from the model developed by Fambro and
Rouphail (6). In review of these models, one can find that most of them were developed on the basis of
the assumption that the subject intersection should not be blocked by queues spilled over from the
downstream intersection, which is not realistic in the real world. Therefore, those models work reasonably
well for isolated intersections, but not as effective when applied to diamond interchanges with two closely
spaced intersections. Several researchers attempted to solve the problem by taking into account signal
coordination in their models, such as Fambro (7) and van Zuylen (4), however, the control delay of two
major external approaches (i.e., the through movements of the arterial street and the left-turn movements
of the frontage roads) has not been satisfactorily formulated to date to reflect the real situation.
Diamond interchanges, like the one shown in Fig.1, are arguably the most commonly used
interchange patterns in North America providing connections between the major highways and arterial
streets in urban and suburban areas (8). One of the ordinarily used diamond interchange patterns is the
tight urban diamond interchange (TUDI), where two traffic signals are installed on the arterial street to
control the interchanging traffic (9). Diamond interchanges are often characterized by limited spacing
between the signals (10). The two traffic signals are usually spaced between 200 ft and 1000 ft apart and
the TTI three-phase and four-phase operations are typically applied (11) (12), as shown in Fig.2 and Fig.3.
Because of the limited interior spacing, when the internal queue is overflowed, spillback occurs and
blocks the external movements, which is shown in Fig.4. For standard TTI four-phase operation, the
overlap time, which is defined when phase 2 and phase 8 are both in green, or phase 6 and phase 4 are
both in green in Fig.1, is normally designed to be less than the time required to travel the internal space of
the diamond interchange. In such a case, no internal queues will be present. If the overlap time is adjusted
for any reason, for instance, to reduce the queue on external roads, the vehicles will be forced to stop
between the two intersections. This situation generates queue in the interior space and may cause queue
spillback.
The delay model in Chapter 16 of HCM 2000 is used to estimate control delays at diamond
interchanges by many traffic analysis tools, such as Synchro 5 and PASSER III. The results, however, do
not realistically reflect the actual situation when spillback of internal queue occurs. For a diamond
interchange operated by the TTI four-phase operation with long overlap time, the internal overflowing
queue blocks the upstream intersection and increases the control delay of external movements
significantly, but the existing delay model in the HCM 2000 doesn’t reflect the change. To address the
effects of internal queue spillback, several methods were developed and employed in traffic analysis tools.
Elefteriadou et al. (13) introduced a method, dubbed as the Elefteriadou model in this paper, to address
this issue. However, this study finds that it tends to overestimate delay at low overlaps. Synchro 7 also
introduced a new series of traffic analysis tools (called Queue Interactions), which looks at how queues
may reduce capacity through spillback, starvation, and storage blocking between lane groups. A new
queue delay factor is introduced to measure the additional delay incurred by the capacity reduction due to
queues on short links. The new models are used for delay calculation of diamond interchanges by
Synchro 7, but the specifics of this model were not published. This study finds that Synchro significantly
over estimates delay for high overlaps.
Xu, Liu, Tian
71 72 FIGURE 1 Diamond Interchange.
73 74 FIGURE 2 TTI Four-phase Operation.
75 76 77 78 79 80 81 82 83 84 85 -3-
FIGURE 3 TTI Three-phase Operation.
The motive and objective of this study is to provide an open-source and accurate analytical delay
formulation model for better operation of diamond interchanges under oversaturated conditions. To this
end, an analytical model was developed with focus on taking into account the effect of internal spillback
on the external movements. Since the effect on the arterial through movements and the ramp left-turn
movements are similar, the study was emphasized on the delay of the arterial through movements under
the TTI four-phase operation. Based on our analysis, it is revealed that the impact of internal spillback on
delay varies depending on several variables including the signal timing, the arrival traffic flow rate of the
arterial, the saturation flow rate of the upstream intersection, the saturation flow rate of the downstream
intersection, the length of the initial internal queue and the distance of the internal space.
Xu, Liu, Tian
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FIGURE 4 Spillback of the Diamond Interchange. In the following section, we will describe in detail the proposed delay formulation model and the
modification of the HCM 2000’s model. The new model was developed based on the classified traffic
movement modes and the calculation of the lost green time and effective green time. All three delays,
namely, the uniform control delay, incremental delay, and the total control delay per vehicle are predicted.
2. THE NEW ANALYTICAL DELAY MODEL FOR EXTERNAL MOVEMENTS OF DIAMOND
INTERCHANGES
2.1 Classification of Traffic Movement Modes and Calculation of the Lost Green Time
In order to calculate the lost green time, the external traffic movements are classified into two modes.
Mode 1 represents the case in which the traffic volume moving through the upstream intersection during
overlap interval is less than the capacity of the internal space, as shown in Fig.5; Mode 2 represents the
case that the traffic moving through the upstream intersection during the overlap interval is more than that
the internal space can contain, as shown in Fig. 6. Calculation of the lost green time of these two traffic
movement modes is introduced in the following two subsections.
FIGURE 5 Spillback Occurs after Overlap Time.
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105 106 107 108 109 110 -5-
FIGURE 6 Spillback Occurs during Overlap Time.
2.1.1 Traffic Movement Mode 1:
The traffic volume (per lane) moving through the upstream intersection during the overlap time is
calculated by the following formula:
(o  r )  q
os
), (
))
3600  n
3600  n
111 VO E  min((
112 113 114 115 116 117 118 119 VO E = lane based traffic volume moving through the upstream intersection during overlap time;
o = overlap time of the external arterial movement;
r = red time per cycle for the external arterial through movement;
q = arrival flow rate of the external arterial through movement;
n = number of lanes of the lane group for the external arterial through movement;
s = the saturated flow rate of the external arterial through movement. 120 121 122 123 124 125 126 127 128 129 130 131 132 133 (1)
Thus, the traffic movement mode 1 occurs when the following inequality is met:
VO E  L  I '
L = the average length of space occupied by one vehicle in the queue;
I ' = the distance between the upstream intersection and the end of the internal queue;
I '  I  QIL
I = length of the internal space of the subject diamond intersection;
QIL = length of the queue left in the internal space at the end of the last green time.
(2)
(3)
The through movement of the external arterial street will not be blocked in its green phase until
the internal space overflows. Therefore, for movement mode 1, the external green time before being
blocked by the internal spillback is longer than the overlap time and equals to the time needed to fill the
internal space, which is expressed by the following formula:
I '  3600  n
I '  3600  n
L
 r ), L
g1 = max((
) q
s
g1 = green time of the external arterial before being blocked by the internal queue.
Since g1 could not be longer than the green time g , the formula (4) is adjusted:
I '  3600  n
I '  3600  n
 r ), L
))
g1  min( g , max(( L
q
s
g = green time of the external arterial movement.
(4) (5)
Xu, Liu, Tian
134 135 136 -6-
During the time interval of g1  o , the vehicles in the internal space are discharged through the
downstream intersection while the external traffic entering the internal space from the upstream
intersection. Therefore, the internal queue length at the end of g1 is
( g1  o)  sd
 L)
3600  n
137 QL  max(0, I 
138 139 Q L = the internal queue length at the end of g1 ;
sd = the saturated discharging traffic flow rate of the downstream intersection.
140 The time used to discharge the internal queue QL is
141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 TI 
QL
L
(6)
 3600  n
sd
(7)
TI = time needed for discharging the internal queue QL ;
To realistically calculate delay, the effect of time needed for a vehicle to travel the internal space
should be considered, so the lost green time is calculated by the following formula:
b  min( g  g1 , TI  TIT )
(8)
b =the lost green time because of internal spillback.
TIT = average time needed for a vehicle to travel the internal space.
The green time of the external arterial after being blocked is
g 2  g  g1  b
(9)
g 2 = green time of the external arterial after being blocked.
If b  0 , then g1  g and g 2  0 .
2.1.2 Traffic Movement Mode 2:
The traffic movement mode 2 occurs when the following inequality is met:
VOE  L  I '
(10)
For traffic movement mode 2, g1 and g 2 are calculated using formula (5) and (9), while the lost
green time b is
I  3600  n
b  min( g  g1 , o  g1  L
 TIT )
sd
(11) 2.2 Calculation of the Effective Green Time and Effective v/c Ratio
With the lost green time b calcuated, the effective green time is obtained by the following formula:
g'  g  b
(12)
g ' = effective green time of the arterial through movement.
Since the green time is decreased from g to g ' , the capacity and v/c ratio of the lane group are
changed.
g'
g
166 c'  c 
167 168 c' = effective capacity of the lane group for the through moment of the external arterial street;
c = capacity of the lane group for the through movement of the arterial street, which is
g
c =sn ;
C
169 (13)
Xu, Liu, Tian
170 171 172 173 174 175 176 177 178 179 180 181 182 183 -7-
C = cycle length of the signal.
q g q
X ' 
c' c  g '
X ' = effective v/c ratio of the through movement of the external arterial street.
(14)
2.3 Calculation of the Uniform Delay d1
This section describes the modification of the delay model in Chapter 16 of the HCM 2000 to reflect the
effect of queue spillback at diamond interchanges. The modification is made based on the changed values
of the effective green time and v/c ratio of the through movement on the external arterial street.
According to the analysis in Section 2.1, the total green time for the through movement of the
external arterial street is divided into two parts with regard to the effective green time (the second
effective green time may be 0) due to the internal spillback. Thus, the delay for uniform arrivals ( d 1 ) is
calculated using the following process:
If g 2  0 , the green time g and v/c ratio X in the formula 16-11 (in Chapter 16 of HCM 2000)
are replaced by the effective green time g ' and effective v/c ratio X ' to get the following formula:
g
g' 2
)
0.5C (1  1 ) 2
C
C
d1 

g'
g
1  [min(1, X ' ) ] 1  [min(1, X ' ) 1 ]
C
C
0.5C (1 
184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 (15)
Else
The uniform arrival flow is expressed by the following formula:
q '  min(q, c' )
(16)
Q1  q 'r
Q1 = the queue length of the through movement on the external arterial street at the
beginning of green time.
If (r  g1 )  q '  g1  S , the external queue is not cleared at the end of g1
Q2  q'(r  g1 )  S  g1
Q2 = the queue length of the external arterial street at the end of g1 .
Q3  Q2  b  q '
(17)
(18)
Q3 = the queue length of the external arterial street at the beginning of g 2 .
d1  (0.5  r  Q1  0.5  (Q1  Q2 )  g1  0.5  (Q2  Q3 )  b
(r  g1  b)  q ' S  g1 2
)  S ) /(C  q' )
S  q'
Else, the external queue is cleared at the end of g1
Q2  0
Q3  Q2  b  q'
 0.5  (
S  r2  S b
C  (S  q' )
If Q3  g 2  ( S  q ' ) , the external queue is not cleared at the end of g 2
d1  0.5 
(19)
(20)
This is the situation shown in Fig.7, which can be considered as an equivalent
situation depicted in Fig.8.
Exchange values of b and r ;
Xu, Liu, Tian
-8-
205 Exchange values of g1 and g 2 ;
206 207 208 209 d1 is calculated by formulas (17), (18) and (19).
End
210 211 FIGURE 7 the External Queue Is Not Cleared at the End of g 2 .
212 213 214 215 216 217 218 219 220 221 222 223 End
End
FIGURE 8 the Equivalent Situation of FIGURE 7.
2.4 Calculation of the Control Delay
The green time g and v/c ratio X in the formula 16-10, 16-12, F16-1 and 16-9 in Chapter 16 of HCM
2000 are replaced by the effective green time g ' and effective v/c ratio X ' to develop the following
formulas to estimate the control delay:
PF 
(1  P) f PA
g'
1 ( )
C
(21)
PF = the uniform delay progression adjustment factor, which accounts for effects of signal
progression.
P = the proportion of vehicles arriving during green time.
f PA = supplemental adjustment factor for platoon arriving during green time.
Xu, Liu, Tian
-9-
8klX '
]
c' T
224 d 2  900T [( X '1)  ( X '1) 2 
225 226 227 228 229 230 d 2 = incremental delay to account for the effect of random arrivals and oversaturation queues,
(22)
adjusted for the duration of the analysis period and type of signal control; this delay component assumes
that there is no initial queue for the lane group at the start of the analysis period (s/veh);
T = analysis duration;
k = incremental delay factor that is dependent on controller settings;
l = upstream filtering/metering adjustment factor.
1800Qb (1  u )t
c' T
231 d3 
232 233 234 235 236 d 3 = initial queue delay, which accounts for delay to all vehicles in the analysis period due to
237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 (23)
initial queue at the start of the analysis period (s/veh);
Qb = initial queue at the start of period T (veh);
u = delay parameter;
t = duration of unmet demand in T(h);
Qb
}
c'[1  min(1, X ' )]
c' T
u  0 if t  T , else u  1 
Qb [1  min(1, X ' )]
t  0 if Qb  0 , else t  min{T ,
The control delay per vehicle (s/veh) is
d  d1 ( PF )  d 2  d 3
(24)
The flow chart of the proposed control delay calculation model is demonstrated in Fig. 9.
3 EFFECTIVENESS OF THE NEW MODEL
This section describes the test of effectiveness of the new delay model. Owing to the difficulty to
reproduce the studied cases in the field, we used Vissim simulation to develop the scenarios and compare
the performance of the proposed model with the Elefteriadou’s model and Synchro 7. The simulation
model used in the test was adapted from a previously developed simulation to evaluate the performance of
diamond and X-pattern interchanges in Texas. Details about the development and calibration of the
simulation model are beyond the scope of the study and can be found in (14). Different overlap time
scenarios were developed, which, in concert with two hypothetical traffic demand profiles representing
respectively moderate and congested traffic conditions, have produced various internal queue situations
for the analysis.
The internal space of the subject diamond interchange is 280 ft. and the signal runs the TTI fourphase plan. The control delay of the through movement on the northbound arterial street (Phase 3 in
Fig.11) was estimated by the new model, Synchro 7 and the Elefteriadou model. The results from these
three methods were compared with the simulation results. Two traffic flow rates, 1577 veh/h and 2000
veh/h, were used in the experiment to examine the performance of the models under normal and high
traffic volumes. For each traffic flow rate, 9 overlap time scenarios varying from 5s to 45s with a step
increment of 5s were applied in the TTI four-phase signal operation. This was intended to produce
enough internal queue cases for the test of model performance under various overlap intervals. Table 1
provides the data collected in the experimental study. The results are also shown in Fig.12 and Fig.13.
As shown in Figs. 12 and 13, the estimated control delay by the new model is very close to the
simulated results as compared to the estimated values by Synchro 7 and Elefteriadou Model. At the flow
rate of 1577 veh/h, the differences between the delay predicted by the three methods and the simulation
are as follows. For the proposed model, the average and largest deviation between the calculated and the
Xu, Liu, Tian
266 267 268 269 270 271 272 273 - 10 -
simulated control delay are 21 s/vehicle and 46.5 s/vehicle, respectively; this is 116.1 s/vehicle and 397.7
s/vehicle for Synchro 7, and 84.6 s/vehicle and 109.7 s/vehicle for the Elefteriadou model. At the traffic
flow rate of 2000 veh/h, the average and largest deviation between the new model and the simulation are
35.8 s/vehicle and 80.2 s/vehicle, respectively. The same deviations are 148.5 s/vehicle and 439.7
s/vehicle in Synchro 7 and 162.5 s/vehicle and 241.2 s/vehicle in the Elefteriadou model.
FIGURE 9 Flow Chart of the New Control Delay Calculation Model.
Xu, Liu, Tian
- 11 -
274 275 FIGURE 10 Traffic Volumes of the Researched Diamond Interchange.
276 277 278 FIGURE 11 TTI Four-phase Signal Timing of the Researched Diamond Interchange.
TABLE 1 Delay Comparison at Different Overlap Times
Overlap time(s)
Delay by New
Model(s/veh)
Delay by Synchro 7(s/veh)
Delay by Elefteriadou
Model(s/veh)
Simulation delay (s/veh)
Overlap time(s)
Delay by New
Model(s/veh)
Delay by Synchro 7(s/veh)
Delay by Elefteriadou
Model(s/veh)
Simulation delay (s/veh)
279 280 281 282 283 284 285 286 Arrival Flow Rate = 1577veh/h
5
10
15
20
25
30
35
40
45
49.2
109.5
105.7
176.8
151.7
264.3
151.7
383.8
151.7
502.9
135.6 138.8 142.1 145.5 148.9
35.1
35.2
36.4
36.1
39.2
Arrival Flow Rate = 2000 veh/h
5
10
15
20
25
152.4
76.3
155.9
108.7
159.4
108.1
163
105.2
30
35
40
45
41.8
56.8
43.8
71.9
48.9
111.4
81.6
161.2
148.8
222.9
241.8
299.7
295.6
398.2
295.6
529.1
295.6
657
275.3
38.4
279.5
38.3
283.8
44.1
288.1
81
292.5
129.1
296.9
190.5
301.4
217.3
305.9
215.4
310.5
217.3
39.0
41.4
39.1
41.7
39.8
43.7
41.3
61
It can also be found from the figures that, with the overlap time increased from 5 seconds to 45
seconds, the trend of the estimated control delay by the new model is consistent with the trend of
simulated results, while the estimation by Synchro 7 and the Elefteriadou model varies depending on low
and high overlap times. It appears that for low overlaps, Elefteriadou’s method tends to over predict delay,
while the new model, the Synchro and Vissim simulation tend to agree. For high overlaps, however, the
new model, Elefteriadou’s method and Vissim simulation tend to agree, while Synchro diverges
significantly by over estimating delay.
Xu, Liu, Tian
- 12 -
287 288 289 FIGURE 12 Delay Calculated by Different Models With an Arrival Traffic Flow Rate of
1577 veh/h. 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 FIGURE 13 Delay Calculated by Different Models with an Arrival Traffic Flow Rate of
2000 veh/h.
5 CONCLUSIONS
An analytical model for calculating control delay of diamond interchanges with consideration of the effect
of internal queue spillback was developed. The performance of the new model was examined in traffic
simulation and compared with the existing methods. The study shows that for low overlaps, the proposed
model tends to agree with the Synchro and Vissim simulation and is better than Elefteriadou’s method,
which tends to over predict delay. For high overlaps, the Elefteriadou’s method, Vissim simulation, and
the proposed model tend to agree, while Synchro diverges significantly by over estimating delay. The
proposed methodological approach, along with the detailed derivation of the formulas and the carefully
designed calculation flow chart, would be helpful for researchers and practitioners to further study and
effectively operate diamond interchanges.
ACKNOWLEDGEMENT
The authors are grateful to Mr. Rhett Dollins and Mr. David Bragg for their constructive comments to the
study and English editing of the document. Xu, Liu, Tian
310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 - 13 -
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