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APPENDIX C DEVELOPMENT OF AN ANALYTICAL MODEL FOR PREDICTING PHASE TIMES AT TRAFFIC-ACTUATED INTERSECTIONS An analytical model for estimating the average cycle length and phase times generated by a traffic- actuated controller was one of the principal products of NCHRP Project 3-48. In this appendix, the mode! will be summarized and examples of its application wait be presented. The entire procedure developed under this project encompasses both the analytical model and a computational structure for implementation of the model. The computational structure win be described in Appendix E. TYPES OF TRAFFIC CONTROL Traffic engineering textbooks describe three types of traffic signal controllers: · Pretimed Controllers, In which a preset sequence of phases is displayed in repetitive order. Each phase has a fixed green time and change interval that are repeated in each cycle to produce a constant cycle length. · Fully-Actuated Controllers, in which the timing on all of the approaches to an intersection is influenced by vehicle detectors. Each phase is subject to a minimum and maximum green time, and some phases may be skipped if no demand is detected. The cycle length for fillly- actuated control win vary from cycle to cycle. Semi-Actuated Controllers, in which some approaches (typically on the minor street) have detectors, arid some do not. The earliest fonn of sern~-actuated control was designed to keep the green on the major street in the absence of a minor street actuation. Once actuated, the minor street green is displayed for a period just long enough to accommodate the traffic demand. While these equipment-based definitions have persisted in traffic engineering terminology, the evolu- tion of traffic control technology has complicated their function from the analyst's perspective. For purposes of capacity andlevel of service analysis, it is no longer sufficient to consider the controller type as a global descriptor of the intersection operation. Instead, an expanded set of these definitions must be applied individually to each lane group because each lane group could fall into any of the above control categories. Each lane group may be served by a phase that is either actuated or non-actuated. Non-actuated phases may be coordinated with neighboring signals on the same route, or they may function in an isolated mode without any influence Dom other signals. Non-actuated phases generally operate with fixed minimum green times which may be extended by reassigning unused green time from actuated phases with low demand, if such phases exist. Appendix C: Page 1
Actuated phases, on the other hand, may be used at intersections where coordinated non-actuated phases exist, but they may not be coordinated themselves. In such cases, the actuated phases are subject to early termination (force off) to provide for system progression. Actuated phases are subject to being shortened on cycles with low demand. On cycles with no demand, they may be skipped entirely, or they may be displayed for their minimum duration. For purposes of analysis, the length of each phase and consequently the cycle length willbe fixed at intersections where all approaches are non-actuated. This denotes the condition ofpretimea, opera- tion mentioned previously. In current practice, one or more phases under this type of control will usually be coordinated. In general, if the intersection is sufficiently removed from its neighbors to operate in an isolated mode, then actuated operation will produce lower delays and a better level of service. The analysis procedures prescribed previously in this report will indicate the degree to which the delays may be reduced by actuated control on any phase. Where all phases at an intersection are actuated, then the length of each phase, and consequently the cycle length, win vary with each cycle. This denotes the condition of fi`lly-actuated operation men- tioned previously. Coordination with neighboring signals is not possible under this control mode. Fully-actuated signals are generally used only at intersections where distances are such that coordination would not be expected to be beneficial. The analysis procedures prescribed previously in this report will support an evaluation of the comparative benefits of coordinated operation versus actuated operation. The semimctuated control mode includes all of the cases that do not fall under either operation, pretimed nor fully-actuated. The majority of coordinated arterial systems must be treated as systems of semi-actuated controllers with coordinated non-actuated phases serving the arterial approaches and isolated actuated phases serving the cross street approaches. The cross street approaches include minor movements such as protected left turns from all approaches. The cycle length is constant at coordinated sem~-actuated intersections and variable at isolated sem~-actuated intersections. The analysis procedures presented in HEM Chapter 9 are based on the assumption of a fixed sequence of phases, each of which is displayed for a predictable time. In the case of pretimed control (i.e., no actuated phases), the length of each phase is assumed to be fixed, and constant from cycle to cycle. Actuated phases must be approximated for analysis purposes by their average green time, recognizing that the actual time may differ from cycle to cycle. For a given timing plan (i.e., constant or average green times), the differences between actuated and non-actuated phases are recognized by the parameters used in the incremental tea ofthe delay equation. PlIASE PLANS Two-phase control is the most straightforward and simplest ofthe available phase plans. Each oftwo intersecting streets is given a green phase during which all movements on the street are allowed to proceed. All left and right turns are made on a permitted basis against an opposing vehicle or pedestrian flow. This phase plan is generally used wherever the led turn movements do not require Appendix C: Page 2
. protected phasing. Multi-phase control, on the other hand, is applied at intersections where one or more led turns are determined to require protected phasing. Multi-phase control can be provided in a wide variety of ways, depending on the number of turns requiring protected phasing and the sequence and overlaps used. It is common in capacity and level of seance analysis to use a "single-ring, sequential" representation of the phase plan in which a single phase is used to indicate the combination of aD movements that are proceeding at a given point in time. Modern tra~c-actuated controllers do not use this scheme. Instead, they implement a "dual-ring, concurrent" phasing in which each phase controls only one movement, but two phases are generally being displayed concurrently. The dual-nng, concurrent concept is illustrated In Figure C-~. Note that eight phases are shown, each of which accommodates one of the through or leg turning movements. A "barrier" separates the north-south phases Dom the east-west phases. Any phase in the top group (ring I) may be displayed with any phase in the bottom group (ring 2) on the same side of the barrier without introducing any traffic convicts. For simplicity, the right turns are omitted and assumed to proceed with the through movements. {in E~ ~ 5 EBL 6 WET . - Left side of barrier E - W Movements Ring 2 Barrier 7 I SOL 8 in. 4 J SAT 1 Right side of barrier ( N - S Movements ) Figure C-1. Dual-ring concurrent phasing scheme with assigned movements ~1 Appendix C: Page 3
The definition of a "phase" as presented in Figure C-1 is not consistent with the definition accepted by most traffic analysts nor with the definition given in the introduction to HCM Chapter 9. It is, however, a definition that is universally applied in the traffic control industry. It is the responsibility of the analyst to recognize which definition is applicable to any given situation. For purposes ofthe capacity and delay analysis procedures presented in the HCM Chapter 9, each lane group is con- sidered to be controlled separately by a phase with specified red and green times, so either definition could apply. The examples shown throughout the HCM Chapter 9 are based on the single ring sequential concept. However, the dual ring definition must be used for estimating the timing plan at traffic-actuated intersections using the mode! described in this appendix. The advantage ofthe dual-ring concept is its ability to generate the optimal phase plan for each cycle in response to the traffic demand. Pretimed controllers and earlier versions of traffic-actuated controllers are more constrained in this regard. The maximum flexibility is provided by allowing the first (usually led turns phases in ring ~ and 2 to terminate independently after their respective demands have been satisfied. It is also possible to constrain these phases to terminate simultaneously to emulate the older, and less efficient equipment. For example, independent termination of the two left turns would introduce an "overlap" between the left turn phase and the through movement phase. The overlap phase would accommodate the heavier of the two left turns together with the concurrent through movement, thereby making more effective use ofthe green time. Simultaneous termination of these phases would eliminate the benefits of the phase overlap. The degree of benefit obtained from phase overlaps of this nature depends on the dissimilarity of opposing leD turn volumes. ALLOCATION OF GREEN TIME The allocation of green time is an important input to the methodology presented in the HCM for the estimation of delay. It is necessary to know the average cycle length and effective green time for each lane group to be analyzed. The most desirable way to obtain these values is by field measurement, however, there are many cases when field measurement is not possible. For example, the comparison of hypothetical alternatives precludes field measurements. Even for the evaluation of existing condi- tions' the required data collection is beyond the resources of many agencies. A procedure for estimating the signal timing characteristics is therefore an important traffic analysis tool. Such a procedure is also useful in designing timing plans that will optimize some aspect of the signal operation. In this respect' pretimed and actuated control must be treated differently' because the design and analysis objectives are different. For pretimed control, the objective is to design an implementable timing plan as an end product. With traffic-actuated control, the timing plan is generated by the controller itself, based on operating parameters that are established for each phase. This creates two separate objectives for traff~c-actuated control. The first is to determine how the controller will respond to a specified combination of operating parameters and traffic conditions. The second is to provide some indication of the optimal values for the key operating parameters. Appendix C: Page 4
Functional Requirements of the Model A practical traffic-actuated control mode} must be functionally capable of providing reasonable estimates of the operating characteristics of traff~c-actuated controllers under the normal range of design configurations at both isolated and coordinated intersections. It must also be sensitive to com- mon variations in design parameters. Examples of design parameters include: Tra~c-actuated controller settings (initial interval, allowable gap, maximum green time) 2. Conventional actuated vs. volume-density control strategies 3. Detector configuration (length and setback) 4. Pedestrian timing (Walk, and Flashing Don't Walk) 5. LeR-turn treatment (permitted, protected, permitted and protected, not opposed) 6. LJefc-turn phase position (leading or lagging) Data Requirements The information that is already required by the HEM Chapter 9 procedure is used to the extent possible to avoid the need for new data. Most of the additional data items relate to the operation of the controller itself The model structure is based on the standard eight-phase dual-r~ng control scheme previously illustrated in Figure C-1. This scheme is more or less universally applied in the U.S.A. From a capacity and level of service point of view, less complex phasing concepts (including simple two-phase operation) may generally be represented adequately as a subset of the dual-ring scheme. For purposes of this discussion' the scheme for assignment of movements to phases presented in Figure C-1 will be adopted. This will greatly simplify the illustration of all modeling procedures without affecting the generality of the results. Appendage presents worksheets that describe each step ofthe computational process. The primary objective of these worksheets is to provide a clear and concise description of the computations for software development purposes. The process is highly iterative, and productive application ofthe worksheets is not possible. The computational process has therefore been implemented in software. The following data are required as input to the model described in this appendix. A detailed explanation of each of these data items is presented in Appendix E. Appendix C: Page 5
Approach-Specific Data The following items are specific to each of the approaches to the intersection: · Left Turn ~T) Treatment Codes · Position Codes · Sneakers · Free Queue · Approach Speed, SP · Termination of Rings ~ and 2 Phasing and Detector Design Parameters The following data items are specific to each phase controlled by a traffic-actuated controller: · Phase Type Phase Reversal · Detector Length, DL · Detector Setback, DS 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, 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: Maximum Initial Interval, MxI · Added Initial Per Actuation, AI · Minimum Allowable Gap, MnA · Gap Reduction Rate, OR Appendix C: Page 6
· Pedestrian Walk plus Don't Walk, WDW · Maximum Green, MxG Intergreen Time, I · Recall Mode Minimum Phase Time for Vehicles, MnV A Basic Green Time Estimation Mode! The determination of required green time is a relatively straightforward process when the cycle length is given. However, traffic-actuated controllers do not recognize specified cycle lengths. Instead, they determine, by a mechanical analogy, the required green time given the length of the previous red interval and the arrival rate. They do this by holding the right-of-way until the accumulated queue has been serviced. The basic principle underlying all signal timing analysis is the queue accumulation polygon (QAP), which plots the number of vehicles queued at the stop line over the cycle. The QAP for a simple protected movement is illustrated in Figure C-2 represented in this very simple case as a triangle. The queue accumulation and discharge is The accumulation takes place on the left side of the triangle (i.e., effective red) and the discharge takes place on the right side of the triangle (i.e. effective green). More complex polygons are generated when permitted movements occur and when a movement proceeds on more than one phase. Chapter 9 of the HEM includes an extensive discussion on this subject. Two methods ofdetermining the required green time, given thelength ofthe previous red, areillus- trated in Figure C-2. The first employs a "Target v/c" approach. This is the basis for the planning method described in HCM Chapter 9, and for some pretimed control timing plan designs. Under this approach, the green time required is determined by the slope of the line representing the target v/c ratio. The target v/c ratio will be achieved if the phase terminates when the queue has dissipated. The second method recognizes the way a traff~c-actuated controller really works. It does not deal explicitly with v/c ratios; in fact it has no way of determining the v/c ratio. Instead, it terminates each phase when a gap of a particular length is encountered at the detector. Good practice dictates that the gap threshold must be longer than the gap that would be encountered while the queue is being serviced. Assuming that gaps large enough to terminate the phase can only occur after the queue service time (based on v/c = 1.0), the average green time may be estimated as the sum of the queue service time and the phase extension time as shown on Figure C-2. Each of these components will be discussed separately. Appendix C: Page 7
8 3 CD - C~ ._ 4 o o At 2 Green time based on phase extension time i_ it, Green time based on target v/c ratio Time (seconds) Green extension time: Figure C-2. Queue accumulation polygon illustrating two methods of green time computation Queue Service Time The queue service time, as, can be estimated as where qr r g = f qr, qg = red arrival rate (veh/sec) and green arrival rate, respectively (veh/sec); r = effective red time (see); saturation flow rate (vehlsec), and fq = I.oS O.] (G/G~)2 Appendix C: Page 8 (2)
The queue calibration factor, fq' was described by Ak~elik [1] as a factor required to account for randomness in arrivals in determining the average queue service time. Green Extension Time To estimate the extension time analytically for a particular phase, it is necessary to determine the expected waiting time for a gap of a specific length, given the average inter-vehicular headway, and some assumptions about the headway distribution. An analytical model for this purpose was des- cnbed by Ak,celik [1,2]. This model made use of Lin's earlier work [3~4]. The average green exten- sion time is estimated by the following formula which is based on the bunched exponential arrival headway distribution: e A(eO+tO-~) (pq ,~ where (3) e0 = the unit extension time setting, MnA on Worksheet 1. to = the time during which the detector is occupied by a passing vehicle to = (Ld, + Ev) / V (4) where L`, = vehicle length, assumed for purposes of this discussion to be 25 it. Ld = detector length, DL on Worksheet 1, and v = vehicle approach speed, SP on Worksheet 1 = minimum amval (intra-bunch) headway (seconds), (p = proportion of free (unbunched) vehicles, and A= a parameter calculated as: i_ ~q - 1 -Aq (5) where q is the total arrival flow (veh/s) for all lane groups that actuate the phase under consideration. Appendix C: Page 9
The bunched exponential distribution of amval headways was originally proposed by Cowan [51. A detailed discussion of this mode! and the results of its calibration using real-life data for single-lane traffic streams and simulation data for multi-lane streams are given in Ak~celik and Chung [63. The following relationship was originally proposed by Briton t7] for estimating the proportion of free (unbunched) vehicles in the traffic stream (`p): ~ = e_b~q (6) where b is a bunching factor. The recommended parameter values based on the calibration of the bunched exponential mode} using real-life and simulation data are: Single-lane case: Multi-lane case (number of lanes = 2~: Multi-lane case (number of lanes > 2~: = 0.5 s and b = 0.5 = 0.5 s and b = 0.8 Computational Structure for Green Time Estimation = 1.5 s and b = 0.6 (7) (8) (9) Although this green time estimation mode} is not difficult to implement, it does not lead directly to the determination of an average cycle length or green times, since the green time required for each phase is dependent on the green time required by the other phases. Thus, a circular dependency is established which is solved by an iterative process. With each iteration, the green time required by each phase, given all the green times required by the other phases, may be determined. The logical starting point for the iterative process is the minimum times specified for each phase. If these times turn out to be adequate for all phases, the cycle length will simply be the sum of the minimum phase times for the critical phases. If a particular phase demands more than its minimum time, then more time must be given to that phase. Thus, a longer red time must be imposed on all of the other phases. This, in turn, will increase the green time required for the subject phase. A Simple Two-Phase Example This circular dependency will converge quite reliably through a series of repeated iterations. The convergence may be demonstrated easily using a trivial example. More complex examples will be Appendix C: Page 10
introduced later to examine the effects of controller settings and traffic volumes in a practical situation. Consider an intersection of two streets with a single lane in each direction. Each approach has identical characteristics, and cames 675 vehicles per hour with no leD or right turns. The average headway is 2.0 seconds per vehicle and the lost time per phase is 3.0 seconds. The actuated controller settings are: Initial interval: Unit extension: Maximum green: Intergreen: 10 seconds 3 seconds 46 seconds 4 seconds The maximum phase time for each phase will be (46+4) = 50 seconds. The rn~n~mum phase time will be (10 + 3+ 4) = 17 seconds, which will be the starting point for the timing computations. So, the first iteration will use a 34 second cycle with 17 seconds of green time on each approach. Allowing for lost time, the elective red time win be 20 seconds, and the effective green time will be 14 seconds for each phase. The total lost time is the sum oftwo components, including the starting lost time and the ending lost time. In the HCM Chapter 9 procedure for estimation of capacity and delay, all of the lost time is assumed to be concentrated at the beginning of the green. This is a valid approximation for delay estimation because the lost time is only used in the computation of effective green time, and its position in the phase is irrelevant. However, for purposes oftrafflc-actuated timing estimation, the apportionment of lost time between the begging and the end of the phase will have a definite influence on the results. The lost time at the beginning ofthe phase willinfluence the length ofthe phase. The lost time at the end ofthe phase will influence the delay, but it will have no effect on the phase duration. For purposes of this discussion' assuming a specified lost time of n seconds, ~ second is assigned to the end of the phase; n-l seconds is assigned to the beginning. The computational process may be described as follows: I. Compute the arrival throughout the cycle, q: q = 675 / 3600 = 0.IS75 vehicles per second. 2. Compute the net departure rate (departure headway - arrival rate): (s-q) = 0.5 - 0.~875 = 0.3125 vehicles per second. Appendix C: Page 11
3. Compute the queue at the end of 20 seconds of effective red time: r * q = 20 * 0.1875 = 3.75 vehicles. 4. Compute the queue calibration factor fq from Equation 2: fq = 1.08 - 0.1 (13 / 46)2 = 1.07 5. Compute the time required to service the queue, as: gs = 2.0 + 1.07 (3.75 / 0.3125) = 14.86 seconds So, after 14.86 seconds of green time, the queue will have been serviced and gaps wait start to be observed at the detector. The question now is how long would we expect to wait for a gap of 3.0 seconds. Determine the parameters of Equation 3 as follows: ~ = 1.5 and b = 0.6, from Equation 7 ~ = e_b~q = e~o.6x~.sXoi9)=0 81 A- (Pq 1 -Aq = (0.81 x 0.19) / [1- (1.5 x .19)] = .215 7. Deterrrune the occupancy time of the detector for a vehicle length of 1 8 tic., a detector length of 30 flc., and an approach speed of 30 mph: to = 1.47 * 30 (30+18)= 1.09 seconds 8. Apply Equation 3 to determine the expected waiting time: g = e Appendix C: Page 12 e A(eO+tO-~) (pq ~ [(e 2~5 (3.0 + 1.09 - I.05) / (0.81~0. 19)] - (] / 0.215) 10.63 seconds
9. Compute the total phase time: G= (14.86 + 10.63 + 4) = 25.49 10. Compute the phase time deficiency as the difference between the trial phase time and the computed phase time, or 25.49 - 17.0 = 8.49 seconds. This indicates that the trial phase time was not adequate to satisfy rules under which the controller operates. It also suggests new trial green times of 25.49 seconds and a cycle length of 50.98 seconds for the next iteration. The next iteration will still produce a Teen time deficiency, because the red time has been increased. However, this deficiency will be smaller. Successive iterations will produce successively smaller Teen time deficiencies until eventually the solution converges. This process is illustrated In Figure C-3. The solution converged (i.e., the green time deficiency became negligible) at a phase time of 37.5 seconds, producing a cycle length of 75 seconds. This was based on a threshold of 0.1 seconds difference in the computed cycle length between iterations. In other words the process terminated when the cycle lengths on two successive iterations fell within 0.1 seconds of each other. Phase Time Deficiency ~ Computed Phase Time 40 30 o CQ - 20 E~ 10 O n go 0.s4 0.29 0.15 o.og 0 04 0 00 1 10 11 Iteration Number Figure C-3. Convergence of green time computation by elimination of the green time deficiency Appendix C: Page 13
In this example, the green time for both phases was determined by the sum of the queue service time and the extension time. Phase times will also be constrained by their specified maximum and minimum times. If the maximum phase times had been set at a value less than the 38 seconds computed above, then the iterative procedure would have terminated before the computed times were reached. This condition would generally produce an oversaturated operation for the movement in question. As a master of interest, consider the effect of reducing the urut extension time, eO, from 3.0 seconds to 2.0 seconds. This would be expected to reduce the green extension time, ge, for both phases and to shorten the resulting cycle length. The extent of the reduction may be estimated by repeating all of the steps described above with the new value for go. In the first iteration, the queue service time win remain the same, but the green extension time will be reduced from the value of 10.63 seconds computed above to 8.42 seconds. Repeated iterations with this lower unit extension time would converge to a cycle length of 65.3 seconds. Minimum Phase Times The whole question of minimum phase time requires more attention. The specified minimum green time constraints are valid only for pretimed phases and phases that are set to recall to the minimum time regardless of demand. The real significance of minimum phase time for an actuated phase lies In the fact that the phase must be displayed for its specified minimum time unless it is skipped due to lack of demand. This situation may be addressed analytically by determining the probability of zero arrivals on the previous cycle. Assuming a Poisson amval distribution this may be computed as: pod= em where: q = the vehicle amval rate (veh/sec) and C = the cycle length for the current iteration (10) So, assuming the phase will be displayed for the minimum time, except when no vehicles have arrived, the adjusted minimum phase time then becomes: AVM = MnV(l - PO\,) where: AVM = The adjusted vehicle minimum time and MnV = The specified minimum green time. This relationship also has circular dependencies because, as the adjusted minimums become shorter, the probability of zero arrivals also becomes higher, which further reduces the adjusted minimums. Fortunately, the solution fits well into the iterative scheme that was just described. Appendix C: Page 14
The use of adjusted minimum green times offers a practical method for dealing with phases that are not displayed on each cycle. This concept applies equally to pedestrian minimum times. Multi-Phase Operation Three important concepts have been introduced for estimating the timing plan at traffic-actuated signals: (1) a model for predicting the green time for any phase, given the length of the previous red; (2) an iterative computational structure that converges to a stable value for the average cycle length and green times; and (3) a procedure to account for minimum green times with low volumes. These concepts were illustrated in a trivial example, but fortunately they are robust enough to deal with the practical complexities oftrafflc-actuated control. These complexities include multi-phase operation, (both single and dual ring), permitted led turns (both exclusive and shared lanes) and compound left turn protection (both leading and lagging). Two extensions to the methodology presented to this point are required to deal with more complex situations. The first is the extension of the QAP from its simple triangular shape to a more complex shape that represents different arrival and departure times at different points in the cycle. The second is a procedure to synthesize a complete single ring equivalent sequence by combining critical phases in the dual ring operation. The QAP extensions will be considered first. Figure C-2 presented the familiar triangular QAP for a protected movement from an exclusive lane. There are four other cases to be considered, Including (~) permitted left turns from an exclusive lane, (2) permitted left turns from a shared lane, (3) protected plus permitted left turns and (4) permitted plus protected left turns. The QAP shapes for each of these cases are illustrated in Figures C-4a through C-4d. Each of the figures conforms to a common terminology with respect to its labeling. Intervals are illustrated along the horizontal axis as follows: r indicates the effective red time gq indicates the portion of the permitted green time blocked by a queue of opposing vehicles gu indicates the portion ofthe permitted green time not blocked by a queue of opposing vehicles gs indicates the portion of the protected green time required to service the queue of vehicles that accumulated on the previous phases. ge indicates the extension to the protected green time that occurs while the controller waits for a gap in the amving traffic long enough to terminate the phase. gf indicates the portion of the green time in which a through vehicle in a shared lane would not be blocked by a left turn vehicle waiting for the opposed movement to clear. This condition occurs only at the beginning of the permitted green when one or more through vehicles are at the front of the queue. Appendix C: Page 15
· - ~n - v · - ~s o o Qq REV qgy ..... ` /\ i . ! i it, Sp - qg Time (seconds) Figure C-4a. Queue accumulation polygon for permitted left turns from an exclusive lane :S ._ v, - ._ > o o A Qr . ... ... / 1 / qr: r . ..^ /l~s-qg Q ~............ Qf ' .. ... ....... ! . . 1 gf I ~' . . . · , \ ' 1 Rasp - qg Time (seconds) Figure C-4b. Queue accumulation polygon for permitted left turns from a shared lane Appendix C: Page 16
· :, ~ - · ~a - · o o he Qq 7' I'm' Ire Qr = Q ga .. ... ... ... ... ... \ \ i \ 1 Is -qg Time (seconds) l Figure C-4c. Queue accumulation polygon for protected plus permitted left turn phasing with an exclusive left turn lane Qq . ... . Qr / ll /~Sp- qr s:1 Qp=Qga .. ..; ... ... ... ... ... ... ... .. \ 1 , ~s-qg Time (seconds) Figure C-40. Queue accumulation polygon for permitted plus protected left turn phasing with an exclusive left turn lane Appendix C: Page 17
Note that, in each case, the phases are arranged such that the protected phase is the last to occur. The length ofthis phase will be determined by its detector actuations. The actual length wait be the sum of the time required to service the queue that exists at the beginning of the phase plus the extension time. Points in the cycle at which the queue size is important to the computations are also identified as follows: Qr indicates the queue size at the end of the effective red Qq Indicates the queue size at the end of the interval gq Qp indicates the queue size at the end of the permitted green period Q'p Indicates the queue size at the end of the permitted green period, adjusted for sneakers Qga indicates the queue size at the beginning ofthe protected green (green arrow) period Qf indicates the queue size at the end ofthe interval gf The shape of each of the QAPS iS based on termination by a gap that exceeds the unit extension, avowing the fig extension time, ge to be displayed. When a phase terminates on the maximum green time, the extension time may be reduced or eliminated. If a perrn~tted led turn phase terminates before the queue has been serviced, a maximum oftwo sneakers wait be dismissed from the queue at that point. These extensions to the QAP analysis win accommodate ad of the practical conditions covered by the procedures presented inHCM Chapter 9. The remaining issue to be dealt with is the synthesis ofthe complete cycle by combining critical phases in a dual ring operation. This procedure may be earned out using the worksheet shown In Figure C-5. The structure of this worksheet is compatible with the dual-ring concurrent phasing illustrated in Figure C-~. The east-west movements (left side of the barr~er)~are shown in the first three columns. The first column, labeled as "A" represents the first phase in each ring (l or 5~. The second, or "B" column represents the second phase (2 or 6~. The third column will 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). There are three rows for each ofthe two rings. The first row indicates whether or not the phase pair is swapped. This information was entered on the input data worksheet presented earlier in this appendix. The next two rows give the movements and phase times for their respective phases. If the phases are not swapped, the assignments win be as shown in the dual-r~ng configuration of Figure C- I. 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. Note that the movements in a given phase pair cannot be swapped if the left turn is not protected. The order of the phases In the pair does not affect the total phase time entered in the "Total" column. Appendix C: Page 18;
TRAFFIC-ACTUATED TIMING COMPUTATIONS East-~\'est Movements North-South Movements l A | B | Total | A | B | Total Rings: Phases Swayed? | No IYes | No byes t=. ~: ~No IYes | No IYes | l .............................. . Move e ~WBL I EBT EBT I WBL GEL I SBT SBT I NBL Phase Times l l l l T 1 I Ring2: Phases Swapped? Movements ~ ~ T EBL WBT ~ WBT EBL a | SBLINBT | NBTISBL Phase Times l I l l l l DIFFERENCE: ABS(Ring1 - Ring2) I I ~ ~ I I CYCLE THE COMPONENTS. | ? ? . ~I Ad rid t T nation ............................................ ................................... ...... ........................ ................................. Simultaneous TerTrunation ~ ~ i ~ Figure C-S. Traff~c-actuated timing computations worksheet 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 circle time must now be determined and entered in the "Cycle Time Compo- nents" row. The procedure will depend on whether the first phase termination is simultaneous or independent. For simultaneous termination, enter the maximum value of each phase in the A and B columns (ring ~ or 2, whichever time is greater3. For independent termination, enter the maximum value of the total time (A+B) from hog ~ or 2. So, for each side of the barrier, either the A and B columns or the "Total" column wait have an entry, but not all three columns. This procedure should be carried out for both sides ofthe 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. If the computed cycle length agrees with the cycle length determined on the previous iteration, then no filrther action will be necessary. If not, then this timing plan will serve as the starting point for the next iteration. COORDINATED SEMI-ACTUATED OPERATION It has been pointed out that non-actuated phases under sem~-actuated control may be coordinated with neighboring intersections. In the most common coordination scheme, a background cycle length is imposed. The actuated phases receive their allotment of green time In the usual manner, except that their maximum green times are controlled externally to ensure conformance to the specified cycle Appendix C: Page 19
length. If the actuated phases require all of their nominal green time allotment, the intersection oper- ates in a more or less pretimed manner. Otherwise, the unused time is reassigned to the coordinated phase. The computational structure described in Appendix E is able to approximate this operation quite effectively. The analysis of coordinated operation requires another iterative loop which executes the procedure, adding more green time incrementally to the coordinated phases until the design cycle length has been reached. The result is a timing plan that approximates the operation of the controller In the field. The procedure for timing plan estimation in coordinated systems requires that a design timing plan be established first' with phase splits that add up to the design cycle length. This becomes the starting point for the iterative procedure that involves the following steps: 1. Set up the controller timing parameters for the initial timing plan computations. The coordi- nated arterial phases (usually 2 and 6) should be set for recall to mc~cimum. This would not be normally done In the field, however, it provides a simple way to impose the desired green time on the model. The maximum green times for all phases should be determined by their respective splits in the pretimed timing plan. No recall modes should be specified for any of the actuated phases. 2. Perform the timing computations to determine the resulting cycle length. If the maximum green times have been specified correctly in step I, then the computed cycle length will not exceed the specified cycle length. 3. If the computed cycle length is equal to the specified cycle length, then there is no green time available for reassignment. In this case the procedure will be complete and the final timing plan will be produced. 4. If the computed cycle length is lower than the specified cycle length, then some time should be reassigned to the arterial phases. This is accomplished by increasing the maximum green times for the coordinated phases. The recommended procedure is to increase the length of the coordinated phases by one half ofthe difference between the specified and computed cycle lengths on each iteration. This procedure provides a reasonable speed of convergence arid assures that the target cycle length will be met. Repeat steps 2 through 4 iteratively until the computed and specified cycle lengths converge. The results of these computations were illustrated in an example problem presented previously. Appendix C: Page 20
REFINEMENT OF THE ANALYTICAL MODEL FOR VOLUME-DENSITY OPERATION Volume-density operation is one version of traffic actuated control best suited for intersections with heavy volumes on all approaches and high approach speeds. Under this type of operation, the con- troller employs a more complex set of cr~tena for allocating time and terminating a phase. Minimum green, passage time, maximum green, vehicle interval (unit extension) volume-density operation, includes variable initial and gap reduction features. Also, detectors are usually placed at a distance from the stop line. Volum - Density Control Descrintion Under volume-density operation, the minimum initial interval (MnI) is the minimum assured green time displayed. This minimum green time must be long enough to assure that vehicles stopped between the detector and the stop line are able to start moving and clear the intersection. On some controllers, the minimum green includes a minimum initial time plus the passage time. Under low volume conditions, the minimum initial interval should be small since few vehicles are expected. In addition, a variable initial interval (or added initial interval) wig be added to the minimum initial interval. The length ofthe variable initial interval shown in Figure C-6 is equal to the product of the number of vehicle arrivals on red and clearance intervals and the specified time interval for each vehicle actuation. This feature increases the minimum assured green so it wail be long enough to serve the actual number of vehicles waiting for the green between the detector and the stop line. o c, - ._ E ._ Maximum Initial T Variable Initial Rarlge l 1 r . _ row . Seconds per Actuation Minimum Green Vehicle Actuations Figure C-6. Variable initial feature for volume-density operation Appendix C: Page 21
1 a o c, - · - E" - w Another volume-density feature is gap reduction. The purposes of gap reduction are to reduce the probability of "maxout" and to prevent phase termination with vehicles in the "dilemma zone". The gap reduction feature shown In Figure C-7 is accomplished by the foDow~ng Functional settings: time before reduction, passage time, magnum gap and time to reduce. These terms are defined below. The time before reduction period beg as when the phase is green and there is a serviceable conflicting cad (e.g. at time "t" In Figure C-7). The passage time is the time required for a vehicle moving at the average approach speed to travel Dom the detector to the stop line. The average approach speed usually is assumed to be the 85th percentile speed. Upon completion of the time before reduction period, the linear reduction ofthe allowable gap begins from the passage time to the minimum gap. The specified time for gap reduction is called time to reduce. Thus, the gap reduction rate is equal to the difference between the passage time and minimum gap settings divided by the setting of time to reduce. The purpose of gap reduction Dom the passage time to the minimum gap is to reduce the probability of maxout. When a phase is terminated on the allowable gap, the unused portion of the passage time is always displayed to avoid dilemma zone problems. Time Before Reduction 1~ It; Baa, \,~um Gap Retime To Reduce t Green Time (seconds) Figure C-7. Gap reduction feature for volume-density operation Appendix C: Page 22
Modeling of Volume-Densitv Control The previously developed analytical model is used to predict the phase time of any movement in a fully actuated operation. However, it can also be applied to estimate the phase duration for through vehicles and protected left turns under volume-density control if proper refinements of the analytical model are done. Before the analytical model is refined for volume-density operation, the time before reduction will be assumed to be zero for simplicity and the gap reduction win begin at the green phase. The proposed refinements to the model will be introduced next and a simple example applica- tion will follow. The main differences between volume-density and basic actuated operations are the minimum green settings (variable initial)' the detector configuration' the gap reduction feature and the passage time setting for the last vehicle actuation. The refinement of the analytical model for volume-density operation focused on these areas. The initial interval (II) (adjusted minimum green) for volume-density operation is equal to the specified minimum green (Smn) plus variable initial interval (added initial) subjected to the constraint of specified maximum initial settings where the specified minimum green here is equal to the minimum initial interval Am) plus starting gap (SG). It is common to choose a value for the starting gap which is equal to the passage time from the detector to the stop line. The variable initial interval is the product of the number of vehicles arrival on the red phase and the specified time for each vehicle actuation. It can be computed as follows: Va~ableInitialInterval(VII) = Qr * tact (12) where Qr = number of arrivals during previous red and clearance intervals taC' = specified time per actuation Then, the initial interval (II) is equal to the sum of the specified minimum green (Smn) and variable initial interval (Vim. However, this value cannot exceed the specified maximum initial interval (Smx). Therefore, Initial Interval (II) = Min. (Smn + VII, Smx) (13) where Smn = specified minimum green time, and Smx = specified maximum initial interval The analytical mode! applies the QAP to compute the queue service time and the bunched arrival model to estimate the vehicle extension time after queue clearance. Based on the computation of total queue service time (QST) and total vehicle extension time (ge), the final phase time can be obtained. To estimate the phase time for the volume-density operation, the computation of the QST needs to be modified. Appendix C: Page 23
The detector placement, for the fully-actuated operation analytical model, is assumed to be at the stop line. However, it is common with volume-density control that the detector setback is greater than zero. In the queue discharge process when the signal turns green, the length of the moving queue will decrease gradually if the volume to capacity ratio (v/c) is less than I.0. The last moving vehicle in queue will pass through the upstream detector first and then the stop line. The queue service time is defined as the time required to serge the queue beyond the detector. There- fore, the queue service time for volume-density control, in general, is shorter than that for conven- tional actuated operation. The proposed model uses the total queue service time of the fully actuated operation minus a passage time to estimate the queue service time for volume-density operation. This will be called computed queue service time (CQST). The initial interval (adjusted minimum green) is the assured green time that will be displayed. If, at the end of the initial interval, the length of the queue from the stop line does not reach the upstream detector, the final queue service time should be equal to the minimum value of the initial interval and computed queue service time. For example, if the initial interval is less than the computed queue service time, the final queue service time is the initial interval because, for the upstream detector, the queue has been cleared at the end of initial interval. If the queue extends past the upstream detector, the final queue service time is just equal to the computed queue service time. An easy way to determine whether or not the queue at the end of the initial interval, QII, reaches the upstream detector is to compare the QII and the maximum storage of vehicles (MSV) between the upstream detector and stop line. A vehicle is assumed to occupy 25 ft. so MSV is equal to the distance between the upstream detector and stop line divided by 25. The value of QII may be esti- mated Dom the average vehicle arrivals during the red and initial interval using the following formula: QII = q' * (R + lI) The computation for QST may be summarized as follows: (14) QST = Min. (CQST, II) if QII<MSV (15) QST = COST if Q1I > MSV (16) Note that the queuing process is assumed to be deterministic and the values represent average queue lengths. REFINEMENT OF THE ANALYTICAL MODEL TO INCORPORATE THE "FREE QUEUE" PARAMETER The Dee queue parameter indicates the number of led turning vehicles that may be stored in a shared lane awaiting gaps in the opposing traffic without blocking the passage of through vehicles. The Appendix C: Page 24
current HCM procedure assumes that the first waiting leR turn will block all of the following vehicles in the shared lane. This produces pessimistic results in some cases. Both the through vehicle equivalence of a left turn ~ ~ ~ and the lane group saturation flow rate are affected. At this point, only the SIDRA model considers the free queue explicitly. Because of its importance to traffic-actuated control, it is essential that the proposed analytical model recognize this phenome- non. A set of curves was developed to illustrate the effect of the free queue parameter on the estimated phase time as a function of the approach volume. Because NETSIM does not recognize the free queue explicitly, it was not possible to test this model by simulation. Computation of New Throu~h-Vehicle Equivalents Left turning vehicles will select gaps through the opposing flow after the opposing queue clears. During this unsaturated period, the HCM Chapter 9 procedure assigns through-vehicle equivalents, Era, for each left turning vehicle. The maneuver time required for a left turning vehicle with "n" through-vehicle equivalents is equivalent to that for "n" through vehicles. The permitted saturation flow rate can be computed based on the assigned value of E~, and the proportion of left turns, Pa, in the shared lane. However, if free queues exist, this permitted saturation flow rate will become larger because the blocking effect of led turning vehicles is reduced. The through-vehicle equivalent for a led turning vehicle needs to be modified to account for the effect of the free queue parameter. The new En, for a left turnung vehicle on a shared lane can be determined by the combination of vehicle arrival type (through or leD turn) after the left turning vehicle and the probability of the combination during the maneuver time required for the leg turning vehicle. If En l is equal to n, n-1 vehicles will follow the leD turning vehicle. These following vehicles can be led turns or through vehicles including right turns. There are a total of 2<n~~' combinations. In each combination, the through-car equivalents of the left turning vehicle and its probability can be computed. By applying the concept of expectation value, the modified through-car equivalents for the left turrung vehicle in a shared lane will be obtained. A simple example for this En ~ computation is shown as follows: . ~ Example: ELI= 3 and PL =0.2 Total combinations = 2 (n-1) = 2 (3-1) = 4 Combination: 1 2 3 4 L L L L L L T T L T L T Probability: 0.04 0.16 0.16 0.64 ELI: 3 3 2 1 New EL1 =(0.04x3)+(0.16x3)+(0.16x2)+(0.64x 1)= 1.56 Appendix C: Page 25
The range of En ~ values shown in the HCM Chapter 9 is from 1.05 to 16 for permuted left turns in a shared lane. These EL} values do not consider any Dee queue parameter. The proportion of left turns in a shared lane, PL., is an important factor to the ELI value when a free queue exists. The reasonable range for PL is Dom O to I.0. Therefore, PL and old EL] values will be taken into account In the computahonofnewEL~ values for Dee queue settings. The reasonable maximum value ofthe n~rnm~ter con.~idered in this stud is 2.0 vehicles. Based on the proposed method, the new Eat values for one and two vehicle Dee queues are shown in Tables C-] and C-2, respectively. If the old ELI or PL. is not a specified value in the tables, the new EL! can be estimated by interpolation. If the Dee queue value is not an integer, Interpolation may be used. ,. __ ~ _ ___ ~ _ ~ , Table C-1. Through-car equivalents, ELI, for permitted left turns in a shared lane with one free queue 0 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 I 05 1.05 1.05 1.05 1.05 1.05 Appendix C: Page 26 0.1 1.05 1.10 1.29 1.56 1.90 2.31 2.78 3.30 3.87 4.49 5 14 5.82 6.54 7.29 8.06 8.85 0.2 1.05 1.20 1.56 2.05 2.64 3.31 l 4.05 4.84 . 5.67 6.54 7.43 8.34 9.27 10.22 11.18 12.14 0.3 1.05 1.30 1.81 l 2.47 l 3.23 4.06 . 4.94 5.86 6.80 7.76 8.73 9.71 10.70 11.69 . 12.68 13.68 The Proportion of Left Turns ~ the Shared Lane, Pa ~. 0.4 1.05 1.40 2.04 2.82 3.69 4.62 5.57 6.54 . 7.53 . 8.52 9.51 10.51 11.50 12.50 13.50 14.50 0.5 .05 .50 - 2.25 3.13 4.06 5.03 6.02 7.01 8.80 9.00 10.00 11.00 12.00 13.00 14.00 . _ 15.00 0.6 1.05 1.60 2.44 3.38 4.35 5.34 6.34 7.33 8.33 9.33 l 10.33 11.33 12.33 13.33 14.33 1 1 15.33 1 I O.7 I 1.05 1 I 1.70 1 1 2.61 1 1 3.58 1 1 4.S7 I S.S7 1 1 6.57 1 1 ~ ~ - 1 1 8.57 W:; 10.57 1 1 11.57 1 12.57 1 1 13.57 1 1 14.57 15.57 I 0.8 I 1.05 I 1.80 1 1 2.76 1 1 3.7S 1 4.75 1 . , 6.75 1 7.7S 1 8.75 9.75 10.75 11.75 12.75 13.75 14.75 15.75 o.9 1 1 1.05 1.90 2.89 3.89 4.89 . 5.89 . 6.89 7.89 . 8.89 I 9.89 1 I 10.89 1 I 11.89 1 I 12.89 I 13.89 1 I 14.89 I 15.89 . 1.05 1 2.00 1 3.OO 1 1 4.OO 1 1 1 S.00 6.00 1 1 1 7.00 1 . . 1 8.00 1 1 1 9.oo 1 ~:~ . . I 12.00 1 14.00 ., I 15.00 1
Table C-2. Through-car equivalents, ELI, for permitted left turns in a shared lane with two free queue ~ . O 1 05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 0.1 0.2 05 1.05 1.1 ~ 1.2 .2 14 13 16 1.4 1.8 1.5 2.0 1.6 2.2 .7 2.4 .8 2.6 1.9 2.8 20 3.0 21 3.2 2.2 3.4 2.3 3.6 2.4 3.8 2.5 4.0 Th ~ 0.30.40.50.6 title 1.05 1.05 1.05 1.051.051.05 1.05 1.05 13 14 15 1617i8 i 9 2.0 1.6 1.8 2.0 2.22.42.6 2.8 3.0 1.9 2.2 ~ 2.5 2 83 13 4 3.7 4 0 2.2 1 2.6 1 3.o 1 3.41 3-81 42 1 46 1 5-0 2.5 3.0 3.5 4.04.55.0 5.5 6.0 2.8 3.4 40 465258 64 7.0 3.1 3.8 4.5 5.25.96.6 7.3 8.0 3.4 4.2 5.0 5.86.67.4 8.2 9.0 3.7 4.6 5 5 6 47 38.2 9.1 10.0 4.0 5.0 6.0 7.08.09.0 10.0 1 1.0 4.3 5.4 6.5 7.68.79.8 10.9 12.0 4.6 5.8 7.0 8.29.410.6 11.8 13.0 4.9 6.2 7.5 8.810.11 1.4 12.7 14.0 5.2 6.6 8.0 9.410.812.2 13.6 15.0 55 70 85 10.011.5130 14.5 160 When the new through-car equivalent of a lefc-turn vehicle is computed, the permitted saturation flow of asharedlane curing the period of unsaturated opposing flow can tee computed as follows, if a free queue exists: s 1 PL (ELI (new) 1) where s = protected saturated flow rate (veh/sec) Pi = the proportion of left turns in the shared lane. Appendix C: Page 27
Estimation of Free Green (gf) for Free Queue Settines The free green, gf, originally described in the HCM needs to be modified when the free queue param- eter is considered. When the green time is initiated' the opposing queue begins to move. While the opposing queue clears, left turns from the subject shared lane are effectively blocked. The portion of effective green blocked by the clearance of an opposing queue of vehicles was referred to as gq in the HCM. Until the first led turning vehicle arrives, however, the through vehicles on the shared lane are unaffected by left sums. The portion of effective green before the arrival of the first left turning vehicle was referred as free green, gf, in the HCM. Basically, Bee green represents the time during which the through vehicles in the shared lane are not affected by left turns when the green begins. Generally, signal timing models assume that the first permitted left 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 left turns or led turning vehicles waiting to turn left. A reasonable free queue setting is between zero and two. The proper value for the free queue setting can be observed from the field. The leD turning vehicle in the free queue will not block the through vehicle in the shared lane, so the free green will increase. The proposed method to estimate the new free green for the free queue setting is describe as follows. Based on the number of opposing lanes nOpp and the relationship between gq and gf, the permitted sat- uration flow rate sp can be determined directly from the current HCM Chapter 9 procedure: s = _ _ if n0pp= 1 and gq>gf (17) ~ - P 1 + PL (EL2 1 ) s so = ~others - 1 + P[, (EL,} - 1) where (18) protected saturation flow rate, proportion of led turns in the shared lane, through-car equivalents for each lefc-turn vehicle during unsaturated green through-car equivalents for each leD-turn vehicle during the period of (gq- go, when g q greater than gf and the number of opposing lanes nOpp is equal to one. Assume the average waiting time for a LT vehicle in the free queue to leave is x seconds. This waiting time can be derived as follows: ~ (LOPE) + pL x sp s Appendix C: Page 28 (19)
Therefore. s - s + S P x = P P ~(20) p L If the number of opposing lanes is greater than one and gq is greater than gf, the required time x be- comes s -s +S P s s P gq gf (21) p L Assuming the free queue is n vehicles, (O<nc2) and the required time for each vehicle in the Dee queue is t seconds, t can be expressed as : t = x / n if n > ~ t=x if O<n<] Note that when n is less than one, the Dee queue win be assumed to be I.0. Later, the free green wall be modified based on the Dee queue setting. This process win avoid a large t when n is small. During t seconds, the probability of leR-turn vehicle arrivals is P. According to the bunched amval distribution, the probability P can be computed as the product of bunched cumulative distribution and the probability of le~c-tum arnvals: where P = ~ ~ - up e~A(e - ^y ~ p (22) = minimum arnval (intra-bunch) headway (seconds), p = proportion of free (unbunched) vehicles, and = a parameter calculated as: where A = (P q subject to q < 0.98/A ~ - ~ q q = total arrival flow (vehicles/second) for all lane groups that actuated the phase under consideration. Appendix C: Page 29
During the waiting time x seconds, the probability of block effect on the through vehicles due to left- turn arrivals is assumed P'. Then P' can be computed as: p, = pn if no! P' = P if O<n<l The new free green, gf, will be: gf (new) gf (I) p / x if n >! gf (new) gf (0~) n p / if 0 < n ~ ~ The degree of difference between the phase times with free queues and the phase times without free queues for a shared lane is mainly dependent on the combined effects of free green and through-car equivalent. The effect of free queue setting is especially important for single shared lane. The phase time analysis with free queues is best illustrated with a simple example. Consider a trivial intersection with four single-lane approaches. All approaches are configured identically and carry the same traffic volume. In each approach, the portions of left turns, through vehicles and right turns are 0.2, 0.7 and 0.1, respectively. This is a simple two-phase fully actuated operation. The minimum phase time for each approach is 15 seconds and the maximum phase time is 80 seconds. Detector is 30 feet long placed at the stop line. The allowable gap is 3 seconds. In each phase, Intergreen is 4 seconds, and 3 seconds of lost time is assumed. Each approach volume varies from 100 vph to 800 vph, while the range of free queue is set from 0 to 2. The value ofthe free queue is not necessarily an integer. Based on the proposed method, the phase time estimation with free queues is shown in Figure C-8. In this figure, the x axis represents the approach volume, y axis shows free queue values, and the vertical axis is the estimated phase times by the proposed method. The effect of the free queue value may be easily observed from this three dimensional surface plot. In this example, left turns only account for 20 percent of the total approach volume. When the approach volume is low, the headway between vehicles is large, and a left-turn maneuver is relatively easy to make. Therefore, the blocking effect caused by left turns should be very small. Theoretically, there will be nearly no difference in phase times between different free queue settings. This phenomenon can be seen in Figure C-8 when the approach volume is below 400 vph. Left turns will become more and more difficult when the approach volume becomes heavier, because the saturated portion of opposing queue increases and the number of gaps for left turns during unsaturated portion of opposing queue decreases. In other words, the through-car equivalent, ELI, becomes large with the increase of approach volume. Appendix C: Page 30
c) In ~ To ~ is ~o~ Figure C-~. Effect of the free queue on phase times for the example problem For the same free queue value, the phase times will increase due to the increasing blocking effect by left turns In the shared lane. On the other hand, if the value of the Dee queue increases, the blocking effect is reduced because left tarrying vehicles are able to wait in the free queue before turning left. The larger the Dee queue, the smaller the blocking effect. These two phenomena can be clearly seen in Figure C-8 when the volume is between 400 vph and 720 vph and the value of the free queues is between 0 and 1.2. It is clear that the phase times will depend on the combined effects of EL} and the value of the free queue. In Figure C-8, it is also easy to observe that the effect of the free queue is reduced when the approach volume becomes large. Art this example, when the volume exceeds 720 vph and the free queue exceeds 1.2, the phase times are always equal to the maximum chase time (84 seconds in this case). This indicates that the effect . ~ ofthe Dee queue is too small to prevent the phase time from reaching its maximum. The computational structure of the proposed analytical model has been modified to incorporate the free queue value. While it would be difficult to verify the model satisfactorily, either by simulation, or in the field, it is suggested that the analysis is robust and the value of the model will be enhanced if the free queue is included. Appendix C: Page 31
APPENDIX C REFERENCES I. Ak~elik, R. Analysis of Vehicle-Actuated Signal Operations. Australian Road Research Board. Working Paper WD TE 93/007, ~ 993. Ak~celik R Estimation of Green Times and Cycle Time for Vehicle-Actuated Signals. Paper No. 94-0446, 73rd Annual Meeting of Transportation Research Board, Washington, January 1994. Fin, F.B. Estimation of Average Phase Durations for FuD-Actuated Signals. Transportation Research Record S8l, TRB, National Research Council, Washington, DC, 1982, pp. 65-72. 4. kin, F.B. Predictive Models of Traff~c-Actuated Cycle Splits. Transportation Research 16B (5), 1982, pp.65-72. 5. Cowan, RJ. Usefill Headway Models. Transportation Research 9 (6), 1975, pp. 371-375. 6. Ak~celik, R. and Chung, E. Calibration of the Bunched Exponential Distribution of Arrival Headways. Road and Transport Research 3 Ail, 1994, pp.42-59. 7. Briton, W. Recent Developments in Calculation Methods for Unsignalized Intersections in West Germany. Intersections Without Traffic Signals, Ed. W. Briton, 1988. Appendix C: Page 32
