a capacity model for all-way stop-controlled intersections based on stream interactions

A capacity model for all-way stop-controlled intersectionsbased on stream interactions

Michael Kyte a,*, George List b

aUniversity of Idaho, College of Engineering, Moscow, ID 83884-0901, USAbRensselaer Polytechnic Institute, Troy, NY, USA

Abstract

This paper presents a methodology for analyzing the capacity of all-way stop-controlled intersectionsand the data that have been collected to validate and support the methodology. The rate of departure fromone approach of an all-way stop-controlled intersection is controlled by the presence or absence of vehicleson the other approaches. This degree of con¯ict is classi®ed into a set of unique cases. Field measurementscovering over 20,000 vehicle headways were classi®ed into ®ve degree of con¯ict cases. The model forecaststhe mean departure headway based on the probability of occurrence of each degree of con¯ict case. # 1999Elsevier Science Ltd. All rights reserved.

1. Introduction

Throughout the world, all-way stop-controlled (AWSC) intersections are used to controlintersecting tra�c when the tra�c ¯ows are nearly balanced on all approaches, or as an interimtransition from stop-control to signalization. AWSC intersections require each driver to stop,judge whether it is safe to enter, and ®nally, accomplish a turning maneuver.This paper presents a new capacity analysis methodology for AWSC intersections. It re¯ects

both recent research and ®eld studies that were conducted to develop the capacity analysismethodology for the 1997 Highway Capacity Manual (HCM). The ®rst part of this paper pre-sents a summary of prior research, and second part describes the new theory, as well as the modeldeveloped from that theory. Supporting ®eld data are also presented.The capacity of all highway facilities depends on how they operate. For AWSC intersections,

``yield to the right'' is the nominal operating rule, but the actual decision making is more complex.More relevant are the probabilities that speci®c con¯ict situations will arise, and the fact that time

TRA 215p/No disk used

0965-8564/99/$Ðsee front matter # 1999 Elsevier Science Ltd. All rights reserved.

PII: S0965-8564(98)00043-3

TRANSPORTATION

RESEARCH

PART A

Transportation Research Part A 33 (1999) 313±335

* Corresponding author. Tel.: +1-208-885-6002; fax: +1-208-885-2877; e-mail: [email protected]

is ``lost'' while these con¯icts are resolved. In contrast, at signalized intersections, right-of-way issignal controlled, and queuing models can be employed. At two-way stop-controlled (TWSC)intersections, hierarchical relationships allow priority queuing relationships to determine capacityand delays. At roundabouts, a similar situation pertains in that vehicles on the entryways mustwait for gaps in the circulating roadway ¯ows before being serviced.

2. Previous work

Six e�orts comprise the major initiatives directed toward AWSC intersections to date. The ®rst,Hebert (1963), focused on developing capacity guidelines. Based on three intersections inChicago, Hebert's work became the basis for the AWSC material presented in the 1985 HCM.Richardson (1987) developed a capacity model based in queuing theory and the headway datafrom Hebert. Zion et al. (1990) investigated the e�ects of various factors on delay and capacity.Kyte and others [Kyte and Marek, 1989; Kyte, 1990; Transportation Research Board (TRB),1991] developed empirically-based models for capacity and delay that were the basis for the 1994HCM update. Subsequent work of Kyte et al. (1993,1994) supports the models presented later inthis paper.

2.1. Hebert (1963)

Hebert (1963) proposed a capacity model for AWSC intersections. He theorized that driversface one of two con¯ict cases: no con¯icting vehicles or one con¯icting vehicle. Service times of4.0 s and 7.6 s were developed for these cases based on three sites in Chicago. The sites were stu-died during periods of continuous queueing. Saturation headways for these two cases form thebasis for the 1985 HCM capacity procedures.Hebert presented equations to forecast the capacity of an approach based on the volume split

between the two intersecting streets and the number of lanes on each street. Eq. (1) gives thecapacity for an intersection with one lane on each approach.

c � 3600

10:15ÿ �p �1�

where �p is the proportion of the intersection volume on the higher volume street. The 1985 HCMused Hebert's work to create capacity guidelines that could be used to assess the performance ofan AWSC intersection based on demand splits for 2-lane by 2-lane (i.e. one lane in each direc-tion), 2-lane by 4-lane, and 4-lane by 4-lane intersections.Tables 1 and 2 show Hebert's capacity estimates for various intersection con®gurations and

demand splits. Capacity trends for 2 by 2 intersections are shown in Table 1, including the in¯u-ence of shifts in demand split. Maximum intersection capacity is achieved when the ¯ows arebalanced; and in this case the capacity for each approach is 475 vph (1900/4). Table 2 suggeststhat capacity increases with more approach lanes, which implies that simultaneous stopbardepartures occur.

314 M. Kyte, G. List/Transportation Research Part A 33 (1999) 313±335

2.2. Richardson (1987)

Richardson (1987) proposed an iterative method to compute the approach capacity of AWSCintersections. The service time for each approach is made a function of the likelihood that vehi-cles are present on the opposing and con¯icting approaches. One of two states can exist: (1) novehicles on the opposing or con¯icting approaches and (2) one or more vehicles on the con¯ictingapproaches. This method converges quickly to a capacity estimate and is easily implementedusing computer software. The method's major limitation is that it only considers a small numberof states.Richardson used the Pollaczek-Khintchine formula to estimate delays, based on an M/G/1

model of the queuing process. The resulting delay equation is shown below:

Dq �xW 1� c2w

ÿ �2 1ÿ x� � �2�

W is the average service time (or time spent in the ®rst position in queue), x is the tra�c intensity,and Cw is the coe�cient of variation of service times. The total average delay is the sum of thetime in queue, Dq, and the time spent in the ®rst position, W.

2.3. Zion et al. (1990)

Zion et al. (1990) conducted ®eld studies of AWSC intersections. Generally, they found thatdelay increased as the intersecting volumes increased, intersections with balanced volumes hadlower delays than those without, and the percentage of left turns had a noticeable e�ect on delay.

Table 1Intersection capacity as a function of demand split

Demand split Capacity (vph)

50/50 190055/45 1800

60/40 170065/35 160070/30 1500

Demand split is the proportional split of tra�c between the intersecting streets.

Table 2Intersection capacity as a function of intersection type

Intersection type Capacity (vph)

2-lane by 2-lane 19002-lane by 4-lane 2800

4-lane by 4-lane 3600

Capacities are based on 50/50 demand split.

M. Kyte, G. List/Transportation Research Part A 33 (1999) 313±335 315

Statistical analyses suggested that Richardson's model might provide a credible estimate ofdelay.Data were obtained from three intersections near Albany, NY. In selecting the sites, three cri-

teria were applied. First, there had to be two-lane streets on all four approaches; second, the siteshad to have a variety of volume splits so that variations in that parameter could be considered;and third, the sites' volumes had to be high enough that delays of signi®cance would be observed.The data collected were processed to provide ®ve-minute records of turning movement volumesand average delay.While not directly germane to estimates of capacity, subsequent stepwise linear regression

generated equations that predicted delay as a function of the total volume passing through theintersection and the total number of left turns. The adjusted R2 values ranged from 0.58 to 0.93.The best model predicted a delay contribution of 8.5 s per intersecting vehicle and 16.7 s per left-turning vehicle. These results clearly illustrate the incremental impact of left turning vehicles,which will be seen again later.Further analysis of the data, breaking the observations down into groups based on their

volume splits suggests that delays did decrease as the demand split approached 50/50. This againsupported the ®ndings of both Hebert and Richardson.

2.4. Kyte and Marek (1989) and Kyte (1990)

Kyte et al. (Kyte and Marek, 1989; Kyte, 1990; Transportation Research Board, 1991) per-formed empirical studies of AWSC intersections in Iowa and the Paci®c Northwest between 1987and 1989. The objective was to learn about the factors that in¯uence capacity and delay at AWSCintersections. Saturation headways were measured, and a capacity equation was developed.The resulting capacity equation, and its supporting data, were published as part of Transpor-

tation Research Circular 373 (Transportation Research Board, 1991). The Committee on High-way Capacity and Quality of Service (HCQS) proposed using this method on an interim basissince it provided a signi®cant improvement over the method in the 1985 HCM (see Transporta-tion Research Board, 1985). After several years of use by the tra�c engineering community, theHCQS Committee approved the procedure for common use and incorporated it into the 1994HCM update (seeTransportation Research Board, 1994).The 1994 HCM Update describes the basis for tra�c operations at an AWSC intersection:

The headway between vehicles departing from a STOP line of an AWSC intersection is afunction of the conditions present on the other intersection approaches. Minimum saturationheadways are achieved if there is not tra�c on any of the other intersection approaches.Maximum saturation headways result if tra�c is present on all of the other approaches.

This microscopic perspective has a direct analogy at the macroscopic level. The maximum¯ow rate on a given approach can be achieved if there is not tra�c on any of the otherapproaches. The minimum ¯ow rate on a given approach results if tra�c is evenly distributedamong all of the intersection approaches. Thus the key variable in the determination ofapproach capacity is the relative distribution of tra�c volumes among the approaches. Thisvariable is called volume distribution.


When tra�c is evenly distributed among the approaches, the capacity of a single-laneapproach is approximately 525 vph and the capacity of a four-leg intersection is approximately2,100 vph under ideal conditions. Research has found the capacity of a single-lane approach,when there is no tra�c on any of the other approaches, is 1,100 vph under ideal conditions.

Data published in Kyte and Marek (1989), Kyte (1990), TRC 373 (Transportation ResearchBoard, 1991) and the 1994 HCM Update suggest that a model with four distinct con¯ict cases isneeded to describe the conditions faced by the subject approach driver at a 2 by 2 AWSC inter-section. These cases are de®ned in Table 3.Each case represents a speci®c con¯ict between vehicles on the ``subject'' approach and the

con¯icting and opposing approaches. Variations in turning percentages, vehicle mix, and otherfactors, cause variations in these baseline values. In a Case 2 headway, for example, the subjectvehicle faces one opposing vehicle. Thus, the 5.5 s value covers a range of combinations, includ-ing: opposing through vehicles, one through vehicle opposed by a left turning vehicle, and onethrough vehicle opposing by a right turning vehicle. While the capacity equation given in the 1994HCM Update provides an adjustment for turning movements, it is based only on the overallproportion of turning movements and not on the microscopic or vehicle-by-vehicle interactionsthat actually re¯ect the degree-of-con¯ict resulting from turning vehicle con¯icts.The capacity equation is as follows:

c � 1000Vps � 700Vpo � 200Ls ÿ 100Lo ÿ 300LTpo � 200RTpo ÿ 300LTpc � 300RTpc �3�where,

c = capacity of the subject approach,Vps = proportion of intersection volume on subject approach,Vpo = proportion of intersection volume on opposing approach,Ls = number of lanes on subject approach,Lo = number of lanes on opposing approach,LTpo = proportion of volume on opposing approach turning left,RTpo = proportion of volume on opposing approach turning right,LTpc = proportion of volume on con¯icting approaches turning left, andRTpc = proportion of volume on con¯icting approaches turning right.

Table 3

Saturation headway data for AWSC intersections (1994 HCM Update)

Condition Mean saturation headway (s/veh)

Case 1 Case 2 Case 3 Case 4

All data 3.5 5.5 6.5 9.0Single lane 3.9 5.6 6.5 9.0

Multi lane 1.5 4.3 6.3 9.3

Single lane notes single lane approach sample sites. Multi lane notes multi lane approach sample sites. Case 1 occurs when the subject

vehicle does not face either opposing or con¯icting vehicles. Case 2 occurs when the subject vehicle faces only an opposing vehicle.

Case 3 occurs when the subject vehicle faces only con¯icting vehicles. Case 4 occurs when the subject vehicle faces both opposing and

con¯icting vehicles.


2.5. Kyte et al. (1994)

Kyte et al. (1994) collected saturation headway data for a single approach at an AWSC inter-section. The intent was to determine if a subdivision of the four basic cases in Table 3 wouldprovide better results.Two tests were conducted. In the ®rst, the e�ects of directional movements on the subject

approach vehicle were examined for each of the four cases. In the second, subsets of Cases 3 and4 were studied.Table 4 shows the saturation headways for the through movements and right turns for each of

the four cases. Distinguishing among the headways by turning movement resulted in considerablydi�erent headway and capacity estimates, with capacity di�erences ranging from 27 percent to 50percent between the through movement capacity and the right turning movement capacity. A``di�erence in means'' test con®rmed that these results were signi®cant at the 0.05 level.There was also an interest in determining if the four cases should be divided into subsets that

better re¯ected the conditions faced by the subject vehicle. For example, Case 3 states that thesubject vehicle is faced by vehicles on the con¯icting approach and not on the opposing approach.But this case can include one or two con¯icting vehicles, one from the left or right, or both.Several subdivisions of Cases 3 and 4 were considered to determine if this ®ner level of distinctionhad value. Table 5 lists the six subsets considered. Table 6 shows the saturation headways mea-sured for each.Statistically, no signi®cant di�erence was found between Cases 3a (5.1 s) and 3b (5.6 s). This

means that drivers on the subject approach do not perceive a di�erence between con¯ictingvehicles approaching on the left or right, as long as only one con¯icting vehicle is present.

Table 4

E�ect of turning movement on approach capacity of AWSC intersection

Case Headway (s) Capacity (vp/h) Percent di�erence

TH RT TH RT

1 3.0 2.1 1200 1714 +43%2 4.2 2.8 857 1286 +50%

3 6.3 4.9 571 735 +29%4 7.9 6.2 456 581 +27%

TH is the through movement. RT is the right turn movement.

Table 5Subsets for saturation headway cases

Subset Description

3a One con¯icting vehicle from the right3b One con¯icting vehicle from the left3c One con¯icting vehicle from both the left and the right

4a One con¯icting vehicle from the left and one opposing vehicle4b One con¯icting vehicle from the right and one opposing vehicle4c One con¯icting vehicle from both the left and right, and one opposing vehicle


But a signi®cant di�erence did exist between Case 3c and Cases 3a and 3b (a value of 6.8 s vs5.1 and 5.6 s respectively). The implication is that if a con¯icting vehicle is present on both the leftand the right approaches, the saturation headway for the subject vehicle is di�erent, in this caselonger, than if the subject vehicle faces only one con¯icting vehicle.Also, there were di�erences in the Case 4 subsets. Similar to the results for Cases 3a and 3b,

while no signi®cant di�erence exists between Cases 4a (6.9 s) and 4b (6.8 s), a di�erence does existbetween Case 4c (8.4 s) and Cases 4a and 4b. Thus, while the direction of approach for the con-¯icting vehicle does not matter, the number of con¯icting vehicles does.

2.6. Kyte et al. (1993)

During Phase I of the National Cooperative Highway Research Project (NCHRP) 3-46,Kyte etal. (1993) measured saturation headways for a total of 4863 vehicles across all four cases inTable 3. The results are summarized in Table 7. While the speci®c headway measurements aresomewhat di�erent than before, the relative trends are similar.As discussed above, one of the shortcomings of the TRC 373 classi®cation system was the

limited number of cases considered, particularly the lack of consideration given to (1) the numberof opposing or con¯icting vehicles faced by the subject approach vehicle, (2) the directionalmovement of the subject approach vehicle, and (3) the e�ect of vehicle type.For a more accurate assessment of the e�ects of opposing and con¯icting vehicles, eight cases

were proposed, as depicted in Table 8. The saturation headways measured during the pilot studywere classi®ed into these eight cases.Several conclusions were drawn from the results:

Table 6Saturation headways for subsets for AWSC intersections

Subset Mean headway (s) SD Obs

3a 5.1 1.60 313b 5.6 1.38 20

3c 6.8 1.67 364a 6.9 1.62 324b 6.8 1.57 62

4c 8.4 2.39 82

SD is the standard deviation. Obs is the number of observations.

Table 7

Saturation headway data

Case: Vehicle on subject approach faced by. . . Pilot study data TRC 373 data

1. No other vehicle on any approach 3.2 3.92. One or more vehicles on the opposing approach only 4.9 5.63. One or more vehicles on the con¯icting approaches only 6.3 6.54. One or more vehicles on both the opposing and con¯icting approaches 8.0 9.0


(a) The approach direction of a con¯icting vehicle (either from the left or from the right) doesnot a�ect the saturation headway of the subject approach vehicle.

(b) The saturation headway values for Cases 3 and 4 are nearly equal; the values for Cases 6and 7 are also nearly equal.

(c) The number of vehicles on the con¯icting approach (either one or two) makes a signi®cantdi�erence in the saturation headway of the subject approach vehicle. Note that the valuesfor Cases 3 and 4 are lower than for Case 5.

(d) While the speci®c approach makes a di�erence if the subject vehicle faces only one oppos-ing or con¯icting vehicle (the headway for Case 2 is less than the headway for either Cases 3or 4), it does not make a di�erence if the subject vehicle faces two opposing or con¯ictingvehicles (Cases 5, 6, and 7 are nearly the same).

The saturation headways for each of the three directional movements of the subject approachvehicle are shown in Table 9. The following conclusions can be drawn from these values and theirtrends.

(a) The turning movement direction has a signi®cant e�ect on the saturation headway of thesubject approach vehicle. In seven of the eight cases, the saturation headway for left turningvehicles is higher than for through vehicles, and both of these exceed the value for rightturning vehicles.

(b) If the through movement is taken as the base case, adjustment factors can be computed forthe e�ect of turning vehicles on the saturation headway. For left turning vehicles, this adjust-ment factor is 0.90, or the capacity reduction due to left turning vehicles is 10 percent over thethrough vehicles. For right turning vehicles, the adjustment is 1.36, or a 36 percent increase.

(c) The assessment of saturation headways without respect to subject vehicle turning move-ment did not show a di�erence between Cases 3 and 4, and Cases 6 and 7. Such a di�erence,however, might be expected when the directional movement of the subject vehicle is con-sidered. The data presented in the table do not support this expectation.

Table 8Saturation headway data for AWSC intersections, pilot study

Case The subject vehicle faces on. . . Saturation headway (s/veh)

The opposing approach The con¯icting approaches All sites 4LSLA sites

1 No other vehicles No other vehicles 3.0 3.0

2 One vehicle No other vehicles 4.7 4.83 No other vehicles One vehicle from the left approach 5.7 5.94 No other vehicle One vehicle from the right approach 5.7 5.6

5 No other vehicle One vehicle from both the left and right approaches 7.3 7.16 One vehicle One vehicle from the left approach 7.3 7.27 One vehicle One vehicle from the right approach 7.4 7.3

8 One vehicle One vehicle from both the left and roght approaches 9.1 9.2

The values are based on the mean of the mean values measured for each of the 13 subject approaches of this pilot study. 4LSLA is 4-

leg single lane approach sites.


The saturation headways for passenger cars and heavy trucks show a signi®cant di�erence.Table 10 shows the computed headways for these two vehicle types. In every case, the passengercars have a lower value than the heavy trucks. The di�erence averages 1.5 s, or 22 percent. Theseheadways can be translated into capacities of 563 vp h for passenger cars and 462 vp h for trucks.This implies a passenger car equivalent of 1.2 for trucks at AWSC intersections.

3. Theory and models

3.1. Capacity

The HCM de®nes the capacity of a transportation facility as:

the maximum hourly rate at which persons or vehicles can reasonably be expected to traversea point or a uniform section of a lane or roadway during a given time period under prevailingroadway, tra�c, and control conditions.

The capacity of an AWSC intersection is dependent on the interplay between ¯ows on thevarious approaches. As is evident from Tables 3±10, service times increase (service rates decrease)as the number of con¯icts grows. At some point, use of the intersection becomes so intense thatone or more movements achieves a tra�c intensity equal to one. At that point, capacity isreached. A marginal increase in the ¯ow rate on any movement pushes the fully saturatedmovement(s) into an oversaturated condition, thus exceeding capacity.Moreover, because of the interaction between the tra�c streams on the various approaches,

and because this interaction governs the maximum ¯ow rate on each approach, two capacityconcepts can be considered:

Table 9Saturation headways by subject approach movement direction

Case Subject approach saturation headways (s/veh)

LT movement TH movement RT movement

Hdwy Obs Std dev Hdwy Obs Std dev Hdwy Obs Std dev

1 4.0 85 1.7 3.7 200 1.3 2.8 71 1.32 6.2 71 1.8 5.0 288 1.5 3.8 60 1.3

3 5.9 60 1.6 5.6 149 1.5 5.2 42 1.84 6.2 20 1.4 6.0 69 1.3 4.5 19 1.15 7.1 39 1.5 7.3 107 1.6 6.4 18 1.16 8.0 134 2.0 7.0 308 1.8 6.3 37 1.8

7 8.9 37 2.1 7.5 151 1.8 5.4 11 1.18 10.7 70 4.6 9.2 203 2.7 8.0 22 2.1

Mean Total 7.1 516 6.4 1475 4.6 280

Hdwy is the saturation headway; Obs is the number of observations; st dev is the standard deviation.


(a) The capacity of a given lane or approach: This is the maximum possible throughput on agiven lane or approach assuming that the ¯ow rates elsewhere in the intersection remainconstant. Hence the question is: how much can the ¯ow on the subject approach beincreased, if the ¯ows on the other approaches remain ®xed?

(b) The capacity of the intersection: This is the maximum possible throughput for the facilityassuming that the distribution of ¯ows on all intersection approaches remains constant.Here the question is: how much can the ¯ows on all the approaches be increased simulta-neously if the initial distribution remains ®xed?

The former is of interest when just one leg of the intersection is the focal point. It is also theappropriate capacity measure to use when estimating and calibrating delay equations. The secondis of interest when the analyst wants to know how much tra�c growth can occur before thefacility's overall processing capacity is exhausted.

3.2. Description of intersection operations

To understand how an AWSC intersection functions, consider the juncture of two one-waystreets. At or near capacity, a pattern of alternating discharge from the two approaches occurs:®rst a vehicle from the one approach enters the intersection followed by a vehicle from the otherapproach. This two-phase, single vehicle discharge pattern is also observed at a standard four-legAWSC intersection where drivers from opposing approaches enter the intersection at roughly thesame time during capacity operations. Some interruption of this pattern occurs when there arespeci®c con¯icts between turning maneuvers (e.g. a northbound left turn and a southboundthrough), but in general the north-south streams alternate right-of-way with the east-weststreams. A four-phase pattern emerges at multi-lane four-leg intersections. Here the developmentof the right-of-way consensus is more di�cult. Thus drivers from each approach enter the inter-section together until the queue is exhausted or a preemption occurs. Then, right-of-way passes tothe next approach, and each is served in turn.

Table 10Saturation headways for passenger cars and trucks

Case Passenger cars Heavy trucks

Headway Obs Headway Obs

1 3.2 798 4.6 15

2 4.8 758 5.9 193 5.7 355 7.6 54 5.7 341 8.7 3

5 7.4 521 8.3 126 7.2 624 7.9 157 7.6 668 7.9 17

8 9.2 694 11.6 18Mean Total 6.4 4759 7.8 104

Obs is the number of observations.


While these patterns are useful in describing intersection operation in general, we muststudy the process more thoroughly, analytically, and rigorously, if we are to develop modelsfor capacity and delay. Our choice is to focus on a ``subject approach'' and how it operates.We are interested in seeing how the interval between departing vehicles on the subjectapproach (i.e. the headway) is a�ected by the degree of con¯ict experienced in interactingwith vehicles on the other approaches. Our hypothesis is that this degree of con¯ict increaseswith two factors: the number of vehicles on the other approaches and the complexity of theintersection geometry.At the intersection of two one-way streets, for example, conditions fall into one of two cases. If

a vehicle on the subject approach ®nds no vehicle waiting on the con¯icting approach, it can enterthe intersection immediately after stopping. However, if there is a vehicle waiting on the con-¯icting approach, agreement must be reached with the con¯icting vehicle about which one willenter the intersection next. Thus, the headway between vehicles departing consecutively on thesubject approach is shorter for the ®rst case than for the second.Two other factors a�ect the departure headway for vehicles on the subject approach: vehicle

type and turning movement. Heavy vehicles will have a longer headway than passenger cars.Further, left turning vehicles will have longer headways than through vehicles, which in turn willbe longer than right turning vehicles.In summary:

(a) AWSC intersections often operate in either two-phase or four-phase patterns, based pri-marily on the complexity of the intersection geometry. Flows are determined by a con-sensus process in which the right of way alternates between and among the approaches(north-south and then east-west for a single-lane approach and each approach in turn for amulti-lane intersection).

(b) The headway between the nth vehicle and the n-1st vehicle on the subject approach dependson the degree of con¯ict the nth vehicle faces. This degree of con¯ict is a function of thenumber of other vehicles that vehicle faces and the number of lanes on the intersectionapproaches.

(c) The headway for a subject approach vehicle is also dependent on its vehicle type and itsturning maneuver.

3.3. Realization 1. The intersection of two one-way streets

This ®rst realization of the AWSC model pertains to the operation of two one-way streets, eachstop-controlled (see Fig. 1). The equations tie directly to Richardson's work. For simplicity, weassume that vehicles only travel straight through the intersection.Here the vehicle service times can take on two values: s1, the service time if no vehicle is waiting

on the con¯icting approach and s2, the service time if a vehicle is waiting on the con¯ictingapproach. This means the service time distribution is bi-valued for both approaches. For thenorthbound approach, the mean service time is:

sN � s1 1ÿ �W� � � s2�W �4�


where �W is the probability of ®nding at least one vehicle on the westbound approach (i.e. thetra�c intensity for that approach) and 1ÿ �W is the probability of not ®nding a vehicle on thewestbound approach.By symmetry, the mean service time for the westbound approach is:

sW � s1 1ÿ �N� � � s2�N �5�

If we substitute Eq. (4) into Eq. (5) and note that the tra�c intensity � is the product of thearrival rate l and the mean service time s, then the service times for each approach can be solveddirectly in terms of the bi-valued service time distributions and the arrival rates on eachapproach, as in Eq. (6) and Eq. (7).

sN � s1 1ÿ lN s1 � s2� �� 1ÿ lNlW s21 ÿ s22

ÿ � �6�

sW � s1 1ÿ lW s1 � s2� �� 1ÿ lWlN s21 ÿ s22

ÿ � �7�

Fig. 1. Intersection of two one-way streets.


3.4. Realization 2. The intersection of two two-way streets

This realization is also based on Richardson's work. As before, the service time for a vehiclecan have two values, s1 or s2. The mean service time for vehicles on an approach is again theexpected value of this bi-valued distribution. However, computing the service time is more com-plex than before. A northbound vehicle will have a service time of s1 only if both the eastboundand westbound approaches are both empty. This event re¯ects the joint probability that there areno vehicles on both approaches. The mean service time for the northbound vehicle is given in Eq.(8) (see Fig. 2):

sN � s1 1ÿ �E� � 1ÿ �W� � � s2 1ÿ 1ÿ �E� � 1ÿ �W� �� 8�

Unlike realization 1, it is not possible to solve directly for the mean service times as a function ofthe arrival rates and bi-valued service time distributions. Each service time is dependent upon ordirectly coupled to the tra�c intensity on the two con¯icting approaches. This coupling makes adirect solution di�cult. However, it is possible ®nd a solution through an iterative procedure,based on the system of equations for all approaches, as implicitly described by Eq. (8).

Fig. 2. Subject, opposing, and con¯icting approaches.


3.5. Realization 3. An extended model for single lane sites

This third realization extends the theory's implementation to the general 2 by 2 case. Throughit we resolve four limitations previously encountered. First, more than two cases can be con-sidered. This accounts for the more complex conditions that arise when more than two vehiclesare simultaneously trying to enter the intersection. Second, we enable vehicle type to a�ect thesaturation headway. Third, we allow more realistic operating conditions than just the driver-on-the-right rule: vehicles on opposing approaches can enter the intersection at the same time,regardless of which vehicle arrives ®rst at the stop line. Fourth, we gain consistency between ®eldmeasurements and model parameters.Five categories of saturation headway enable us to represent the di�erent ``degree of con¯ict''

situations faced by a vehicle on a given subject approach. Table 11 speci®es the conditions forthese cases and the probability of occurrence for each. The probabilities are based on the tra�cintensities for the opposing and con¯icting approaches. The way the model functions, and itscomplexity, is evident when one realizes that the tra�c intensity for one approach is computedfrom its capacity, which in turn depends on the tra�c intensity of the other approaches. Thiscircularity is the essence of the interdependencies that exist, and illustrates why an iterative pro-cedure is useful to obtain the set of departure headways and service time that ®t a given situation.From Table 11, the probability, P Ci� �, for each degree-of-con¯ict case i can be computed. The

tra�c intensities on the opposing approach, the con¯icting approach from the left, and the con-¯icting approach from the right are given by �O; �CL, and �CR, respectively.

P C1� � � 1ÿ �o� � 1ÿ �CL� � 1ÿ �CR� � �9�

P C2� � � �o� � 1ÿ �CL� � 1ÿ �CR� � �10�

P C3� � � 1ÿ �o� � �CL� � 1ÿ �CR� � � 1ÿ �o� � 1ÿ �CL� � �CR� � �11�

P C4� � � �o� � 1ÿ �CL� � �CR� � � �o� � �CL� � 1ÿ �CR� � � 1ÿ �o� � �CL� � �CR� � �12�

Table 11

Probability of degree of con¯ict case

Degree of con¯ict case Vehicles on. . .approach Probability of occurrence

Sub Opp Con-L Con-R

1 Y N N N 1ÿ �o� � 1ÿ �CL� � 1ÿ �CR� �2 Y Y N N �o� � 1ÿ �CL� � 1ÿ �CR� �3 Y N Y N 1ÿ �o� � �CL� � 1ÿ �CR� �4 Y Y N Y �o� � 1ÿ �CL� � �CR� �4 Y Y Y N �o� � �CL� � 1ÿ �CR� �4 Y N Y Y 1ÿ �o� � �CL� � �CR� �5 Y Y Y Y �o� � �CL� � �CR� �Sub is the subject approach. Opp is the opposing approach. Con-L is the con¯icting approach from the left. Con-R is the con¯icting

approach from the right.


P C5� � � �o� � �CL� � �CR� � �13�

Further, the departure headway for a given approach is the expected value of the saturationheadway distribution, or

hd �X5i�1

P Ci� �hsi �14�

where P Ci� � is the probability of the degree of con¯ict case Ci and hsi is the saturation headwayfor that case, given the tra�c stream and geometric conditions of that approach.The service time required for the calculation of delay is computed based on the departure

headway and the move-up time.

s � hd ÿm �15�s is the service time, hd is the departure headway and m is the move-up time.As was discussed before, two perspectives on capacity can be taken. The ®rst focuses on the

maximum possible throughput for a subject approach, given that the ¯ows on the other approa-ches remain constant. The second focuses on the maximum throughput for the intersection as awhole given that the volume distribution remain ®xed. In either case, when the saturation rate forone or more movements equals 1.0, capacity has been reached.The steps involved are as follows:

Step 1. For each approach, determine the ¯ow rates for each turning movement as well asthe proportion of heavy vehicles and the geometric con®guration.

Step 2. Compute the base saturation headway for each degree of con¯ict case for eachapproach, given the proportion of left and right turns, the proportion of heavyvehicles, and the geometric con®guration.

Step 3. Establish the starting value of the departure headway for each approach. Aninitial value of 4.0 s can be used although a reasonable starting value will produceconvergence.

Step 4. Compute the tra�c intensity for each approach based on the product of the arri-val ¯ow rate (in vp/s) for the approach and the initial value of the departureheadway.

Step 5. Compute the revised expected value of the departure headway for each approachbased on the computed degrees of saturation for all of the approaches, using Eq. (14).

Step 6. If the revised and adjusted departure headway for any approach has changed bymore than a small increment (e.g. 0.01 s), go to step 4. Otherwise, go to step 7.

Step 7. Compute the service time based on the departure headway and the move up time.Step 8. Compute the capacity based on one or both of the capacity concepts described

above. For the ®rst, increase the ¯ow rate on the subject approach until the tra�cintensity for one of the approaches equals one, keeping the ¯ows constant on theother approaches. For the second, increase the ¯ow rates on all approaches, whilekeeping the percentage distribution ®xed, until the tra�c intensity on one of theapproaches equals one.


3.6. Example calculation

An example calculation illustrates how this procedure is used. Let's examine two intersectingone-way streets (see Fig. 3). For simplicity we will assume no serial correlation is present betweenthe headways (which means no serial correlation adjustment is required). With no heavy vehiclesand no turning movements, the saturation headways are 3.9 s/veh for Case 1 and 5.9 s/veh forCase 3. The computations are summarized in Table 12.

Step 1. Initial conditions. The northbound ¯ow rate is 300 vph while the westbound ¯ow rate is200 vph. Ideal conditions are assumed.Step 2. Compute the base saturation headways. Two degree of con¯ict cases pertain: Case 1Ðno

opposing or con¯icting ¯owsÐwith a saturation headway of 3.9 s/veh and Case 3Ðone con¯ict-ing ¯owÐwith 5.9 s/veh. No adjustments are required to either value since we have no turningmovements or heavy vehicles.Step 3. Starting value. The mean departure headway for each approach can be set initially to 4 s.Step 4. Initial tra�c intensity values. The tra�c intensity for each approach is the product of

the mean arrival rate and the service time. For the northbound and westbound approaches, theinitial values are given computed in Eqs. (16) and (17).

Fig. 3. Example situationÐtwo one-way streets.


�WB � �WBSWB

3600� 200� � 4:0� �

3600� 0:22 �16�

�NB � �NBSNB

3600� 300� � 4:0� �

3600� 0:33 �17�

Step 5. Compute the departure headways based on the revised tra�c intensities. The expected valuefor the departure headway on each approach is computed based on the tra�c intensities initiallyestimated for each approach, as per Eq. (14).

hd;NB �X5i�1

P Ci� �hsi � hs1 1ÿ �WB� � � hs3 �WB� � �18�

hd;NB � 3:9� � 0:78� � � 5:9� � 0:22� � � 4:3 �19�

hd;WB �X5i�1

P Ci� �hsi � hs1 1ÿ �NB� � � hs3 �NB� � �20�

hd;WB � 3:9� � 0:67� � � 5:9� � 0:33� � � 4:6 �21�

Step 4a. Recompute the tra�c intensity based on the new value of headway.

�WB � �WBSWB

3600� 200� � 4:6� �

3600� 0:25 �22�

�NB � �NBSNB

3600� 300� � 4:3� �

3600� 0:36 �23�

Table 12Example calculations for headway and rea�c intensity

Iteration Departure headway Tra�c intensity

WB NB WB NB

1 4.0 4.0

0.22 0.332 4.6 4.3

0.25 0.36

3 4.6 4.30.25 0.37

4 4.6 4.4

0.25 0.37


Step 6. Repeat steps 4 and 5 until the departure headway values remain unchanged. Table 12 showsthat the departure headway values converge after only four iterations, resulting in 4.6 s for thewestbound approach and 4.4 s for the northbound approach.It is important to note that these values are not the saturation headways for either approach.

Rather, they are the expected values of the departure headways for the vehicles departing fromthe stop lines, given the ¯ows on both approaches.A numerical analysis helps make this last point clear. We ®rst need to convert each headway to

an equivalent hourly ¯ow rate. In this case, we obtain 815 vph (3600/4.4) for the northboundapproach and 777 vph (3600/4.6) for the westbound approach. Now, if we increase the volumes tothese values, the mean headways become 5.9 s on both approaches. But this condition is impos-sible! Notice that if the volumes really were 815 and 777 vph and the headways were 5.9 s/veh,then we would need 4808 s in one hour to process the northbound vehicles, and 4584 s for thewestbound approach. But there are only 3600 s in an hour, so the processing capacity of theintersection would be exceeded. This means the values we found are only the mean departureheadways if there are 300 vph on the northbound approach and 200 vph on the westboundapproach. However, they do let us compute the tra�c intensity for each approach. Here, thosevalues are 0.37 for the eastbound approach and 0.26 for the westbound approach.Step 7. Compute the service time for each approach. If the move-up time is 2.0 s, the service

times are 2.6 and 2.4 s respectively, as indicated by Eqs. (24) and (25).

sW � 4:6ÿ 2:0 � 2:6 s �24�

sn � 4:4ÿ 2:0 � 2:4 s �25�

Step 8. Compute the capacity. Both capacity de®nitions will be considered to illustrate the dif-ference between them.Step 8a. Compute the capacity of the northbound approach given the ¯ow rate on the westbound

approach and compute the capacity of the westbound approach given the ¯ow rate on the northboundapproach. This illustrates the ®rst capacity de®nition. We will answer two questions: (1) howmuch can the ¯ow on the northbound approach be increased, if the westbound volume is ®xed at200 vph, and (2) how much can the ¯ow on the westbound approach be increased, if the north-bound volume is ®xed at 300 vph? We consider the northbound approach. Leaving the westbound¯ow ®xed at 200 vph, we increase the westbound ¯ow to 790 vph before reaching a tra�c intensityequal to 1.0 on the northbound approach. The results are shown in Table 13.Step 8b. Compute the capacity of the westbound and northbound approaches given the initial

volume distributions on each approach. This illustrates the second capacity de®nition. First we notethat the volume distribution is 40% westbound and 60% northbound. These percentages must bemaintained. Given that fact, one question must be answered: how much can the total ¯ow on allapproaches be increased without fully saturating one or more approaches? The answer is1118 vph (671+447) as shown in Table 14.Note that the saturation ¯ow rates for both approaches have been found, given this distribu-

tion of tra�c ¯ows. The ¯ow on the northbound approach cannot exceed 671 vph without forcingthe northbound tra�c intensity to exceed one. Similarly, even though the tra�c intensity for thewestbound approach is not equal to 1.0, if the westbound ¯ow increases above 447 vph, the


northbound tra�c intensity will exceed one. So the maximum rate of ¯ow on the westboundapproach, for this distribution of tra�c ¯ows, has also been found.

3.7. Realization 4. Multi-lane sites

For multi-lane sites the saturation headways are longer than at single-lane sites, all other fac-tors being equal. This is the result of two factors. Larger intersections (i.e. a larger number oflanes) require more time to cross, thus increasing the saturation headway. Also, additional lanesmean more driver confusion and an increasing degree of con¯ict with opposing and con¯ictingvehicles, again increasing the saturation headway.By contrast, some movements do not con¯ict as signi®cantly as at single-lane sites. For exam-

ple, northbound right turning vehicles can depart at the same time as eastbound through vehicles,if they occupy separate receiving lanes. This means the saturation headway can be lower at multi-lane sites.The theory described earlier proposes that the saturation headway is a function of the direc-

tional movement of the vehicle, the vehicle type, and the degree of con¯ict faced by the subjectvehicle. This theory is extended here for multi-lane sites with respect to the concept of degree ofcon¯ict: saturation headway is a�ected to a large extent by the number of opposing and con-¯icting vehicles faced by the subject driver. For example, in degree of con¯ict case 2, a subjectvehicle is faced only by a vehicle on the opposing approach. At a two-lane approach intersection,there can be either one or two vehicles on the opposing approach. It is proposed here that eachdegree of con¯ict case be expanded to consider the number of vehicles present on each of theopposing and con¯icting approaches. These cases are de®ned in Tables 15 and 16, for two-laneapproach and three-lane approach intersections.For multi-lane sites, separate saturation headway values have been computed for the number

of vehicles faced by the subject vehicle for each of the degree of con¯ict cases. This requires afurther extension of the service time model to account for this increased number of sub-cases.

Table 13Example capacity calculationÐ®rst de®nition

Northbound Westbound

Volume 790 200Mean headway 4.6 5.9

Seconds consumed 3599 1180Tra�c intensity 1.00 0.33

Table 14Example capacity calculationÐsecond de®nition

Northbound Westbound

Volume 671 447Mean headway 5.4 5.9

Seconds consumed 3597 2637Tra�c intensity 1.00 0.73


Table 17 lists the 28 possible combinations of the number of vehicles on each approach foreach degree of con¯ict case for intersections with two lanes on each approach. When consideringlane distributions, there are a total of 64 combinations.As before, the probability of a vehicle at the stopline in a given lane is �, the tra�c intensity.

The product of the six tra�c intensities (encompassing each of the six lanes on the opposing orcon¯icting approaches) gives the probability of any given case occurring.The departure headway of the approach is the expected value of the saturation headway

distribution.

hd �X12i�1

P Ci� �hsi �26�

Ci represents each of the twelve degree of con¯ict sub-cases and hsi is the saturation headway forthat case.The iterative procedure to compute the departure headways and capacities for each approach

as a function of the departure headways on the other approaches is the same as described earlier.The additional sub-cases clearly increase the complexity of this computation, however.

Table 15Degree of con¯ict cases for two-lane approach intersections

Degree of con¯ict case Approaches with vehicles Number of opposing andcon¯icting vehicles

Opposing Con¯icting left Con¯icting right

1 02 x 1,23 x 1,2

x 1,24 x x 2,3,4

x x 2,3,4

x x 2,3,45 x x x 3,4,5,6

Table 16Degree of con¯ict cases for three-lane approach intersections

Degree of con¯ict case Approaches with vehicles Number of opposing and

con¯icting vehiclesOpposing Con¯icting left Con¯icting right

1 0

2 x 1,2,33 x 1,2,3

x 1,2,3

4 x x 2,3,4,5,6x x 2,3,4,5,6

x x 2,3,4,5,6

5 x x x 3,4,5,6,7,8,9


4. Summary

This paper presents a methodology for analyzing the capacity of an AWSC intersection. Itdescribes the theory upon which the methodology is based and illustrates its use through a seriesof example problems.As with all other highway facilities, the capacity characteristics of AWSC intersections are a

re¯ection of how they operate. While ``yield-to-the-right'' is the nominal operating rule, ®eldstudies suggest that a much more complex decision making process is involved. It involves theprobabilities that speci®c con¯ict situations arise and the fact that time is ``lost'' while thesecon¯icts are resolved. For example, when the juncture of two one-way streets is at or near capa-city, a pattern of alternating discharge from the two approaches occurs: ®rst a vehicle from the

Table 17Probability of degree of con¯ict case-multilane AWSC intersections (two-lane approach intersections)

DOC case vehicles Number of vehicles on approach

Subjectapproach

Opposingapproach

Con¯icting leftapproach

Con¯icting rightapproach

1/0 1 0 0 02/1 1 1 0 02/2 1 2 0 0

3/1 1 0 1 01 0 0 1

3/2 1 0 1 1

1 0 2 01 0 0 2

4/2 1 1 0 1

1 1 1 01 0 1 1

4/3 1 2 1 01 1 2 0

1 0 1 21 0 2 11 2 0 1

1 1 0 24/4 1 2 2 0

1 2 0 2

1 0 2 25/3 1 1 1 15/4 1 1 2 1

1 2 1 11 1 1 2

5/5 1 2 2 11 2 1 2

1 1 2 25/6 1 2 2 2

DOC case/vehicles is the degree of con¯ict case and the number of vehicles on the opposing and con¯icting approaches.


one approach enters the intersection followed by a vehicle from the other approach. A four-phasepattern emerges at multi-lane four-leg intersections. Drivers from each approach enter the inter-section together until the queue is exhausted or a preemption occurs. Then, right-of-way passes tothe next approach, and each is served in turn.More generally, the headway between successive vehicles on the same approach depends on the

degrees of con¯ict faced by each vehicle. This degree of con¯ict depends on the number of othervehicles that a given vehicle can face on a given the other approaches and the number of lanes onthe various approaches. Vehicle type and turning maneuver are also important.We present an eight-step procedure for estimating the capacity of a given situation. It starts

with adjustment of general parameters to re¯ect site-speci®c conditions, and then uses an iterativeprocedure to develop a set of simultaneous and consistent tra�c intensity values for the variousapproaches. Capacity is reached when the tra�c intensity for one or more of the approachesreaches 1.0.Two AWSC capacity de®nitions are de®ned and described. The ®rst focuses on the maximum

¯ow for a given approach, assuming the other ¯ows remain ®xed. The ¯ow on the approach ofinterest is increased until one or more movements it reaches a tra�c intensity equal to one. Thesecond focuses on maximum ¯ow for the intersection as a whole, assuming the ratios among the¯ows remain constant. The ¯ow on all approaches is increased simultaneously, while maintainingthe proportionality among the ¯ows, until one or morethe subject movements reaches a tra�cintensity equal to one. The former is most useful in calculating delay, and assessing the maximumgrowth potential on a single approach. The latter is useful in determining how much overallgrowth in tra�c can occur before full intersection utilization is reached.While this paper provides a signi®cant advance in the capacity treatment of such facilities,

much opportunity for further advancement exists. Not yet known is the impact of pedestrians,which are often the reason why AWSC intersections are installed. We also have only limitedknowledge about the in¯uence of variations in geometry, such as ¯ared approaches, and whethervariations in the timing of arrivalsÐserial correlation among the discharge headwaysÐin¯uencesthe performance achieved.From a methodological standpoint, enhancements such as those mentioned above can be

added to the existing model. In addition, as the use of simulation becomes more re®ned andcommonplace, there is also a need to develop simulation-based models of such facilities thatincorporate the behavioral characteristics reported here.

References

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Kyte, M., 1990. Estimating the capacity of an all-way stop-controlled intersection. Transportation Research Record1287, 70±81.

Kyte, M., Marek, J., 1989. Estimating capacity and delay at an all-way stop-controlled intersection. Transportation

Research Record 1225, 73±82.Kyte, M., Kittelson, W., Brilon, W., Troutbeck, R., Tian, Z., Mir, Z., Robinson, M., Vandehey, M., 1993. Phase IReport-Capacity Analysis of Unsignalized Intersections, National Cooperative Highway Research Project 3±46.

Kyte, M., Tian, Z., Kuhn, J., Po�enroth, H., Butorac, M., Robertson, B., 1994. Saturation headways at stop-con-

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Richardson, A., 1987. A delay model for multiway stop-sign intersections. Transportation Research Record 1112, 107±

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a capacity model for all-way stop-controlled intersections based on stream interactions

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