00-1652 QUEUE ESTIMATION ALGORITHM FOR REAL-TIME CONTROL POLICY USING DETECTOR DATA
Jinil Chang Polytechnic University 6 Metrotech Center Brooklyn, NY 11201 Civil Engineering Department Telephone: (718) 460-1291 e-mail:
[email protected]
Edward B. Lieberman, P.E. KLD Associates, Inc. 300 Broadway Huntington Station, NY 11746 Telephone: (516) 549-9803 Fax: (516) 351-7190 e-mail:
[email protected]
Dr. Elena Shenk Prassas Polytechnic University 6 Metrotech Center Brooklyn, NY 11201 Civil Engineering Department Telephone: (718) 260-3788 e-mail:
[email protected]
Paper prepared for presentation at the Transportation Research Board’s 79th Annual Meeting in Washington, DC, January 2000 and Publication in the Transportation Research Record Series.
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
1
ABSTRACT Real-time traffic control policies often need to estimate queue lengths. Since the large installed base of computerized traffic control systems relies almost exclusively on a [generally sparse] system of loop detectors, estimation of queue length on an approach to a signalized intersection must depend upon “point” measures obtained from a detector. This paper describes a queue length estimation algorithm designed for use with a highly-responsive real-time signal control system. The algorithm requires only a single passage detector, set back from the stop-bar; it utilizes detector counts and occupancy, the kinematic properties of vehicles traveling on a signalized approach and knowledge of the [varying] signal state. This algorithm was designed to support the new RT/IMPOST on-line real-time control policy which is designed to responsively adjust signal timing to provide optimized service in congested (i.e. oversaturated) traffic environments. The WATSim microsimulation model was interfaced with the RT/IMPOST control policy containing this queue estimation algorithm to provide a test and evaluation platform. In this context, results obtained with the queue estimation algorithm operating on simulated detector data were compared with “actual” simulated queue lengths. These comparisons of estimated queue lengths with actual simulated queue lengths over a range of conditions are presented. Field deployment of this algorithm is anticipated in a few months in support of a field demonstration of the RT/IMPOST policy.
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
2
INTRODUCTION Real-time traffic control policies often need to estimate queue lengths. Since the large installed base of computerized traffic control systems relies almost exclusively on a [generally sparse] system of loop detectors, estimation of queue length on an approach to a signalized intersection must depend upon “point” measures obtained from a detector. This paper describes a queue length estimation algorithm designed for use with a highly-responsive real-time signal control system. The algorithm requires only a single passage detector, set back from the stop-bar; it utilizes detector counts and occupancy, the kinematic properties of vehicles traveling on a signalized approach and knowledge of the [varying] signal state. This algorithm was designed to support the new RT/IMPOST on-line real-time control policy which is designed to responsively adjust signal timing to provide optimized service in congested (i.e. oversaturated) traffic environments. The WATSim microsimulation model was interfaced with the RT/IMPOST control policy containing this queue estimation algorithm to provide a test and evaluation platform. In this context, results obtained with the queue estimation algorithm operating on simulated detector data were compared with “actual” simulated queue lengths. These comparisons of estimated queue lengths with actual simulated queue lengths over a range of conditions are presented. Field deployment of this algorithm is anticipated in a few months in support of a field demonstration of the RT/IMPOST policy. Design Concepts The RT/IMPOST signal control policy relies upon accurate estimates of the standing queue length on each approach for each signal cycle. The standing queue is formed on an approach prior to the arrival of the lead vehicle of the primary (usually, through) incoming platoon. This standing queue is extended in length when the incoming platoon joins it to create the longer extended queue. The RT/IMPOST real time traffic control policy achieves its objectives by limiting this extended queue to avoid spill-back into the upstream intersection. When this extended queue discharges, it provides maximum productivity at the downstream intersection. The control policy adjusts signal timing based largely on knowledge of the standing queue length; also, it is able to predict the length of the extended queue only if it is provided an accurate estimate of the standing queue, as defined. The standing queue on an approach is composed of two groups of vehicles. The first group is from the “primary” flow which entered the approach during the green phase which services the through movement on the approach to the upstream intersection; the second group is the “secondary” flow which is turn-in traffic from the cross street approach(es) to the upstream intersection.
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
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A standing queue is created when a vehicle in a traffic stream responds to the beginning of a RED signal indication and is the first to decelerate to stop at the stop-bar. The following vehicles in the lane likewise decelerate to stop behind their respective lead vehicles in a sequential process. The algorithm seeks to determine when this lead stopped vehicle crosses the detector located upstream of the stop-bar by estimating its trajectory between the detector and the stop-bar. Referring to Figure 1, the algorithm A estimates the time, TA, in advance of the known start of the RED phase, TR , by calculating an estimate of this vehicle’s travel time, tA, from the detector to the stop-bar. The algorithm also seeks to estimate the earliest time, TB, that the first vehicle in the secondary platoon (i.e., turn-in traffic) will arrive at the detector. This estimate is B based on the known start of the RED phase at the upstream intersection, TR , and on a calculation of this vehicle’s travel time from the upstream stop-bar to the detector. Finally, the algorithm seeks to estimate the time, TC, that the first vehicle in the incoming primary platoon will arrive at the detector, assuming that the standing queue length does not extend past the detector. (The subject of “detector blackout” when the queue does extend past the detector is discussed later.) In the absence of “detector blackout”, TC may be estimated based on the known B start of the GREEN phase upstream, TG , and a calculated estimate of this vehicle’s travel time, tC, to the detector. If subsequent to time, TA, the queue length extends beyond (and over) the detector, an information “blackout” will occur and vehicle arrivals at the tail of the standing queue cannot be recorded by detector crossings. The algorithm treats conditions with and without blackout, as discussed subsequently. Estimate Time, TA, That Lead Queued Vehicle Arrives at the Detector A
As shown in Figure 1, TA = TR – tA, where tA = LD / VA . We estimate VA as the mean speed of traffic crossing the detector over the latter portion of the GREEN phase. Specifically, the mean speed over a detector is: 1 N Vo = ( Lv + WD ) /( ∑ TP ,i ) (1) N i =1 where Lv = Average vehicle length reflecting traffic composition, ft; WD = Width of detector, ft; TP, i = Elapsed time for vehicle, i, to cross the detector, sec.; A N = Number of vehicles crossing the detector during the sampling period, τ R , during which time the occupancy over the detector is computed. (When N = 0, Vo is set to “free-speed”)
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
But occupancy over the sampling period is: A Occ]τ R A = ∑ TP ,i / τ R
4
(2)
i
where Occ]τ R A = Measured occupancy at the detector over the sampling period,
τRA. Solving (2) for
∑T
P ,i
and substituting into (1) yields:
i
Vo =
( Lv + W D ) N
(3) A τ R × Occ]τ Two issues must be addressed: (1) How does the estimate of mean speed of travel from the detector to the stop-bar, VA, relate to the “point” estimate of mean speed over the A detector, Vo.?; (2) How is the sampling period, τ R , determined? Clearly, the speed of traffic between the detector and the stop-bar can vary widely depending on whether these vehicles are in a congested state or are free-flowing. We identify several flow scenarios: A
R
(1)
A
A [discharge] queue exists during the GREEN phase terminating at time, TR . The tail of this queue extends beyond the detector causing blackout at some time prior to TRA. This blackout condition is detected by the algorithm, as discussed later. Knowing the duration of the GREEN phase, GA, we can estimate the “residual queue” (if any), Qo, of that portion of the queue which was originally downstream of the detector, that can not be serviced during the green phase: L (G − s ) BL Qo = max D − A , 0 (4) S h v where GA = Green phase duration at intersection, A, sec. s = Start-up lost time, sec. h = Mean queue discharge headway, sec./veh. LD = Distance between detector and stop-bar, ft. Sv = Mean vehicle space in a queue, ft. BL Qo = Number of vehicles in the original queue downstream of the detector that could not be serviced during GA. BL Here, the residual queue, Qo = Qo . (a) If Qo = 0, all queued vehicles downstream of the detector were serviced. Then TA is computed as described in the section, “Estimation of Vehicle Travel Times”. (b) If Qo > 0, this residual queue constitutes the front end of the standing queue that is stored during the next RED phase. Then TA is the time that the queued vehicle that was stopped over the detector starts to discharge: L A TA = TR − G + D ; u where u = Speed of the queue discharge wave speed, ft/sec. We have estimated u = 20 ft./sec.
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
(2)
5
The queue during the GREEN phase does not cause blackout. That is, the [unknown] queue length, Qˆ < LD / S v (a)
(G B − s ) L D − S v , then a queue that extended to within one vehicle < h Sv length of the detector could not be fully serviced by the GREEN phase. Thus, the maximum value of the residual queue is: L − S v (G B − s ) max = D − . Qo Sv h
If
Since the actual residual queue which is in the range,
[ 0, Q ], is not max
o
known, then the best estimate of the residual queue is Qo = 1 2 Qo . This residual queue will form the front end of the standing queue. The value of TA is the earliest time that a vehicle can cross the detector and arrive at the tail of the [estimated] residual queue. The estimated queue length at the beginning of the GREEN phase, in vehicles, is G−s Qˆ = + Qo (5) h Qˆ S v ( LD − Qˆ S v ) A (6) and TA = TR − G + − u Vo where Vo is obtained from equation (3). The value of Vo is used rather than VA due to the relative short travel distance, LD − Qˆ S v . (G − s ) L D − S v ≥ , then any queue would be fully serviced. Then Qo = If h Sv 0 and TA is computed as described later in the section, “Estimation of Vehicle Travel Times”. max
(b)
For flow scenarios 1a and 2b, it is necessary to calculate Vo from (3) for use in A equations (7a), (10a) and (11a). To apply equ. (3), the sampling period, τ R , should A
extend over the latter portion of the GREEN phase duration, ending at time TR . The duration of τ R depends on which flow scenario applies: A
Flow Scenario:
1a
2b
LD ]] GA/2 u The procedures to estimate the standing queue length at time, TC, differ depending on whether “detector blackout” occurs between times, TA and TC.
Duration, τ R : A
min [GA/2 ,max[10, G A −
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
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Queue Length Estimation When There is No Detector Blackout Estimate TB and TC : Times when first vehicle in secondary and primary platoons, respectively, arrive at the detector Refer to Figures 1 and 2. We assume that the trajectory of the first vehicle in the secondary platoon traveling from the upstream intersection to the stopped position at the tail of the queue of length, LQ , takes one of the forms shown in Figure 2. We seek an estimate of the time, TB, that this vehicle arrives at the detector located a distance, LD from the stop bar. To estimate this vehicle’s travel time, tB, from the upstream intersection to the detector, we must calculate its travel time until it stops at the rear of the standing queue. For the trajectory shown in Figure 2A, the lead vehicle accelerates at a constant rate, A, from a stopped condition, to a speed Vm < Vf, then decelerates at a constant rate, D, to stop at the tail of the queue. Calculate the maximum attainable speed Vm: Vm =
2 A( L − LQ )
(7)
(1 + A / D) 2
V The distance traveled while the vehicle is accelerating to speed, Vm, is S A = m . If 2A S A ≥ L − L D , then the vehicle will arrive the detector while it is accelerating. In this case, t B = 2( L − L D ) / A (8) If S A < L − L D , then the vehicle will arrive at the detector while it is decelerating from speed, Vm, to stop at the rear of the queue. In this case, 2( LD − LQ ) ( A + D) (9) t B = Vm − AD D
If equ. (7) yields Vm > V f , the vehicle will attain its free speed. (See Figure 2B). Calculate S A =
Vf
2
2A
, the distance traveled while the vehicle is accelerating to speed, Vf.
If S A ≥ L − L D , then the vehicle will cross the detector while it is accelerating. In this case, t B is given by (8). If, however, S A < L − L D and L − L D ≤ L − LQ −
Vf
2
, then the 2D vehicle will arrive at the detector while it is travelling at its free speed, Vf. In this case, calculate: tB =
Vf A
+
( L − LD − S A ) Vf
(10)
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
Finally, if S A < L − L D and L − L D > L − LQ −
7
V f2
, then the vehicle will arrive at the 2D detector while it is decelerating. In this case, calculate
tB =
V f ( A + D) ( L − LQ ) 2( LD − LQ ) + − Vf D 2 AD
(11)
Since the standing queue length, LQ, is not known at the time that these calculations are made, we need to estimate this value. A sensitivity study indicated that the values of tB are not sensitive to reasonable variation in LQ. We have found that estimating LQ ≈ LD / 2 for this purpose is a satisfactory approach. Then, B TB = TR + t B (12) The same formulation may be used to calculate tC. Here, we estimate LQ ≈ 2LD /3, and calculate B TC = TG + t C (13) 2 2 We have found that values of A = 4 ft/sec and D = 8 ft/sec produce good comparisons with simulated trajectories. This value of A reflects the expectation that the lead vehicles in the incoming platoons would be unimpeded on their travels until they decelerate to join the queue. Estimation of Vehicle Travel Times This analysis details the calculation of estimated travel time, from the detector to the stop-bar for the “flow scenarios” (1a) and (2b). This travel time, tA, Figure 1, is needed to estimate the time, TA, which is when the algorithm starts counting detector crossings in order to estimate the length of the standing queue. The vehicle trajectories are assumed to take the same forms shown in Figure 2, except that the initial speed is Vo as computed from (3) instead of zero. Thus, these equations are variations of those presented above: 2 A( LD + S o ) (7a) Vm = (1 + A / D) V − Vo V m If Vm ≤ Vf , + (10a) tA = m A D 2 ( A + D)V f V V 1 tA = + ( LD + o ) − o (11a) If Vm > Vf , 2 AD 2A Vf A where Vo is calculated from equ. (3) and, based on simulation results, A = (2, 0.5) ft/sec2 if Occ. = ( < 70%, ≥ 70%); D = 8 ft/sec2 2 Vo So = 2A
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
Then
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TA = TR − t A A
Queue Estimation The standing queue length can be estimated by adding the number of vehicles crossing the detector from time, TA, to time, TC , to the residual queue (if any), Qo: LQ = [Qo + N Q
TC TA
] Sv
(14) where N Q
TC TA
is the count of vehicles crossing the detector from time TA to time TC on a
per lane basis, and Sv is the mean vehicle spacing in a queue, ft. Note that S v = Lv + H , the mean vehicle length plus separation (front-to-rear) , H, in a queue state. To estimate the queue length at any other time, TB < TQ < TC,, apply a variation of (14): LQ = [Qo + N Q
TQ TA
(14a)
Sv
The queue estimation procedure differs for the case where the standing queue extends beyond the detector prior to time, TC, as discussed below. Queue Length Estimation When There is Detector Blackout The algorithm monitors occupancy over the detector continually, using a sequence of short sampling periods. Detector blackout is identified when the occupancy over the detector rises above a specified threshold at a time, TA < TBL < TC, which marks the beginning of detector blackout. Refer to Figure 3. It is necessary to determine the time, TK, when the vehicle that caused this blackout at time, TBL, entered the approach. The equations (7a), (10a) and (11a) will yield this vehicle’s travel time, tK, from the upstream stop-bar to the detector. This vehicle’s speed at this stop-bar, Vo, is not obtained from equ. (3) but must be estimated as follows, based on a study of simulated vehicle trajectories: • •
If time, TK, occurs when the GREEN phase services the [through] feeder approach, set Vo = 20 ft/sec and A = 1 ft/sec2. Otherwise, if turn-in traffic is serviced at time, TK, set Vo = 15 ft/sec and A = 1 ft/sec2. B
Then TK = TBL -tK. Next, compute tR = TR - TK. Three conditions must be explored to estimate the standing queue length at time, TC > TBL. (1)
tR > GB. The full incoming primary and secondary flows form the portion of the standing queue that is upstream of the detector: (15) LQ = LD + (GB / h + NC ) Sv
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
(2)
(3)
9
where N C = Estimated number of vehicles turning onto the subject approach from the cross street approaches to the upstream intersection, B, during a signal cycle, divided by the number of lanes on the subject approach. 0 ≤ tR < GB: A portion of the incoming primary flow plus all of the secondary flow form the portion of the standing queue that is upstream of the detector: (16) LQ = LD + (tR / h + NC ) Sv tR < 0: A portion of the incoming secondary platoon forms the portion of the standing queue that is upstream of the detector: (17) LQ = LD + (tR /RB) NC Sv B B where RB = TG - TR
Discussion of Standing Queue Length In this paper “standing queue length”, LQ, is defined as the distance from the stopbar to the rear of the last [stopped] queued vehicle. (This distance is also called “queue reach”). Under some signal offset conditions, the front of the standing queue will be discharging at time, TC. Given our definition of queue length, the status of the lead vehicles at that time is not relevant. Of course, if the entire standing queue (i.e., the last queued vehicle) is in motion at time, TC, then, by our definition, the standing queue no longer exists at that time. The time, To , when the queue ceases to exist can be estimated as follows: LQ A (18) To = TG + u where To = Time when the standing queue ceases to exist, sec A
TG = Time at the beginning of the green phase at the downstream intersection, A. LQ = Estimated standing queue length at the time the last queued vehicle starts in motion, ft. u = Queue discharge wave speed, ft/sec. (≈ 20 ft/sec.)
The algorithm estimates the standing queue length, LQ, at time, TC, using the applicable procedure described previously. Calculate To from (18) using this value for LQ. If To < TC, there is no standing queue at time, TC. When the [real-time] clock advances beyond TC, the queue estimation algorithm “resets” and repeats the process for the following signal cycle. Simulation Testing of Queue Estimation Algorithm Simulation testing of the queue estimation algorithm was conducted on a test twoway arterial using simulation data generated by the WATSim model. The simulation model provides the occupancy and vehicle crossing times for every detector at every second over the simulation period. WATSim also provides the “actual” simulated standing queue length every second. Thus, we compared the standing queue length
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
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estimated by the algorithm based on these detector data, with the “actual” simulated queue length calculated by WATSim, for each signal cycle sampled, at time, TC. A section of the two-way test artery is shown in Figure 4. The simulation run extends over one hour consisting of three time periods with different demand volumes for each period: Test Period 1 2 3
Duration (min.) 15 30 15
Demand Volume 75% of Period 2 See Figure 4 60% of Period 2
Since the RT/IMPOST policy controls the signal offsets and green phase durations, this signal timing changes each cycle. This policy maintains constant offsets for some period of time until it determines that they must be updated to reflect changing conditions. For this case study, the control policy updated signal offsets three times (approximately at times, 3100, 4500 and 5200, as shown on Figures 5, 6 and 7). The policy’s signal transition algorithm implemented the changing signal offsets. The green phase durations are changed every cycle at each intersection. The following Table indicates the variation in these control parameters over time. Figures 5, 6 and 7 also display the cycle-by-cycle changes in arterial green phase durations, except when the control transitions from one offset pattern to the next.
Variation of Signal Timing Parameters over Time Signal Offset Plans: Range of Green Phase Intersection Durations (sec.) 1 2 3 4 1 0 0 0 0 40 (fixed) 2 77 78 1 77 17 - 48 3 71 73 76 71 17 - 50 4 79 67 7 79 26 - 46 Note: Offset values indicate the start of the GREEN phase servicing through movements. An offset value of 77 indicates that the GREEN phase starts 77 seconds after the start of the GREEN phase at Intersection 1. The cycle length is 80 sec.
The two-way arterial system extends from node 1 to node 7; only half is shown in Figure 4. The cross street approaches to nodes 1 and 4 are two-way; the others are oneway. All arterial approaches and the cross street approaches to Nodes 1 and 4 have two lanes with a left-turn pocket (if that movement is serviced). Detectors are located 300 ft from the downstream stop-bar for approaches (2,1) and (1,2), and 350 ft for the other arterial approaches. The signal cycle length is fixed to 80 seconds and the green phase duration for Intersection 1 is fixed at 40 seconds.
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
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Discussion of Simulation Results Figures 5-7 compare estimated queue lengths obtained by the algorithm with the actual simulated queue lengths. These Figures also display the Arterial GREEN phase durations for the three approaches. One of the RT/IMPOST policy objectives is to fully utilize approach storage capacity on oversaturated approaches, subject to constraints. Thus, as shown in the Figures, the queues are longer on those approaches that are of greater length. As a result, detector blackout occurs frequently on Approach (4,3), a few times on Approach (3,2) and not at all on Approach (2,1), Figures 7, 6 and 5, respectively. Figures 8, 9 and 10 display the queue estimation errors. Not surprisingly, the errors increase with approach length and with increasing distance between detector and stopbar. This is due to the greater difficulty of accurately estimating travel times over longer distances and to the detector blackouts repeated on the longest approach. Nevertheless, the errors are limited in extent and display no inherent bias, as shown in Table 1. This queue estimation algorithm enabled the RT/IMPOST policy to yield impressive results in comparison with other policies, as documented in Formulation of a Real-Time Control Policy for Oversaturated Arterials by E. Lieberman to be presented at the 79th Annual Meeting of TRB. CONCLUSIONS Congestion is growing in the nation’s urban areas, stimulating the search for signal control policies which can effectively manage such traffic environments. Such policies seek to manage unstable queue growth, which must be estimated in real time. At the same time, managers of computerized traffic control systems seek to extract as much reliable information as possible from their surveillance while minimizing the cost of installation, maintenance and operation. The queue estimation algorithm presented here is designed to satisfy both needs. Despite its reliance on a single passage detector, simulation testing has demonstrated that it provides reliable queue length estimates over a range of representative approach lengths, signal timing and traffic demand. In support of the RT/IMPOST policy development, it will be deployed in White Plains, NY early this year. ACKNOWLEDGMENTS The authors gratefully acknowledge the support provided by the following agencies: • •
ITS-IDEA Program with funding provided by FHWA. Mr. Keith Gates is the Contract Monitor New York State Energy Research and Development Authority (NYSERDA).
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
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REFERENCES 1.
“Internal Metering Policy for Oversaturated Networks”, Volume 1 and 2, NCHRP Report 3-38(3), June 1992.
2.
Tarko,A., Rouphail,N., “Distribution-free model for estimating random queues in signalized networks”, Transportation Research Record 1457, pp 192-197
3.
Rouphail, NM., Akcelik, R., “Oversaturation delay estimates with consideration of peaking”, Transportation Research Record 1365, pp 71-81
4.
Luk, JYK., esley, M., Dougthy, BW., “The box hill study: queue length estimation SCATS – 14th ARRB conference, 28 August – 2 September, 1988”, Transport and Road Research Laboratory, Vol 14, pp 157-72
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
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tA
A
TR
A
TG
A
Stops at tail of queue (not shown)
VA
LD
L TA
TB
TC Detector Location
Vo
TR
B
TG
B
B
Time
tB
tC
TA : Estimated time when a vehicle that reaches the stop-bar at the start of the RED phase at intersection A, arrives at the detector TB : Estimated earliest time that the lead vehicle in the incoming secondary platoon arrives at the detector TC : Estimated earliest time that the lead vehicle of the incoming primary platoon arrives at the detector tA : Travel time from the detector to arrive at the stop-bar on the approach to intersection A, at time, TRA tB, tC : Travel times from the stop-bar on the feeder approach to intersection B, to arrive at the detector at times, TB, TC, respectively. VA : Mean speed of traffic that travels detector to the stop-bar on the approach to intersection A, ft/sec. Vo : Mean speed of traffic crossing detector at the detector, ft/sec. LD : Distance from stop-bar to the detector, ft. L : Approach length (stop-bar to stop-bar), ft. TXY : Time of beginning of signal phase X at intersection Y: X = R, G; Y= A, B, sec. Note: The latter portion of the “primary” flow crosses the detector in the time interval from TA to TB. The entire “secondary” flow crosses the detector in the time interval from TB and TC.
FIGURE 1 A Typical Signal Control Pattern used by the Queue Estimation Algorithm
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
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Speed Vf
Vm A
D
Time Vf Vm A D
= = = =
Free flow speed, ft/sec Maximum attainable speed, ft/sec Mean acceleration rate, ft/sec2 Mean deceleration rate, ft/sec2
FIGURE 2A Representative Vehicle Trajectory (Free-speed not attainable)
Speed Vf A
D
Time
FIGURE 2B Representative Vehicle Trajectory (Free-speed attainable) Note: These trajectories apply when the vehicle accelerates from rest, then decelerates to a stop.
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
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A
LD
TBL TA
TC Detector Location
TK TR
B
B
tR tK
Here, TBL = Blackout time based on measured Occupancy over detector TK = Estimated time that the vehicle that produces the detector blackout, crosses the stop-bar on the approach to the upstream intersection, B. tK = Estimated travel time of this vehicle as it travels from the upstream stop-bar to the detector, tR = Estimated remaining green time at the upstream intersection after the discharge of this vehicle. FIGURE 3
Detector Blackout Scenario
75 400 75
150 500
4
2
50 2200
100 400 100
100 100 50
250 200 50
FIGURE 4
Queue Lengths (ft)
1 100 2200 100
2250 50
800 ft
250
200 500 200
600 100
3
150 1800 50
300
100 400 100
50 100 150
16
50 200 200
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
600 ft
400 ft
Volume and Turning Movements
Estimated Queue Length Actual Queue Length Green Phase Duration at Node 1 (sec).
Approach Length : 400 ft Distance from Detector to Stop-Bar : 300 ft
200
150 No Detector Blackout 100
50
0
2000
2500
3000
3500
4000
4500
5000
5500
6000
6500
7000
Time (sec.)
FIGURE 5
Variation of Estimated and Actual Queue Lengths and of Green Phase Duration : Approach (2,1)
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
450 400
17
Approach Length : 600 ft Distance from Detector to Stop-Bar : 350 ft
Estimated Queue Length Actual Queue Length Green Phase Duration at Node 2 (sec).
Queue Lengths (ft)
350 Detector 300 250 200 Detector Blackout When Queue Length Exceeds 350ft. 150 100 50 0
2000
2500
3000
3500
4000
4500
5000
5500
6000
6500
7000
Time (sec.)
FIGURE 6
700 600
Variation of Estimated and Actual Queue Lengths and of Green Phase Duration : Approach (3,2)
Estimated Queue Length Actual Queue Length Green Phase Duration at Node 3 (sec).
Approach Length : 800 ft Distance from Detector to Stop-Bar : 350 ft
Queue Lengths (ft)
500 Detector 400 300 200 Detector Blackout When Queue Length Exceeds 350 ft. 100 0
2000
2500
3000
3500
4000
4500
5000
5500
6000
6500
7000
Time (sec.)
FIGURE 7
Variation of Estimated and Actual Queue Lengths and of Green Phase Duration : Approach (4,3)
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
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150 Approach Length : 400 ft Distance from Detector to Stop-Bar : 300 ft
Queue Length Error (ft)
100
50
0
2000
2500
3000
3500
4000
4500
5000
5500
6000
6500
7000
-50
-100
-150
Time (sec.)
FIGURE 8
Variation of Differences between Estimated and Actual Queue Lengths : Approach (2,1)
150 Approach Length : 600 ft Distance from Detector to Stop-Bar : 350 ft
Queue Length Error (ft)
100
50
0
2000
2500
3000
3500
4000
4500
5000
5500
6000
6500
7000
-50
-100
-150
Time (sec.)
FIGURE 9
Variation of Differences between Estimated and Actual Queue Lengths : Approach (3,2)
J. Chang; E. Lieberman, P.E.; E. Prassas, Ph.D.
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150 Approach Length : 800 ft Distance from Detector to Stop-Bar : 350 ft
Queue Length Error (ft)
100
50
0
2000
2500
3000
3500
4000
4500
5000
5500
6000
6500
7000
-50
-100
-150
Time (sec.)
FIGURE 10
Variation of Differences between Estimated and Actual Queue Lengths : Approach (4,3)
Table 1. Queue Estimation Statistics
Approach (2,1) (3,2) (4,3)
Length (ft.) 400 600 800
Detector Distance from Stop-Bar (ft) 300 350 350
Estimated Queue Length Error Distribution Percent of Sample Points Mean Std. Dev. With Error Within (ft.) (ft.) 50 ft. 100 ft. 150 ft. 7.6 20 100 100 100 -14.2 38 73 100 100 1.5 53 64 95 100
