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11 Chapler Three THEORY AND MODELS This chapter desonbes the operation of an AWSC intersection and documents how this descriptor can be translated into computational procedures for senice time, capambr, and delay. Candidates models are identified and evaluated. A recommended procedure is descnbed. CONCEPTS OF CAPACITY AND DELAY Capacity The capacitor mode] for signalized intersections in Be 1994 HCM Update is based on He saturation headway Tomato for a given approach. The saturation headway is computed from an ideal value (~.9 seconds) that is modified based on intersection geometry, traffic control parameters, and traffic flow conditions. The estimation of the saturation headway is often complex. For example, several models have been developed to forecast the saturation headway for a left turn stream at a signalized intersection depending on whether the steam movement is permitted or protected, and whether it occurs from a shared lane or an exclusive lane. The capacity flow rate is computed from the saturation flow rate and tile proportion of the signal cycle that is allocated to this stream, as Shown in Equation 5. g c = s C (A where c is He approach capacity, s is He saturation flow rate, g is the effective green time for the subject approach, and C is the cycle length. The capacity of a lane at an AWSC intersection is also dependent on the saturation headway of that lane. Since there is no traffic signal controlling the stream a. . ~.. ~. ~ The HCM defines He capacitor of a transportation facility as the maximum hourly rate at which persons or vehicles can reasonably be expected to traverse a point or a uniform section of a lone or roadway during a given time period under prevailing roadway, tragic, and control conditions. . For an AWSC intersection prevailing conditions mean the geometry of the intersection and the distribution of flow rates on each of He intersection approaches. Because of the interaction between the traffic streams on each approach, and because it is this interaction Hat governs the maximum flow rate on each approach, two capacity concepts must be considered: what is the capacity of a given lane or approach given He flow rates on the other intersection approaches. Here the question is: how much can the flow on the subject approach be increased, if the flows on He other approaches remain fixed. what is the capacity of each lane or approach, given a distribution of flows on all of the intersection approaches. Here is the question is: how much can the flows on each approach be increased if He initial distnbution remains fixed. Delay Delay is defined as He time that a motorist spends traveling at less than his or her desired free flow speed. Slowing or stopping at an intersection, usually the result of the red phase of a signal, the presence of a stop sign, or the presence of a queue, is counted as delay. At an AWSC intersection, all vehicles must stop before proceeding through the intersection. The delay inherent in the operation of an AWSC intersection includes four components: deceleration, time in the moving queue, service time, and acceleration time. The deceleration and acceleration components of delay are often referred to as geometric delay. movement, or allocating the n~t-of-way to each conflicting traffic stream, He rate of departure is controlled instead by the interactions between the traffic streams ~emseIves. There is a degree of conflict that can be observed Hat increases with the number of approaches Cat are loaded simultaneously. To a lesser extent, He geometry of the in~rsechon itself condors this Standard queueing models assume that vehicles form a rate of departure. vertical stack as they wait In queue for service at the stop
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12 line. While the service Patton at some traffic facilities can be modeled as either deterministic or random, the pattern of departure from the stop line at an AWSC intersection is controlled by the degree of conflict exerted by He presence of vehicles on the opposing and conflicting approaches. The results presented later in this report show that the service times can be assumed to be a finite collection of values, based on the intersection geometry, He composition of vehicles in the traffic stream, end the flow Dies on each intersection approach, with the probability of each value based on a specific degree of conflict or likelihood of vehicles on the opposing and conflicting approaches. . Stopped delay has been used as the primary measure of effectiveness for detennnung the level of senice at signalized intersections since He introduction of the 1985 edition of the HEM. Stopped delay provided several advantages over the previous measure, load factor. It was directly measly able in the field and it related directly to the motonst's expenence, thus providing a strong link to perceived level of senice. The 1994 HCM Update included, for the first time, a delay-based measure of effectiveness for both TWSC and AWSC intersections. When accounting for the difference between total delay and stopped delay, this provides a common measure that can tee used to compare the performance and operation of an intersection under different kinds of control. DESCRIPTION OF INTERSECTION OPERATIONS between certain turning maneuvers (such as a northbound led ton vehicle and a southbound through vehicles, but In generalthe north-south shams alternate right-of-way with the east-west streams. A four-phase pattern emerges at multi-lane Dogleg intersections where the development of the r~ght-of-way consensus is more difficult. Here drivers from each approach enter the intersection together as nght-of-way passes from one approach to the next and each is served In turn. AWSC Intersections require Divers on ad approaches to stop before proceeding into the intersection. While the priority to the driver on the right is a recognized mle in some areas, it is not a sufficient descriptor of intersection operations. Whalen fact happens is the development of a consensus of right-of-way that alternates between the Divers on the intersection approaches, a consensus that is dependent primarily on the intersection geometry and the arrival patterns at He stoplme. Consider an in~s=hon composed of two one-way sheets. Here, Divers alternately proceed into the intersection, one vehicle from one approach followed by one vehicle from the other approach. This same two-phase pattern is observed at a standard four-leg AWSC intersection where Divers from opposing approaches enter the intersection at roughly the same time during capacity operations. Some in~ruptionofthis pattern occurs when there are conflicts While these patterns are useful to describe the overall intersection operation, we must next consider how the pawns affect the capacity of an approach, which we win describe as He subject approach. At the intersection of two one-way streets, the headways of vehicles departing from the subject approach fall into one of two cases. If there are no vehicles on any of He other approaches, subject approach vehicles can enter the intersection immediately after stopping However, if there are vehicles waiting on the conflicting approach, a vehicle from the subject approach cannot enter the intersection immediately after the previous subject vehicle but must wait for consensus writhe next conflicting vehicle. The headways between consecutively departing subject approach vehicles win be shorter for the first case than for the second. Thus the headway for a departing subject approach vehicle is dependent on He degree of conflict experienced In interacting with vehicles on the over intersection approaches. This degree of conflict increases with two factors: He number of vehicles on the other approaches and the complexity of the intersection geometry. Two other factors affect the departure headway of a subject approach vehicle: vehicle type and turning movement The headway for a heavy vehicle will be longer than for a passenger car. Further, the headway for a left turn vehicle wid be longer Han for a through vehicle, which in turn will be longer than for a right turn vehicle. ~ summary: . . AWSC intersections operate un either two-phase or four-phase patterns, based primarily on the complexity of He intersection geometry. Flows are determined by a consensus of right of way that alternates between He north-south and east-west streams (for a single-lane approach) or proceeds In ton to each intersection approach (for a multi- lane approach intersection). The headways between consecutively departing
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13 data set and a more extensive measurement of saturation headways. The model, published as an interim procedure in TRC 373 (1991) and now included in the 1994 HCM Update, relates capacity to the relative distribution of volumes on each of the intersection approaches. c = E~a~,vp`+E,Joc L,l+~a3LTp~+~a4RTp ~(~) . subject approach vehicles is dependent on We degree of conflict between these vehicles and the vehicles on the other~nter~ion approaches. The degree of conflict is a function of the number of vehicles faced by the subject approach vehicle (with whom he or she is competing for right-oŁ way) and by the number of lanes on the intersection approaches. The headway of a subject approach vehicle is also dependent on its vehicle type and its horning maneuver. This description of intersection operations must be translated into computational models or procedures that can be used to calculate the service lime, capacity, and delay for given conditions of traffic flow rates and intersection geometry. A search of the literature yielded six potential models Hat were considered as candidates for a new version of the. HEM include both emp~ncal and queueing models. CANDIDATE CAPACITY MODELS Capacity Model 1 Hebert (1963) proposed an emp~ncal model for the capacity of an AWSC intersection. This mode! is included as part of~el985HCM. 7200 c = ,^ v (10.15 - sv ) i", pm pm where vp', is the proportion of volume once main street. Capacity Mode] 2 Richardson (1987) developed a mode! for estimating capacity using an M/G/l queuing mode! based on the service times measured by Hebert 3600 c - ~ [~l - PC1) (} - PC2 )3 bm - Tc] + TC JO where p is He traffic intensity (the ratio of the arrival rate to the service rate), tm is He minimum saturation headway, and Tc is He saturation headway for vehicles faced by a conflicting approach vehicle. Capacity Model 3 Kyte and Marek (1989) and Kyte (1990) extended Hebert's empirical approach for capacity based on a larger where vpi is He volume on tiLe ith approach, Ej is the number of lanes once jib approach, LTpk is He proportion of leR-tuming traffic, and RTp,: is the proportion of rigl~t- bwming tragic, and It's are He regression coefficients. CANDIDATE DELAY MODELS Delay Model 1 Tro~beck and Akcelik (1991) developed a mode} for delay based on queuing theory. This mode} includes an approximation for the time-dependence of delay on queue formation and clearance during a peak period, where T is He length of He peak penod. It is shown in Equation 9. d = s + gOOT [ (x - I) + 4(x I) ~ 4soT3] where T is He length of the study period or congested penod, x is the degree of saturation, and s is the service time. Delay Mode! 2 Kyte (1990) developed an empirically based delay model teased on the vol~e/capacilyraiio. This model,published as an interim procedure in TRC 373 and now included in He 1994 HCM Update, relates He delay on an approach to He degree of saturation. The model is given in Equation 10. d = e3sx `103 Delay Mode! 3 Richardson (1987) proposed a delay estimation model for AWSC intersections based on He MlG/1 queueing theory model. [2A(l-p)] (11) where p is the traffic intensity, ~ is the arrival rate, and
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14 is He standard deviation of the service time. MODEL EVALUATION AND SELECTION Framework for Mode! Evaluation A primary objective of NCHRP 346 is to develop new methodologies for computing capacity and level of service for s~conkoDed intersections based on data Hat are representative of U.S. conditions. At the heart of these methodologies must be models that produce reasonably accurate forecasts of capacity and delay (assuming Hat delay is He basis for determining level of serviced ailing input data normally available to practicing traffic engineers. How do we determine that a mode] is able to meet this objective? Consider He following statements: · Specification. We must be able to specie a mode! using standard traffic engineering parameters. Theory. The model specification must represent a sound underlying theory of traffic flow. Calibration. We must be able to estimate the mode} parameters using the data that have been collected. This process of determining He ~ O numencal values of the mode] parameters is caned mode! calibration. Range. The mode] must be able to account for a wide range of traffic flow and geometric conditions likely to be encountered by the practicing traffic engineer. Validation. We must be able to verily the accuracy ~ of file mode] forecast over a uncle range of operating conditions. When forecasts are verified using data that was not used to calibrate the model, this process is called moa7e! valid. Quality. The mode! must produce better forecasts than other competing models. . . If the standards inherent ~ these statements can be met, a mode} can be put forward as the core of a new methodology to forecast capacitor or delay. These six statements have been translated into five evaluation criteria that are used to provide an initial assessment of the candidate models. The criteria used to assess each mode! is listed below. Ideally, a mode} should be (~) theoretically sound, (2) easily validated with field data, (3) practical and easily applied by the practitioner, (4) produce sufficient and appropriate measures of effectiveness as output, and (5) relevant~n terms of common situations encountered by the practitioner. Each of the models described above were evaluated using these cnteria. The results of this evaluation are given In Table 12. Table 12. Model Evaluation =.................................... , . ............ ........... . .; ............................................................................................ ..................... .................. ..................... ................. ................... ...................................................................................................... ., , ., , , , ~ ~ ~ ~ ~ P Y Y Y Y Y Y P Y Y P Y Y Y Y P Y Y Y Y Y Y Y __ AWSC: Capacity Models Model 1 Model 2 Model 3 AWSC Delay Models Model 1 Model 2 Model 3 Notation Y = Yes, meets criterion N = No, does not meet criterion P = Partially meets criterion AWSC Intersection Capacity Models Cap acid Mode} ~ considered only two cases faced by the subject approach driver and is thus somewhat limited In its ability to deal with a broad range of traBic conditions. Capacity Model2 provides a sound theoretical base but is limited to the two cases of Capacity Mode! I, Capacity Mode} 3 is empiricalEy-based and includes a number of practical conditions faced by a Diver at a stop-controlled approach While the present version of Capacity Mode} 2 is limited to Only two cases, it has one distinct advantage overate other two models: if it can tee extended to consider a large number of cases, it can provide capacity estimates based on a sound theory of ~ntersechon operations. Capacity Mode} 2 was recommended for final testing. AWSC Intersection Delay Models Delay Mode! 3 is based on sound theory but it has two limitations. It does not directly account for oversaturated conditions and one of its parameters, the variance of the service time, may be difficult to estimate for complex conditions. Delay Mode} 2 is empirically based but does
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~ r not directly consider oversaturated conditions. Delay Mode} ~ meets ad of the established criteria and was recommended for final testing. DESCRIPTION OF RECOMMENDED CAPACITY MODEL Capacity Mode} 2 is the basis for the procedure that is recommended for the computation of capacity of an AWSC intersection ~ this section, the mode] is descnbed In detail. Then, the required extensions to the mode! are explained and illustrated with a numerical example. Formulation I. Intersection of Two One-way Streets The first formulation of the mode} is based on the intersection of two one-way streets, each stop-controlled. Vehicles on either approach travel only straight through the~ntersection. Seeiigurel. _ _. Figure 1. Configuradon-Formulabon 1 , Conflicting Approach Subject Approach The service time for a vehicle assumes one of two values, based onHebert's headway measurements: so is the service time if no vehicle is waiting on the conflicting approach and s2 is the service time if a vehicle is waiting on He conflicting approach. The mean service time for vehicles on an approach is the expected value of this bi-valued distnbution. For the northbound approach, the mean · · - service time Is sAr = s! (1 - Ply) + s2 Pw (12) where Pw is the traffic intensity for the westbound approach and is equal to the probability of finding at least one vehicle on that approach Thus ]~ s the probability of finding no vehicle on the westbound approach. By symmetry, the mean service time for He westbound approach is sW = sl, (} - Pat) ~ S2 PN (13) When Equation 12 is substituted into Equation 13, and noting that the traffic intensity p is the product of the arrival rate ~ and He mean service time s, the service times for each approach can be solved directly in terms of He bi- valued service times and the arrival rates on each approach, as in Equations 14 and 15. S], [1 ~ AN (S] + shy] [1 ~ AI! AW (S] - sit] (14) S] [1 - AW (Sit + s2~3 w [1 - AW AN (sl2 - sit] ~ 5) Formulation 2. Intersection of Two Two-way Streets As before, He service time for a vehicle assumes one of two values, so or s2. The mean service time for vehicles on an approach is again the expected value of this bi-valued OlstnOutlon. AS exl?ected In tms case, computing the service time is more complex than In formulation 1. A northbound vehicle win have a service dine of s1 if both the eastbound and westbound approaches are empty simultaneously. The probability of tills event is the product of the probability of an empty westbound approach and the probability of an empty eastbound approach. The mean service time for the northbound vehicle is given in Equation 16. See Figure 2. sn sit (1 - pE) (1 - PW3 + s2 ~ - (1 - pay (1 - PA Unlike formuladon I, it is not possible to directly solve for He mean service time in terms of a combination of arrival rates end the bi-valued senice times. The service time on any approach is dependent upon or directly coupled web the traffic intensity on the two conflicting approaches.
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16 This coupling prevents a direct solution. However, it is possible to solve iteratively for the service time on each approach, based on a system of equations for each intersection in the form shown in Equation 16, above. Opposing Approach Cow - gapped 1 ., . Figure 2. Configurabon-Fo~mulation 2 I Conflicting Approach Tom Right Subject Approach Extended Mode] for Single Lane Sites There are four problems USA these two formulations of the mode} that must now be rectified. First, the limitation of only to cases is severe. Drivers face much more complex condidons based on the interaction of two or more vehicles simultaneously Eying to enter the intersection. Second, vehicle type significantly affects the saturation headway. Third, the drrver~n-the-right rule often does not correctly describe actual intersection operations. It is more likely Table 13. Probabilitr of Degree of Conflict Case that, during conditions of continuous queueing, vehicles on opposing approaches wid enter the intersection at the same time, regardless of which vehicle arrives first at the stop line. Fourth, there is a difference In the parameter ford by the mode} and He parameter that is measured In the field The first two problems reflect the limitation in Hebert's database. The third problem reflects an incorrect description of He operation of traffic at AWSC intone containedin~e 1985 HEM, namely that each approach is served iteratively, proceeding counterclockwise to the next vehicle on the right. The fourth problem requires a cIanficabon of the correspondence between the measured serv~c;*e time values and the forecasted values of headway. This will be discussed in greater depth later ~ this section. To resolve these problems, an extension of the mode! has been developed. The new mode! is based on five Satan headway values, each reflecting a different level or degree of conflict faced by the subject approach driver. Table 13 specifies the conditions for each case and He probability of occurrence of each. The probability of occurrence Is based on He traffic intensity on the opposing end conflicting approaches. The essence ofthemodel,and its complexity, is evident when one realizes that the traffic intensity for one approach is computed Tom its capacity, which in turn depends on He traffic intensity on the other approaches. This circularity is based on the interdependence ofthe traffic flow on all of the intersection approaches, and shows the need for iterative calculations to obtain stable estimates of departure headway and service time, and thus, capacity. , ...................................................................... 1 ~ . If ................... ....... . i""'''"" P ''''' '"'I'""''''''' ''"'"""''I''''''''' ~""''''''""'~""''''''"''"''] 1 Y N N . . .. . . 2 Y Y N ~ 3 N ~ ~N 4 Y Y Y 4 ~Y N 5 Y Y Y N N N y (l-poXl~p=Xl-PCR) (PoXl-Pa)(l~PcR) (l~PoXPaXl~P< (l-poXl~P=xp~ y N y y (PoXl-Pcs)(Pce) (Pod) (l-PoXP~(P`~) (POX1-PC:LXP~ Note: Sub ifs the subject approach Opp is the opposing approach Con-L is the conflicting approach from the left Con-it is the conflicting approach firm the right
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17 From Table 13, the probabili~, P[CJ, for each degree-of- conflict case can be computed. The ~a~c ~ntensities on dle opposing approach, ~e conflictirlg approach ilom ~e lefl;, and the conflichng approach fLom ~e nght are g~ven by PO, PGL, and PCR' respectively. P[C1] = (1 - po) (1 - PCL) (1 - PCR) (17) P[C2] = (po) (1 - PCI) (1 - Pc~.) (~) P[C3] = (1 - Po) (PC:L) (1 - PCR) 19 + (1 - po) (1 - PCE,) (PCR) P[C~ = (po) (] - P=) (P=) + (Po) (Pcz) (1 ~ PCR) + (l - Po) (PCL) (Pc~) (20) P[C~ = (po) (p=) (p~) (2~) The departure headway for an approach is ~e expected value of the saturation headway distribution, or h~ = ~ P[Ci] h$' (22) f ~ ~ where P[C] is the probability of the degree of conflict case Ciandh~, is ~e sah~ion headway for that case, g~ven the ~ffic skeam and geometnc conditions of the ~ntersection approach The service time requ~red for the calculation of delay is computed based on the departure headway and the move- up dme. s = hd ~ m (23) where s is the service bme, h~ is the departure headway and m is the move-up Ome. The computation of capacity is based on two different concepts. If it is desired to compute the capacity of the subject approach, g~ven the flows on~e other approaches, the volume on the subject approach is ~ncreased incrementally unti! ~e degree of saturation on any one approach exceeds one. This flow rate is tiLe maximum possible flow or throughput on the subject approach under prevailing conditions. Or, it may be desired to compute the capacity of the approach assuming a fixed proportion of h~icon all approaches based on~e g~ven conditions. Here, the volumes on all approaches are increased incrementally (maintaining ~e same proportion of traffic on each approach) unti! the degree of saturation on any one approach exceeds 100 percent. Aga~n, this flow is ~e max~mum throughput under prevailing conditions. Bo~ capacity calculations can be considered correct, depending on the question that ~e engineer or analyst must answer. The computational method is descnbed below. Step ]. For each approach, detennine flow rates for each turning movement, proportion of heavy vehicles, and geometnc configuration. Step 2. Compute the base saturation headway for each degree of conflict case for each approach, g~ven ~e proportion of left anfd nght turns, proportion of heavy vehicles, and geometnc configuration. Step 3. Establish the starting value of ~e departure headway for each approach. A value of 4.0 seconds is typically used (note: almost any stardng value w~! converge). Step 4. Compute the degree of saturation for each approach based on ~e product of the arrival flow rate (in veh/sec) for the approach and ~e initial value of the departure headway. Step 5. Compute the revised expected value of the departure headway for each approach based on ~e computed degrees of saturation for all of ~e approaches, using Equation 22. Step 6. If ~e revised departure headways for any approach has changed by more than a smaD increment (e.g., 0.01 seconds), go to step 4. If the revised departure headways for all approaches have changed less than this increment, go to step 7. Step 7. Compute anadjus~nentto the forecasted values of the fdeparture headways to account for ~e dependence in the headway forecasts.
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18 Step 8. Compute Me service time based on the departure headway and We move up time. Step 9. Compute Me capacity based either on concepts one or two. For concept one, increase the flow rate on the subject approach until the degree of saturation on any approach exceeds one, maintaining the flows constant on the other approaches. For concept two, increase the flow rates on each approach, maintaining the same proportion on each approach, until the degree of saturation on any approach exceeds one. Example Calculation An example calculation illustrates the application of this procedure. To simper the explanation, Me intersection of two one-way streets wall be assumed. See Figure 3. For standard conditions, win no heavy vehicles and no fuming movements, assume Mat Me saturation headways are 3.9 seconds/vein for cases ~ and 5.9 seconds/vein for case 3. The computations for Me example are summanzed in Table 14. 200 vph _ 1 1 ~ 1 _ _ 1 :.~4 ~ 3. Example calculation Step 2. Compute the base sal;uration headways. There are only two degree of conflict cases, for which Me saturation headways are 3.9 sec/veh (case I) and 5.9 sec/veh (case 3~. No adjus~nents are required since Mere are neither turning movements nor heavy vehicles. Step 3. Starting value. Starting values for the mean departure headways for each approach are assumed to be 4 seconds. Step 4. Initial values of degree of saturation. The degree of saturation for each approach is Me product of the mean arnval rate and Me service time. For Me northbound and westbound approaches, Me initial values are given computed In Equations 24 and 25. ply = ID ID = (200Xoo-o) = 0.22 (24) = M M = (300X4 ~ = 0.33 < 3600 Step 5. Compute the departure headways based on the revised degrees of saturation. The expected value for the departure headway on each approach Is computed based on the Melees of saturation initially estimated for each approach, as per Equation 22. Note Mat Me geometric and traffic flow conditions for this example I~m~t Me degree of conflict cases to case ~ aIld case 3. h`,,NB = ~ P[C,] he = h`, (I - P.") ~ hs3 it) (2Q . ~ ~ he = (3.9X.78) ~ (5-9X-22) = 4-3 (2~ h4" = ~ P[C,] he = ha, (1 - PM) ~ he (Pa) (28) , . ~ him = (3.9X.6 ~ (S.9X.33) = 4.6 as' Step 4a Recompute the degree of samranon based on the Step 1. Initial conditions. The northbound flow rate is new value of headway. 300 vph and the westbound flow rate is 200 vph. Ideal conditions are assigned. v`rB S`,B `200x4.6) = 0.2s (30) 3600 3600 PUB = 3~6 ~ = (3O3O6XO4 3) = 0.36 01) Step 6. Repeat steps 4 and 5 until the departure headway values remain unchanged. Table 14 shows that Me
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19 depart headway values have nearly stabilized after only four iterations to 4.6 seconds for the westbound approach and 4.4 seconds for the northbound approach. What is important to note is that this expected value is not the saturation headway for Me approach: rather it is the expected value of the departure headway for We vehicles departing from the stop line, given the flows on each of the other approaches. Again, note that these headways cannot be used to compute capacity flows. They are not sa~ionheadways unless Were is a continuous queue on the subject approach. Here's why. Suppose each headway is converted to an equivalent hourly flow rate. In this case, these flow rates would be 3600/4.4, or 815 vph, for the northbound approach and 3600/4.6, or 777 vph, for the westbound approach K the volumes are increased to the values given as maximum flow rate, the mean headways are re- calculated to 5.9 seconds on bow approaches! What has been computed is the mean value of We departure headway if there are 300 vph on the northbound approach and 200 vph on the westbound approach: 4.4 seconds on the northbound approach and 4.6 seconds on the westbound approach. What We expected values of the headway on each approach do allow is the computation of He propordonofthe hour that is consumed by He vehicles on each approach. For He above example, He proportion of the hour that is utilized on He northbound approach is 0.37, for He westbound approach it is 0.26. Thus there are she some resources of He intersection that are not Ally utilized How this translates into a calculation of capacity wiD be shown in step 9 of this example. Table 14. Example Calculations for Headway and Traffic Intensity Step 8. Compute the service time for each approach. If the move-up time is 2.0 seconds, He sentence times are given in Equations 32 and 33. s" = 4.6 - 2.0 = 2.6 see `32, An = 4.4 - 2.0 = 2.4 see <33' Step 9. Compute the capacity. First it is important to cianii the notion of capacity? The capacity for each approach cannot be computed directly. But given the volume on each approach, the mean headway for each vehicle can be computed (again, not the saturation headway, but simply the headway between consecutively departing vehicles) and, based on this headway, the proportion of the hour that the resources of a given approach are utilized can also be computed. Two alternative definitions will be offered here for consideration. Capacity definiffon #I: maximum volume increase on one approach with volumes on other approaches remaining unchanged!. For a given set of flows, how much can the flow rate be increased on one specific approach, with all other flows remaining unchanged. Capacity definition #2: constant proportion on each approach. For a given percentage distnbution of flows by approach, what is He maximum throughput for each approach and for the intersection. Step 9a Compute the capacity of the northbound approach given theflow rate on the westbound approach and compute the capacity of the westbound approach given the flow rate on the northbound approach. Consider the example of two intersecting one-way streets described above. How much can the flow on tile northbound approach be increased, if the westbound volume is fixed at 200 vph? Or, put another way, what limit does a flow rate of 200 vph on the westbound approach place on the northbound approach? The maximuln flow rate on the northbound approach, given a 200 vph flow rate once westbound approach, is 790 vph. Or, wad a limibug flow of 200 vph on the westbound approach, the maximum throughput on the northbound approach is 790 vph. The reverse is also true: if the flow rate on He nor~bo~d approach is 790 vph, the maximum flow on the westbound approach is 200 vph. A flow rate
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20 of 200 vph has been maintained on the westbound approach and the flow rate of 300 vph on the northbound approach teas been increased incrementally, until the degree of saturation on either of Me two approaches reaches one. The resets are shown In Table 15. Note that the seconds consumer! is the product of the volume and the mean headway. It is literally the time during which the approach i~dunng the hour. This value, divided by 3600, is equal to the degree of saturation. Table 15. Example Capacity Calculation-Concept 1 _< ~ - · a_ Volume ~790 200 Mean headway 4.6 S.9 Seconds consumed 3599 1180 Degree of saturation 1.00 0.33 Step 9b. Compute the capacity of the westbound and northbound approaches given the initial volume distributions on each approach. In the example from above, We volume distnbution is 0.40 for the westbound approach and 0.60 for the northbound approach. How much can the volume on each approach be Increased, maintaining We given distribution, until one or both approaches reach a degree of saturation of one? The final values are shown In Table 16. Table 16. Example Capacity Calculation-Concept 2 . .......................... ,, 1 my .. ............... Volume 671 Mean headway 5.4 5.9 Seconds consumed 3597 2637 Degree of salvation 1.00 0.73 ' Again, this shows that We mean headway for We original case does not represent the saturation headway, but only the mean departure headway for the given flow rates. There is a rule emerging here, looking at the two examples presented above, Mat might help define capacity. When the seconds consumed on any one approach reaches 3600, the flow rates are Me maximum possible flow rates or throughput. Even Tough Were is seeming spare capacity on the westbound approach, increasing the flow on the westbound approach causes the northbound approach to consume more Can 3600 seconds and Bus Me volcanoes on its approach must be reduced. Suggested definition of capacityflow rate: theflow rate on each approach such that the degree of saturation of any of the intersection approaches exceeds one. Extension of Degree of Conflict Cases It is expect Mat saturation headways at multi-lane sites Could be longer than at single-lane sites, ad over factors being equal. This is the result of two factors. A larger intersection geometry (i.e., a larger number of lanes) requires more travel time through the ~ntersecdon, thus Increasing He sah~radon headway. Additional lanes also means more driver concision and an increasing degree of conflict undo opposing and conflicting vehicles, again increasing the saturation headway. By contrast, some movements may not as readily conflict with each other at multi-lane sites as they might at single- lane sites. For example, a northbound vehicle turning right may be able to depart simultaneously as an eastbound through movement, if they are able to occupy separate receiving Am when departing to He east. This means in some cases that the saturation headway may be lower at multi-lane sites. The theory described earlier proposed that the saturation headway is a fimcdon of He directional movement of the vehicle, He vehicle type, and He degree of conflict faced by the subject vehicle. This theory is extended here for multi-lane sites with respect to the concept of degree of conflict: saturation headway is affected to a large extent by the Amber of opposing and conflicting vehicles faced by He subject driver. For example, In degree of conflict case 2, a subject vehicle is faced only by a vehicle on the opposing approach. At a two-lane approach intersection, dlere can be either one or two vehicles on the opposing approach. It is proposed here Hat each degree of conflict case be expanded to consider the number of vehicles present on each of the opposing and conflicting approaches. These cases are defined in Tables 17 and IS, for two-lane approach and three-lane approach intersections.
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21 Table 17. Degree of Conflict Cases for Two-Lane Approach Intersections or - x x x .: ~ .. ~ ~ ~1 .................................................... , ............................................................ .................................................................... .......................... - . t ~. ~ 'I"""'' At" '"' 1"" " "'it'" ' 1 4 x x 2,3,4 S x x x 3,4,5,6 Table 18. Degree of Convict Cases for Three-Lane Approach intersections ::5 n 1 2 1.2 x x x x 2.3.4 3,4,5,6 l ~52 ~... l.'.'''.' '.'', '.'.' ,. , . ~l[ ~O ~ 2 x 3 _ x am' _ 5 I x I x I Extended Mode! for Multilane Sites For multi-lane sites, separate saturation headway values have been computed for Me number of vehicles faced by the subject vehicle for each of the degree of conflict cases. This requires a further extension of Me service time mode! to account for this increased number of sub-cases. Table 19 lists Me 28 possible combinations of Me number of vehicles on each approach for each degree of conflict case for intersections with two lanes on each approach. 1,2,3 1,2,3 x x x x 2,3,4,S,6 3,4,S,6,7,8,9 These combinations can tee further sub-divided if a vehicle can be on either one of the lanes on a given approach Tables 20 and21 lists the 64 possible combinations when alternative lane occupancies are considered, where a "1" indicates that a vehicle is in the lane while a "O" indicates Mat a vehicle is not in a lane. As before, the probability of a vehicle at the stopline in a given lane is p, the traffic intensity. The product of the six
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22 traffic intensities (encompassing each of the six lanes on Me opposing or conflicting approaches) gives the probability of any given case occurring The departure headway of We approach is Me expected value of the saturation headway distribution. 12 he = ~P[C,J ho' f=1 where Ci represents each of Me twelve degree of conflict sub-cases and h,i is Me saturation headway for that case. The iterative procedure to compute the departure headways and capacities for each approach as a function of Me depar~eheadways on the other approaches is the same as described earlier. The alimony sub-cases clearly increase Me complexity of this computation, however. Table 19. Probability of Degree of Conflict Case-Multilane AWSC Intersections (Two-Lane Approach Intersections) ~ :: ~.~ :~ ~ ~... : ::::::::::: ~:-:-:-:-:-:-:-:-:-:-:-:-:-:-:-:~ 1 , ig i ~: ~ ~ ~ 1 - ~ S my- 1 Hi i ' 1' ""~i 1/0 1 0 0 0 . 2/1 1 r 1 1 1 1 0 1 0 2/2 1 1 1 2 1 0 1 0 3/1 1 1 1 1 1 7 o 1 1 O O 1 3t2 1 1 1 0 1 1 1 O O 2 . 4/2 1 1 O 1 1 1 1 O 1 O 1 1 4/3 1 1 1 2 T 2 1 0 0 1 2 1 2 1 2 O 1 1 1 O 2 4/4 1 2 2 O 1 2 O 2 1 . O 2 2 S/3 1 1 1 1 S/4 1 1 2 1 1 2 1 . 1 1 1 1 2 5/5 1 1 1 2 1 2 1 1 1 2 1 2 1 1 2 2 .. . l 5/6 1 1 1 2 1 2 1 2 Notes: DOC Case/lrehicles is the degree of conflict case and the number of vehicles on the opposing and conflicting approaches. r' ~1 ~1 ~1 n 2 . 2 2
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23 T"b~ 2D. Pm~li~ of D~me of ConOio1 C"~l~e ~SC Intersootions (1~ ~e ~pm~ ~ns) 1 1 ?1171~71ilIIl~l~l~I~!~l~!l#~l~#l6~_~} i~.~ ~l~.~l~ I I ! !!~T~i~!!~I!!!!!!!I~Il!l!!!ll~ll#~Ill~l!i!l!i!ll~!~l!l~!~!~!~!~!I!l!l~!!l lll~l~)Il~l~l~l~!~l~ll~!~!~l!l!l!l!l!~!l!l!l!lI!l~l!l!i!lIIl!~
Representative terms from entire chapter: