- Email: [email protected]
A STOCHASTIC MODELLING APPROACH TO ACTUATED CONTROLS
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Abstract: This paper proposes a method to estimate the d istribution of travel times at actuated signals by computing the probability distribution of green times and the eventual overflow queues, which may occur in saturated conditions. This method enables one to analyse the stochastic and dynamic character of travel times at such signals. Due to the required elaborate computations, this methodology can be applied in two ways: it can be used to derive approximate formulas of expected green times and queue lengths and their standard deviation, or it can be used to validate existing formulas. The results on a toy scenario show that actuated control s work well only when the flows approaching the intersection are small, while a large o verflow queue can occur in oversaturated conditions. &RS\ULJKW,)$& Keywords: Stochastic modelling, actuated controls, trav
1. INTRODUCTION Delays at signalized intersections represent a major contributor to travel times at urban networks. Unfortunately, traffic volumes have in these systems a strong dynamic and stochastic character; therefore an optimal regulation of traffic is very difficult. The dynamic and stochastic nature of traffic flows in a daily scenario in fact reflects itself in the variab ility of delays (Van Zuylen and Viti, 2003). Pretimed controls have become an exception. Most intersections are controlled by equipment that measures the traffic and adapts the traffic control t o the actual traffic. In congested conditions this often results in a fixed time control, because the length of green phases is limited by a maximum. Actuated control is most common at present, but most theory about this control is still based on fixed time control theory, and the effect of this dynamic mechanism is simply approximated by using a multiplicative factor (e.g. TRB, 2000). This is certainly due to the complex character of the process of adaptively controlled traffic. Macroscopic models, which are based on average conditions, do not fully catch this service mechanism, which is based on actual vehicle headways. Microscopic models are at present used most often to analyse this kind of traffic control. Although traffic performance improvements have been reported and the benefits of such control methods have been demonstrated analysing field data, © 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006
el time estimation
it is still difficult to evaluate the delay travellers experience at these systems. The fluctuations of the demand within a cycle are influenced by the way signals are set, which depends on all traffic streams arriving at the intersection. This paper describes a method to compute signal settings and overflow queues in time using a stochastic modelling approach and assuming known the probability distribution of the arrivals. The knowledge of the probability distribution of the phase plans and the eventual residual queue when the intersection is saturated enables the traffic analys t to compute the distribution of delays in time under very general assumptions. 2. PROBLEM STATEMENT The dissemination and the different schedule of the activities in an urban area result in a large variabilit of flows, which can be observed both within a day and in between days. Figure 1 shows a collection of traffic counts in an urban road located in Delft, the Netherlands during an entire day and for 90 days and using data collected from camera observations. The large spread of data gives an idea of how traffic can change from one day to another at the same time and in the same location. Moreover peaks of traffic are frequently observed during the morning and in the afternoon, showing the dependency of the demand with the schedule of the activities.
Page 478
y
As long as the traffic is below the capacity of the signal travel times remain small. When oversaturated conditions occur, travel times increase considerably, and they become rather unpredictable.
Figure 1: Traffic observations from loop detectors Figure 2 shows how variability of traffic is may affect only a small period of the day. Travel times have been computed using camera detections again in Delft, the Netherlands. Travel times in this period have a very large standard deviation.
usually constrained to be within minimum and maximum values, which are mainly determined by the geometry of the intersection. Since the stochastic nature of the arrivals, different headway distributions and different flow rates can be observed from cycle to cycle. Therefore, the assigned green times and the delay incurred are variable too. The assigned green times are thus variable according to the variability of queues forming during the red phase and to the variability of vehicle headways. If one considers that the queue formed during the red phase depends on the number of vehicles arriving during that phase and to the length of the red phase itself, which depends on the green time extensions given to all conflicting streams, the mathematical formulation of the expected travel time experienced by the travellers is quite complex, as it is shown in this paper. Rigorous mathematical solutions have been proposed in the past, but they are characterised by limiting assumptions (e.g. average demand and service rates). Other models have been derived from heuristics to better represent the results of field data but they cannot give clear insight of the real effects of this service type on the propagation of flows across the network. The problem becomes more complex if one considers that this process has a strong dynamic character when overflow queues occur, i.e. the number of vehicles arriving during the red phase exceeds the maximum number allowed by the maximum green extension. This overflow queue creates a delay, which propagates in time and affects also vehicles arriving at later periods. This contribution has not been considered in any past study. Immediate consequence of this approximation is that delays in periods that follow congested cycles are underestimated. This paper proposes a stochastic model formulation, which enables one to compute the probability distribution of expected greens and overflow queues in time and their effect on later green time extensions. Knowing the distribution of green splits in a cycle and the overflow queue one can compute the distribution of delays by applying standard delay formulas like the one proposed by the Highway Capacity Manual (TRB, 2000). 3. STATE OF THE ART
Figure 2: Travel times computed using cameras One possible way to relieve this unbalanced use of the roads is to adapt their capacity to the actual demand by e.g. dynamic control schemes. Dynamic signal phase plans are designed to regulate traffic giving attention to this dynamic and stochastic propagation of flows. Green and cycle times are adapted on the basis of the actual traffic states in order to improve the efficiency of the signal. Vehicle actuated control is determined by the headway measurement of vehicles in time. Green time is given to a traffic stream in order to serve all vehicles waiting at the signal during the previous red phase. The signal switches as long as the detector records a headway distance, which is longer than a pre-defined threshold value. This green time is © 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006
The first model of delay with a simple traffic actuated signal was proposed by Morris and Pak-Poy (1967) with an application on a signal coordinating two oneway streets. For each traffic condition they computed the optimal vehicle interval to minimize the total delay. Newell (1969) studied the same problem under the assumption of stationary arrival process in undersaturated conditions but near capacity. He developed approximate formulas of delays and expectation values of green and red extensions in function of the average arrivals at fully-actuated signals. Dunne (1967) proposed a delay model derived by assuming a binomial arrival process and green times determined by the queue length detected. Cowan (1978) used bunched exponential distribution
Page 479
for the arrivals to compute expected green, red times and delay. Webster’s (1958) delay formula was adapted by Courage and Papapanou (1977) to compute the average cycle length with actuated signals. Optimal cycle lengths were used to compute pre-timed controls using the formula:
&2
=
1.5 ⋅ / + 5
1 − ∑ \L L
where / is the total lost time in the cycle and \ = T / V is the volume to saturation flow ratio. Fully-actuated signals are instead computed considering average cycle length. The HCM 2000 (TRB, 2000) considers a discount factor of 0.85 to multiply to the uniform delay component. The manual also gives an approximate expression of the average signal cycle: L
L
L
({&} =
[F ⋅ /
[F − ∑ \L L
where [F = TLF / VL is the critical volume to capacity ratio (which in the HCM 2000 is set to 0.95). The effective green is given by:
({ J } = L
\ ⋅ ({&} [ L
L
Lin and Mazdeysa (1983) proposed an extension of the Webster’s delay formula by including two extra parameters, .1 and . 2 , in the form: 3600 ⋅ . ⋅ [ & (1 − . ⋅ J / & ) + 2 ⋅ [1 − . ⋅ ( J / & ) ⋅ . ⋅ [ ] 2 ⋅ T ⋅ (1 − . ⋅ [ ) 2
G
= 0.9
2
1
1
2
2
2
The two heuristic parameters reflect the sensitivity of actuated signals to the degree of saturation, and they have been calibrated with simulation. Li et al. (1994) proposed to include a discount factor N in the Australian Capacity Manual time-dependent overflow delay model to account for fully actuated signals:
8⋅ N ⋅ [
F7
G (7 ) = 900 ⋅ 7 ⋅ [ − 1 + ( [ − 1) 2 +
very important. This paper contributes to the modelling of these characteristics. 4. METHODOLOGY Green phases at actuated controls are primarily determined by the headway in the arrivals. The basic assumption in the estimation of green times is to extend the green time length until the gap between a vehicle and its follower in the flow stream is larger than a pre-defined gap (XQLWH[WHQVLRQ). Let τ be the unit extension, assumed known and constant. This unit extension is determined by the length of the passage loop and the speed at which vehicles drive on it. The length of the loop detector is usually fixed in such a way that the queue formed during the red phase is completely served, avoiding that the signal changes while the queue is not yet fully served or unless the maximum green extension is met. Let assume that all vehicles in queue drive across the loop with headways shorter than τ . If 4 U (τ ) is the total number of vehicles queuing up and waiting at the signal at the cycle τ during the red phase U (τ ) the expected green time given to serve these vehicles is given by: Jmin U J (τ ) = 4U (τ ) / V Jmax
© 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006
U
min
U
min
max
(1)
U
max
where V is the saturation flow, assumed in this paper known and constant (uniform service times). The queue length 4 U (τ ) is determined by the number of vehicles arriving during the red phase. If one assumes uniform arrivals, the expected green time is computed with formula (1) using simply the average flow rate. The probability distribution of green times at one traffic stream L is computed knowing the distribution of arrivals at stream L and the distribution red times at the previous phase, which depends on the distribution of arrivals at all conflicting streams1. If D (τ ) is the arrival rate (in vehicles per second), and U (τ ) is the red time at the previous cycle one can compute the probability of a certain number of vehicles N queuing up during the red phase of a certain length ρ as: L
L
Umax
3[4 (τ ) = N ] = ∫ 3[4 (τ ) = N | U (τ ) = ρ ] = U
This discount factor has been later adopted also in the Highway Capacity Manual formula. Although several – more or less heuristic - delay formulas have been proposed to account for the effects of traffic actuated control, the complex behaviour of such systems needs still to be fully explored. This paper contributes to the state of the art of delay estimation at actuated signals by analysing the dynamic and stochastic character of delays. This can be valuable information for control design, realtime management systems and ATIS systems. In these applications the expected delay is an important characteristic, but the variability of the delay is a lso
4 (τ ) / V ≤ J if J ≤ 4 (τ ) / V ≤ J if 4 (τ ) / V ≤ J
if
U
L
L
(2)
L
ρ = Umin
=
Umax
∫ (3(D (τ ) ⋅ ρ = N ) ⋅ 3 (U (τ ) = ρ ) )G ρ L
L
ρ = Umin
The formula is obtained by assuming that the arrival rate and the red time are independent stochastic variables. The formula to compute the probability of red time length is given later in this section.
1
In practice the computation previously determines the critical paths, which are the all conflicting streams wi largest demand. Page 480
th the
The probability of a green time J (W ) needed to clear the queue at the end of the red phase to be a value O is given by the following condition: U
3 ( J (τ ) = O ) = U
∑ 3(4 (τ ) = N )
(3)
U
L
N
/ V =O
While clearing the queue formed during the red phase, other vehicles may join the queue. Formula (4) computes the probability for these vehicles similarly to formula (2). The probability of red time is simply replaced by the probability of green time computed with formula (3)2:
3(4LJ (τ ) = N ) = 3[4LJ (τ ) = N | JLU (τ )] = Jmax
∫
ς = Jmin
(4)
3 (D (τ ) = N | ς ) ⋅ 3 (JLU (τ ) = ς ) Gς L
Accordingly, formula (3) is adapted to compute this extra queue with the following formula (5):
3 ( JLJ (τ ) = O ) =
∑ 3 (4
J L
N / V =O
(τ ) = N )
(5)
The probability of green time due to all vehicles in queue J 4 is thus given by the following relationship:
3L ( JL4 (τ )
= P) =
∑
N +O =P
3L ( JLU (τ ) = N ) ⋅ 3L ( JLJ (τ ) = O )) (6)
The total queue to dissipate within this time is computed accordingly:
3L (4L (τ ) = P) =
∑ 3 (4 (W) = N ) ⋅ 3 (4 (W ) = O))
N +O =P
L
U L
L
J L
(7) Apart from the green time assigned to clear this queue, one should take into account that the green phase is extended as long as a vehicle passes the detector within the unit extension, which it can happen also when the queue has been fully served but vehicles are still arriving with short distances. To account for this extra-time one can use the distribution of arrival headways instead of the number of vehicles arriving within the time period. The Poisson distribution can describe for example the probability of observing Q arrivals in a period from 0 to W with the following expression:
3Q (W ) =
(λ ⋅ W )
Q!
Q
⋅H
− λ ⋅W
(8)
This equation gives information about how the probability is distributed over a time interval in terms of number of vehicles. In a sequence of Q arrivals one can observe vehicles passing with a random headway distance. If no arrivals are observed within a time τ < τ the signal will switch to yellow. This probability is given by the following formula (9): 2
Formula (4) is an approximation of the actual number of vehicles queuing up, since other vehicles may arrive at the J back of the queue during the clearing of 4L (τ ) . These higher order terms are neglected in this paper. © 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006
30 (τ ) = H− λ ⋅τ
(9)
If one computes the probability distribution of a Q sequence of vehicles at times 0 < W1 < W2 < ... < WQ = W the probability of observing this sequence with W2 − W1 < τ , W3 − W2 < τ , etc. is given by the following formula (10): Qmax
3 (WH[WQ = W ) = ∑ 3(W1 < W2 < ...WQ = W ) ⋅ 3 (Q, W ) Q=0
V.W.
W1 < τ , W2 − W1 < τ ,..., WQ − WQ −1 < τ (10)
Even if green time extension is needed, one should compute the probability that this extension time is actually available. The probability of a certain number of seconds available for eventual green time extension before the maximum green extension is easily derived from the probability of green time due to the formed queue, formula (6). The probability of having an extension of exactly W seconds is given by the following relationship:
3L ( JLH (τ ) = W ) = ∑ 3 (WH[WQ = W ) ⋅ 3 ( J max − JL4 ≥ W )
(11)
The average total green time is finally given by computing the joint probability of guaranteed green time and extended green time J WRW (W ) = J 4 (W ) + J H (W ) . The probability of a total green time J WRW is thus given by the following relationship, analogous to formula (6):
3L ( JLWRW (W ) = P) =
3L ( JL4 (W ) = N ) ⋅ 3L ( JLH (W ) = O )) ∑ N O P + =
(12)
Green times are computed using this method for each flow stream of the intersection and the total cycle is computed by summing up all green times at each conflicting stream, together with the corresponding lost times. Knowing then the green times and the cycle length one can finally compute the uniform delay component using for example the Highway Capacity Manual formula. If the green time assigned during the previous cycle is smaller than the maximum green extension no overflow queue is supposed to be present and only the arrivals during the red phase should be served. Overflow queues are likely to occur only when the intersection is oversaturated and the maximum green extension is met. If the signal assigns the maximum green extension, one should calculate the eventual overflow queue, which will have to wait for the next green phase. Overflow queue occurs then only when J 4 = J max . In this case the overflow queue is computed by formula:
42L (W) = 4L (W ) − J
max
⋅ VL
(13)
with V the assumed saturation flow of the road section L . The corresponding probability is computed by the following formula: L
Page 481
3 (42L (W ) = T ) =
∑ 3(4L (W ) = N )
N − Jmax ⋅ V = T
(14)
L
Since an eventual overflow queue should be cleared in the next green phase, formula (3) should also consider that, apart from the arrivals, also the eventual overflow queue should be served. Formula (3) is thus reformulated as:
3 ( JL4 (W ) = O ) =
3(4L (W ) = N ) ⋅ 3 (42L (W ) = T ) ∑ N T V O
( + )/ =
(15) Last step to compute formula (3) is to derive the probability distribution of the red times at the previous cycle. The red time is determined by the sum of all the green times given to the conflicting strea and the total time lost 7/of the signal:
U (W ) = ∑ J (W − 1) + 7/
ms
WRW
M
≠L
M
(16)
The corresponding probability of a red time to be a certain value U (W ) = V is thus computed with the following formula:
3 (U (W ) = V) = 3 (∑ J (W − 1) + 7/ =V) WRW
M
≠L
M
(17)
Assumption should be made on the initial red phase probability and on the initial overflow queue in order to compute the distribution in time.
Figure 3: Expected green time in the example The expectation value is the minimum green time only when both flows are zero. The green time is sensitive to both the increase of each flow stream, but it has a steeper increase if increasing D$% of one unit. The expected value is equal to the maximum green time for a large overall demand. Overflow queues are more likely to occur in these conditions, as figure 6 shows, while its value is nearly zero for low values of the demand, especially if D$% is small. When one of the two streams has a very large demand overflow queue is likely to reach the maximum value assumed in the example (70 vehicles) within the total period of analysis.
5. NUMERICAL EXAMPLE In the following of the section the vehicle actuated control is modelled with the probabilistic approach in a simple case of two traffic streams crossing an intersection.
Let D$% and D&' be the average flows (expressed in vehicles per second) and V $% = V&' = 1800[YHK / KU ] . Let assign the first green time to the direction $% and an initial red time U $% (0) = 30V . Initial overflow
queue is zero 42$% (0) = 0 . The total lost time of the intersection is assumed 7/ = 12V and the unit extension is W = 3V . Finally minimum and maximum green time extensions are respectively set to Jmin = 10 V and J max = 60V . Let assume the arrival rate distributed as Poisson and let the process have a stationary arrival rate for a period of 30 minutes. Using formulas (2) - (15) one can compute at each time step the distribution of the green times and the eventual overflow queue. Expected green times and overflow queues have been computed for each couple (D $% , D&' ) , each ranging from a value from 0.1 to 0.6 of the saturation flow. Figure 5 shows the expectation value as function of the couple (D $% , D&' ) at the end of the period of stationary conditions.
© 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006
Figure 4: Expected overflow queue in the example The overflow queue in conditions of moderate saturation starts assuming a strong dynamic character. For example if D$% = 0.5 ⋅ V $% and D&' = 0.2 ⋅ V&' the expected value of the overflow queue is quite large, as figure 7 shows.
Page 482
Dunne M.C., 1967. LQWHUVHFWLRQ
7UDIILF 'HOD\ DW D VLJQDOL]HG ZLWK
%LQRPLDO
Transportation Science, 1, pp. 24-31. Li J., Rouphail N. and Akcelik R., 1994. GHOD\
HVWLPDWLRQ
IRU
LQWHUVHFWLRQV
.
DUULYDOV
2YHUIORZ ZLWK
IXOO\
. Presented at the 73rd TRB Annual Meeting, Washington D.C. Lin F.-B., Mazdeysa F., 1983. (VWLPDWLQJ DYHUDJH DFWXDWHGVLJQDOFRQWURO
F\FOHOHQJWKVDQGJUHHQLQWHUYDOVRIVHPLDFWXDWHG
. Transportation Research Record 905, pp. 33-37. Morris R.W.T., Pak-Poy P.G., 1967. ,QWHUVHFWLRQ FRQWURO E\ YHKLFOH DFWXDWHG VLJQDOV. Traffic Engineering and Control, No. 10, pp. 288-293. Newell G. F., 1969. 3URSHUWLHV RI 9HKLFOH $FWXDWHG 6LJQDOV , 2QH:D\ 6WUHHWV. Transportation Science, 3, pp. 30-52 Rouphail, N., A. Tarko, J. Li (2000). 7UDIILF IORZ DW VLJQDOL]HG LQWHUVHFWLRQV. In: Revised Monograph on Traffic Flow Theory, update and expansion of the Transportation Research Board (TRB) Special Report 165, "Traffic Flow Theory," published in 1975. Transportation Research Board, 2000, +LJKZD\ &DSDFLW\ 0DQXDO, special report 209, National Research Council, Washington D.C., TRB. Van Zuylen and Viti, 2003. 8QFHUWDLQW\ DQG WKH '\QDPLFV RI 4XHXHV DW 6LJQDOL]HG ,QWHUVHFWLRQV. Proceedings CTS-IFAC conference 2003, 6-8 August, Tokyo. Elsevier, Amsterdam Webster F.V., 1958. 7UDIILF 6LJQDO 6HWWLQJV. Road Research Lab, Technical Paper No. 39. Her Majesty Stationary Office, London, England. VLJQDO RSHUDWLRQV IRU OHYHORIVHUYLFH DQDO\VLV
Figure 5: Expected overflow queue and standard deviation in time The standard deviation is also very large, in the example is the double of the expected value thus in these conditions vehicle actuated controls are characterised by uncertain conditions even larger than fixed controls. 6. CONCLUSIONS Although vehicle actuated controls are widely and successfully adopted in practice there is still little knowledge of how these systems work in theory. This paper gives insight in the traffic flow operations at such signals by proposing a methodology that enables the estimation of expectation values and distributions of green times and the eventual overflow queues in time. The method assumes known the probability distribution of arrivals and it computes the probability that a green time extension is assigned to each flow stream. Green times are therefore determined by the flow rate and the headway distribution of vehicles. The probability that a non-zero overflow queue occurs is considered only when maximum green time extension is met, assuming that queues formed during the red phase are completely served if the total servic time needed is smaller than the maximum. This method has two possible applications: 1) it can inspire the development of approximate analytic formulas or 2) it can be used to validate existing ones. According to the results shown in the two-stream example vehicle actuated controls are flexible and efficient in mild conditions of traffic, while it is affected by a large uncertainty as long as one or more flow streams are near capacity.
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REFERENCES Courage K.G., Papapanou P.P., 1977. (VWLPDWLRQ RI GHOD\ DW WUDIILF DFWXDWHG VLJQDOV. Transportation Research Record, 630, pp. 17-21. Cowan R., 1978. $Q LPSURYHG PRGHO IRU VLJQDOL]HG LQWHUVHFWLRQV ZLWK YHKLFOH DFWXDWHG FRQWURO. Journal of Applied Probability, 15, pp. 384-396.
© 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006
Page 483
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Abstract: This paper proposes a method to estimate the d istribution of travel times at actuated signals by computing the probability distribution of green times and the eventual overflow queues, which may occur in saturated conditions. This method enables one to analyse the stochastic and dynamic character of travel times at such signals. Due to the required elaborate computations, this methodology can be applied in two ways: it can be used to derive approximate formulas of expected green times and queue lengths and their standard deviation, or it can be used to validate existing formulas. The results on a toy scenario show that actuated control s work well only when the flows approaching the intersection are small, while a large o verflow queue can occur in oversaturated conditions. &RS\ULJKW,)$& Keywords: Stochastic modelling, actuated controls, trav
1. INTRODUCTION Delays at signalized intersections represent a major contributor to travel times at urban networks. Unfortunately, traffic volumes have in these systems a strong dynamic and stochastic character; therefore an optimal regulation of traffic is very difficult. The dynamic and stochastic nature of traffic flows in a daily scenario in fact reflects itself in the variab ility of delays (Van Zuylen and Viti, 2003). Pretimed controls have become an exception. Most intersections are controlled by equipment that measures the traffic and adapts the traffic control t o the actual traffic. In congested conditions this often results in a fixed time control, because the length of green phases is limited by a maximum. Actuated control is most common at present, but most theory about this control is still based on fixed time control theory, and the effect of this dynamic mechanism is simply approximated by using a multiplicative factor (e.g. TRB, 2000). This is certainly due to the complex character of the process of adaptively controlled traffic. Macroscopic models, which are based on average conditions, do not fully catch this service mechanism, which is based on actual vehicle headways. Microscopic models are at present used most often to analyse this kind of traffic control. Although traffic performance improvements have been reported and the benefits of such control methods have been demonstrated analysing field data, © 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006
el time estimation
it is still difficult to evaluate the delay travellers experience at these systems. The fluctuations of the demand within a cycle are influenced by the way signals are set, which depends on all traffic streams arriving at the intersection. This paper describes a method to compute signal settings and overflow queues in time using a stochastic modelling approach and assuming known the probability distribution of the arrivals. The knowledge of the probability distribution of the phase plans and the eventual residual queue when the intersection is saturated enables the traffic analys t to compute the distribution of delays in time under very general assumptions. 2. PROBLEM STATEMENT The dissemination and the different schedule of the activities in an urban area result in a large variabilit of flows, which can be observed both within a day and in between days. Figure 1 shows a collection of traffic counts in an urban road located in Delft, the Netherlands during an entire day and for 90 days and using data collected from camera observations. The large spread of data gives an idea of how traffic can change from one day to another at the same time and in the same location. Moreover peaks of traffic are frequently observed during the morning and in the afternoon, showing the dependency of the demand with the schedule of the activities.
Page 478
y
As long as the traffic is below the capacity of the signal travel times remain small. When oversaturated conditions occur, travel times increase considerably, and they become rather unpredictable.
Figure 1: Traffic observations from loop detectors Figure 2 shows how variability of traffic is may affect only a small period of the day. Travel times have been computed using camera detections again in Delft, the Netherlands. Travel times in this period have a very large standard deviation.
usually constrained to be within minimum and maximum values, which are mainly determined by the geometry of the intersection. Since the stochastic nature of the arrivals, different headway distributions and different flow rates can be observed from cycle to cycle. Therefore, the assigned green times and the delay incurred are variable too. The assigned green times are thus variable according to the variability of queues forming during the red phase and to the variability of vehicle headways. If one considers that the queue formed during the red phase depends on the number of vehicles arriving during that phase and to the length of the red phase itself, which depends on the green time extensions given to all conflicting streams, the mathematical formulation of the expected travel time experienced by the travellers is quite complex, as it is shown in this paper. Rigorous mathematical solutions have been proposed in the past, but they are characterised by limiting assumptions (e.g. average demand and service rates). Other models have been derived from heuristics to better represent the results of field data but they cannot give clear insight of the real effects of this service type on the propagation of flows across the network. The problem becomes more complex if one considers that this process has a strong dynamic character when overflow queues occur, i.e. the number of vehicles arriving during the red phase exceeds the maximum number allowed by the maximum green extension. This overflow queue creates a delay, which propagates in time and affects also vehicles arriving at later periods. This contribution has not been considered in any past study. Immediate consequence of this approximation is that delays in periods that follow congested cycles are underestimated. This paper proposes a stochastic model formulation, which enables one to compute the probability distribution of expected greens and overflow queues in time and their effect on later green time extensions. Knowing the distribution of green splits in a cycle and the overflow queue one can compute the distribution of delays by applying standard delay formulas like the one proposed by the Highway Capacity Manual (TRB, 2000). 3. STATE OF THE ART
Figure 2: Travel times computed using cameras One possible way to relieve this unbalanced use of the roads is to adapt their capacity to the actual demand by e.g. dynamic control schemes. Dynamic signal phase plans are designed to regulate traffic giving attention to this dynamic and stochastic propagation of flows. Green and cycle times are adapted on the basis of the actual traffic states in order to improve the efficiency of the signal. Vehicle actuated control is determined by the headway measurement of vehicles in time. Green time is given to a traffic stream in order to serve all vehicles waiting at the signal during the previous red phase. The signal switches as long as the detector records a headway distance, which is longer than a pre-defined threshold value. This green time is © 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006
The first model of delay with a simple traffic actuated signal was proposed by Morris and Pak-Poy (1967) with an application on a signal coordinating two oneway streets. For each traffic condition they computed the optimal vehicle interval to minimize the total delay. Newell (1969) studied the same problem under the assumption of stationary arrival process in undersaturated conditions but near capacity. He developed approximate formulas of delays and expectation values of green and red extensions in function of the average arrivals at fully-actuated signals. Dunne (1967) proposed a delay model derived by assuming a binomial arrival process and green times determined by the queue length detected. Cowan (1978) used bunched exponential distribution
Page 479
for the arrivals to compute expected green, red times and delay. Webster’s (1958) delay formula was adapted by Courage and Papapanou (1977) to compute the average cycle length with actuated signals. Optimal cycle lengths were used to compute pre-timed controls using the formula:
&2
=
1.5 ⋅ / + 5
1 − ∑ \L L
where / is the total lost time in the cycle and \ = T / V is the volume to saturation flow ratio. Fully-actuated signals are instead computed considering average cycle length. The HCM 2000 (TRB, 2000) considers a discount factor of 0.85 to multiply to the uniform delay component. The manual also gives an approximate expression of the average signal cycle: L
L
L
({&} =
[F ⋅ /
[F − ∑ \L L
where [F = TLF / VL is the critical volume to capacity ratio (which in the HCM 2000 is set to 0.95). The effective green is given by:
({ J } = L
\ ⋅ ({&} [ L
L
Lin and Mazdeysa (1983) proposed an extension of the Webster’s delay formula by including two extra parameters, .1 and . 2 , in the form: 3600 ⋅ . ⋅ [ & (1 − . ⋅ J / & ) + 2 ⋅ [1 − . ⋅ ( J / & ) ⋅ . ⋅ [ ] 2 ⋅ T ⋅ (1 − . ⋅ [ ) 2
G
= 0.9
2
1
1
2
2
2
The two heuristic parameters reflect the sensitivity of actuated signals to the degree of saturation, and they have been calibrated with simulation. Li et al. (1994) proposed to include a discount factor N in the Australian Capacity Manual time-dependent overflow delay model to account for fully actuated signals:
8⋅ N ⋅ [
F7
G (7 ) = 900 ⋅ 7 ⋅ [ − 1 + ( [ − 1) 2 +
very important. This paper contributes to the modelling of these characteristics. 4. METHODOLOGY Green phases at actuated controls are primarily determined by the headway in the arrivals. The basic assumption in the estimation of green times is to extend the green time length until the gap between a vehicle and its follower in the flow stream is larger than a pre-defined gap (XQLWH[WHQVLRQ). Let τ be the unit extension, assumed known and constant. This unit extension is determined by the length of the passage loop and the speed at which vehicles drive on it. The length of the loop detector is usually fixed in such a way that the queue formed during the red phase is completely served, avoiding that the signal changes while the queue is not yet fully served or unless the maximum green extension is met. Let assume that all vehicles in queue drive across the loop with headways shorter than τ . If 4 U (τ ) is the total number of vehicles queuing up and waiting at the signal at the cycle τ during the red phase U (τ ) the expected green time given to serve these vehicles is given by: Jmin U J (τ ) = 4U (τ ) / V Jmax
© 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006
U
min
U
min
max
(1)
U
max
where V is the saturation flow, assumed in this paper known and constant (uniform service times). The queue length 4 U (τ ) is determined by the number of vehicles arriving during the red phase. If one assumes uniform arrivals, the expected green time is computed with formula (1) using simply the average flow rate. The probability distribution of green times at one traffic stream L is computed knowing the distribution of arrivals at stream L and the distribution red times at the previous phase, which depends on the distribution of arrivals at all conflicting streams1. If D (τ ) is the arrival rate (in vehicles per second), and U (τ ) is the red time at the previous cycle one can compute the probability of a certain number of vehicles N queuing up during the red phase of a certain length ρ as: L
L
Umax
3[4 (τ ) = N ] = ∫ 3[4 (τ ) = N | U (τ ) = ρ ] = U
This discount factor has been later adopted also in the Highway Capacity Manual formula. Although several – more or less heuristic - delay formulas have been proposed to account for the effects of traffic actuated control, the complex behaviour of such systems needs still to be fully explored. This paper contributes to the state of the art of delay estimation at actuated signals by analysing the dynamic and stochastic character of delays. This can be valuable information for control design, realtime management systems and ATIS systems. In these applications the expected delay is an important characteristic, but the variability of the delay is a lso
4 (τ ) / V ≤ J if J ≤ 4 (τ ) / V ≤ J if 4 (τ ) / V ≤ J
if
U
L
L
(2)
L
ρ = Umin
=
Umax
∫ (3(D (τ ) ⋅ ρ = N ) ⋅ 3 (U (τ ) = ρ ) )G ρ L
L
ρ = Umin
The formula is obtained by assuming that the arrival rate and the red time are independent stochastic variables. The formula to compute the probability of red time length is given later in this section.
1
In practice the computation previously determines the critical paths, which are the all conflicting streams wi largest demand. Page 480
th the
The probability of a green time J (W ) needed to clear the queue at the end of the red phase to be a value O is given by the following condition: U
3 ( J (τ ) = O ) = U
∑ 3(4 (τ ) = N )
(3)
U
L
N
/ V =O
While clearing the queue formed during the red phase, other vehicles may join the queue. Formula (4) computes the probability for these vehicles similarly to formula (2). The probability of red time is simply replaced by the probability of green time computed with formula (3)2:
3(4LJ (τ ) = N ) = 3[4LJ (τ ) = N | JLU (τ )] = Jmax
∫
ς = Jmin
(4)
3 (D (τ ) = N | ς ) ⋅ 3 (JLU (τ ) = ς ) Gς L
Accordingly, formula (3) is adapted to compute this extra queue with the following formula (5):
3 ( JLJ (τ ) = O ) =
∑ 3 (4
J L
N / V =O
(τ ) = N )
(5)
The probability of green time due to all vehicles in queue J 4 is thus given by the following relationship:
3L ( JL4 (τ )
= P) =
∑
N +O =P
3L ( JLU (τ ) = N ) ⋅ 3L ( JLJ (τ ) = O )) (6)
The total queue to dissipate within this time is computed accordingly:
3L (4L (τ ) = P) =
∑ 3 (4 (W) = N ) ⋅ 3 (4 (W ) = O))
N +O =P
L
U L
L
J L
(7) Apart from the green time assigned to clear this queue, one should take into account that the green phase is extended as long as a vehicle passes the detector within the unit extension, which it can happen also when the queue has been fully served but vehicles are still arriving with short distances. To account for this extra-time one can use the distribution of arrival headways instead of the number of vehicles arriving within the time period. The Poisson distribution can describe for example the probability of observing Q arrivals in a period from 0 to W with the following expression:
3Q (W ) =
(λ ⋅ W )
Q!
Q
⋅H
− λ ⋅W
(8)
This equation gives information about how the probability is distributed over a time interval in terms of number of vehicles. In a sequence of Q arrivals one can observe vehicles passing with a random headway distance. If no arrivals are observed within a time τ < τ the signal will switch to yellow. This probability is given by the following formula (9): 2
Formula (4) is an approximation of the actual number of vehicles queuing up, since other vehicles may arrive at the J back of the queue during the clearing of 4L (τ ) . These higher order terms are neglected in this paper. © 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006
30 (τ ) = H− λ ⋅τ
(9)
If one computes the probability distribution of a Q sequence of vehicles at times 0 < W1 < W2 < ... < WQ = W the probability of observing this sequence with W2 − W1 < τ , W3 − W2 < τ , etc. is given by the following formula (10): Qmax
3 (WH[WQ = W ) = ∑ 3(W1 < W2 < ...WQ = W ) ⋅ 3 (Q, W ) Q=0
V.W.
W1 < τ , W2 − W1 < τ ,..., WQ − WQ −1 < τ (10)
Even if green time extension is needed, one should compute the probability that this extension time is actually available. The probability of a certain number of seconds available for eventual green time extension before the maximum green extension is easily derived from the probability of green time due to the formed queue, formula (6). The probability of having an extension of exactly W seconds is given by the following relationship:
3L ( JLH (τ ) = W ) = ∑ 3 (WH[WQ = W ) ⋅ 3 ( J max − JL4 ≥ W )
(11)
The average total green time is finally given by computing the joint probability of guaranteed green time and extended green time J WRW (W ) = J 4 (W ) + J H (W ) . The probability of a total green time J WRW is thus given by the following relationship, analogous to formula (6):
3L ( JLWRW (W ) = P) =
3L ( JL4 (W ) = N ) ⋅ 3L ( JLH (W ) = O )) ∑ N O P + =
(12)
Green times are computed using this method for each flow stream of the intersection and the total cycle is computed by summing up all green times at each conflicting stream, together with the corresponding lost times. Knowing then the green times and the cycle length one can finally compute the uniform delay component using for example the Highway Capacity Manual formula. If the green time assigned during the previous cycle is smaller than the maximum green extension no overflow queue is supposed to be present and only the arrivals during the red phase should be served. Overflow queues are likely to occur only when the intersection is oversaturated and the maximum green extension is met. If the signal assigns the maximum green extension, one should calculate the eventual overflow queue, which will have to wait for the next green phase. Overflow queue occurs then only when J 4 = J max . In this case the overflow queue is computed by formula:
42L (W) = 4L (W ) − J
max
⋅ VL
(13)
with V the assumed saturation flow of the road section L . The corresponding probability is computed by the following formula: L
Page 481
3 (42L (W ) = T ) =
∑ 3(4L (W ) = N )
N − Jmax ⋅ V = T
(14)
L
Since an eventual overflow queue should be cleared in the next green phase, formula (3) should also consider that, apart from the arrivals, also the eventual overflow queue should be served. Formula (3) is thus reformulated as:
3 ( JL4 (W ) = O ) =
3(4L (W ) = N ) ⋅ 3 (42L (W ) = T ) ∑ N T V O
( + )/ =
(15) Last step to compute formula (3) is to derive the probability distribution of the red times at the previous cycle. The red time is determined by the sum of all the green times given to the conflicting strea and the total time lost 7/of the signal:
U (W ) = ∑ J (W − 1) + 7/
ms
WRW
M
≠L
M
(16)
The corresponding probability of a red time to be a certain value U (W ) = V is thus computed with the following formula:
3 (U (W ) = V) = 3 (∑ J (W − 1) + 7/ =V) WRW
M
≠L
M
(17)
Assumption should be made on the initial red phase probability and on the initial overflow queue in order to compute the distribution in time.
Figure 3: Expected green time in the example The expectation value is the minimum green time only when both flows are zero. The green time is sensitive to both the increase of each flow stream, but it has a steeper increase if increasing D$% of one unit. The expected value is equal to the maximum green time for a large overall demand. Overflow queues are more likely to occur in these conditions, as figure 6 shows, while its value is nearly zero for low values of the demand, especially if D$% is small. When one of the two streams has a very large demand overflow queue is likely to reach the maximum value assumed in the example (70 vehicles) within the total period of analysis.
5. NUMERICAL EXAMPLE In the following of the section the vehicle actuated control is modelled with the probabilistic approach in a simple case of two traffic streams crossing an intersection.
Let D$% and D&' be the average flows (expressed in vehicles per second) and V $% = V&' = 1800[YHK / KU ] . Let assign the first green time to the direction $% and an initial red time U $% (0) = 30V . Initial overflow
queue is zero 42$% (0) = 0 . The total lost time of the intersection is assumed 7/ = 12V and the unit extension is W = 3V . Finally minimum and maximum green time extensions are respectively set to Jmin = 10 V and J max = 60V . Let assume the arrival rate distributed as Poisson and let the process have a stationary arrival rate for a period of 30 minutes. Using formulas (2) - (15) one can compute at each time step the distribution of the green times and the eventual overflow queue. Expected green times and overflow queues have been computed for each couple (D $% , D&' ) , each ranging from a value from 0.1 to 0.6 of the saturation flow. Figure 5 shows the expectation value as function of the couple (D $% , D&' ) at the end of the period of stationary conditions.
© 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006
Figure 4: Expected overflow queue in the example The overflow queue in conditions of moderate saturation starts assuming a strong dynamic character. For example if D$% = 0.5 ⋅ V $% and D&' = 0.2 ⋅ V&' the expected value of the overflow queue is quite large, as figure 7 shows.
Page 482
Dunne M.C., 1967. LQWHUVHFWLRQ
7UDIILF 'HOD\ DW D VLJQDOL]HG ZLWK
%LQRPLDO
Transportation Science, 1, pp. 24-31. Li J., Rouphail N. and Akcelik R., 1994. GHOD\
HVWLPDWLRQ
IRU
LQWHUVHFWLRQV
.
DUULYDOV
2YHUIORZ ZLWK
IXOO\
. Presented at the 73rd TRB Annual Meeting, Washington D.C. Lin F.-B., Mazdeysa F., 1983. (VWLPDWLQJ DYHUDJH DFWXDWHGVLJQDOFRQWURO
F\FOHOHQJWKVDQGJUHHQLQWHUYDOVRIVHPLDFWXDWHG
. Transportation Research Record 905, pp. 33-37. Morris R.W.T., Pak-Poy P.G., 1967. ,QWHUVHFWLRQ FRQWURO E\ YHKLFOH DFWXDWHG VLJQDOV. Traffic Engineering and Control, No. 10, pp. 288-293. Newell G. F., 1969. 3URSHUWLHV RI 9HKLFOH $FWXDWHG 6LJQDOV , 2QH:D\ 6WUHHWV. Transportation Science, 3, pp. 30-52 Rouphail, N., A. Tarko, J. Li (2000). 7UDIILF IORZ DW VLJQDOL]HG LQWHUVHFWLRQV. In: Revised Monograph on Traffic Flow Theory, update and expansion of the Transportation Research Board (TRB) Special Report 165, "Traffic Flow Theory," published in 1975. Transportation Research Board, 2000, +LJKZD\ &DSDFLW\ 0DQXDO, special report 209, National Research Council, Washington D.C., TRB. Van Zuylen and Viti, 2003. 8QFHUWDLQW\ DQG WKH '\QDPLFV RI 4XHXHV DW 6LJQDOL]HG ,QWHUVHFWLRQV. Proceedings CTS-IFAC conference 2003, 6-8 August, Tokyo. Elsevier, Amsterdam Webster F.V., 1958. 7UDIILF 6LJQDO 6HWWLQJV. Road Research Lab, Technical Paper No. 39. Her Majesty Stationary Office, London, England. VLJQDO RSHUDWLRQV IRU OHYHORIVHUYLFH DQDO\VLV
Figure 5: Expected overflow queue and standard deviation in time The standard deviation is also very large, in the example is the double of the expected value thus in these conditions vehicle actuated controls are characterised by uncertain conditions even larger than fixed controls. 6. CONCLUSIONS Although vehicle actuated controls are widely and successfully adopted in practice there is still little knowledge of how these systems work in theory. This paper gives insight in the traffic flow operations at such signals by proposing a methodology that enables the estimation of expectation values and distributions of green times and the eventual overflow queues in time. The method assumes known the probability distribution of arrivals and it computes the probability that a green time extension is assigned to each flow stream. Green times are therefore determined by the flow rate and the headway distribution of vehicles. The probability that a non-zero overflow queue occurs is considered only when maximum green time extension is met, assuming that queues formed during the red phase are completely served if the total servic time needed is smaller than the maximum. This method has two possible applications: 1) it can inspire the development of approximate analytic formulas or 2) it can be used to validate existing ones. According to the results shown in the two-stream example vehicle actuated controls are flexible and efficient in mild conditions of traffic, while it is affected by a large uncertainty as long as one or more flow streams are near capacity.
e
REFERENCES Courage K.G., Papapanou P.P., 1977. (VWLPDWLRQ RI GHOD\ DW WUDIILF DFWXDWHG VLJQDOV. Transportation Research Record, 630, pp. 17-21. Cowan R., 1978. $Q LPSURYHG PRGHO IRU VLJQDOL]HG LQWHUVHFWLRQV ZLWK YHKLFOH DFWXDWHG FRQWURO. Journal of Applied Probability, 15, pp. 384-396.
© 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006
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