A probabilistic model for actuated traffic signals
Francesco Viti, Ph.D.
Delft University of Technology
Section Traffic and Spatial Planning
Stevinweg 1, 2600 GA Delft. The Netherlands
f.viti@tudelft.nl
Minwei Li
Delft University of Technology
Section Traffic and Spatial Planning
Stevinweg 1, 2600 GA Delft. The Netherlands
m.li@tudelft.nl
Prof. Dr. Henk J. van Zuylen
Delft University of Technology
Section Traffic and Spatial Planning
Stevinweg 1, 2600 GA Delft. The Netherlands
h.j.vanzuylen@tudelft.nl

Corresponding Author: Tel: +31152784912 - Fax: +31152783179
1
A probabilistic model for actuated traffic signals
Dr. Francesco Viti, Minwei Li, Prof. Dr. Henk J. van Zuylen
Delft Univ. of Technology, Section Traffic and Spatial Planning
Stevinweg 1, 2600 GA Delft. The Netherlands
f.viti@tudelft.nl
Abstract
Vehicle actuated controls are designed to adapt each green phase according to the
variability of the arrivals. Therefore, waiting times for the drivers can be variable too
depending on these time-varying signal settings. This paper presents a new modeling
approach for queues lengths and vehicle actuated control settings based on
probabilistic theory. This methodology allows the computation of the dynamic and
the stochastic behavior of such variables with any arrival distribution.
The results of the probabilistic approach have been compared to the results of
repeated microscopic simulations, showing good consistency between the two
approaches. The smaller number of parameters and computing times than the latter
makes the model suitable for planning and design problems, as well as model-based
travel time predictions.

Corresponding Author: Tel: +31152784912 - Fax: +31152783179
2
1.
Introduction
Traffic control at intersections is done by means of traffic lights that are most of the
times operated automatically using a cyclic sequence of green, amber and red lights.
The timing of the green and red duration can be fixed and predetermined (fixed time
or pre-timed control). Another alternative is semi-actuated control. The main flow is
interrupted after a green time of at least the minimum green time if the detector on the
side road is activated. The green time on the minor road is determined by gap
measurements, i.e. the green phase terminates when the gap between two vehicles is
larger than a certain maximum. Fully actuated control is the mode of operation where
all approaches have detectors and all green phases are controlled by means of detector
information. This paper deals with this last class of signal controls.
The choice of which control type to implement, and the determination of the signal
settings are mainly done with the objective of reducing the delay to the vehicles.
1
Actuated controls work very differently from fixed or pre-phased controls, since
signal settings are not predetermined but they are determined by the headway
distribution of the arrivals at the stop line. The basic mechanism is to extend the
green time reserved to a flow stream until the (time) headway between two vehicles is
larger than a certain threshold. This green time is usually constrained to be within
minimum and maximum values, which are mainly determined by the geometry of the
intersection.
1
Delay is usually defined as the difference between the travel time experienced by a vehicle passing an
intersection and the travel time experienced if the vehicle passes the intersection at cruise speed.
3
Since the stochastic nature of the arrivals, different headway distributions and
different flow rates can be observed from cycle to cycle, leading to different green
time extensions. The assigned green times are therefore variable according to the
arrival time of vehicles at the intersection during the red and green phases. If one
considers that the queue formed during the red phase depends on the number of
vehicles arriving during that phase and on the length of the red phase itself, which
depends on all green time extensions given to the conflicting streams, the relationship
of these variables to the expected delay experienced by one traveler is quite complex.
This paper analytically solves this problem by computing the probability distribution
of effective green and red times, and the eventual overflow queue lengths that can be
observed at vehicle actuated controls with any arrival time distribution. The model
assumes that no overflow queue is observed if the green time is below the maximum
value. The probability of queue lengths at the end of each phase, together with the
assigned green and red times are computed sequentially, since the number of arrivals
of one flow-group during one red phase depends on the length of the red phase itself,
which is determined by the green time given to all other flows of the dominant
conflict group.2
The main contribution of this model is that it can deal with the dynamic and
stochastic character of the arrivals. This effect is particularly important when the
signal operates near the capacity and the green is likely to reach the maximum value.
2
Dominant conflict group is the sequence of conflicting streams, which require the largest cycle time
4
In these situations green times can be extended up to their maximum value because
long overflow queues have occurred previously.
The paper is structured as follows. The next section gives a literature review on
vehicle actuated controls. A description of the probabilistic relationships that connect
arrivals, queues and signal settings, and which form the basis of the probabilistic
model, is given in Section 3. The probabilistic model is later presented in Section 4
and a numerical example follows. Finally comparison with the results of a
commercial microsimulation program and conclusions close this paper.
2.
Literature review
The large number of vehicle actuated control types designed in the past and the
complex architecture of such systems justifies the lack of a general theory of
expectation values of queues and delays with traffic actuated controls (Rouphail,
2000). Garwood (Garwood, 1940) proposed the first model for a one-way street using
probability theory. The model is developed in a similar way as the model presented in
this paper, although it is restricted to Poisson arrivals. The probabilistic approach was
also adopted by Darroch et al. (Darroch, 1964) under the assumption of exponentially
distributed arrival times, and Lehoczky (Lehoczky, 1972), who assumed general
arrival distribution and Markovian dependence among the traffic flow directions. All
these models do not explicitly consider however the time-dependent behavior of these
controls under congested conditions and especially in the presence of overflow
queues from previous cycles.
5
An alternative model was proposed by Morris and Pak-Poy (Morris, 1967) with an
application on a signal coordinating two one-way streets. For each traffic condition
they computed the optimal vehicle interval to minimize the total delay. Newell
(Newell, 1971), Dunne (Dunne, 1967) and Cowan (Cowan, 1978) studied the same
problem under the assumption of respectively uniform, binomial and bunched
exponential distributions for the arrival process and developed approximate formulas
of delays and expectation values of green and red time extensions.
Courage and Papapanou (Courage, 1977) applied heuristics to adapt the well known
Webster’s (Webster, 1958) delay formula for fixed time. Optimal cycle lengths were
used to compute the cycle time in undersaturated conditions using the formula:
tC* 
1.5  L  5
1   yi
(1)
i
where L is the lost time in the cycle and yi is the volume to saturation flow ratio.
The Highway Capacity Manual (TRB, 2000) considers simply a discount factor k in
between 0.04 and 0.5 in the uniform delay component of the fixed control delay
formula to account for vehicle actuated mechanism. The manual also gives an
approximate expression of the average signal cycle:
tC * 
xc  L
xc   yi
(2)
i
where xc is the critical volume-to-capacity ratio ( xc =0.95 in the HCM200). The
effective green is given by considering the equi-saturation method as follows:
6
tg* ,i 
yi *
t
xi C
(3)
where xi and yi are respectively the volume-to-capacity and the volume-tosaturation flow ratios for the approach i .
Lin and Mazdeysa (Lin, 1983) proposed an alternative extension of the Webster’s
delay formula, using the above expressions for the green and cycle times, by
including two extra parameters, K1 and K 2 , in the form:

tC* (1  K1  t g* / tC* )2
3600  K 2  x 2 

W  0.9 


*
*
 2 [1  K1  (t g / tC )  K 2  x] 2  q  (1  K 2  x) 


(4)
The above models were all developed under the assumption of undersaturated
conditions. To overcome this limitation, Li et al. (Li, 1994) proposed to include a
parameter k in the Australian Capacity Manual time-dependent overflow delay
model to account for fully actuated signals:

W (T )  900  T   x  1  ( x  1)2 

8 k  x 

cT 
(5)
Using this formulation, Akcelik et al. (Akcelik, 1997) developed simple heuristic
formulas to estimate average cycle times and green times for various types of
actuated signal control. Analytical models for delay, queue length, clearance time etc.
were also developed and compared with microsimulation.
Although several delay formulas have been proposed the complex behavior of traffic
actuated control systems is not yet fully explored in a dynamic environment. It is in
fact questionable whether the model for fixed control could simply be adapted with a
7
discount factor, as it was done for the Webster’s formula and the HCM2000, since the
service mechanisms are so different. Moreover, these models compute expectation
values, but, as already said in the introduction, queues, waiting times and signal
settings are stochastic and therefore a model that enables one to evaluate their
distribution should be very important too.
3.
The probabilistic nature of vehicle actuated controls
Actuated control phase plans are in general determined by the headway distribution
of the arrivals at the intersection. The basic mechanism is to extend the green phase
until no vehicle is detected by the detection point within a certain time period. This
green time is usually constrained to be within minimum and maximum values, which
are mainly determined by the geometry of the intersection. Figure 1 explains how
vehicle actuated signals work. The figure shows two loop detectors placed underneath
the road pavement and at some distance from the stop line.
[Figure 1]
The closest loop to the traffic signal (presence loop) detects whether at least one
vehicle arrives during one cycle length. If no vehicles arrive the signal will never turn
green if no guaranteed green time is designed. In this paper we consider a minimum
guaranteed green time for each flow stream. The second loop (passage loop) monitors
the time headway between two sequential vehicles. If the loop is not occupied a time
counts and if no vehicles pass within this time (called here unit extension) the signal
switches to amber. The length of the detector determines the minimum headway
8
below which green time is extended. The mechanism is explained more in detail with
Figure 2.
[Figure 2]
Figure 2 (a) represents the case of a queue located over the passage detector. This can
be observed during the red phase, when the queue builds up, and also during the
green phase, when vehicles are being served while other vehicles reach the back of
the queue in the meantime. As long as at least one vehicle is detected by the passage
loop the green time is extended, unless this time is in between a minimum and a
maximum value. Figure 2 (b) and (c) are two cases in which green time is still
extended because vehicles arrive in short times. Figure 2 (b) represents the case of
only one vehicle detected within the unit extension, while Figure 2 (c) shows two
vehicles passing the detector within the same unit extension. Finally Figure 2 (d)
shows when the distance between two consecutive vehicles is larger than the length
of the detector. When the gap between these vehicles is large enough (i.e. larger than
the unit extension) the signal will turn to amber. Other vehicle actuated mechanisms
consider a fixed time extension for each vehicle counted, independently on their time
headway. This mechanism is therefore simpler to be formulated since the green time
extension becomes only a function of the number of arrivals within a cycle.
The queue length shown in Figure 2 (a) can be variable for several reasons.
Therefore, the queue length at the end of this red phase will be variable too and it will
imply different green times. Furthermore the red phase itself can vary, since it
depends on how much green time is given to all conflicting streams.
9
While all vehicles queued up during the red phase are being served since the sequent
green phase starts, more vehicles can join the queue and extend further the green
phase. This number will consequently be dependent on the number of vehicles
queued up during the red phase. If the maximum green time extension is not yet
reached by the above vehicles queued up, the signal will remain green until the (time)
headway between two consecutive vehicles is larger than a certain threshold value
(unit extension), which will also depend on the arrival distribution.
4.
Probabilistic model formulation
The system as it is described in Section 3 applies to single service points. Moreover,
the model described in this section is developed under the assumption of
deterministic service rate, i.e. vehicles will leave in regular time gaps as soon as a
green phase starts. In reality these factors will certainly influence the variability of
this system. A multilane approach, or an intersection where priority for public
transport vehicles is considered, will have different results than the ones presented
later in this paper. A stochastic service rate will also affect the mechanism. We leave
these issues for later studies, since the main scope of this paper is to directly relate the
distribution of the arrivals to the distribution of signal settings and queue lengths.
This model has been recently developed by the first author and presented in earlier
studies (Viti 2006a, Viti 2006b). In this paper we compare it with microscopic
simulations to test its consistency with a more detailed modeling approach. This
comparison is done in Section 5.
10
Since the model requires a sequential computation of the variables green, red and
queue lengths, an initial known green or red time queue length distributions are
required. We suppose an initial red phase known (e.g. set to the minimum value) and
zero queue at the start of this red phase.
Computation of green time extension due to vehicles waiting in the queue
Let  be the unit extension, assumed known and constant. Let Q r ( ) be the total
number of vehicles queuing up and waiting at the signal at the cycle  during the red
phase r ( ) . If we assume constant service rate s for each vehicle the expected green
time given to serve these vehicles is:
g
 min
g r ( )  Qr ( ) / s

 gmax
if Qr ( ) / s  gmin
if gmin  Q r ( ) / s  g max
(6)
if Qr ( ) / s  gmax
The queue length Q r ( ) is determined by the number of vehicles arriving only during
the red phase. If one assumes uniform time headway, the expected green time is
computed with the above formula by using simply the average flow rate. The
probability distribution of green times at flow stream i is computed knowing the
probability distribution of arrivals at i and the distribution of red times at the
previous phase, which depends on the distribution of arrivals at all other streams.
If ai ( ) is the arrival rate (in vehicles per second), and ri ( ) is the red time at the
previous cycle one can compute the probability of a certain number of vehicles k
queuing up during the red phase of length  as:
11
Pr(Qir ( )  k ) 
rmax
 Pr(k |  )Pr(  ) d  
  rmin

rmax

  rmin
(7)
 Pr(ai ( )    k )  Pr(ri ( )   )  d 
Where Pr(.) means the probability distribution assumed for a variable. The formula is
obtained by assuming that the arrival rate and the red time are independent stochastic
variables. The integrals are assumed computed within the maximum, rmax , and
minimum red time, rmin .
The probability of a green time g r (t ) needed to clear the queue at the end of the red
phase to be a value l is given by the following condition:
Pr( gir ( )  l )  Pr(Qr ( )  s  l )
(8)
This value does not yet consider eventual vehicles arriving during the green phase
and while the formed queue is served. While clearing the queue formed during the red
phase, other vehicles may in fact join the queue. The following formula computes the
probability for these vehicles. In this formula the probability of red time used to
calculate the queue length in the red phase is simply replaced by the probability of
green time needed to clear this queue:
Pr(Qig ( )  k ) 
gmax
Pr(ai ( )  k |  )  Pr( gir ( )   )   d


 g
(9)
min
Accordingly, green time is adapted to compute this extra queue with the following
formula:
Pr( gig ( )  l )  Pr(Qig ( )  s  l )
(10)
12
Formulae (9)-(10) are respectively lower bounds of the total number of vehicles
arriving during the green phase and of the green time extension due to these vehicles.
While green is extended other vehicles might join the queue during this period,
extending the green phase further. This process might be repeated until the effects on
the total green and the number of vehicles becomes negligible. The probability of
green time due to all vehicles in queue g Q (assuming so far an empty signal at the
start of the red phase) is thus given by the following relationship:
Pri ( giQ ( )  m) 
Pri ( gir ( )  k )  Pri ( gig ( )  l ))

k l  m
(11)
The total queue to dissipate within this time is computed accordingly.
Green time extension due to vehicles passing with short arrival headways
Apart from the green time assigned to clear the queue, one should take into account
that the green phase is extended as long as vehicles pass the detector within the unit
extension, which it can happen also when the queue has been fully served but
vehicles are still arriving in short headways. 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. A Poisson distribution can describe for example the
probability of observing n arrivals (with average rate a ) in a period t with the
following expression:
Prn (t ) 
(a  t )n  at
e
n!
(12)
In a sequence of n arrivals one can observe vehicles passing with random time
headways. The time headway distribution is therefore the probability of observing 0
13
vehicles within t . For Poisson arrivals this distribution is exponential. If one
computes the probability distribution of a sequence of n vehicles at times
0  t1  t2  ...  tn  t the probability of observing this sequence with t2  t1   ,
t3  t2   , etc. while tn  tn1   is given by the following formula:
Pr(textn  t ) 
s.t.
nmax
 Pr(t1  t2  ...tn  t)  Pr(n, t)
n 0
t1   ,
t2  t1   ,
...
tn  tn1   ,
tn1  tn  
(13)
For Poisson distributed arrivals this formulation can be simplified after some easy
manipulations:
N
Pr(textn  t )  (a  )  ea   n  Pr(n, t )  a 2  (  t )  ea
(14)
n 0
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
(11)). The probability of having an extension of exactly t seconds (expressed as an
integer value) is therefore given by the following relationship:
Pi ( gie ( )  t )   P(textn  t )  P( gmax  giQ ( )  t )
(15)
The probability of a total green time g tot ( ) is finally given by the following
relationship:
14
Pr( gitot ( )  m) 
Pr( giQ ( )  k )  Pr( gie ( )  l))

k l  m
(16)
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.
Computation of the overflow queue length at the end of the green phase
Overflow queues are assumed to occur only when the intersection is oversaturated
and the maximum green extension is met. If the signal phase reaches the maximum
green extension, one should calculate the eventual overflow queue, which will have
to wait for the next green phase. Overflow queue occurs therefore only if
g Q ( )  g max . In this case the overflow queue is given by the simple formula:
QOi ( )  Qi ( )  gmax  s
(17)
The corresponding probability is computed by the following formula:
P(QOi ( )  q) 

[k  gmax s ]q
P(Qi ( )  k )
(18)
The square brackets in the summation limit are used to specify an integer value.
The effect of overflow queues on the following green times
Since an eventual overflow queue should be cleared in the next green phase, the
computation of the queue length at the end of the red phase should also consider that.
Formulas (7) and (8) should therefore be modified to account for this queue. The new
formulas for the queue length distribution at the end of the red phase that includes the
overflow queue carried over the previous green phase is the following:
15
P(Qir Qo ( )  m) 
P(Qir ( )  k )  P(QOi ( )  q)

( k  q )m
(19)
Computation of red time probability
Last step to compute the expected green time and queues is to derive the probability
distribution of red times at the previous cycle. The red time is determined by the sum
of all the green times given to the conflicting streams and the total time lost of the
signal:
r ( )   g tot
j ( 1)  TL
j i
(20)
The corresponding probability of a red time to be a certain value r ( )   is thus
computed with the following formula:
P(r ( )   )  P( g tot
j ( 1)  TL  )
j i
(21)
Numerical example
In the following of the section the vehicle actuated control is modeled with the
probabilistic approach in a simple case of two traffic streams crossing an intersection
as the simple scheme in Figure 3 displays.
[Figure 3]
Let a AB and aCD be the average flows (expressed in vehicles per second) and
sAB  sCD  1800[veh / hr ] . Let assign the first green time to the direction AB and an
initial red time r AB (0)  30s . Initial overflow queue is zero QOAB (0)  0 . The total lost
time of the intersection is assumed TL  12s and the unit extension is t  3s . Finally
16
minimum and maximum green times are respectively set to gmin  10s and
g max  60s . Let assume the arrival rate distributed as Poisson and let the process have
a stationary arrival rate for a period of 30 minutes.
[Figure 4]
Using the model proposed in this paper one can compute at each time step the
probability distribution of green times and the eventual overflow queue in time.
Expected green times and overflow queues have been computed for each couple
(aAB , aCD ) , each ranging from a value from 0.1 to 0.6 of the saturation flow.
Therefore the model is applicable also to oversaturated conditions. Figure 4 shows
the expectation value as function of the couple (aAB , aCD ) at the end of the period of
stationary conditions.
[Figure 5]
The expectation value is the minimum (or guaranteed) 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 a AB 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 5 shows,
while its value is nearly zero for low values of the demand, especially if a AB is small.
When one of the two streams has a very large demand the overflow queue is likely to
reach the maximum value assumed in the example (70 vehicles) within the total
period of analysis.
[Figure 6]
17
The overflow queue in conditions of moderate saturation starts assuming a strong
dynamic character. Figure 6 shows the example of aAB  0.5 sAB and aCD  0.2  sCD .
It should be pointed out that the sequence of arrival probabilities has been chosen
stationary for illustrative reason. The probabilistic model accepts any probability
distribution as input3, also non-stationary.
5.
Comparison with microsimulation
To test the correctness of the model presented in this paper we compare its results
with a commercial microsimulation software package, VISSIM (PTV, 2003). A
simple intersection has been simulated in the program similarly to the simple
schematization drawn in Figure 3. To reserve sufficient space for eventual overflow
queues to grow a buffer of 2 kilometers was reserved upstream the signal control.
Both sections are single lane and only passenger car traffic was loaded, in order to
resemble as close as possible the assumptions of the model developed and presented
numerically with the example in Section 4.
The program has been run 100 times for a given combination for the demand
(aAB , aCD ) with different random seeds in order to obtain a statistically valid
representation of the variability of traffic. The simulation program has the option to
load the input demand using a Poisson distribution according to the model presented.
It has to be pointed out that while the input demand in the system can be controlled,
the arrivals at the intersection should not be distributed anymore as Poisson, since
3
also empirical, which means it may be applied in hybrid data driven-model based travel time
predictions (Lint 2004)
18
car-following processes modify this distribution when vehicles drive onto the
network.
Service rates are, in the microscopic program, variable and, a priori, unknown (due to
e.g. acceleration/deceleration effects and reaction times). In the probabilistic model
they are assumed fixed and known. Therefore, an average service rate has been
calculated from a first simulation with a total demand condition below the saturation
flow of the intersection. Assumed initially that the saturation flow of the intersection
would be at least 1800 veh/h, a demand of 1600 veh/h was loaded for one hour of
simulation time. The demand was split respectively to 1000 veh/h and 600 veh/h to
the two streams.
A vehicle actuated control program was developed at the Delft University of
Technology and implemented in VISSIM, TRAFCOD (Furth, 1999). This program
follows the same control criteria explained in Section 3. Minimum guaranteed green
times have been fixed to 6 seconds while maximum green time extensions have been
computed according to the equi-saturation criterion.
Figure 7 shows the number of vehicles served within a cycle in relationship with the
sum of green and amber time needed to serve this number of vehicles.
[Figure 7]
The spreading of points is the result of the random service times simulated in
VISSIM. It looks that a linear relationship represents well the relationship between
number of vehicles and total discharge time. Therefore a constant service rate for
each vehicle seems a reasonable assumption in the model presented in Section 4.
19
Based on these results an average service rate was estimated for a sequence of green
and amber. This average value was found to be 0.52 seconds per vehicle, which can
represent a saturation flow of around 1900 veh/h.
Figure 8 gives an idea of the variability of green times, which can be observed with
the same average demand condition, due to the variability of the arrivals.
[Figure 8]
Comparison of the model calibrated on this scenario using only the above average
service rate and the same parameters set in TRAFCOD is satisfactorily in these low
demand conditions. Both expected green times and standard deviation for the two
intersections were found very close to the microsimulation results, as it is shown in
Table 1. Each cell in the table shows the expected green time computed with the
microsimulation program and the probabilistic model (top), and it shows the standard
deviation of green times (bottom). The first scenario was used to calibrate the model.
This result has been used to test the prediction capability of the probabilistic approach
under a different demand condition. A large demand condition, close to the total
capacity of the intersection, was simulated. A total demand of 1800 veh/h was loaded
split respectively into 1000 veh/h and 800 veh/h among the two flow streams.
The probabilistic model was found to have again very similar results in terms of
expected green times, as it is shown in Table 1, Scenario 2. To give a visual example
of the similarity of the results between the microscopic program and the probabilistic
model the cumulative distribution of green times was compared in Figure 9.
[Figure 9]
20
The probabilistic model presented has in conclusion very similar results if compared
to a microsimulation program. However, the model has very small computation times
(a few seconds) while the microscopic program needed at least 10 hours of
computations to generate each scenario.
6.
Conclusions
This paper has presented a new analytical approach to calculate the variability of
traffic at vehicle actuated signals. This model assumes a known distribution of the
arrivals at intersections, or approximated by a known probability distribution function
(e.g. Poisson), and deterministic service rate, expressed in number of seconds that
each vehicle needs to be served. The presented model gives various contributions to
e.g. design and planning problems, or travel time predictions:

It shows that travel times at all approaches of vehicle actuated signals are
interdependent, which means that models, which consider these travel times
independent cannot predict well the traffic at such points;

It gives an estimate of the expectation value of queue lengths during the
various signal phases, and also of green and red times;

By considering the probability of overflow queue to occur also in
undersaturated conditions, it better catches the time dependency of signal
settings to the dynamics of these queues;

It gives an estimate also of the variability of queues and signal settings by
computing the evolution of the probability distribution of these variables from
the variability of the arrivals;
21

It predicts these variables as well as a microscopic simulation program, which
however involves far more parameters and criteria, and incomparable
computation times.
Future research will compare this model under oversaturated conditions and with
non-stationary demand conditions to test the validity of the model in catching the
temporal dynamics of congestion. Moreover, the effect of traffic heterogeneity and
of variable service rates should be further investigated.
22
References
Akcelik, R., Chung, E., Besley, M. (1997) ‘Recent research on actuated signal timing
and performance evaluation and its application in SIDRA 5’. Presented at the 67th
Annual Meeting of the Institution of Transportation Engineers. Boston.
Courage, K.G., Papapanou, P.P. (1977) ‘Estimation of delay at traffic actuated
signals’. Transportation Research Record,. 630. pp. 17-21.
Cowan, R. (1978) ‘An improved model for signalized intersections with vehicle
actuated control’. Journal of Applied Probability, 15, pp. 384-396.
Darroch, J.N., Newell, G.F., Morris, R.W.J. (1964) ‘Queues for a vehicle-actuated
traffic light’. Operations Research, 12, pp. 882-894.
Dunne, M.C. (1967) ‘Traffic Delay at a signalized intersection with Binomial
arrivals’. Transportation Science, 1, pp. 24-31.
Furth, P. Muller, T. (1999) ‘A method for stream based control of actuated traffic
signals’. Presented at the 78th Annual meeting of the Transportation Research Board,
Washington, D.C.
Garwood F., (1940) ‘An application of the theory of probability to the operation of
vehicular-controlled traffic signals’. Supplement of the Royal Statistical Society, 7(1),
pp. 65-77.
Lehoczky, J.P. (1972) ‘Traffic intersection control and zero-switch queues under
conditions of Markov Chain dependence input’. Journal of Applied Probability, 9,
pp. 382-395.
23
Li, J., Rouphail, N. and Akcelik, R. (1994) ‘Overflow delay estimation for
intersections with fully actuated signal control’. Presented at the 73rd TRB Annual
Meeting, Washington, D.C.
Lin, F.-B., Mazdeysa, F. (1983) ‘Estimating average cycle lengths and green intervals
of semi-actuated signal operations for level-of-service analysis’. Transportation
Research Record 905, pp. 33-37
Van Lint, J.W.C. (2004) Reliable travel time prediction for freeways. PhD Thesis,
Delft University of Technology, TRAIL Press, Delft, The Netherlands.
Morris, R.W.T., Pak-Poy, P.G. (1967) ‘Intersection control by vehicle actuated
signals’. Traffic Engineering and Control, 10, pp. 288-293.
Newell, G.F. (1971) Applications of queuing theory. Monographs of Applied
Probability and Statistics, Chapman and Hall.
PTV, Planung Transport Verkehr AG (2003) VISSIM version 3.70 user manual.
Germany.
Rouphail, N., Tarko, A., Li, J. (2000) ‘Traffic flow at signalized intersections’. In
Revised Monograph of Traffic Flow Theory, update and expansion of the
Transportation Research Board (TRB) special report 165, "Traffic Flow Theory",
published in 1975. Ed. Lieu H.
TRB (2000) ‘Highway Capacity Manual’. Special Report n. 209. National Research
Council, Washington D.C.
Webster, F.V. (1958) ‘Traffic signal settings’. Road Research Lab, Her Majesty
Stationary Office. Technical paper No. 39, London.
24
Viti, F., Van Zuylen, H.J. (2006a) ‘A stochastic modeling approach to vehicle
actuated control’. In proceedings of the 11th IFAC Conference, Delft, the
Netherlands.
Viti, F, Van Zuylen, H.J. (2006b) ‘The uncertainty of delays at optimal pre-timed and
vehicle actuated signals’. In proceedings of the 13th EURO Mini Conference, Bari,
Italy.
Viti, F. (2006c) The dynamics and the uncertainty of delays at signals. PhD Thesis,
Delft University of Technology, TRAIL Press, Delft, The Netherlands.
25
List of illustrations
Table 1: Results of comparison between microsimulations and the probabilistic model
Figure 1: Loop detectors before a vehicle actuated signal
Figure 2: Vehicle actuated mechanism
Figure 3: Two conflicting streams example
Figure 4: Expected green time length for stream AB for different demand conditions
Figure 5: Expected overflow queue for stream AB for different demand conditions
Figure 6: Expected overflow queue and standard deviation in time
Figure 7: Relationship between number of vehicles and time required to be served
Figure 8: Number of observed green times from VISSIM simulations
Figure 9: Comparison between cumulative distribution of microsimulation results and
of the probabilistic model
26
Table 1
Scenario 1
1000 - 600 veh/h
Scenario 2
1000 - 800 veh/h
VISSIM
Probabilistic Model
19.54 / 13.87
18.84 / 13.23
7.52 / 5.46
7.95 / 5.4
21.89 / 18.52
21.5 / 18.7
8.06 / 7.26
8.16 / 7.03
27
Passage loop
Presence loop
Figure 1
28
(a)
(b)
(c)
(d)
Figure 2
29
C
A
B
D
Figure 3
30
60
55
50
45
E[green]
40
35
q2=360
q2=648
q2=936
q2=1224
q2=1512
q2=1800
q2=2088
30
25
20
15
10
400
600
800
1000
1200 1400 1600
number of arrivals (veh/h)
1800
2000
2200
Figure 4
31
70
q2=360
q2=648
q2=936
q2=1224
q2=1512
q2=1800
q2=2088
60
50
E[Qo]
40
30
20
10
0
400
600
800
1000
1200 1400 1600 1800
number of arrivals on stream AB (veh/h)
2000
2200
Figure 5
32
10
Expected overflow queue
St. dev. overflow queue
9
8
queue length
7
6
5
4
3
2
1
0
5
10
15
cycle number
20
25
30
Figure 6
33
Figure 7
34
number of observations from microsimulation
500
450
400
350
300
250
200
150
100
50
0
10
15
20
25
30
35
green time
40
45
50
55
Figure 8
35
1
0.9
Probability of green time
0.8
0.7
0.6
0.5
0.4
0.3
0.2
Microsimulation VISSIM
Probabilistic model
0.1
0
10
15
20
25
30
35
green time
40
45
50
55
Figure 9
36