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....................
THE GENERALIZED COMBINATION METHOD
FOR AREA TRAFFIC CONTROL
by
Nathan Gartner and John D.C. Little
OR 020-73
August 1973
Operations Research Center
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
Research supported by KLD Associates, Inc., in connection with Department
of Transportation Contract FL-11-7924.
ABSTRACT
A procedure is described for determining optimal signal settings
in a network including cycle time, splits of green time and
ffsets.
A
number of cycle times are scanned in the process. For each cycle, splits are
determined locally at each intersection according to proportions of conflicting traffic loads, and offsets are optimized by the Generalized
Combination Method.
Total cost
for travelling through the signal-controlled
intersections in the network is evaluated as the sum of two components:
deterministic and stochastic.
The deterministic component is a function of
the offsets throughout the network and generally increases with cycle length.
The stochastic component is dependent on the expected overflow queue on each
link
nd decreases with cycle length.
Optimal settings are determined as an
e(ui ibriumi point of minimum total costs resulting from the combined effect
of the two components.
1.
INTRODUCTION
The primary objectives of an area-wide traffic control system
are to provide smooth flow conditions for all traffic streams through the
area and to reduce the delay, or travel time, incurred by the users of the
system.
are:
The variables of each signal program that affect the traffic flow
cycle time, splits of green time and offsets.
A coordinated traffic
signal network requires a common cycle time for all signals in the network,
or a cycle which is a submultiple of some master cycle.
Tn some cases it is
advantageous to partition the network into subnetworks that may operate with
nonsynchronous cycle times.
The conventional procedure for determining the
control variables is to use a sequential decision process:
time is selected for the network first.
A common cycle
Then the splits at each intersec-
tion are determined according to the proportions of demand/capacity ratios
on conflicting approaches.
Finally, linking of the signals is achieved by an
appropriate method for selection of a fundamental set of offsets throughout
the network.
Experience of researchers and practitioners in the urban traffic
control field has shown that cycle time may well be the one most important control variable in a synchronized traffic signal network (1).
selecting a cycle time can be divided into two classes.
the node approach.
The approaches for
The first class is
Since through capacity increases with cycle length, this
approach is based on analysing the capacity requirements of each intersection
in the network.
The common cycle time is determined according to the require-
ments of the most heavily loaded intersection, i.e., the intersection with
2.
the highest sum of demand/capacity ratios on conflicting signal phases.
procedure that is
A
used for a single intersection, such as Webster's
method(2), is then used to calculate the cycle length.
This approach has
been primarily used in conjunction with offset optimization methods such as
COMt3INATION and TRANSYT(3).
The main deficiency in this approach is that the
interaction of flows in the spatial road network structure of the area is
disregarded.
A formula
devised for an isolated intersection,
assumling randomly distributed arrival times of cars, is not necessarily valid
in a network situation where flows are fed from adjacent intersections.
result is generally a cycle time that is too long,
The
thus causing excessive
delays (see also reference 4).
The second class is the network approach.
In this case an attempt
is made to select a cycle time that, while satisfying the capacity requiremnents at each intersection, is also congruent with the particular network
structure at hand.
Simple examples in this category are the arterial pro-
gression schemes in which a cycle that produces maximal bandwidths is selected
according to distance and speed data (e.g., 5,6,7).
The underlying principle
is that tihe optinmal progression (i.e., offsets between signals), for a given
block-length pattern, is strongly dependent on cycle time.
work this approach is principally used by SIGOP(8,9).
ot
ycle timles are scanned in this method.
In a general net-
A predetermined number
For each cycle, offsets are
optilNized by the OPTMIZ subroutine and performance is evaluated by a coarse
siniulation ot
traffic flow through the network (the VALUAT subroutine).
The
optimal set o cycle and offsets is selected according to the results obtained
by VALUAT.
TRANSYT indicates also the possibility to iterate on cycle time
in conjunction with its hill-cliitbing procedure for offset selection(lO0).
3.
However, computational considerations seem to rule out this possibility in
practice.
Two deficiencies of the network approach in SIGOP are apparent:
first, the offset optimization procedure determines a
local optimum rather than
global optimum and, second, stochastic effects
on link performance are ignored.
These effects do not affect the selection
of offsets at a fixed cycle time, but are of prime importance in evaluating
a range of cycle times.
They become pronounced as a signalized intersection
approaches its capacity and in an optimal procedure would deter the cycle
time froin assuming values close to the minimum.
One typical study(ll) has
shown that the lower bound on cycle time was consistently selected as the
optimal value.
Stochastic effects would have conceivably
shifted the result upwards.
In this paper, network settings (including cycle, splits and
offsets) are determined in conjunction with a rigorous synchronization
procedure (i.e.
a procedure capable of determining the global optimum),the
Generalized Combination Method (GCM).
The Combination Method (CM), which
is an offset optimization procedure applicable to series-parallel networks,
was first introduced by Hillier(12).
works of a general structure(13).
It was then extended by Allsop to net-
The method
was later formulated in terms
of Dynamic Programming and applied in conjunction with a computationally
efficient network partitioning algorithm(14).
The Dynamic Programming
pr(cedure for the general network is presented in this paper as a set of two
GCM network operation rules that are a straightforward generalization of the
CM
ules tfor series-parallel networks.
The procedure is further used as a
tool in determining optimal network settings that take into account costs
attributable to both the deterministic traffic flow model and its associated
stochastic fluctuations.
4.
Traffic Flow Model
In order to illustrate the key features of
the traffic flow process, we consider an idealized model.
The discrete nature
of vehicular movement is disregarded and traffic is thought of as a conThe following assumptions are made:
tiluous fluid.
(a) All cars travel with uniform speed between adjacent intersections.
(b) Traffic flow is saturated, i.e., traffic volume at each intersection equals serving capability.
Let i and j denote two adjacent signalized intersections in the
network, such that cars can travel from i to j along the link connecting them.
Referring to Fig. 1 we define the following parameters:
gj(rj)
- effective green (red) time of signal j
C = gj + rj
- network common cycle time
Sij
- offset time between signals i and j
t.
- travel time from i to j
f j(t)
- instantaneous traffic flow (vehicles
per unit of time)
C
f (t)dt - average traffic flow
C, iJ
1ij
In the case that traffic is assumed to have a periodic arrival pattern
ot rt'cLatirlllar shape as illustrated in Fig. 2(a), it can be easily verified
tlht the rate of delay (delay per unit of time)
on link (i,j), d
is
diF. -.3
1
ij(ij)
Fijf
=Fij(ij -t
ij
Oj)
if t.j - gj <
j
ij
f
if
tij
t
.ij
< t
i<
t
tij ij
+j
rj
(.ij),
4a.
,0
E
FI~~-~
co
I 1
auolsj
ID~i~
4b.
A
l
t*- 9 -~
'-
_
i
,,~~~
-
Fij
0
-
-
~~~I~
t
I
t2
t +C
i
t2 + c
a.
I
t + 2C
t
--
._
F
r
b.
oij
so
TA
-
F
C.
2
iGURE
FIGURE 2
5.
and is similarly periodic with respect to ij (see Fig. 2(b)).
Examination
of Fig. 1 indicates that offset variations can be confined to a single cycle
time by introducing the transformation
0 ij
Thus, 0
Fig.
(ij)mod C
Oj..C and the resulting delay function dj(oij) is illustrated in
(c).
To approximate more closely traffic conditions, taking into
account secondary flows, platoon dispersion etc., more elaborate models can
be used (15,16,17).
An example of an actual traffic flow pattern, that has
been measured directly by detectors on the street is shown in Fig. 3(a).
The
link delay function associated with this pattern is obtained by applying
elementary queueing relationships (Fig. 3(b)).
While the discussion in this
paper is confined primarily to delays, the optimization methods can be used
with a more general link performance function combining costs of delays,
stops, acceleration noise, or other measures of effectiveness, by using
appropriate weighting factors (18,19).
Criterion of Optimization
The objective of the network optimization procedure is to determine
signal settings (i.e., cycle time, splits and offsets) that minimize total
delay.
The total delay in the network, D, is
regarded as a sum of two
components:
D =Dd + D
Tlhe first component Dd , is the delay time resulting from the deterministic
talfic flow model described above:
n
·Dd
z
-
n
i=l j=l
d ij
i (i
5a.
_A
2.0
'-
1.5
>
1.0
z
0
°J 0.5
LiL
0
I
5
10
15
20
25
3(0
ASPECT CYCLE IN 2 SECOND INTERVALS
35
4
I
LiJ
-30
z
o
u
C,
>-
20
LD
Lr
LI
>
0
0
10
20
LINK
30
40
50
OFFSET (SECONDS )
FIGURE 3
6 0
70
6.
where n denotes the number of intersections (nodes) in the network.
if link (i,j) does not exist.
function of offsets only.
dj - 0
For given cycle and splits this delay is a
The second component of delay, Ds, is due to the
stochastic nature of traffic flow.
It is taken to be independent of the
choice of offsets in the network but will be important for evaluating the
best choice for cycle time.
The procedure for optimization consists of scanning a number of
cycle times, usually in 0lsec. intervals not exceeding the range of 40 sec.
to 120 sec. For each cycle time, splits at each node are calculated according to proportions of conflicting traffic streams(2), and offsets throughout
the network are optimized by the Generalized Combination Method.
maximum number of offsets
The
ij that can be assigned independent values in a
network of n nodes is n-l and the links across which they are defined must
constitute a tree pattern, i.e., have no loops(20).
The computational procedure for minimizing Dd with respect to offsets involves the division of offsets into N equal intervals.
It is convenient
to consider the link delaysto be a function of an integer number, say k,
k
,1,.,.,N-1.
(X)mod N
To simplify notation we also adopt the convention
(X)N.
Tile Combination Method
The Combination Method is a technique for determination of offsets
thit minimize delays in series-parallel networks(12).
network reduction
The method applies a
sequence to yield a total delay function for the complete
network that is represented by a single equivalent link(21).
The optimizing
offsets of the network are determined by minimizing this function.
reduction sequence is based on the following two rules:
The
7.
CM 1:
Reduction of parallel links.
Where two or more links occur in
parallel, joining a pair of nodes, the delay functions of the
individual links are added with respect to the same offset to
yield a combined delay function represented by a single link
between the two nodes.
Application of this rule is illustrated in Fig. 4(a).
Given dl2(i)
and d 21(j), the combined delay D 12(i) for the equivalent link is
D 12 (i) = d
CM 2:
2 (i) +
d2 1(N-i)
Reduction of series links.
for
i = O,1,...,N-1.
Wherever a node is connected by
two links with two other nodes, that node is eliminated and
the two links are replaced by a single link.
The equivalent
delay function for this link is computed by minimizing the
total delay for each offset between the extremeties of the
two links.
At each step the procedure involves a search
over all possible offsets between one of the extremal nodes
and the common node and selecting the minimum.
Referring to Fig. 4(b), given d 2(i) and d 23(j) tr;e delay function
for the equivalent link is
D13(k) = i
{d12(i) + d23[(k-i)N]} for k,i = 0,1,...,N-1.
Since the three offsets i,j,k form a closedloop the constraining relationship
amollnrl them in this case is
j = (k-i)mod N'
Tie (;eneral ized Combination Method
This mitethod relieves the
eries-parallel restriction imposed on the
strucLture of' networks by the ordinary Combination Method.
By generalizing
7a.
I
@
ft
DA
ltdCD
LLt
zO
f
®D
d
--
a-~~~.
0
8.
the rules stated in the preceding section it is possible to optimize networks of arbitrary layout
GCM 1:
(subject only to computational considerations).
Combination of partial networks. Delay functions that
pertain to separate parts of a network and depend on
offsets between the same set of nodes are added to produce an equivalent delay function for the combined parts
of the network.
GCM 2:
Elimination of interior nodes.
An equivalent delay function
for a partial network is calculated for.,all offsets betweenits boundary nodes (i.e., the nodes which disconnect it from
the remainder of the network) by eliminating from the optimization process the offsets relative to its interior nodes.
The values of the function are determined by minimizing the
total delay of the partial network for all offsets between
the boundary nodes.
At each step the calculation is effected
by searching over all possible offsets associated with the
interior nodes and selecting the minimum.
Recursive application of these rules defines a total delay function
for the complete network, for offsets between a certain final set of nodes.
Optimizing offsets are determined
y minimization of this function.
Applica-
tion of the method is illustrated in the following two examples.
Exa"iPle 1:
parallel
The network to be optimized is illustrated in Fig. 5(a).
Series-
combination produces the v-Y configuration shown in Fig. 5(b), that
cannot be further reduced.
At this stage the network is disconnected into
two parts and a delay function is calculated for each (Fig. 5(c)).
Following GCM 2,
8a.
'U
c
k
\h
a.
)
FIGURE 5
9.
Dl(h,i)
mi n
k {d25 (k) + d 75[(k-h)N] + d5 4[(h+i-k)N]}
This partial minimization also yields the relation k*(h,i) where k* is the
optimizing value of offset k for each combination of h and i.
Following GCM 1 we obtain,
D 2(h,i) = d27 (h) + d 74(i) + d 24 [(h+i)N]
and the total delay D,
D(h,i) = D(h,i) + D2 (h,i)
Minimization of D(h,i) determines the optimizing offsets h* and i*. Backtrack computation yields the optimizing offsets for all links of the
original network.
Example 2:
The original signal network is shown in Fig. 6(a).
After series-
parallel reductions the compressed network of Fig. 6(b) is obtained.
Optimizing
offsets are calculated by stagewise partitioning of this network and recursive
application of the GCM rules at each stage. A partitioning plan that minimizes
number of operations and storage requirements for this network is given in
this table:
Eliminated
Interior
Nodes
Stage
Number
Disconnecting
Nodes
1
2,3,4
1
2
5,3,4
2
3
5,6,4
3
4
5,6,7
4,8
The detailed minimization process is outlined below and illustrated in Fig. 6(c).
9a.
=
i
A I
A
----- 9
--A
I
)-----I
i
,m'm
b.
a.
h
i
---®-- ®:
)
4
q
i
n
r
,E)- -"FIGURE 6
C.
10.
mi n
D (h i) = in{dl2[(j-h)N] + d13(j) + d 4 [(i+J)N]}
+
[d2 3(h) + d 34 (i)]
min
k {d25 (k) + Dl[(k+Z)N,i]}
D2 (Q'i)
D3 (n,p) = mn {d36(m) + D2[(n-m)N,(m+P)N]}
D4 (n,r) = q {d47 (q) + D3 [n,(r-q)N]}
+
[d56 (n) + d 67 (r)]
+ nin{d58 [(n+s)N] + d68 (s) + d78 [(s-r)N]}
S
The delay function obtained at stage 4 represents total delay in the network
for each possible combination of offsets n and r. The terminal optimization
stage consists of minimizing this function with respect to n and r and
calculation, by backtracking, of an independent set of optimal offsets (in
this case offsets j*,k*,m*,q*,n*,r*,s*).
The Network Cycle Time
The traffic flow pattern on a signalized link can be regarded as
the combination of a periodic component imposed by the preceding signal and
a random component arising from variations in driving speeds, marginal fric-
tion, and turns.
The latter component causes additional delay owing to the
occurrence of an overflow queue at the signal's stop line.
While this effect
is negligible at low degrees of saturation, its importance becomes predominant
at. igh
values (22,23,24).
At each nrde in the network we have the relation
gj
C - L
J
i.e., the sum of effective green times on all phases equals the net green
11.
tine available for movement through the intersection (cycle time less lost
time).
Rearranging we obtain
~X.:=
1
j
LC
where xj = -C denotes the green split j (fraction of cycle time alloted to
phase j).
The split, in turn, is determined as follows:
j=
F.
Yj
(1-
L)
.
SJ is the representative ratio of flow to saturation flow of a particular
phase and Y =
X yj
is the sum of y-values over all phases of the intersection.
The y-values depend only on flow and saturation flow, but not on the signal
settings themselves.
The total lost time L is usually a fixed quantity at a
particular intersection.
Therefore, a change in C alters the total net green
time available for passage through the intersection and, consequently, its
allotment to the phases--the green splits.
This eventually brings about a
change in the degree of saturation and with it in the size of the overflow
queue.
An estimate of the expected overflow queue, based on the capacity
of the signal's approach and the degree of saturation was given by Wormleighton.
le considered the traffic behavior along the link as a non-homogeneous Poisson
process with
periodic intensity function (25).
A typical relationship
between overflow queue and split is shown in Fig. 7. Denote the expected
overflow queue at the phase with split xj by Qo(xj).
expected delay, D
The network wide
associated with these waiting vehicles is:
lla.
O
c0
O
O
U)
w
I
0-J
O
t
0
u-_
ow
LU
CD
I-
z
Q1
o
0
o~~~~~~--
l)
0.,_
~ 0)
(STI 31 H 3A) 3n3no MOJ:I3AO- 0 0
0
L
LL
Ds =
Qo(x)
where the sum is to be taken over all phases at all nodes of the network.
This gives the second component of the network objective function,
D
=
Dd + D·
Recursive application of the GCM for different cycle times, taking
into account both deterministic and stochastic effects, produces
rsults
typical
s shown in Fig. 8. These curves were calculated for the network
shown in Fig. 6(a).
It should be noted that input links must be also
included in the calculation.
While they do not affect signal coordination
(i.e., calculation of offsets), they play an important role in evaluating
the total delay for selecting the cycle time.
It is evident that the optimal
cycle time for the network constitutes an equilibrium point between delays
caused by deterministic effects and delays caused by stochastic effects.
While the first delays usually increase with cycle length, the latter
decrease with it owing to the decrease in the degree of saturation (or the
load factor).
They approach asymptotically from above the minimal cycle
time for the network, which is the theoretical minimal cycle time for the most
heavily loaded intersection if all flows were deterministic.
These
characteristics are in complete analogy with the behavior of delay with
respect to cycle time at a single intersection (2).
However, the results are
different and a1 single intersection analysis would virtually never give
optimuill cycle time for the network.
the
12a.
0)
OJ
Ci)
0
O o
LmJ
U)
a)
ui
00
LL
F
ow0
0
L0
no
0q
0
0
Gj~
0
O
0
co
0
w
0
do-
0
CM
(8H/H X H3A) SAVq3G >I0.M13N -1VL01
0
LL
-
c
13.
SUMMARY
Traffic signal settings in a network involve cycle time, splits
and offsets.
The methods currently used ignore some of the important
factors that affect the choice of an optimal set of settings.
In this
paper determination of settings in a synchronized signal network is
effected by scanning a number of cycle times.
For each cycle splits are
determined locally at each intersection according to proportions of
traffic loads on conflicting approaches.
Offsets throughout the network
are optimized by the Generalized Combination Method which constitutes the
basic building-block of the optimization procedure.
The additional costs incurred by travellers through the network
owing to the signalized intersections are regarded as a sum of two
components:
deterministic and stochastic.
These components of costs are
attributable to corresponding components of the traffic flow model.
For
given cycle time and splits, the deterministic costs component is a function of the offsets in the network and generally increases with cycle
length.
The stochastic costs component is determined by the size of the
expected overflow queue at each signal-controlled approach which is a
function of the capacity at the approach and the degree of saturation.
ThTis component is taken to be independent of the offsets and decreases
with decreasing rate when cycle time increases.
It is shown that, with
tis objective function optimal signal settings in a network, including
the cycle time and the set of splits and offsets associated with it,
are determined as an equilibrium point of minimum costs resulting from
the combined consideration of the two components.
14.
References
1. Wagner, F.A., Barnes, F.C., and Gerlough, D.R. Improved Criteria for
Traffic Signal Systems in Urban Networks. NCHRP Report 124, 1971
2. Webster, F.V., Traffic Signal Settings. Road Research Technical Paper
No. 30, H. M. Stationery Office, London, 1958
3. Holroyd, J., The Practical Implementation of Combination Method and
Transyt Programs. Colloquium on Area Traffic Control, IEE, 1972.
4. Munjal, P.K., and Hsu, Y.S., Comparative Study of Traffic Control
Concepts and Algorithms. Highway Research Record 409, 1972, pp. 64-80.
5. Matson, T.M., Smith, W.S., and Hurd, F.W., Traffic Engineering,
McGraw-Hill, New York, 1955.
6. Baerwald, J.E. (Editor). Traffic Engineering Handbook.
Institute of Traffic Engineers, Washington, D.C., 1965.
Third Edition.
7. Little, J.D.C., The Synchronization of Traffic Signals by Mixed-Integer
Linear Programming. Operations Research, Vol. 14, 1966, pp. 568-594.
8. SIGOP:
Traffic Signal Optimization Program, A Computer Program to
Calculate Optimum Coordination in a Grid Network of Synchronized
Traffic Signals. Traffic Research Corp., New York, 1966, PB 173 738.
9. SIGOP:
Traffic Signal Optimization Program Users Manual.
Peat,
Marwick, Livingston and Comp., New York, 1968, PB 182 835.
10.
Robertson, D.I., TRANSYT:
A Traffic Network Study Tool,
Road Research
Laboratory Report LR 253, Crowthorne, Berkshire, 1969.
11.
Wagner, F.A., SIGOP - TRANSYT Evaluation: San Jose, California.
Research Report DOT-FH-11-7822, July, 1972.
12.
Hillier, J.A., Appendix to Glasgow's Experiment in Area Traffic Control.
Traffic Engineering and Control, Vol. 7, No. 9, 1966, pp. 569-571.
13.
Allsop, R.E., Selection of Offsets to Minimize Delay to Traffic in a
Network Controlled by Fixed-Time Signals. Transportation Science,
Vol. 2, 1968, pp. 1-13.
15.
14.
Gartner, N., Optimal Synchronization of Traffic Signal Networks by
Dynamic Progranming. Traffic Flow and Transportation (G.F. Newell,
Editor), Anerican Elsevier Publishing Comp., New York, 1972, pp. 281-295.
15.
Wagner, F.A., Gerlough, D.L., and Barnes, F.C., Improved Criteria for
Traffic Signal Systems on Urban Arterials. NCHRP Report 73, 1969.
16.
Huddart, K.W., and Turner, E.D., Traffic Signal Progressions - GLC
Combination Method. Traffic Engineering and Control, Vol. 11,
1969, pp. 320-322,327.
17.
Seddon, P.A., The Prediction of Platoon Dispersion in the Combination
Methods of Linking Traffic Signals. Transportation Research, Vol. 6,
1972, pp. 125-130.
18.
Huddart, K.W., The Importance of Stops in Traffic Signal Progressions.
Transportation Research, Vol. 3, 1969, pp. 143-150.
19.
Chung, C.C., and Gartner, N., Acceleration Noise as a Masure of
Effectiveness in the Operation of Traffic Control Systems. Working
Paper OR 015-73, Operations Research Center, M.I.T., Cambridge, 1973.
20.
Gartner, N., Constraining Relations Among Offsets in Synchronized
Signal Networks. Transportation Science, Vol. 6, 1972, pp. 88-93.
21.
Robertson, D.I., An Improvement to the Combination Method of Reducing
Delays in Traffic Networks. Road Research Laboratory Report LR 80,
Crowthorne, 1967.
22.
Hillier, J.A., and Rothery, R., The Synchronization of Traffic Signals
for Minimum Delay. Transportation Science, Vol. 1, 1967, pp. 81-94.
23.
Jacobs, W., A Study of the Averaged Platoon Method for Calculating the
Delay Function at Signalized Intersections. B.A.Sc. Thesis (unpublished),
Dept. of Civil Engineering, University of Toronto, 1972.
24.
Gartner, N., Microscopic Analysis of Traffic Flow Patterns for Minimizing
Delay on Signal Controlled Links. Highway Research Records (to appear),
1973.
25.
Wormleighton, R., Queues at a Fixed Time Traffic Signal with Periodic
Randomii Input. Canadian Operations Research Society J., Vol. 3, 1965,
pp. 129-141.
16.
LIST OF FIGURES
Fig. 1:
Link and signal parameters.
Fig. 2:
Idealized traffic flow pattern and link delay functions.
Fig. 3:
Actual platoon profile and link delay function.
Fig. 4:
Series-parallel network reduction rules.
Fig. 5:
Signal network for Example 1.
Fig. 6:
Signal network and optimization sequence for Example 2.
Fig. 7:
Overflow queue vs. green time split.
Fig.
Variation of total network delays with cycle time.
: