Capacity Analysis of Traffic-Actuated Intersections: Final Report (1996)

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APPENDIX E A COMPUTATIONAL FRAMEWORK AND ILLUSTRATED EXAMPLES FOR TRAFFIC-ACTUATED CONTROLLER MODELING INTRODUCTION The procedure for modeling traffic-actuated control developed under NIP Project 3~S contains a number of analytically complex and iterative steps that do not lend themselves to manual computa- tions. Therefore, an analysis program called ACT3-48 was developed in this study as a too! to provide for demonstration and testing of the model. A computational Damework involving five worksheets has been incorporated into the ACT3~8 program. The worksheets presented in this report propose extensions to the original worksheets proposed by Courage and Ak~elik [~] for modeling simple through and protected left turn phases. The worksheets play a very important part in overcoming the "black box" image of complex models. They provide a structure for presenting the results of intermediate computations in a common form that is compatible with the proposed techniques. Thus, the format for the intermediate outputs pro- duced by ACT348 was designed to be identical to the format of the worksheets. The computational process is illustrated with two examples. The details of the analytical mode! and procedure used to predict the trafflc-actuated signal timing plan [l, 2, 3, 4] and delay estimate [5] were described in the indicated references. These details wall not be repeated here. COMPUTATIONAL PROCESS Each worksheet has its own purpose. The purpose of Worksheet ~ is to enter the data required speci- fically for traffic-actuated control. Worksheet 2 is used to perform lane group data computation. Worksheet 3 sets up the traffic-actuated timing computation. Worksheet 4 computes the green times required to service the queues accumulated on the red phase. Worksheet 5 determines the green extension times based on the unit extension time settings. It is important to note that the analytical mode} developed in this study uses an iterative procedure to predict the tra~c-actuated signal timing plan. Thus, while the worksheets themselves are quite simple, the overall procedure contains iterative loops. The complete procedure involving the work- sheets is illustrated in Figure EM In addition to the five worksheets, two iterative loops are indicated as "Loop A" and "Loop B". ~AIL - Loop A. Required Time: This is an iteration between Worksheets 3 and 4. The purpose of this iteration is to make the phase times converge to a stable cycle length. Worksheet 4 must also refer to Worksheet 5 if phase time extensions are required to compute the required phase times. Appendix E: Page 1

Loop B. Phase Extension Time: This is an internal iteration within Worksheet 5. It is only required when gap reduction features are employed. When the allowable gap is a function of the phase time, the phase time cannot be computed without an iteration process. 4. Required Times (.,~,, r ~ -I *-a. 5. Phase Extension Times 1. Data I ~1 2. Lane Input ~ Groups .~. ~ 3. Cycle Time Adjustments ., A. ., ~ B ~ SYMBOLS Main Information Flow _ Worksheets Iterative Loops ~J '4,,,,,,,ei Figure E-1. Iterative loops in the phase time and cycle length computation procedure The computational worksheets are conceptually simple, but must be applied iteratively to arrive at usefill results. The program is required because the iterative nature of the procedure makes it totally impractical for manual implementation. The program is now fully functional and able to evaluate the analytical models using a variety of data. It also provides a flexible computational tool for examining other models that may be proposed. WORKSHEET 1: lllAFFIC-ACTUATED CONTROL INPUT DATA This first worksheet, presented in Figure E-2, gathers together all data required by the proposed analytical model. As a convention that will be used in the presentation of all worksheets, the row identification headings at the left side of the worksheets will be shown in lower case text if they are input data and In UPPER CASE text if they belong to intermediate computations based on input data. Appendix E: Page 2

l WORKSHEET 1: TRAFFIC-ACT UATED CONTROL INPUT DATA APPROACH-SPECIFIC DATA Northbound Southbound Eastbound Westbound | Left Turn: Treatment Codet 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 Position | Lead Lag N/A | Lead Lag N/A | Lead Lag N/A | Lead Lag N/A || Max Sneakers, S~, (vein) Free Queue, Qf (vein) ; Approach Speed, SP (mph) Ring 1 and 2 T~ation Simultaneous Independent N/A Simultaneous Independent N/A ~_ F_ DESIGN PARAMETERS | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 WBL EBT NBL SBT EBL WBT SBL NBT l Ph~eType(LT,G,N,X) l l l l l l I 1 l Phase Reversal? Y N Y N Y N Y N Detector Location: Length, DL Setback, DS i . i . Max Initial Interval, MxI Added ~itial per AcWation, AI Minimum Allowable Gap ~A , Gap Reduction Rate, GR Pedestri ~ 1 1 1 1 Maximum Green, MxG IntergreenTime,l Recall Mode (M~ Max, Ped, None) ! ! ! ! ! ! ! MIN VEH PHASE TIME, MnV: | STARTING GAP, SG | MIN INITIAL INTERVAL, MnI | MAXIMUM PHASE TIME,MxT: MxG+I | DETECTOR OCCUPANCY TIME, DT 1 (DL+18)/1~47SP I I I I | TRIALPHASE TIME: Max(MnV,WDW+I) Notes: 1. LT Treatment codes: (0) Prohibited, (1 ) PermiKed' (2) Protected, (3)Protected ~ Permitted, (4) Not Opposed. Figure E-2. Worksheet I: Traff~c-actuatecl contro! input data = _ T I 1 T Appendix E: Page 3

Approach-Specific Data The top portion of Worksheet 1 summarizes the approach-specific information. A separate column is used for each of the four approaches. The following items are required: Left Turn (:LT) Treatment Codes: The logic of the proposed mode} requires that the LT treatment be identified explicitly. The codes used here are consistent with the HCM Chapter 9 planning method worksheets. These codes are defined in Section ~ of HCM Chapter 9. 0 = LT does not exist ~ = Permitted 2= Protected 3 = Protected plus permitted or permitted plus protected 4 = Not Opposed The term "simple leR-turn protection"will refer to treatment number 2, in which the left turn moves only on the protected phase. The terser "compound leflc-turn protection" will be used to denote either protected plus permitted or permitted plus protected treatments. This is not an attempt to replace any HCM terminology, but to augment the terminology with more concise and easily understood references. Position Codes: These are required to distinguish between leading and lagging led turn protection. The terms "leading" and "lagging" apply equally to the cases of simple and compound le~c-turn protection. These codes do not apply if the leg turn is not protected. The worksheet offers a simple choice of "Lead," "Lag" or "N/A". The definition is very simple: leading left turns precede the movement of the opposing through traffic and lagging left turns follow it. Sneakers: This describes the number of left turns per cycle that are dismissed at the end of a permitted phase. An implicit default of two sneakers per cycle is built into the supplemental permitted left-turn worksheets for purposes of determining the mirumum saturation flow rate. Since any vehicle that rests in the detection zone wait extend its respec- tive phase, a more detailed treatment of sneakers will be required for traffic-actuated control. Free Queue: The current pretimed mode! assumes that the first perrn~tted leg turn at the stop line will block a shared lane. This is not always the case, as the through vehicles in the shared lane are often able to "squeeze" around one or more led turns. This is a definite defi- ciency of the pretimed model, but it is especially critical with traffic-actuated control, because vehicles In the free queue do not occupy the detector and therefore do not extend the green phase. A permitted left turn stopped on the detector must be treated entirely differently in the modeling process than one that is stopped beyond the detector. Appendix E: Page 4

Approach Speed, SP: The speed of vehicles on a signalized intersection approach is not important to the current HCM Chapter 9 methodology. It is, however, required in the analysis of tra~c-actuated operation. It determines the passage time between the detector and the stop line as well as the portion of inter-vehicle headways during which a presence detector is occupied. When modeling the operation of vehicles at traffic signals, it is common to assume a single value for speed throughout the cycle. Termination of Rings ~ and 2: The nature of dual-ring control requires that the second phases of rings ~ and 2 terminate simultaneously because they yield control to approaches with convicting traffic. However, control may pass Tom the first phase to the second phase of either ring without causing any conflict. Independent termination of the first phases improves efficiency in the allocation of time among competing movements and is generally exploited for this reason. The type of operation created by independent termination is sometimes referred to as "phase overlap." It is, however, not essential that the phases terminate independently. Older single-ring con- troller operation may be approximated by requiring that the first phases of each ring (i.e., phases ~ and 5 or 3 and 7) terminate simultaneously. In some situations involving coordi- nated arsenal routes, it is common to force both rings out of their first phase simultaneously. The model developed considered simultaneous or independent termination as legitimate alter- natives. It is possible that one or more of the first phases will not be used, because its associated left turn is not protected. In this case, the question of simultaneous or independent termination will not apply. This is another multiple-choice entry on the worksheet. The alternatives are "Simultaneous," "Independent" and "N/A". Phasing and Detector Design Parameters The bottom portion of the worksheet includes all the data items specific to each of the eight phases represented on Figure E-~. A separate column is provided on the worksheet for each phase. The first group includes the design parameters relating to phasing and detector placement. The following data items are included: --I- r Phase Type: This is the first of several phase-specific inputs that are required. A phase that is not active will not be recognized in any of the subsequent computations. Inactive phases are indicated by an "X" in the appropriate column of the worksheet. A leflc-turn phase will be considered active only if it accommodates a protected left turn. A through phase will be considered active whenever it accommodates through, left or right traffic. Active phases will be designated by: Appendix E: Page 5

"it" If the phase accommodates a protected ieDc turn on a green arrow. t'Tt' ttGt' t'N't If the phase accommodates through and right-turning traffic only. In this case, all leD turns are accommodated entireIv on another chase (i.e.. leflc-turn protection). , ~simple If any left turns are accommodated on this phase, opposed by oncoming traffic. This will occur on phases with permitted led turns and those with compound leflc-turn protection. If the phase accommodates, in addition to other movements, led turns that are not opposed at any time in the phase sequence. This will happen at "T" intersections, on one-way streets and in cases in which the phasing completely separates all movements on opposing approaches. Note that right turns are not referenced specifically in these designations. Right turns are assumed to proceed concurrently with the through traffic. Phase Reversal: Normally the first phase in each ring on each side of the battier (odd numbered phase) handles protected left turns and the second phase (even numbered) handles the remaining traffic. This creates a condition of leading left-turn protection. When lagging lefc-tu~n protection is desired, the movements In the first and second phases are interchanged. Most controllers provide an internal function to specify phase reversal. For purposes ofthis discussion, two phases may only be swapped if both phases are active. Detector Length, DL: Defined as the effective distance, measured parallel to the direction of travel, through which a vehicle will occupy the detector. A user-specified design parameter influenced by local practice. The detector length influences the choice of other parameters, such as the allowable gap in traffic that will terminate the phase. Detector Setback, DS: This is the space between the downstream edge of the detector and the stopline. Controller Settings The controller itself has several operating parameters that must be specified for each phase. Collectively, these will be referred to as the "controller settings," because they must be physically set In the controller with switches, alpha-numer~c keypads or some other electrical means. The following settings will exert a significant influence on the operation of the intersection and must therefore be recognized by the analysis methodology: Appendix E: Page 6

Maximum Initial Interval, M0: Used only when the initial interval is extended under volume-density control. Must be long enough to ensure that a queue of vehicles released at the beginning of green will be in motion at the detector before it terminates. Added Initial Per Actuation, Al: Used only when the initial interval is extended under volume density control. This value wall depend on the number of approach lanes. It should be long enough to ensure that each vehicle crossing the approach detector on the red will add an appropriate increment of time to the initial interval. Minimum Allowable Gap, MnA: This is a user-specified controller parameter, the effect of which wid be discussed later as the proposed models are exercised. It is typically set in the range of 2-3 seconds. It establishes the threshold for the length of the gap in traffic that wait cause the phase to terminate. The value of MnA is usually influenced to some extent by local practice. It is important to distinguish between the time gap and the time headway between vehicles. The time headway indicates the elapsed time between the successive arrival of two consecutive vehicles at a detector. The time gap indicates the elapsed time between the departure of the first vehicle from the detector and the arrival of the second. The time gap is what is led of the headway after the detector occupancy time has been subtracted. A tra~c-actuated controller using presence detectors views the passage of traffic in terms of gaps and not headways. Gap Reduction Rate, GR: This determines the rate at which the allowable gap is reduced in volume-density controllers as the green display continues. There are subtle differences in the definition of the gap reduction rate among controllers. For purposes of this project, a linear reduction rate (seconds reduction of gap per second of elapsed green time) will be assumed. Pedestrian Walk plus Don't Walk, WDW: This is m~n~rnum time given to each phase when pedestrian demand is registered or the pedestrian recall is active. It inclucles both the pedestrian walk and flashing don't walk intervals. These are actually entered into the controller as two separate parameters, but wall be combined for purposes of this analysis. If the pedestrian timing function is not implemented in a particular phase, the WDW value should be entered as zero. Maximum Green, MxG: This is a user-specified parameter, the effect of which will be discussed later. Local practice open plays an important part in the determination of maximum green times. Appendix E: Page

Intergreen Interval, I: Another user-specified controller parameter determined in accor- dance with local practice. The intergreen interval consists of a yellow change interval fol- iowed by an all-red clearance interval. These two intervals are entered separately into the controller, but will be combined here to simplify the analysis. Recall Mode: This detenn~nes how a phase will be treated in the absence of demand on the previous red phase. The options are: None: The phase will not be displayed. Max: The phase will be displayed to its specified maximum length. Min: The phase will always be displayed to its specified minimum length, but may be extended up to its specified maximum length by vehicle actuations. Ped: The phase will be given the filet Walk plus Flashing Don't Walk and may be extended farther, up to its specified maximum by vehicle actuations. This function will have a significant effect on the operation of the controller. For example, the maximum recall option wall have the effect of creating a pretuned phase. Preliminary Computations Several items computed Tom the above data will remain fixed, and are not subject to modification as a result of iterative computations. These values are included at the bottom ofthe worksheet for use in subsequent computations. They appear on the worksheet in upper case letters because they are computed values. In some cases, they are actually controller settings, but their values may be computed in terms of data that have already been established. Minimum Phase Time for Vehicles, MnV: This is actually a traffic engineering input, but it is included here as a computed value because it is subject to modification by other control parameters. It is an important input to any process for computing an implementable timing plan. It does not appear in the HCM Chapter 9 data at this time because HCM Chapter 9 does not offer the ability to compute timing plans. On the other hand, it is not possible to deal realistically with traffic-actuated control without recognizing the existence of a minimum phase time. For compatibility with other signal timing programs, the phase times include all intervals, including green, yellow and all-red. For purposes of the worksheet, the minimum phase time must be replaced by the maximum phase time MUG + ~ if the "Recall to Maximum" mode is In effect, and therefore, it is treated on the worksheet as a computed value. Appendix E: Page 8

Starting Gap, SG: If the gap is reduced during the green phase under volume-density control, a starting gap must be specified. It is common to choose a value for the starting gap equal to the passage time Tom the detector to the stop line. The SG may be computed as: SG= DS / 1.47SP If this value is equal to or less than the minimum allowable gap, MnA, then gap reduction would not normally be employed. In this case, SG = ~A. Min Titian Literal, MnI: This determines the minimum green time to be displayed. Given a specified value of the minimum phase time for vehicles, MnV, the minimum initial interval may be calculated as: HI = MT1V - ~ - SG Maximum Phase Time: The maximum time that may be assigned to the phase. This may be computed as (MUG + T). Throughout the iterative process, the computed green times must be checked to ensure that they do not exceed this value. Detector Occupancy Time, DT: This is the time during which a vehicle passing over the detector at the approach speed will occupy the detector. It may be computed as (DL+~) / I.47SP. This assumes an average vehicle length of IS feet. In computing the estimated length of a phase extension, DT must be added to the controller's current allowable gap threshold to determine the vehicle headway value that win cause a phase to terminate. Trial Phase Time: This will provide the starting point for the iterative procedure that will determine the estimated phase times. The procedure, as described later, constructs an initial timing plan based on the minimum acceptable phase times. Using the data already entered on this worksheet, the minimum acceptable phase times would be either the minimum phase time for vehicles, MnV, or the minimum phase time for pedestnans, (WDW + I), whichever is greater. WORKSHEET 2: LANE GROUP ASSIGNMENTS Figure E-3 shows the Lane Group Assignment Worksheet. The purposes ofthis worksheet are: I) To associate each of the lane groups with a signal phase; 2) To identify the movements that occur in each lane group; and 3) To convert the volumes and saturation flow rates entered on previous HCM Chapter 9 worksheets into units of vehicles per second. The timing computations on subsequent worksheets wait require these units. Appendix E: Page 9

; -- WBL EBT NBL SBT EBL WBT SBL NBT Thase Movement~ (T TR) . Opposing Through Phase LT EQUIVALENT S. E~ LG 1: No of!~nes LG Movements (LTR) ~7 ~T Volume(vph) l I T I I 1 1 1 ARRIVAL RATE, ql (vps) ~T I I I I T Platoon Ratio Rp GREEN ARR RATE, q.1 (vps) Sat Flow Rate (vphg) l l l l l l l l SERVICE RATE, sl (vps) l l l l l l l | LG2: No.ofLanes l l l l l l l | Movements (LTR) l l l l l | Volume (vph) ARRIVAL RATE, q2 (vps) Platoon Ratio R GREEN ARR RATE, q,2 (vps) Sat Flow Rate (vphg) SERVICE RATE, s2 (vpsg) l l l l l l LG3: No. of Lanes l l l l l l I Movements (LTR) l l l l l l l Volume(vph) l l l l l I 1. GREEN ARR RATE, q,3 (vps) ARRIVALRATE, q3 (vps) l l l l l | ~ Platoon Ratio Rp l l l l l l l Sat Flow Rate (vphg) l l l l l l 1: SERVICE RATE, s3 (vpsg) T I T T T I VEHICLE RIVAL RATE, VAR PedVolume(pph) l I ~ ~= PED ARRIVAL RATE. PAR: (pps) Figure E-3. Worksheet 2: Lane group data Appendix E: Page 10

The lane group breakdown should come directly from the HCM Chapter 9 Volume Adjustment Worksheet. Typically, the lefc-turn phases will have only one lane group and the through-traffic phases may have up to three lane groups. The first line ofthis worksheet indicates ad the movements ~,T,R) accommodated by the phase. The next line deals with progression quality. Basically, the arrival type (AT) from the HCM Chapter 9 Input Data Worksheet is repeated here. Arrival type 3 is commonly used at ~ly-actuated isolated intersections, where progression is not a factor and the green arrival rate is assumed to be equal to the red arrival rate. In coordinated operation, the AT input value will determine the green and red arrival rates ofthe through movements ofthe arterial. The next line is the phase number of the opposing through phase which opposes the permitted left turns of the subject phase. For example, NEMA phase 6 is the opposing through phase of NEMA phase 2 because the movements for NEMA phase 6 will oppose the permitted leD turns for NEMA phase 2. The last line for this top portion is left turn equivalence, Eat, which is the through-car equivalent for each permitted le~c-turning vehicle during the unsaturated opposing flow. The values of Eel can be obtained from Figure 9-7 of HCM Chapter 9. Six subsequent lines are associated with each lane group: Number of Lanes: From the HCM Chapter 9 Input Data Worksheet Volume (vph): From the HCM Chapter 9 Volume adjustment worksheet. The adjust ment for PHE should be applied if a peak IS-m~nute analysis is desired. The lane utilization adjustment should not be applied since this adjustment adds fictitious vehicles. Arrival Rate: Convert hourly volumes to vehicles per second. Platoon Ratio: tne platoon ratio, Rp, is obtained from HCM Table 9-3. This value is based on the arrival type. Rp will be used to determine amval rates on the red and green phases. In HCM Table 9-3, the default values for amval types I, 2, 3, 4, 5 and 6 are of 0.333, 0.667, 1.000, 1.333, 1.667 and 2.000, respectively. Green Arrival Rate: The arrival rate during the green phase (veh/sec) which is equal to the product of arrival rate and platoon ratio. Sat Flow Rate: From the HCM Chapter 9 Saturation Flow Adjustment Worksheet (vphg). Service Rate: Convert the hourly rate to vehicles per seconds of green. Appendix E: Page 11

The rates computed above win be used to determine the arrival and departure characteristics of each lane group. The combined arrival rate for all lane groups seined by the phase, i.e., the Vehicle Arrival Rate (VAR) may then be computed as the sum of the arrival rates for each of the active lane groups. The vehicle amval rate will be used to estimate the green extension time for each phase. It wait be assumed that ad vehicles using the phase will activate a single detector input terminal for the phase. The last two rows on the worksheet deal with pedestrian volumes, repeated Tom the HCM Chapter 9 Input Data Worksheet, and the computed arrival rate (peds/second). These items apply to through phases only. It will be assumed that the pedestrian timing features will be assigned to the phase in which the right turns conflict with the pedestrian movement (e.g., the east crosswalk will be asso- ciated with the northbound through phase). WORKSHEET 3: TRAFlIC-ACTUATED TIMING COMPUTATIONS _ The structure of Worksheet 3 is compatible with the dual-r~ng concurrent phasing displayed In Figure Em. Worksheet 3 is divided into two portions. Worksheet 3a shows the tra~c-actuated timing com- putations arid Worksheet 3b presents the timing plan sensitive capacity parameter estimates. These worksheets are presented In Figures E-5a and E-5b, respectively. The columns in the worksheet are arranged to match the dualizing format. Bamer Ring 1 Ring2 1 ~ WBL EBT EBL WBT Left Side of Ba'Tier E-W Movements ~ Barrier 47; I SBL 8 ~ T NET Right Side of Barrier ( N-S Movements ) 1 Figure E-4. Dual ring concurrent phasing scheme with assigned movements The east-west movements (led side of the barrier) are shown in the first three columns. The first column, labeled as "A" represents the first phase in each ring (1 or 5~. The second, or "B" column represents the second phase (2 or 6~. The third column wail contain the total of the phase times for the movements in the first two columns. The same format is repeated in the second three columns for the north-south movements (right side of the barrier). Appendix E: Page 12

WORKSHEET 3a: TRAFFIC-ACT UATED TIMING COMPUTATIONS - East-West Movements Rings: Phases Swapped? Movements Phase Times Ring2: Phases Swapped? Movements Phase Times DIFFERENCE: ABS(Ringl-Ring2) CYCLE TIME COMPONENTS: Independent Termination Simultaneous Termination No ~ Yes l WBL ~ EBT EBL I WET North-South Movements Total No I Yes llllllllll No I Yes No IYes EBT I WBL llllllllll NBL I SBT SBT I NBL WBT I EBL IIIIIIIRI SBL I NBT NBT I SBL 11111111111 11111111111 11111111111 11111111111 Illllllllil Il!llllilll lllllllllll 1111 11111111111 Figure E-5a. Worksheet 3a: TrafO'c-actuated timing computations ~, 1 ~- ~ ~ Total Inure are Free rows tor each of the two nngs. The first row indicates whether or not the phase pair is swapped. This information was entered on Worksheet 1. The next two rows give the movements and phase times for their respective phases. If the phases are not swapped, the assignments will be as shown in Figure E-4. If they are swapped, then the movements and times for the phase pair will be reversed. When a phase pair is reversed, the through movements will appear in column A and the left turns in column B. As indicated in the Worksheet 1 discussion, the movements in a given phase pair cannot be swapped if the left turn is not protected. The order of the chases in the nnir rim not affect the total phase time entered in the "Total" column. ~ ~~~~ Rae ^-^ ant- ~^ ~ 41V~ The next row contains the absolute value ofthe phase time difference between the two rings. Values are entered for each of the six columns. The components of the cycle time must now be determined and entered in the "Cycle Time Components" row. The procedure will depend on whether the first phase termination, entered on Worksheet 1, is simultaneous or independent. For simultaneous termination, enter the maximum value of each phase in the A and B columns (nnn; ~ or 2, whichever time is greater. For independent - r ~ termination, enter the maximum value of the total time (A+B) from ring ~ or 2. So, for each side of  the barrier, either the A and B columns or the "Total" column will have an entry, but not all three columns. This procedure should be carried out for both sides of the barrier. Remember that the termination treatment may be different on either side. The cycle length may now be determined as the sum of all the entries in the "Cycle Time Components" row. Appendix E: Page 13

WORKSHEET 3b: TIMING PLAN SENSITIVE CAPACITY PAF AMETER ESTIMATION OLD CYCLE LENG1 ~| 1 | 2 | 3 | 4 | 5 | 6 | , NEW CYCLE LENGTH VVBL EBT NBL SBT EBL WBT SBL Unad'usted Ph Time (Above J ) PHASE TIME ADJUSTMENT INTERIM PHASE TINIE, IPT INTERIM EFFECTIVE GREEN TIME, g l l l l l I I _ INTERIM RED TIME,IRT I r ~I ~I I _ INTERIM EFFECTIVE RED TIME, r RED TIME R R=IRT except for the LTpro~ =m 1 I T I I I 1 compound LT protection seenanos . STARTUP TIME T I T I I I I _ Complete the remainder ofthe items only if another iteration is required Recall Mode: Wksht 1 . Veh Arrival Rate, VAR: Wksht 2 T T T ~T ~ AVERAGE VEH ARRIVALS ON RED: VAR * R PROB(Oamvalson red) P0: q~e ~at &, 1 ~T 1 T I I = 1, ~ = VAR and ~ = 0 for the first iteration Min Veh Tune, MnV: Wksht 1 ADJUSTED VEHMIN,AVM: MnV(I-P0v) l l l l l I I Ped Amval Rate, PAR Wksht 2 I I T I T I 1 AVERAGE PED ARRIVALS ON RED: PAR $ R _ PROB (0 Peds on red), P0~,: e - ~AR-R) Ped Walk plus Don't Walk WDW: Wksht 1 l l T 1 1 1 1 ADJUSTED PEDMIN,APM: WDW(l-PoP) l I ~l l I I _ LAINEGROUPl I T T I T I I _ Movements~TR) I I I T I I I GREEN ARRIVAL RATE, q,l: ql * Rp l l l l l I I _ RED ARRIVAL RATE' q,1: [ql (r+g) - (q1 *Rp*g)] / r = SECOND LT EQUIVALENTS, E~2' l T 1 I T I I _ FREE GREEN TIME, g,12 l I ~l l I I _ OPPOSrNG QUEUE SERVICE TIME, gqll l I I ~I I _ QUEUE AT END OF EFFECTIVE RED r, Ql l I I ~| | QUEUE AT END OF gl1, Q~1 QUEUE AT END OF gql' Qql QUEUE AT BEGINNING OF SERVICE, Q1 Q51 = Q1 = Q`1= Qql for TH or Protected LT Movement Figure E-5b. Worksheet 3b: Timing plan sensitive capacity parameter estimation Appendix E: Page 14 8 NRT = = _ = = 1 = = _ = _ _ - _ = _ l

WORKSHEET 3b: (Continued for Lane Groups 2 and 3) Old Cycle Length 1 1 1 2 1 3 1 4 1 5 1 6 1 1 WBL EBT NBL SBT EBL_ WBT SBL LANE GROUP 2 l l l l T I I Movements(LTR) l l l l l I 1 = GREEN ARRIVAL RATE, q82: q2 * Rp RED ARRIVALRAIE,42:[q2(r+g)-(q2*Rp*g)]/r I I I I T I I = SECOND LT EQUIVALENTS, E=' I I I I T I T = FREE GREEN TIME, gr2Z l l l l l l I = OPPOSING QUEUE SERVICE TIME, gq21 I I I T T I T = QUEUE AT END OF EFFECTIVE RED, Q2 I I I T I I T = QUEUE AT END OF g,2, Ql2 1 1 ~T II T I 11 QUEUE AT END OF gq2, Qq2 QUEUE AT BEGINNING OF SERVICE, Qq2 Q,2 = Q2 = Qf2 = Qq2 for TH or Protected LT Movement LANE GROUP 3 Movements(LTR)~ ~ ~ ~ ~ I ~ GREEN ARRIVALRATE,q,l: ql *Rp~ T 11 I RED~VALRA=~:tq3(r+g)-(q3*~*g)]/r T T 11 | SECOND LT EQUIVALENTS, E,2' FREE GREEN TIME, gr32 | OPPOSING QUEUE SERVICE TIME, g'3 ~I T T I I T I T 11 | QUEUE AT END OF EFFECTIVE RED, ~l l 11 | QUEUE AT END OF g,3, ~l l 11 QUEUE AT END OF gq3, Qq3 QUEUE AT BEGrNNING OF SERVICE, Q,3 Q'3 = Q,3 = Q,3 = Q~3 for TH or Protected LT Movement 8 NBT - - Notes: 1. Refer the computation to the HCM Chapter 9 2. Refer the computation to the methodology in the 1985 HCM Chapter 9 Figure ¢5b. Worksheet 3b: Timing plan sensitive capacity parameter estimation (continued) Appendix E: Page 15

If the computed cycle length agrees with the cycle length determined on the previous iteration, then no further action wid be necessary on this worksheet. Otherwise, it is necessary to proceed with the minimum phase time adjustments. This brings us to the phase-specific section, worksheet 3b, shown In Figure E-5b. The first two rows ofthe phase-specific data have also been added to deal with dual- r~ng controllers. These are descnbed as follows: 01d Cycle Length: On the first iteration, this should be entered as zero. New Cycle Length: For the time being, this is simply the sum of the two phase times. The practical implications of eight-phase dual-r~ng control wall be considered later. If the new cycle length agrees with the old cycle length, then the iterative process uall have converged, and no further action wall be required. If not, the following information must be determined for the new cycle length: Unadjusted Phase Times: The phase times originally entered in the box above should now be placed here. Each phase time should be placed in the column for its respective phase regardless of any phase reversals that may have been imposed in the top part of the work- sheet. Phase Time Adjustment: The phase time adjustments are required here to align the phases in both rings. When the phases in two rings must terminate together, the controller must wait until the longest of the two has terminated before passing control on to the next phase. The adjustment procedure is also influenced by the question of simultaneous and independent termination. With simultaneous termination, the ring I-2 difference for the "Total" column must be added to the second phase in the ring with the smallest total phase time. With independent termination, the ring I-2 difference for both the "A" and "B" columns must be added to the shortest phase in each column. These adjustments will ensure that all of the phase times will fit together into the format of a dual-ring controller. Interim Phase Time, IPT: This tune is computed as the sum of the unadjusted phase time and the phase time adjustment in the rows directly above. On the first iteration, this wall be the trial phase time from Worksheet I. Interim Effective Green Time, g: IPT minus the lost time per phase. Interim Red Time, IRT: The cycle length minus the interim phase time. Interim Effective Red Time, r: TRT plus the lost time per phase. Appendix E: Page 16

Red Time, R: It is equal to the interim red time, TRT, except for the protected left turn phase In the compound led turn protection case where R equals TRT minus the phase time for the permitted led turn movement, because the real red time for the protected phase does not include the phase time for the permitted left turn movement. startup Time: This is the startup time of the phase with zero reference at the beg~nn~g of the first phase in the left side of the bamer. The following items are necessary only if another iteration is required. Recall Mode: From Worksheet I. It is important for the minimum time adjustments. Vehicle Arrival Rate, VAR: From Worksheet 2. Average Veh Arrivals on Red: (VAR x R). Probability of Zero Vehicles on Red, Pov Assuming a bunched exponential headway distnbution, based on the probability of arrivals on the previous red, Pov can be computed as p ~ era ~ ~ A) In this equation, up, ~ and ~ are ad bunched parameters. For the first iteration, ~p=l, \=VAR and A=0. On the consequent iterations, (p and ~ can be obtained Dom Worksheet 5 and may be estimated by .1 - tPq 1-~q subject to q < 0.98/A It is necessary to note that q is the total equivalent through volume (veh/sec) for all large groups that actuate the phase under consideration. If the Recall to Minimum or Recall to Maximum feature is in effect a zero probability must be entered on the worksheet. Minimum Time for Vehicles, MnV: From Worksheet I. Adjusted Vehicle Minimum, AVM: REV (] - Pal)] Appendix E: Page 17

The same procedure is repeated for pedestrians to compute the following items: Ped Arrival Rate, PAR Average Ped Arrivals on Red Probability of No Pedestrians on Red, POp Ped Walk plus Don't Walk, WDW Adjusted Pedestrian Minimum, APM The probability of no pedestrian arrivals dunng the effective red wall also be influenced by the recall mode. In this case, POp must be set to zero if the Pedestrian Recall feature is in effect. For the POp, the random amval mode! is used instead of the bunched arrival model. Ten subsequent lines are associated with each lane group: Movements CITE): Green Arrival Rate (q g ): Red Arrival Rate (qr): The movements accommodated in the lane group. (qi * Rp) where qi is the arrival rate for lane group I. ([qi (r + g) - (~ * Rp * g)] / r3 2nd LT Equivalence (Em) From HCM Chapter 9 Equation 9-22. Free Green (gr3 The portion of effective green until the arrival of the first leR-tLlrT ing vehicle. The computation for gf is based on HCM Chapter 9 methodology (1985). QSTOpp (gq): The portion of elective green blocked by the clearance of an opposing queue of vehicles. The computation for gf is based on the methodology in the HCM Chanter 9. A.' QatEoR (Q): The accumulated queue at the end of effective red, r, based on its queue accumulation polygon (QAP). QatEoF (Qr): The accumulated queue at the end of gf based on its QAP QatEOQ (Qq) The accumulated queue at the end of gq based on its QAP. QatBOS (Qs) The accumulated queue at the beginning of actual queue service (as) based on its QAP. Appendix E: Page 18

WORKSHEET 4: REQUIRED PHASE TIMES Worksheet 4 is presented in Figure E-6. Four global items must be entered first: Red Time, TRT: From Worksheet 3b. Target v/c Ratio, XT: This should be set to T.0 to determine the actual queue service time if a phase extension time will be computed. Otherwise, use the desired value of XT. Lost Time per Phase, t,: From the original HEM input data worksheet. Startup Lost Time per Phase, tS' Estimated by Ott -if. The phase specific items may now be entered. Note that each phase may have up to three lane groups that must be analyzed separately to determine total queue seance times. The Service Rate, s, and Left Turn Equivalence, EII are copied Dom Worksheet 2 for each lane group. The Movements, Green Arrival Rate, qg, Red Arrival Rate, it, and QatBOS, ~ are copied from Worksheet 3b for each lane group. Proportion of left turns in a Shared Lane, PI : Based on HEM Chapter 9 methodology. Equivalent Service Rate, es: es = s / (0.95 *EL]) es = s / [~! + PI, GEL, -all es=s Net Service Rate, ns: (es - qg / XT) Green Time, G: G= (IPT - I) G=IPT I(~~POV) for permitted led turns Tom exclusive lanes for a shared lane for through or protected led turn movements if there is no phase skip if there exists phase skip Queue Length Factor, fq: A queue length calibration factor proposed by Ak~elik to allow for variation in queue service time. The fq can be computed using the following equation: fq = 1.08 - 0. ~ (G / MxG)2 Lane Utilization Factor, fu : A lane utilization factor for unbalanced lane usage. It is esti- mated based on the HCM Chapter 9 Table 9-4. Appendix E: Page 19

WORKSHEET4a: REQUIRED PHASE TIMES WORKSHEET Tar et v/c Ratio, ~: WBL EBT NBL SBT EBL WBT SBL NBT 1 1 1 1 1 1 1 1 11 Red Time R Wlc~ht3h P , Start-u Lost Time er Phase t~ P P , LANK GROUP1 Movements (LTR) Green Anival Rate, q~l: Wksht 3b Rate A'TivalRate,c~l: Wksht3b LT Equivalents, ET ~Wksht 2 PROPORTION OF LT IN A SHARED LANE, PL1 Senrice Rate, sl: Wksht 2 EQUIVALENT SERVICE RATE,esl: esl = sl / (0.9S * E~) for pennitted LT with exclusive lanes esl = sl / [ (1 + P: 1 (ET.3 - 1) ] for a sh~ed lane esl = sl for TH or protected LT movements NET SERVICE RATE nsl: e51-41/X;r I I T T I T I :1 Queue at Beginriing of Service, Qql: Wksht 3b GREEN TIME, G1: IPT1 - I1 no phase skip IPTl - I1 (1 - Po`,l) phase skip QUEUE LENGTH FACTOR, fql: 1.08 - 0.1 (G1 / MxGl) 2 LANE VTILIZATION FACTOR, f.l: HCM Table 94 SERVICE TIME, t~1 + g,l: t,,1 + fql * ful * (Qql / X~) / nsl WAITING TIME BEFORE g,l, g.1 . TOTALQUEUE SERVICE TIME,QST1: t~1 +g.1 +g,1 . PERMl-l-ll;D PHASE TERM~AL QUEUE, Qpl . ADJ. PERMITTED PHASE TERMINAL QUEUE, Qp'1 . TOTAL QUEUE SERVICE TIME, QST2 Wksht 4b OTAL QUEUE SERVICE TIME, QS 13 Wksht 4b l l l l l l I 1[ | MLK Phace Time, MxT: Wksht 1 l T I I I I I I 1~ BASE PHASE TIME, BPI Min (Max (QST1, QST2, QST3), MxT) : Refer to Worltsheet S to determine phase extension times if the base phase time < the ma~nmum phase time. I Phace Fytencion Time FYt Wksht 5 l l l l l I I _ . | REQUIRED PHTIME,RPT: BPT+Ext ADJ MIN PHASE TIME, MnP COMPUTED PHASE TIME. CPT Max (MnP. RPT) Figure E-6. Worksheet 4: Required phase time computations Appendix E: Page 20

] WORKSHEET 4b: REQUIRED PHASE TIMES WORKSHEET (Supplement for Lane Groups 2 and 3) Target V/C Rati° XT: ~, WBL EBT NBL SBT EBL WBT SBL NBT RedTime R Wk~ht3h 1 1 1 1 1 1 1 1 Lost Time per Phase. t, I ~T I I I ~I Start-up Los[T~me per Ph~ce, ~I T r LANE GROUP, (N = 2 OR 3) Movements (LTR) Green Arrival Rate, q~N: Wksht 3b Red A'Tiva1 Ra$e, q,N: Wksht 3b LT Equivalents, E~ ~: Wksht 2 PROPORTION OF LT IN THE SHARED LANE, PLN Service Rate, sN: Wksht 2 EQUIVALENT SERVICE RATE, esN: esN = sN / (0.9S * E: ,) for permitted LT with exclusive lanes esN = sN / [ (1 + P~N (E~ - 1) ] for a shared lane esN = sN for TH or protected LT movements NET SERVICE RATE,nsN esN-q,N/X; Queue at Beginrung of Service, QqN: Wksht 3b GREEN TIME, GN: IPTN - IN no phase skip IPTN - IN (1 - Po.N) phase skip QUEUE LENGTH FACTOR"fqN 1.08-0.1 (GN/MxGN) LANE OTILIZATION FACTO}< f~N: HCM Table 9 - SERVICE TIME, t+~+g,N: t'N+fqN * £,N * (QqN / X~) / WAITING TIME BEFORE g.N, g-N TOTAL QUEUE SERVICE TIME, QSTN: t,,N + g-N + g,N PERMIIlED PHASE TERMINAL QUEUE, QpN ADJ. PERMITTED PHASE TERMINAL QUEUE. Q - Figure E-6. Worksheet 4: Required phase time computations (continued) Appendix E: Page 21

Service Time, to + gs This is the sum of startup lost time (to) and actual queue service time (go ). The gs is the service time required to discharge the accumulated queue with the net service rate (ns). Therefore, gs may be computed as [fq * fu * (Qs / XT) / ns] Waiting Time Before as, go: This is the green time before the queue service time (not including the startup lost time). The go is equal to zero for the through phase and protected leR turn phase from an exclusive lane. For permitted left turns from an exclusive lane, gw is equal to gq. For permitted left turns Tom a shared lane, the value of gw may be determined by the relationship among g q, gf, arrival rate and departure rates. Total Queue Service Time, QST: (to + gw ~ as) Permitter! Phase Terminal Queue, Qp: This is the number of vehicles accumulated at the end of the permitted phase before sneakers have been released. Adjusted Permitted Phase Terminal Queue, Qp': Defined as the number of vehicles accumulated at the end of the permitted phase after sneakers have been released. Based on the QST of each lane group, the QST of the subject phase can be determined. Maximum Phase Time, MxT: From Worksheet I. Base Phase Time, BPT: This is the maximum lane group total queue service time, QST, without exceeding MxT. Phase Extension Time, Ext.: This must be obtained from Worksheet 5. The extension time should be set to zero without using Worksheet 5 if the phase time computations are based on a target v/c ratio (Appendix ~ method) instead of the phase extension method described in this paper. The extension time wid also be zero if the BPT has already reached the Maximum Phase Time. Worksheet 5 win determine the extension time analytically in a single pass if the allowable gap is constant. If gap reduction features are in effect, multiple iterations will be required for each phase. Required Phase Time, RPT: ~ BET + Ext). Adjusted Minimum Phase Time, 5InP: Max(AVM, APM). Computed Phase Time, CPT: This is the final product of Worksheet 4. It is the Requirec} Phase Time, RPT or the Adjusted Minimum Phase Time, MOP, whichever is greater. In the iterative procedure, the CPT's for each phase win be enterer! once more on Worksheet 3. The cycle length win be cleterm~ned and each phase time will be checker! again with respect to its · e e e own maximum anc minimum times. Appendix E: Page 22

WORKSHEET 5: EXTENSION TIMES BASED ON ALLOWED GAPS This worksheet is shown in Figure E-7. It differs from previous worksheets in that the columns represent computed values instead of phases. This is because each phase may require a different number of iterations to amve at a final value for the phase extension. The purpose of this worksheet is to estimate the vehicle extension time after accumulated queues have been served. Each row on worksheet 5 represents one iteration for one phase. If the gap settings for a particular phase are fixed (i.e., no gap reduction features are used) then the extension time may be determined with a single iteration. Otherwise, the allowable gap will be a Unction of the extension time and several iterations may be required to resolve this circular dependency. The specific term definitions and their computations are described as follows: Headway Distribution Model: Users can select either bunched exponential headway distr~- bution or simple negative exponential headway distribution. If the HCM Chapter 9 Appendix ~ method is selected, no headway distribution mode! win be used to compute phase extension time. Arrival Rate, q (vps): This is the total equivalent through arrival flow in vehicles per second for all lane groups that actuate the phase under consideration. For example, for the simple phase (not compound left turn protection) with both through and permitted leD turn move- ments, the equivalent through flow is equal to (VT {VL *E] I) instead of the actual volume (VT +V~ ). Note that VT and VI are the through flow rates (veh/sec) including right turns and per- m~tted left turn flow rate (veh/sec), respectively. Delta Value, /` (seconds): This is the minimum headway between arrival vehicles. For the simple negative exponential model, ~ is equal to zero seconds. For the bunched exponential headway model, the recommencled values based on the calibration by Ak,celik and Chung [4] are: Single-lane case: Multi-lane case (number of lanes =2~: Multi-lane case (number of lanes > 2~: A= I.5 seconds = 0.5 seconds = 0.5 seconds Phi Value, q,: This is the proportion offree vehicles which represents the unbunched vehicles with randomly distributed headways. For the simple negative exponential model, (p is equal to one. For the bunched exponential headway model, the following relationship suggested by Ak,celik and Chung t4] can be used for estimating the proportion of unbunched vehicles in the traffic stream (`p) ~ = e~b ~ q Appendix E: Page 23

WORKSHEET 5: PHASE EXTENSION TI5IE WORKSHEET | Hend~vay Distabution Model: PHASE | ARlUVAL | DELTA | PHI | MAXIMUM | MINIMUM | COMPUTED | ADJUSTED | NEW RATE VALUE VALUE HEADWAY HEADWAY E:'mNSION EXTENSION ~:ADWAY (vp S) (see) (see) (see) (see) (see) (see). 1 1 1 1 1 1 1 1 ~ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . 1 1 1 1 1 1 ~1 l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ~1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ~ I T T 1 I T ~T I I 1 1 1 1 1 1 1 1 1 ~ 1 1 1 1 1 1 1 : 1 ~=r l l l l l l l l l I I I rl 1 1 1 1 1 1 1 1 ~ 1 1 1 1 1 1 1 1 ] 1 1 1 1 1 1 1 1 ~ r I l l I I I I 1 Figure E-7. Worksheet 5: Phase extension time worksheet Appendix E: Page 24

The recommended parameter values for ~ and b are as follows: Single-lane case: Multi-lane case (number of lanes =23: Multi-lane case (number of lanes > 2~: Maximum Headway (seconds): (SG + DT) A= I.5 seconds and b = 0.6 = 0.5 seconds and b = 0.5 = 0.5 seconds and b = 0.8 Minimum Headway, ho (seconds): MA +DT). MA and DT are also expressed as eO and to, respectively, so ho can also be expressed as (eO + to). Computed Extension, EXT (seconds): This is the average extension time (ye) plus the intergreen time (I). The average extension time, Oc, by gap change can be estimated Dom the following fonnula [l, 3] based on the bunched exponential headway distribution. The simple negative exponential mode} is a special case of the bunched arrival mode! with ~ equal to zero and up equal to one. e x(eO+~O-~y ~ = _ _ He Be A If the phase is skipped, the original computed extension needs to be modified by multiplying the factor of hi- Pov) Adjusted Extension, AEXT (seconds): The above computed extension needs to be adjusted if the sum of the base phase time (BPT) and the computed extension time (EXr) is greater than the maximum phase time (^T). Then, AEXT wait be equal to (BPT + EXT - MET). Otherwise, AEXT is equal to EXT. New :EIeadway, hneW (seconds): The new headway will be equal to the maximum headway. Under fudy-actuated control, without gap reduction feature, the new headway wait be equal to the magnum headway. For volume-density control, with gap reduction feature, the vehicle headway needs to be computed by an iterative procedure until it converges to a stable value. For volume-density control, in each iteration, the new headway is computed as: knew = SG- GR * (QST+EXT)+DT Subject to how > ho If the difference between the new headway and the old headway exceeds 0. ~ seconds, the iterative procedure will continue until the convergence is reached. Appendix E: Page 25

SUMMARY WORKSHEET: SUMMARY OF CAPACITY AND DELAY Figure E-S shows the summary worksheet for capacity and delay. The major items in the worksheet include the final phase times, final cycle length, capacity, v/c ratio, uniform delay, incremental delay and tote delay. Al! these items are the major components for defining the level of service at a traffic- actuated signalized intersection. Each term in this summary worksheet is described as follows: Final Cycle Length, C: This is the final cycle time in the iterative computation procedure which can be obtained from the new cycle length of the final Worksheet 3b. Final Phase Time: This is the final phase time in the iterative computation procedure which can be obtained from the interim phase time (IPT) of the final Worksheet 3b. Saturation Green Time: Defined as the proportion of phase the with saturated traffic flow. The summary of capacity and delay are associated with each lane group. Movements(LTR): From Worksheet3b. Traffic Volume, v: This is the traffic volume for each lane group based on the original input data. The unit for traffic volume is vehicles per hour. Traffic volumes of shared lane movements need some modification based on its original `~1~a arena the nrmn^~timn It t11~Q; - At OhO-~ lo - = ~ I- "~ ~- ~AV~V1~1V11 V' 1~L L"111O 111 a O11~" I"11~, ~ L . The "LT" movement tragic volume for the shared lane is equal to the original lePr turn volume divided by PL . If there is no exclusive right lane, the traffic volume for the lane group with movement "my" is equal to the total traffic volume minus the computed traffic volume of the shared lane. It is necessary to note that the assignments of total leR turn traffic volume to permitted and protected movements for compound led turn protection are basest on their associated portion of time. This is best illustrated with an example. Assuming there exists a protected plus permitted phasing, the total left turn traffic volume is 200 vehicles per hour. Red time is 40 seconds. The protected phase time and permitted phase time are both equal to 20 seconds. Based on the associated portion of time, the traffic volume for the protected movement is equal to [200 * (40 t20) / 80 = ~ 50 vph] and the traffic volume for the permitted movement is [200 * 20/80 - 50 vph]. :Left turn factor, f,': For the through movement, fit, is equal to I.0 and for the protected movement, it is 0.95. For the permitted leR turn movement, the computation of fit is based on the methodology described in the HCM Chapter 9 with the supplemental worksheet for permitted left turns. Appendix E: Page 26

CAPACITY AND DELAY COMPUTATIONS (Lane G FINAL CYCLE LENGTH roup 11) WBL EBT NBL SBT EBL W~T SBL FINAL PHASE TIME ~ SATURATION GREEN TI\fE l l l l l l I _ Summaw of lane group capacity and delay LANK GROUP1 l l T I I I I Movements (LTR) I ~l l I _ Trafftc Volume (vph), vl = LEFT TURN FACTOR.41 l l I ~l l I = FINALSATURATION FLOW RATE, Satl (vphg) I I T T I I T _ EFFECTIVE GREEN,gl l l l l l l l FrNAL GREEN / CYCLE RATIO, g/C I | I T T I T ~= CAPACITY, cl: Satl * g/C1 1 1 T T I T T VOLUME / CAPACITY RATIO, vi/cl UNIFORM DELAY, d,,1 (sec/veh): under QAP area ~ CRlTICALDEGREEOF SATURATION, xol I I I I I ~I I = NON-OVERFLOW TERM PARAMETER, f. 1 I I T T I T T 1 11 OVERFLOW TERM PARAMETER, kdl l l l l l l l l 11 NON-OVERFLOW DELAY, d,1 (sec/veh) r I I T I T T I 1[ OVERFLOW DELAY, d21 (sec/veh) l l l l l l l l 1[ TOTAL DELAY, dl LeR Tum Volume, v~ LEFT TURN CAPACITY, CL I I I I I I I 1 LEFT TURN VOLUME / CAPACITY RATIO, VL / CL I I I I 1[ LEFT TURN UNIFORM DELAY, due LEFT TURN CRITICAL DEGREE OF SATURATION, XOL 1 1: I LEFT TURN OVERFLOW TERM PARAMETER, ~L LEFT TURN NON-OVERFLOW DELAY, dlL l I T I T I 1' LEFT TURN OVERFLOW DELAY, d2L I ~1 T I T I I 1 LEFT TURN TOTAL DELAY, dL l T I T I T I I 1 LANE GROUP 2 v/c RATIO Wksht 6b | T I I I l I I 1 LANE GROUP 3 v/c RATIO Wksht 6b l l l l l l I 1: CRITICAL v/cRADO Max (v/cl, v/c2, v/c3) 1 1 1 1 1 1 1 L~ Figure E-8a. Worksheet 6a: Capacity and delay computations (lane group 1) 8 NBT _ _ . = . _ . _ . = Appendix E: Page 27

CAPAC1 1 ~r AND DELAY COMPUTATIONS (Lane GrOUPS 2 and 3) FINALCYCLE LENGTH | 1 | 2 | 3 | 4 | 5 | 6 | 7 VVBL EBT NBL SBT EBL WBT SBL FINAL PHASE TIME LANE GROUP 2 _ Movements (LTR) l l l l I _ Traffic Volume(vph),v2 LEFT TURN FACTOR, f,'2 FINAL SATURATION FLOW RATE, Sat2 (vphg) l l l l l l EFFECTIVE GREEN,g2 l l l l l I I _ FINAL GREEN / CYCLE RATIO, g/C2 CAPACITY, c2: SaU * g/C2 I _ l l l l I I VOLUME / CAPACITY RATIO, v2 / c2 _= I l == UNIFORM DELAY, d,,2 (sec/veh): under QAP area CRITICAL DEGREE OF SATURATION, xo2 NON-OVERFLOW TERM PARAMETER, f~n2==== === OVERFLOW TERM PARAMETER, kd2_ = = NON-OVERFLOW DEALY, dl2 (sec/veh)== == === OVERFLOW DELAY, d22 (sec/veh)_ TOTAL DELAY, d2 l l l l l | | _ LANE GROUP 3 Movements (LTR) l l l l l | | _ Traffic Volume (vph), v3 . LEFT TURN FACTOR, f.3 1 1 ~l l T EFFECTIVE GREEN, g3 FINAL SATUR~TION FLOW RATE, Sat3 (vphg) == LEFT TURN FACTOR, f,l3 = CAPACITY, c3: Sat3 * g/C3 VOLUME / CAPACITY RATIO, v3 / c3 l l l l I I _ UNIFORM DELAY, d`,3 (sec/veh): under QAP area _ _ = CRITICAL DEGREE OF SATURATION, ~,3 NON-OVERFI,OW TERM PARAMETER" f~3 OVERFLOW TERM PARAMETER, kd3 l l l l I _ NON-OVERFLOW DEALY, dl3 (sec/veh)_= == === OVERFLOW DELAY, d23 (sec/veh)_ == TOTALDELAY,d3 I I r ~I I I NBT Figure E-8b. Worksheet 6b: Capacity and delay computations ~ lane groups 2 and 3) Appendix E: Page 28

Final Saturation Flow Rate, Sat (vph): The final saturation flow is recorded here, assuming the original saturation flow is S (vph). For the through or right turn movement, the final saturation flow is equal to S because fit is ~ .0. For permitted leD turns Tom an exclusive lane, the saturation Dow rate, Sat, is equal to (S * A, / 0.95~. For the movements Tom a single shared lane, the final saturation flow, Sat, is equal to (S * fib. For the shared lane "LT" movement with ~ total lanes greater than one, Sat is equal to (S * fit / N). Note that this A, applies only for shared Varies. For the "TR" movement, Sat can be computed as [S * (my) *0.91 / N]. The 0.91 factor accounts for the shared lane effect described in HCAl Chapter 9. Effective Green, g: In general, this can be computed as the phase time of the movement minus the lost time. Some slight modification may be required for the protected and permitted movements with compound left turn protection. For example, the effective green for a permitted left turn movement with protected plus permitted phasing is equal to its filll phase time, as explained in HCM Chapter 9. Final Green Ratio, g/C: (g / C). Capacity, c: (Sat * g / C). Volume/Capacity Ratio, v/c: (v / c). QAP Uniform Delay, du: This is computed as the area under the queue accumulation polygon. For the permitted leg turn movement or protected left turn movement with compound leg turn protection, ~ is equal to its associated QAP area divided by its arnvals within the period. For coordinated phases, ~ is multiplied by the progression factor to account for the progression effect. Critical Degree of Saturation, xO: If the degree of saturation is below xO, the average overflow queue is zero. The equation for the computation of xO is expressed as follows: x0 = 0.42 hod ~ MxG0 2 Non-overflow Term Parameter, fat: t} + 0.40 (Sat * g / 3600~35 (v / Sat)° ~] OverQow Term Parameter, kd: [0.40 (Sat * g /3600~° Is (v / Sat)i i] Non-overflow Delay, Do: This is also called the first term of the delay formula. DO is equal to (fill * du) if the degree of saturation, x, is less than the critical value, xO. Otherwise, D, is computed as (f~`x=~' * du ). The fd'~X=~' is the value of fat when x is equal to one. Appendix E: Page 29

Overflow Delay, D2: This is also called the second term of the delay formula. If x is less than or equal to xO, D2 will be equal to zero. If the value of x is greater than xO , D2 can be computed by the following formula: where D2 = 900 Tp six - I) + ](x _ I)2 + 8 1~: ~) j Tp = Peak flow period (analysis periods in hours. The default value is 0.25. c = Hourly capacity (veh/hr). Total Delay, D: The total delay per vehicle is the sum of non-overflow delay and overflow delay (D = DO + Dig. Level of service for a signalized intersection is mainly based on the value of total delay. To assess the capacity and delay for left turns, this summary worksheet provides a left turn section under lane group one. Included in this section are, left turn volume, led turn capacity, left turn v/c Ratio, leR tum unifonn delay, leR turn non-overDow delay, left turn overflow delay and left turn total delay. Except for the case of compound leg turn protection, the value for each item in the leD turn section is the same as the one shown in lane group one (left turn movement). For compound protection, the values for the leR turns are presented in the leD turn section. For compound left turn protection, the computation of each item in the left turn section is addressed as follows: Left Turn Volume, VL: Sum of permitted and protected lane group volumes which is equal to the adjusted leg turn volume on the WHICH movement screen. Left Turn Capacity, cat: Sum of permitted and protected lane group capacity. Left Turn Volume/Capacity Ratio, (v/c)I: vI / cat. Left Turn Uniform Delay, due: This is computed as the area under the queue accumulation polygon for the left turn movement including both protected and permitted left turn move- ments. Left Turn Critical Degree of Saturation, COO If the degree of saturation is below xO, the average overflow queue is zero. The equation for the computation of xO~ is expressed as: x0L = 0.42 hoist ~ ~GL.O2. For the compound left turn protection, the values of hoI and MxGL adopt the values for the protected left turn movement. Appendix E: Page 30

Left Turn Overflow Term Parameter, ~L: 040 (Sate * gL /36001°-7s (VL / Sated 11. For compound led turn protection, use the total (Say * g0. For the leD turn movement, (Sate * gL) = ~ (Sat * go, use VL for total left turn volume per hour, vI = vent + vpe~m, and use the aver- age saturation flow, Sate, for the two green periods, Sate = (Sat * gel / Z~ = (SatL * pi) / by. Left Turn Non-overtlow Delay, dlL: [dl~pro`' * vprOt + dame * vend / VL Left Turn Overflow Delay, 02~ Use the same formula as for the through movement except using the values for overall left turn movement instead of individual permitted or protected movement. Left Turn Total Delay, do. (d,L + d2I) Critical Volume/Capacity Ratio, (v/c~c: The maximum v/c ratio among three lane groups. IMPLEMENTATION OF COMPUTATIONAL WORKS MEETS The implementation of computational worksheets to predict average phase times, cycle length, capacity and delay is best illustrated with examples. In this working paper, two examples are designed to cover a basic two-phase operation with simple permitted left turns and a more complicated multi-phase operation with compound leR turn protection. For each example, Worksheets ~ and 2 win be presented to show the input data and actuated design parameters. The main signal timing computation are included in Worksheets 3, 4 and 5. Since the computation procedure is an iterative one, only the results of the last two iterations are shown followed by the summary worksheet. EXAMPLE FOR A PERMITTED LEFT TURN FROM AN EXCLUSIVE LANE AND A SHARED LANE The first example presented in this working paper is for through movements and permitted left turns Rom an exclusive lane or a shared lane. The led turn on each approach is permitted. The intersection in this example, as shown in Figure E-9, covers a variety of lane configurations and vehicle movements. The eastbound approach is designed for two shared lanes (one for the left turn and through and the other for through and right turn). The westbound approach includes only a single shared lane. The left turns on both the northbound and southbound approaches take place from an exclusive left turn lane. The northbound approach has two through lanes but the southbound approach has only one. Appendix E: Page 31

FIRST S1~= SS 1 1 1 i tlPl 1 1 1 1 FIRST STREET NB Figure E-9. Intersection layout for the permitted left turn example For simplicity, a saturation flow rate of 1800 vphgpl, corresponding to a headway of 2.00 seconds per vehicle, wait be assumed. The remainder of the input data is shown in reproductions of Worksheets ~ and 2, presented In Figure E-l0. Each phase has been assigned the following constant parameters: Detector length: Detector setback: Intergreen: Lost torte per phase: Allowable gap: Minimum phase time: Maximum green time: No pedestrian timing features No volume-density features 30 feet 0 feet (placed at the stop line) 4 seconds 3 seconds 3 seconds 15 seconds 60 seconds The data for this example were entered using WHICH, and following the process outlined In Working Paper NCHRP 3-48-16. The information in Worksheets ~ and 2 shown in Figure E-10 is mostly based on the input from WHICH. The computation of phase time and cycle length is mainly through Worksheets 3, 4 and 5. Worksheet 3a implements the dual-ring operation and computes the traffic-actuated signal timing. If the cycle length converges to a stable value, the iterative procedure terminates and the summary worksheet of capacity and delay is produced. Otherwise, Worksheet 3b will continue to perform the timing plan sensitive parameter computations by constructing the queue accumulation polygons. If the Base Phase Time (BPT) from Worksheet 4 does not exceed the Appendix E: Page 32

NCHRP PROJECT 3-48: CAPACITY ANALYSIS OF TRAFFIC ACTUATED SIGNALS Worksheet 1: Traffic-Actuated Control Input Data filename EXAMPLE1 (Perm~tted Phases) FIRST STREET at FIRST AVENUE APPROAC H NORT HBOUND _ _ _ _ - - - - - - - -T- SOUTHBOUND EASTBOVND WEST80UND N/A 2.00 0.00 30 Max Initial Add Initial Min Gap Reduce Gp By Walk + FDU Max Green Change T Clr Recall Mode +_ _ _ _ - _ - - - - - -+- - - - - - -T- VEH MINIMUM STARTING GAP MIN INITIAL MAX PH TIME DT OCC TIME TRIAL TIME + + . 1 N/A 2.00 0.00 30 LT Treatment 1111 LT Position N/AN/AN/AN/A Sneakers 2.002.002.002.00 Free Queue 0.000.000.000.00 Speed 30303030 Termination -------- N/A -------- -------- N/A ------- + + PHASE RING 1 RING _ DATA 1(WBL)2(EBT) 3(NBL) 4(SBT) 5(EBL) 6(WBT) 7(SBL) 8(NBT) Phase Type XG X G ~G X G PH Reverse? NoNoNo No Det Length 303030 30 SetBack 000 0 + + + + + + + + + + 8 0.00 3.00 0.00 o 60 4.00 N + + 15 3.00 8.00 64  1.09 15.00 . + + Worksheet 8 0.00 3.00 0.00 o 60 4.00 N + + 15 3.00 8.00 64 1.09 15.00 + + Lane Group Data 8 0.00 3.00 0.00 o 60 4.00 8 0.00 3.00 0.00 o 60 4.00 N . + + + 15 15 3.00 3.00 8.00 8.00 64 64 1.09 1.09 15.00 15.00 + + + PHASE RING 1 RING 2 DATA 1(WBL)2(EBT) 3(NBL)4(SBT) 5(EBL)6(~8T) 7(SBL) PH Movements LTR LTR LTR Arr Type Opp PH LT Equiv 8(NBT) LTR 3 ~ Lanes Mov'ts Vo(ume Arr Rate Rp Grn Arr Sat Flow Svc Rate - ~ 2 LTR 400 0.11  1.00 0.11 3420 0.95 275 0.08 1.00 0.08 1710 0.48 LTR 300 0.08 1.00 0.08 1579 0.44 150 0.04 1.00 0.04 1710 0.48 _ _ _ - - - - - -T ~ Lanes Mov'ts Volume Arr Rate Rp Grn Arr Sat Flow Svc Rate o o 0.00 0.00 0.00 o 0.00 TR 400 0.11 1.00 0.11 1732 0.48 o o 0.0Q 0.00 0.00 o 0.00 600 0.17 1.00 0.17 3465 0.96 T-- - - - - - - - - - _T- - - - - - _T- - - - - - _T- - - - - - _T-- - -- - _T- - - - - - _T- - - - - - _T ~T- -~ VEH ARR RATE 0.11 0.19 0.08 0.21 Ped Volume 0 0 0 0 PED ARR RATE 0.00 0.00 0.00 0.00 Figure E-IO. Worksheets ~ and 2 for the permitted left turn example  Appendix E: Page 33

maximum phase time, Worksheet 5 will determine the phase extension time for each phase after the accumulated queue has been served. Based on the results from Worksheet 3 and 5, Worksheet 4 computes the required phase time for each phase. Then, the dual-ring operation will be constructed based on the BPT's Dom Worksheet 4 and presented in Worksheet 3 again to check whether the convergence of cycle length has been reached. Thus, an external iterative process between Worksheets 3 and 4 is necessary. The signal timing computations of the last two iterations for the permitted led turn example are illustrated on Figures E-11, E-12 andE-13. Figure E-14 shows the summery worksheet for capacity and delay. The final cycle length for this example is 97.5 seconds with 39. 18 seconds assigned to the east-west phases and 58.33 assigned to the north-south phases. EXAMPLE FOR COMPOUND LEFT TURN PROTECTION This is a more complex example With compound leg turn protection. The intersection layout for this example is shown In Figure E-15. All led turns are made Dom an exclusive left turn lane. The north- bound approach has protected plus permitted phasing for led turns, while the southbound approach has a permitted plus protected phasing. A dual leading led turn phasing is used for both eastbound and westbound approaches. Each of the southbound movements has twice the volume of the corresponding northbound movement. The relationship is the same for the eastbound and westbound through traffic volumes. The eastbound "LT" volume is 1.5 times greater than the westbound "LT" volume. The detailed traffic volumes are shown in Worksheet 2. For simplicity, a saturation flow rate of 1800 vphgpl, corresponding to a headway of 2.00 seconds per vehicle, will also be assumed. The remainder of the input data is shown In reproductions of Worksheets 1 and 2, presented in Figure E- 16. Each phase has been assigned the following constant parameters: Detector length: Detector setback: Intergreen: Lost time per phase: Allowable gap: Minimum phase time: Maximum green time: Pedestrian recall: Walk plus Flashing Don't Walk: Pedestrian hourly volume: No volume-density features .4ppendixE: Page 34 30 feet 0 feet (placed at the stop line) 5 seconds 3 seconds 2.5 seconds 1 5 seconds (led turn phase) 20 seconds (through phase) 1 5 seconds (lest turn phase) 45 seconds (through phase) Active 20 seconds (through phase) 50 pedestrians

Iteration 14 Ring 1 PH Swap? Movts PH Time Ring 2 PH Swap? Movtsw~, -------- nv~ ~ PH Time38.87 38.87 36.91 36.91 RING 1-2 DIFF-14.57 -14.57 21.65 21.65 CYCLE COMPONENTS38.87 0.00 58.56 0.00 Worksheet 3b: Timing Plan sensitivity Capacity Parameter Estimation Worksheet 3a: Traffic-Actuated Timing Computations B Total No EBT 24.30 No UBT 38.87 -14.57 38.87 No SBT ------- 58.56 58.56 No NBT 36.91 21.65 58 56 Old Cyc 96.9 New Cyc 97.4 Unadj PH Time Adjustment New Ph Time Eff Green 9 Int Red Time Eff Red r Red Time Startup Time Recall Mode Veh Arr Rate Av Vehs on Red P rob (No Vehs) Veh Min Time Adj Veh Min Ped Arr Rate Av Peds on Red P rob (No Peds) Ped Min Time Adj Ped Min LG 1 Mov'ts Grn Arr Red Arr 234 UBLEBTNBLSBT 24.30 14.57 38.87 35.87 58.56 61.56 58.56 0.00 N  0.11 6.51 0.00 15.00 15.00 0.00 0.00 1.00 4.00 0.00 SBL NBT _ _ _ _ 36.91 21.65 58.56 55.56 38.87 41.87 38.87 38.87 N 0.21 8.10 .00 .00 .00 1.00 1.00 .00 .00 1.00 _ _ ~ 58.56 0.00 58.56 55.56 38.87 41.87 38.87 38.87 N 0.19 7.29 0.00 15.00 15.00 0.00 0.00 1.00 4.00 0.00 . 38.87 0.00 38.87 35.87 58.56 61.56 58.56 0.00 N 0.08 4.88 0.00 15.00 15.00  0.00 0.00 1.00 4.00 0.00 LT L 0.04 0.08 0.04 0.08 EL2 2.68 0.00 Free G gf 0.95 0.00 QSTOpp gq 11.98 5.37 QatEOk Qr 2.51 3.20 QatEOF Qf 2.10 3.20 QatEOQ Qq 0.10 3.61 QatBOS Qs 0.10 3.61 + _ _ _ + + _ + + + LG 2 Mov'ts TR TR Grn Arr 0.07 0.11 ~ 0.00 Red Arr 0.07 0.11 0.00 EL2 0.00 0.00 0.00 Free G gf 0.00 0.00 0.00 QSTOpp gq 0.00 0.00 0.00 QatEOR Qr 4.33 4.65 0.00 QatEOF Qf 4.33 4.65 0.00 QatEOQ Qq 4.33 4.65 0.00 QatBOS Qs 4.33 4.65 0.00 Cycle convergence has not been ach~eved. Difference = .4 LTR 0.08 0.08 0.00 2.46 4.69 5.13 4.26 4.44 4.44 0.04 0.04 0.00 0.00 B.96 1.74 1.74 2.12 2.12 + + + + TR 0.17 0.17 0.00 0.00 0.00 6.98 6.98 6.98 6.98 Figure E-11. Worksheet 3 of iteration 14 for the permitted left turn example Appendix E: Page 35

Worksheet 4: Required Phase Times Based on 1 PHASE DATA Int Red Time Lost Time tl Lost Time tsl LG 1 Mov'ts Grn Arr Red Arr ELI PL Svc Rate Eqiv Svc Net Svc QatBOS Qs Green G fq Land Utl Svc Time Wait Time Tot Time ResQ Qp ResQ Qp' 2 Mov'ts Grn Arr Red Arr ELI PL Svc Rate Eqiv Svc Net Svc QatBOS Qs Green G fq Land Utl Svc Time Wait Time Tot Time ResQ Qp ResQ Qp' Max Ph Time BASE PH TIME Ph No. 4 6 Q --------____l________________________________ v/c Ratio RING 1 RING 2 1(WBL) 2(EBT) 3(NBL) 4(S8T) 5(EBL) 6(WBT) 7(SBL) 8(NBT) 58.56 38.87 58.56 38.87 3.00 3.00 3.00 3.00 2.00 2.00 2.00 2.00 LT L LTR 0.04 0.08 0.08 0.04 0.08 0.08 2.65 2.90 2.60 0.68 1.00 0.33 0.48 0.48 0.44 0.22 0.17 0.29 0.18 0.10 0.20 0.10 3.61 4.44 34.87 54.56 34.87 1.05 1.00 1.05  1.00 1.00 1.00 2.59 39.48 24.94 11.98 5.37 4.69 14.57 44.86 29.63 0.00 0.00 0.00 0.00 0.00 0.00 TR 0.07 0.07 2.65 0.00 0.43 0.43 0.36 4.33 34.87 1.05 1.00 14.51 0.00 14.51 0.00 0.00 . 64.00 14.57 . 0.08 0.08 2.90 1.00 0.48 0.17 0.10 3.61 54.56 1.00 1.00 39.48 5.37 44.86 0.00 0.00 TR 0.11 0.11 2.90 0.00 0.48 0.48 0.37 4.65 54.56 1.00 1.00 14.54 0.00 14.54 0.00 0.00 64.00 44.86 Arr Delta Rate Value 0.04 0.04  2.60 1.00 0.48 0.19 0.15 2.12 54.56 1.00 1.00 16.02 8.96 24.99 0.00 0.00 TR 0.17 0.17 2.60 0.00 0.96 0.96 0.80 6.98 54.56 1.00 1.05 11.18 0.00 11.18 0.00 0.00 64.00 24.99 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 64-00 29.63 . Worksheet 5: Green Extension Times: Max Min Comp Adj New Hdwy Hdwy Ext Ext Hdwy 4.09 9.82 9.82 4.09 4.09 13.48 13.48 4.09 4.09 9.55 9.55 4.09 4.09 11.81 11.81 4.09 . 0.16 0.50 0.33 0.50 0.13 1.50 0.28 0.50  Base Ph Time Ph Ext Time REQ PH TIME MIN PH TIME COMP PH TIME 0.96 0.92 0.89 0.90 4.09 4.09 4.09 4.09 Computed Phase Times Including Extension Intervals , Phase ~12345 MovementsUBLEBTNBLSBTEBL , 44.86 13.48 58.33 15.00 58.33 14.57 9.82 24.39 15.00 24.39 29.63 9.55 39.18 15.00 39.18 78 SOLNBT 24.99 11.81 36.80 15.00 36.80 Figure E-12. Worksheets 4 of iteration 14 for the permitted left turn example Appendix E: Page 36

blorksheet 3a Traffic-Actuated Timing Computations Iteration 15 A B Total A B Total Ring 1 PH Swap? Movts PH Time Ring 2 PH Swap? Movts PH Time RING 1-2 DIFF CYCLE COMPONENTS _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ No-------- Ho EBT-------- SBT 24.3924.39 58.33 No-------- No UBT-------- NBT 39.1839.18 36.80 36.80 -14.79-14.79 21.54 21.54 _ 39.180.00 58.33 0.00 Uorksheet 3b: Timing PLan sensitivity Capacity Parameter Estimation , Old Cyc 97.5 1 234 56 78 New Cyc 97.5 UBL EBT NBL SBT EBL ~JBT SBL NBT 58.33 39.18 36.80 0.00 0.00 21.S4 58.33 39.18 58.33 55.33 36.18 55.33 39.18 58.33 39.18 42.18 61.33 42.18 38.87 58.56 38.87 39.18 0.00 39.18 Unadj PH Time Adjustment New Ph Time Eff Green 9 Int Red Time Eff Red r Red Time Startup Time 24.39 24.39 14.79 39.18 36.18 58.33 61.33 58.56 0.00 , 58.33 Cycle convergence has been achieved at 97.5 Figure E-13. Worksheet 3a of the last iteration for the permitted left turn example Appendix E: Page 37

EXAMPLE 1 Cycle: 97.5 1,JBL EBT 39.18 14.57 Phase T i me Sat Green + + + + + + + + + + ~56 SBTE3L~JBT SBL 58.33 39.18 44.86 29.63 + 7 8 NBT 58.33 24.99 LG 1 Mov' tsLTLLTRL Volune147.37275.00300.00150.00 f lt0.480.310.640.32 Finat Sat828.26560 651003.10580.62 Eff g36.1855 3336.1855.33 Final g/C0.370 570.370.57 Capacity307.30318 14372.17329.48 v/c Rat'o0.480.860.810.46 QAP du22.8720.8227.8417.12 xo0.830.830.830.83 fdl1.161.181.161.16 kd0.290.920.600.47 Delay d126.5424.4732.2319.90 Delay d20.002.710.000.00 Tot Delay26.5427.1832.2319.90 + + + + + + + + + + LT Vol0.00 275.00 0.00 150.00 LT Cap0.00 318.14 0.00 329.48 LT v/c0 . 00 0 .86 0 .00 0 .46 LT QAPdu 0.00 20.82 0.00 17.12 LT xo0.00 0.00 0.00 0.00 LT kd0.00 0.00 0.00 0.00 LT d10.00 24.47 0.00 19.90 LT d20.00 2.71 0.00 0.00 LT Delay0.00 27.18 0.00 19.90 + + + + + + + + + + LG 2 Mov'ts TR TR TR Volune 252.63 400.00 0.00 600.00 f lt 1.00 1.00 0.00 1.00 Final Sat 1556.00 1732.00 0.00 3465.00 Eff g 36.18 55.33 0.00 55.33 Final g/C 0.37 0.57 0.00 0.57 Capaci ty 577.30 982.83 0.00 1966.23 v/c Rat, o 0.44 0.41 0.00 0.31 QAP du 23.03 1 1 86 0.00 1 1 .03 xo 0.83 0 83 0.00 0.83 fd1 1.13 1.11 0.00 1.08 kd 0.43 0.94 0.00 1.15 Delay d1 25.97 13.16 0.00 11.95 Delav d2 0.00 0.00 0.00 0.00 Tot belay 25.97 13.16 0.00 11.95 _ _ _ v/c Ratio 0.48 0.86 0.81 0.46 + + + + + + + + + + Figure E-14. Summary worksheet for the pe~mitted left turn example  Appendix E: Page 38

~ N Nor t h-Bout h Or SB 19 Forth-south Or FIB Figure E-IS. Intersection layout for the example of compound! left turn protection 4ppendixE: Page 39

NCHRP PROdECT 3-48: CAPACITY ANALYSIS OF TRAFFIC ACTUATED SIGNALS Worksheet 1: Traffic-Actuated Controt Input Data Filename: EXAMPLE2 (Conpound LT Prot) Museum Rd at North-south Dr + + APPROACH NORTHBOUND SOUTHBOUND DATA EASTBOUND ~JESTBOUND LT Treatment3333 LT PositionLeadLanLeadLead Sneakers2.002.002.002.00 Free Queue0.000.000.000.00 Speed30303030 Termination ---- Independent - ~- Simultaneous PHASE RING 1 RING 2 DATA 1(WBL) 2(EBT) 3(NBL) 4(SBT) 5(EBL) 6(WBT) 7(SBL) 8(NBT) Phase Type PH Reverse? Det Length SetBack L G L G L GLG No No No No No NoYesYes 30 30 30 30 30 303030 O O O O O OOO ~+ + + + + + + + + Max Initial 8 18 8 18 8 18 8 18 Add Initial 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Min Gap 2.50 2.50 2.50 2.50 2.50 2.50 2.50 2.50 Reduce Gp By 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 ilalk + FDW 0 20 0 20 0 20 0 20 Max Green 15 45 15 45 15 45 15 45 Change + Clr 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 Recall Mode N P N P N P N P . . . . . . . . . . VEH MINIMUM 15 25 15 25 15 25 15 25 STARTING GAP 2.50 2.50 2.50 2.50 2.50 2.50 2.50 2.50 MIN INITIAL 7.50 17.50 7.50 17.50 7.50 17.50 7.50 17.50 MAX PH TIME 20 50 20 50 20 50 20 50 DT OCC TIME 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 TRIAL TIME 15.00 25.00 15.00 25.00 15.00 25.00 15.00 25.00 + + + + + + + + + + tJorksheet 2: Lane Group Data + + PHASE RING 1 RING 2 DATA 1(~BL) 2(EBT) 3(NBL) 4(SBT) 5(EBL) 6(\JBT) 7(SBL) 8(NBT) PH Movements L LTR Arr Type 3 3 Oop PH 0 6 Lt Equiv 0.00 2.55 + + + + LG1 # Lanes 1 1 Mov'ts L L Vol~ne 200 300 Arr Rate 0.06 0.08 Rp 1.00 1.00 Grn Arr 0.06 0.08 Sat Flow 1710 1710 Svc Rate 0.48 0.48 + ~. LG2 ~ Lanes Mov'ts Vol~ne Arr Rate Rp  Grn Arr Sat Flow Svc Rate VEH ARR RATE Ped Volume PED ARR RATE +. o ; 3 o 0.00 L LTR L LTR L LTR 3 3 3 3 3 3 0 8 0 2 0 4 0.00 2.18 0.00 8.20 0.00 5.75 + + + + + + 1 1 1 1 1 1 L L L L L L 50 100 300 200 100 50 0.01 0.03 0.08 0.06 0.03 0.01 1.00 1.00 1.00 1.00 1.00 1.00 0.01 0.03 0.08 0.06 0.03 0.01 1710 1710 1710 1710 1710 1710 0.48 0.48 0.48 0.48 0.48 0.48 , + + + + + 10 1 0 1 0 1 TR TR TR TR 0 6000 500 0 300 0 250 0.00 0.170.00 0.14 0.00 0.08 0.00 0.07 0.00 1.000.00 1.00 0.00 1.00 0.00 1.00 0.00 0.170.00 0.14 0.00 0.08 0.00 0.07 0 17470 1737 0 1747 0 1737 0.00 0.490.00 0.48 0.00 0.49 0.00 0.48 + + + + + + + + 0.06 0.25 0.01 0.17 0.08 0.14 0.03 0.08 0 50 0 50 0 50 0 50 0.00 0.01 0.00 0.01 0.00 0.01 0.00 0.01 L LTR 3 3 0 2 0.00 8.20 0.17 0.08 50 0 0.01 0.00 + + . Figure E-16. Worksheets ~ and 2 for the example of compound left turn protection Appendix E: Page 40

The signal timing computations for the last two iterations of the compound led turn protection example are illustrated on Figures E-17, ENS and E-19. Figure E-20 shows the summary worksheet for capacity and delay. Note that in the computation of green extension time in Worksheet 5, the arrival rate is an equivalent through flow rate for the whole lane group. However, the extension time computation for the through phases does not include the equivalent through flow rate of the permitted left turns with compound left turn protection because the permitted left turns clo not actuate the detector to extend the permitted phase. The final cycle length for this example is 130.2 seconds. Both eastbound phases, through and protected left turn, are at their maximum times. Movements with overflow delays include the eastbound through and left turn movement, the southbound through and the westbound left turn movement. Appendix E: Page 41

Worksheet 3a: Traffic-Actuated Timing Computations A Ring 1 PH Swap?NoNo MovtsUBLEBT PH Time20.0050.00 Ring 2 PH Swap?NoNo MovtsEBLUBT PH Time20.0029.09 RING 1-2 DIFF0.0020.91 CYCLE COMPONENTS20.0050.00 . B TotalA B Total No No ------- NBL SBT ------- 70.0012.71 47.40 60.11 ------Yes Yes ------- -------NBT SBL ------- 49.0930.26 13.44 43.69 20.91-t7.55 33.96 16.42 0.000.00 0.00 60.11 Uorksheet 3b: Timing Plan sensitiv~ty Capac~ty Parameter Est~mation Old Cyc 129.212 345 6 7 New Cyc 130.1UBLEBT NBL SBTEBL UBT SBL Unadj PH Time20.0050.00 12.71 47.4020.00 29.09 13.44 30.26 Adjustment0.000.00 0.00 0.000.00 20.91 16.42 0.00 New Ph Time20.0050.00 12.71 47.4020.00 50.00 29.85 30.26 Eff Green 916.0046.00 8.71 43.4016.00 46.00 29.85 26.26 Int Red Time110.1180.11 117.40 82.71110.11 80.11 100.26 99.85 Eff Red r114.1184.11 121.40 86.71114.11 84.11 100.26 103.85 Red Time60.1180.11 99.85 82.7160.11 80.11 82.71 99.85 Startup Time0.0020.00 70.00 82.710.00 20.00 100.26 70.00 Recall ModeNP N PN P N P Veh Arr Rate0.060.25 0.01 0.170.08 0.14 0.03 0.08 Av Vehs on Red6.1220.03 1.63 13.799.18 11.13 2.78 8.32 Prob (No Vehs)0.030.00 0.25 0.000.01 0.00 0.10 0.00 Veh Min Time15.0025.00 15.00 25.0015.00 25.00 15.00 25.00 Adj Veh Min14.4925.00 11.23 25.0014.92 24.97 13.51 24.98 Ped Arr Rate0.000.01 0.00 0.010.00 0.01 0.00 0.01 Av Peds on Red0.000.00 0.00 0.000.00 0.00 0.00 0.00 Prob (No Peds)1.000.00 1.00 0.001.00 0.00 1.00 0.00 Ped Min Time5.0025 .00 5 .00 25 .005 .00 25 .00 5 .00 25 .00 Adj Ped Min0.0025 .00 0.00 25 .000 .00 25 .00 0.00 25 .00 + _ + + + + + + + + + LG 1 Mov'ts L L L L L L L L Grn Arr 0.06 0.08 0.01 0.03 0.08 0.06 0.03 0.01 Red Arr 0.06 0.08 0.01 0.03 0.08 0.06 0.03 0.01 EL2 0.00 0.00 0.00 0.00 0.00 0.00 0 .00 0.00 Free G gf 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 QSTOoD gq 0 .00 20.82 0.00 0 .04 0.00 46. 06 0.00 17.55 QatEb~ Qr 4.10 0.00 1.44 2.41 5.34 0.00 0.00 0.00 QatEOF Qf 4.10 0.00 1.44 2.41 5.34 0.00 0.00 0.00 QatEOQ Qq 3.56 1.74 1.44 2.41 5.34 2.56 2.78 0.24 QatBOS Qs 4.10 1.74 1.44 2.41 5.34 2.56 0.00 0.24 + + + + + + + + + + LG 2 Mov'ts TR TR TR TR Grn Arr 0.00 0.17 0.00 0.14 0-.00 0.08 0.00 0.07 Red Arr 0.00 0.17 0.00 0.14 0.00 0.08 0.00 0.07 EL2 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Free G gf 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 QSTOpP gq 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 QatEOk Qr 0.00 14.02 0.00 12.04 0.00 7.01 0.00 7.21 QatEOF Qf 0.00 14.02 0.00 12.04 0.00 7.01 0.00 7.21 DatEOQ Qq 0.00 14.02 0.00 12.04 0.00 7.01 0.00 7.21 QatBOS Qs 0.00 14.02 0.00 12.04 0.00 7.01 0.00 7.21 8  NBT Cycte convergence has not been achieved. Difference = .8 Figure E-17. Worksheet 3 of iteration 5 for the example of compound left turn protection Appe~zdix E: Page 42

Worksheet 4: ~red Phase Times Based on 1 Target Ratio PHASE DATA1(1JBL) Int Red Time110.1t Lost Time tl4.00 Lost Time tsl3.00 LG 1 Mov'ts Grn Arr Red Arr ELI PL Svc Rate Eqiv Svc Net Svc QatBOS Qs Green G fq Land Utl Svc Time Wait Time Tot Time ResQ Qp ResQ Qp' RING 2 6(WBT) 7(SBL) 8(NBT) 80.11 100.26 99.85 4.00 4.00 4.00 3.00 3.00 3.00 0.01 0.03 0.08 0.06 0.03 0.01 0.03 0.08 0.06 0.03 0.00 2.18 0.00 8.20 0.00 0.00 1.00 0.00 1.00 0.00 0.48 0.48 0.48 0.48 0.48 0.48 0.23 0.48 0.06 0.48 0.46 0.20 0.39 0.01 0.45 1.44 2.41 5.34 2.56 0.00 8.97 42.40 15.00 45.00 15.00 1.04 0.99 0.98 0.98 0.98 1.00 1.00 1.00 1.00 1.00 6.27 14.82 16.37 465.63 0.00 0.00 0.04 0.00 46.06 0.00 6.27 14.86 16.37 511.69 0.00 0.00 0.00 0.00 2.54 0.00 0.00 0.00 0.00 0.54 0.00 TR TR TR TR 0.17 0.00 0.14 0.00 0.08 0.00 0.07 0.17 0.00 0.14 0.00 0.08 0.00 0.07 2.55 0.00 2.18 0.00 8.20 0.00 5.75 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.49 0.00 0.48 0.00 0.49 0.00 0.48 0.49 0.00 0.48 0.00 0.49 0.00 0.48 0.32 0.00 0.34 0.00 0.40 0.00 0.41 14.02 0.00 12.04 0.00 7.01 0.00 7.21 45.00 0.00 42.40 0.00 45.00 0.00 25.26 0.98 0.00 0.99 0.00 0.98 0.00 1.05 1.00 0.00 1.00 0.00 1.00 0.00 1.00 46.12 0.00 37.74 0.00 20.09 0.00 21.31 0.00 0.00 0.00 0.00 0.00 0.00 0.00 46.12 0.00 37.74 0.00 20.09 0.00 21.31 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 50.00 20.00 50.00 20.00 50.00 20.00 50.00  46.12 6.27 37.74 16.37 20.09 0.00 21.31 ilorksheet 5: Green Extens i on T i mes . Arr DettaPhi Max Min Comp Adj New Rate ValueValue Hdwy Hdwy- Ext Ext Hdwy 0.06 1.500.95 3.59 3.59 8.69 7.42 3.59 0.17 0.500.96 3.59 3.59 9.99 3.88 3.59 0.01 1.500.99 3.59 3.59 6.50 6.50 3.59 0.14 0.500.97 3.59 3.59 9.70 9.70 3.59 0.08 1.500.93 3.59 3.59 9.19 3.63 3.59 0.08 0.500.98 3.59 3.59 9.19 9.19 3.59 0.03 1.500.98 3.59 3.59 7.91 7.91 3.59 0.07 0.500.98 3.59 3.59 9.08 9.08 3.59 Computed Phase Times Including Extension Intervals RING 1 2(EBT) 3(NBL) 4(SBT) 5(EBL) 80.11 117.40 82.71 110.11 4.00 4.00 4.00 4.00 3.00 3.00 3.00 3.00 L L L L 0.06 0.08 0.06 0.08 0.00 2.55 0.00 1.00 0.48 0.48 0.48 0.20 0.42 0.11 4.10 1.74 15.00 45.00 0.98 0.98 1.00 1.00 12.58 18.08 0.00 20.82 12.58 38.91 0.00 0.00 0.00 0.00 LG 2 Mov'ts Grn Arr Red Arr ELI PL Svc Rate Eqiv Svc Net Svc QatBOS Qs Green G fq Land Utl Svc Time blait Time Tot Time ResQ Qp ResQ Qp' 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  0.00 0.00 0.00 0.00 Max Ph Time 20.00 BASE PH TIME 12.58 l 0.01 0.01 5.75 1.00 0.48 0.09 0.07 0.24 25 .26 1.05 1.00 6.50 17.55 24.04 0.24 0.00 Phase # Movements Base Ph Time Ph Ext Time REQ PH TIME MIN PH TIME COMP PH TIME NBL 12.58 46.12 6.27 37.74 7.42 3.88 6.50 9.70 20.00 50.00 12.76 47.44 14.49 25.00 11.23 25.00 20.00 50.00 12.76 47.44 678 EBLUBTSBLNBT 16.3720.090.0021.31 3 639.197.919.08 20 0029.287.9130.38 14.9225.0013.5125.00 20.0029.2813.5130.38 Figure E-~. Worksheet 4 of iteration 5 for the example of compoun~i left turn protection Appendix E: Page 43

Llorksheet 3a: T raf f i c -Actuated T imi ng Computat i ons I terat i on A B Totat A Tota ~ R i ng 1 PH Swap?No No - - - - - - - -NoNo Movts tJBL EBT -------- NBL SBT PH Time 20.00 50.00 70.00 12.76 47.44 R i ng 2 PH Swap? No No - - - - - - - - Yes Yes Movts EBL ~JBT -------- NBT SBL PH Time 20.00 29.28 49.28 30.38 13.51 RING 1-2 DIFF 0.00 20.72 20.72 -17.62 33.93 CYCLE COMPONENTS 20.00 50.00 0.00 0.00 0.00 Worksheet 3b: Timing PLan sensitivity Capac, OLd Cyc 130.2 1 23 New Cyc 130.2 UBL EBT NBL Unadj PH Time Adjustment New Ph Time Eff Green 9 Int Red Time Eff Red r Red Time Startup Time 20.00 50.00 0.00 0.00 20.00 50.00 16.00 46.00 1 10.20 80.20 1 14.20 84.20 60.11 80.11 0.00 20.00 43.90 16.31 60.20 ity Parameter Estimation 5 6 EBL UBT 4567 SBTEBLUBTSBL 12.7647.4420.0029.2813.51 0.000.000.0020.7216.31 12.7647.4420.0050.0029.82 8.7643.4416.0046.0029.82 117.4482.76110.2080.20100.38 121.4486.76114.2084.20100.38 99.8582.7160. 1 180. 1 182.71 70.0082.760.0020.00100.38 Cycte convergence has been achieved at 130.2 NBT 30.38 0.00 30.38 26.38 99.82 1 03 .82 99.85 70.00 Figure E-19. Worksheet 3a of the last iteration for the example of compoun~i left turn protection  Appendix E: Page 44

+ ~ - - - - - - - + - - - - - - - + - - - - - - - + - - - - - - EXAMPLE2 1 2 3 4 Cycle: 130.2 UBL EBT NBL SBT EBLLIBT Phase Time 20.00 50.00 12.76 47.4420.0050.00 Sat Green 12.58 46.12 6.27 37.74 16.37 20.09 SBL 29.82 0.00 LG 1 Mov' ts Volune fit Final Sat 1710.00 Eff g16.00 Finat g/C0.12 Capaci ty210.13 v/c Rat ~ o0.59 QAP du37.95 xO0.64 fd11.15 kd0.76 Delay d143.68 Delay d24.72 Tot Delay48.40 LLL 123.20115.2043.23 0.950.230.95 411.921710.00 50.008.76 0.380.07 158.18115.11 0.730.38 7.5449.31 0.640.64 1.191.17 1.050.07 8.9857.60 8.990.00 17.9757.60 77. 10184 .8076.8022.906. 77 0.460.950.080.950.23 825 201710.00144.001710.00410.34 13 6216.0050.0029.8217.62 0.100.120.380.230.14 86.31210.1355.30391.6555.52 0.890.881.390.060.12 42.6931.1724.830.008.81 0.640.640.640.640.64 1.211.161.291.101.21 0 181.050.760.180.07 51 :n36.0731.910.0010.64 0.008.994.720.000.00 51.7345.0636.630.0010.64 LT Vot LT Cap LT v/c LT QAP du Ll xO LT kd LT d1 LT d2 LT Delay 0.00300.00 0.00368.31 0.000.81 0.0022.09  0.000.64 0.001.05 0.0025.67 0.008.99 0.0034.66 0.00100.00 0.00477.96 0.000.21 0.0032.91 0.000.64 0.000.18 0.0039.88 0.000.00 0.OO39.88 + + + + + + . LG 2 Mov'ts Volume fit Final Sat Eff 9 Final g/C Capac i ty v/c Rat ~ o QAP du xO fd1 kd DeLay d1 DetaY d2 Tot telay TR 0.00600.00 0.001.00 0.001747.00 0.0046.00 0.000.35 0.00617.20 0.000.97 0.0041.47 0.000.79 0.001.12 0.001.27 0.0046.50 0.0019.04 0.0065.54 ~R 0.00500.00 0.001.00 O. 001737.00 0 0043.44 0.000.33 0.00579.51 0.000.86 0.0040.59 0.000.79 0.001.12 0.00t.OO 0.0045.54 0.003.07 0.0048.60 + ++ ~+ _~ v/c Ratio 0.590.970.38 0.89 + ~+++ + 0.00200.00 0.00265.43 0.000.75  0.0032.91 0.000.64 0.000.76 0.0039.16 0.004.72 0.0043.88 ~+ TR TR 0 00300.000.00250.00 0 001 .000.001 . 00 0.001747.000.001737.00 0.0046.000.0026.38 0.000.350.000.20 0.00617.200.00351.97 0.000.490.000.71 0.0032.870.0048.35 0.000.790.000.79 0.001.110.001.14 0.000.590.000.32 0.0036.590.0054.89 0.000.000.000.00 0.0036.590.0054.89 +++ 0.881.390.06 ++++ 0.0050.00 0.00170.64 0.000.29 0.0043.83 0.000.64 0.000.07 0.0051.25 0.000.00 0.0051.25 Figure E-20. Summary worksheet for the example of compound left turn protection Appendix E: Page 45

APPENDIX E REFERENCES Courage, K. G. and R. Ak~elik, "A Computational Framework for Modeling Traffic- Actuated Controller Operations," Working Paper NCHRP 3-48-1, Transportation Research Center, University of Florida, Gainesville, May 1994. Courage, K. G. and P-S. Lin, "A Computational Framework for Modeling Traffic- actuated Controller Operations," Working Paper NCHRP 3-48 - , Transportation Research Center, University of Florida, Gainesville, August 1994. Courage, K. G., D. B. Fambro, R. Ak~elik, P-S. Lin, M. Anwer, "Capacity Analysis of Traffic-Actuated Intersections," NCHRP 3-48 Draft Interim Report, TRB, National Research Council, Washington D.C., January, 1995. Ak~elik R. arid E. Chung, "Calibration of the Bunched Exponential Distribution of Arrival Headways," Road and Transport Research, Vol. 3, No. I, 1994, pp. 42-S9. Ak~elik, R. "Extension of the HEM Progression Factor Method for Platooned Arrivals," Working Document WD TO 95/! I, ARRB Transport Research, Australia, August 199S. Appendix E: Page 46