Algorithms for the design of traffic-signal progressions for fixed-timecontrol are described.

Least-squares and minimax fits are used to derive solutions for givenvolume requirements within specified limits of speed and cycle time.

The algorithms have been programmed for processing on a digitalcomputer, thus reducing the initial design time considerably andleading to solutions that are superior to manually derived designs.

Algorithms for traffic-signal controlby L. A. Yardeni

the problem

148

Control of vehicular traffic presents a problem of ever-increasingseverity, especially along arteries with large numbers of trafficsignals. Many attempts have been made to set the timing of thesignals for satisfactory traffic flow. However, most of these designsfor fixed-time control require a manual analysis of the traffic sit-uation. Such designs are time-consuming and do not always lead tothe best possible solution.

This paper presents two algorithms that have been programmedfor processing on a digital computer. The initial work in developinggood traffic-signal progressions is reduced from days to minutes, andsolutions superior to those derived manually are realized.'

Major traffic arteries are characterized by a string of signalizedintersections and relatively heavy vehicular flow in either or bothdirections during at least some hours of the day. Ideally, anyonevehicle or group of vehicles should be allowed to proceed along theartery at a suitable speed without stopping or slowing down due tored signal intervals. Although such an ideal situation can beachieved rarely for two-way streets, it can be approached byproperly setting all the traffic signals along the artery.

The problem consists of finding the correct timing for eachsignal in order to arrive at the best overall solution. Presently, citytraffic engineers find a solution (which is not necessarily optimal)by drawing time/space diagrams, as shown in Figure 1. The verticaldistances between the horizontal lines indicate the true-scale

IBM SYSTEMS JOURNAL' VOL. 4 . NO.2' 1965

Figure 1 Time/space diagram

'"!j; 2 -'"::;S BASE~

10:1-

6 -

5_

9 -

i16

24

32

~ 40

300TIME, IN SECONDS_

uiozgc

C48r-...,..------.---.---------.---..,--------,::l

distances of the signalized intersections. The lines themselves rep-resent the red-signal interval times in both directions, whereasthe blanks between reds are the green (plus amber) interval times,called green split (expressed in seconds) or green ratio (percentage ofthe entire signal cycle). The two pairs of parallel diagonal linesenclose the through-bands in opposite directions; their slope rep-resents the speed a vehicle must maintain in order to stay as partof a group, or platoon, of vehicles moving along the major streetwithout signal interference.

The essential input information for a time/space diagram islimited to the distances between the intersections in the system andthe green ratios for these signals. For the conventional drawingboard design, various sets of speeds and cycle times are used withgreen splits precomputed from averaged volume data. The numberof such input sets that can be tried for the best solution is limitedby the time required to draw and evaluate the results. The al-gorithms described in this paper overcome this limitation becausethey can be programmed for digital computer application.

Usually, an attempt is made to design for higher speeds withinthe legal speed limits. Shorter cycle times, such as 60 seconds orless, are also preferred, because they provide shorter average wait-ing time for vehicles stopped at a red signa1. But in view of accel-eration losses due to signal switching, shorter cycles reduce thefeasible volume (vehicles per hour). The algorithms described heretake into account these conflicting requirements: they allow acomputer to select the highest feasible speeds and the lowest cycletime within given ranges. The computer accepts volume data foreach intersection and computes proportional green ratios in prep-

ALGORITHMS FOR TRAFFIC-SIGNAL CONTROL 149

objectives

initialbase bands

aration for the design computations. It also considers input datafor the free-flow volume versus speed relations in computing actualflow capabilities of the through-bands.

The quality of the design may depend on the criterion ofoptimality or measure of effectiveness chosen. Typical designobjectives may be defined as follows: (1) minimize red-phase inter-ference in the through-band, (2) maximize through-band widths,(3) maximize some criteria of flow quality, or (4) select a maximumspeed and minimum cycle combination that meets flow demands.The last design objective listed is used in the approach of thispaper. It is felt that this objective is superior to other optimizationcriteria examined, because it addresses itself explicitly to meetingtraffic throughput requirements.

The algorithmsFrom a set of intersections along an artery, the intersection withthe lowest green ratio is selected as a base. All other intersectionsmay now be considered independently in relation to this base.Through-bands enclosing an entire green split of the base inter-section are initially the widest possible bands for a given set ofgreen ratios. Figure 2 shows the relation of any other intersectionto the base.

Figure 2 Midpoint relations

BASE----

D,

I-<--------T,---------..I

I-<-------nic------~

150 L. A. YARDENI

Consider any intersection [ at a distance J), from the base.Assuming any pail' of speeds VI and V 2 , we may draw them at theirproper slopes through the midpoint of the base's green interval.Transposing this point along both directions to I, we obtain theturn-around time 'I', which is given by the sum of travel times inboth directions:

1). 1). ( 1 1 )t: = V: + V: = », VI + V

2•

Also,

T . = 21)i• V'

where

(0

To assure that the signal timing for intersection I does not inter-fere with the initial base bands, it is required that:

T, = cai.,or

T;-;;: = n;,.

(2)

where ni is an integer that yields a cycle c, within a given allowedrange (e.g., 50 to 100 seconds). This is an ideal cycle time whichwould eliminate any red phase band interference for intersection I.

In a real system, it is seldom feasible to select a system cycletime C that properly satisfies the above requirement simultaneouslyfor all intersections.

For system cycle c, the deviation Ai from the ideal turn-around time for intersection I, 'I', , is then Ai = Ail + Ai 2 ='I', - cn, ,

The optimization objective is now to minimize these deviationsfor all intersections simultaneously. To do this, the conventionalleast-squares fit is applied.

The criterion function A is now defined as:

A = L: (T; - cni)2,;

least-squareslinear fit

or, by using Equation 2,

A = L: (c, - c)2n~. (3)

Minimization with respect to C yields the best cycle time for a givenpair of speeds, VI and V 2 , or a mean speed V.

Some manipulation of variables gives other equivalent formsof this function as follows: Replacing T; from Equation 1,

ALGORITHMS FOR TRAFFIC-SIGNAL CONTROL 151

A ~ (~ D, - en,)" = :2 ~ (D i - kVn;)2

= ~2 ~ cu. - Kni?' (4)

where K = cV /2 is a system constant, the space-periodicity con-stant, to be discussed further.

Equations 3 or 4 may be used to calculate an optimal cyclelength (CO) for given speeds, an optimal average speed (VO) forgiven cycle lengths, or an optimal system constant (KO).

Any pair of c and V satisfying the condition KO = cV/2 isequally optimal with respect to the criteria of minimum A. Asshown later, the actual selection of c and V values is governed byother considerations, whereas the function A is defined in terms ofthe systems constant K only.

Before differentiation and evaluation of a formula for optimalvalues KO of K, the function A is first generalized. According toEquation 4 and Figure 3, KO is the slope of a best fit through thediscrete points relating D, and n; .

Figure 3 Space-periodicity fit

jD

B -------------------------

7 ---------------------

/6 ------------------

BASET----b"

-:l-on_

The first obvious generalization is to provide for an intercept,or a base line, which does not necessarily coincide with the locationof any real intersection, as follows:

152 L. A. YARDENI

A = :2 ~ (D i - b - KnY. (5)

Both K and b are expressed in units of distance, i.e., feet; b isthe distance of an imaginary base intersection from the first inter-section; K is the system's space periodicity constant. In the presentformulation, K depends primarily on the given fixed distances(D, - b); it may possibly have more than one optimal value ifone or more intersections are assigned different values of n, . Forsome intersections, its relative position to the best-fit line of slopeK may be very close to flipping from one value of ni to the next;e.g., intersection 5 in Figure 3 may be assigned n5 = 4, but wouldbe assigned ns = 5 for a very slight change in K. This would yieldanother value of KO. To select the best KO, we can plug the differentKO back into Equation 5 until we find the lowest value of A, usesome other goodness-of-fit test, or test for expected bandwidth.

To improve the fitting even further, we modify the algorithmto consider only relevant deviations. This is done by taking intoaccount all differences in green splits.

If the green interval of intersection I in Figure 2 is larger thanthe base green interval, it is not necessary to have ~il = ~i2 = 0for noninterference in the through-bands. This consideration allowsrelaxation of the requirement expressed by Equation 2. The mod-ified formula is of the form

where

o :s; gi < 0.5,

gi being an appropriate expression for the difference between thegreen ratios of the intersection I and the base. The sign (+ or -),depending on the occurrence of a lead-trail or trail-lead fit (as ex-plained later), tends to decrease the derivations in the least-squaresfit, as shown in Figure 4.

Redefining our integers n, as modified integers m, = n,(l ± gi)we obtain:

algorithmrefinements

A = :2 L (D, - b - Klni)2.,(6)

So far, this criterion function is essentially a sum of squares ofrather rigidly defined deviations. It lends itself readily to thecustomary least-squares fit. Prior to performing such a fit, anadditional generalization is now introduced: the weighting of thedeviations by a weighting coefficient a, so that

A = :2 ~ (D i - b - Klni?a~. (7)

The introduction of this factor obviously renders considerableflexibility to our formulation and can provide the decision makerwith a range of distinct models. Such models depend on the method

ALGORrrHMs FOR TRAFFIC-SIGNAL CONTROL 153

Figure 4 Space-periodicity fit with modified n i , m.

1D

B -----------------------------------

·7 --------------------------------0

"-B6

l':-O---!+-~--~-_+~-~--__!:_-_____,':_~_7_-----'

2 -----0IIIIIIIIo

used for the selection of a/so For example, consider the followingcases:

(1) All a. = 1 (obviously, this trivial case yields Equation 6).

(2) a. = max {F.d,L

where F' L is later defined by Equation 9 as a function of requiredvolume and green ratio for intersection I in direction L. In thismodel, the deviations are thus weighted by coefficients that providea ranking of the intersections with respect to their degree of crit-icality. A critical intersection is loosely considered as one thatimposes design restrictions on the system.

(3) Modification of a/s as a result of an iteration procedure. Forexample, starting with all a. = 1, design through-bands anddetermine the band limiting intersections, and then assign valuesa, > 1 to these intersections and redesign. This may be repeatedany desired number of times. Finally select the design with max-imum band widths.

154 L. A. YARDENI

Differentiating Equation 7 with respect to K and b, and settingaAjaK and aAjab equal to zero, KO and b" are derived:

IC = 2: a~ 2: a~D;m; - 2: a~D; 2: a~m;2: a~ 2: a~m~ - (2: a~m;)2

and

where the summations are taken over all intersections; D; is thedistance of any intersection I from the first intersection or, moregenerally, from any reference point in the system if the m!s arederived and b is measured with respect to that same reference point;m, is the modified integer; and a; is a properly defined weightingcoefficient for intersection I.

As implied earlier, different values of KO and b"may be obtainedby assigning different values to the set {m;}. A simple methodprovides the selection of all feasible sets within the given ranges ofspeeds and cycle times. Clearly, each set of {KO, bO} is an optimumwith respect to a distinct set {m;}. Some possible discriminationand selection criteria for the best {KO, bO} have been mentioned.Another possible criterion is developed next.

Just as we defined an ideal cycle time (c.) for intersection I,an ideal constant (K;) can be considered as follows:

K. = D; - bO. ,m;

so that anyone deviation in the least-squares fit is:

d; = (K; - KO)m;.

This expression, when used in its absolute form, is useful in obtain-ing an estimate of expectable through-band widths, and further indeveloping another selection criteria for KO.

First, the largest deviation is found:

dw = max {d;} = max {IK; - KOI m,},

from which the worst K; , say K; , is obtained.Next it is recognized that the minimum reduction III the

initially feasible through-band width is approximately:

t.B""' ~ [Kw - KOI (1- gw) = :c·t.Kw.(l- gw)'c

= t.Kw e(l _ )K O lIw

where gw is the absolute difference in green ratios as earlier defined.Now let GbL equal the green interval (in seconds) of the im-

aginary base in direction L. This is the initially feasible bandwidth. Then the maximum expected band width is given by:

expectedbandwidthtest

ALGORITHMS FOR TRAFFIC-SIGNAL CON'l'ROL 155

(8)

where

S - GbE7JL -

C

is the green ratio for the base in direction L, and

k = f!.K w

T K"

is the relatively worst deviation from the system's periodicityconstant.

If we set the right-hand side of Equation 8 equal to zero, weobtain the conditions for through-band feasibility, i.e.,

o < k < SbL- T 1 - gw

Thus, an estimate of through-band feasibility and relativemagnitude can be obtained from fixed distance and variable flowinformation only, independent of cycle or interval times and speeds.However, this assumes that green ratios are selected in some fixedrelation to volume ratios.

An alternative algorithm is based on the criterion function

minimaxlinear fit

{ad = {D i - b - Km,las shown in Equation 6. In this approach, K O and b" are evaluatedand alternative values selected by a minimax algorithm as follows:

aO = min {max (a,)l -. K Oand b",/(

Traffic volume and through-bandsAs indicated earlier, a highest-speed and lowest-cycle combinationis selected within allowable limits for both speeds and cycle times.A further restriction for both algorithms described can be imposedby considering the speed/volume relationships.

When considering traffic flow on a freeway or in a tunnel, weobserve that, at relatively high speeds, wider vehicular spacingsreduce the traffic density. Empirical and theoretical studies" ofsuch traffic-flow characteristics show that traffic volume (i.e.,vehicles per hour) decreases with increasing speeds. This relation-ship holds true for normal free flow. At a point of maximum volumeat a reduced speed, flow becomes unstable and finally turns into aforced flow. In this region, speed and volume decrease simulta-neously. Figure 5 shows a typical speed/volume curve. The programaccepts only a single-valued function of volume, so that only onesection, either the normal or forced flow part, can be considered.

Such a speed/volume relation can be established for any urbanartery as if all signalized intersections were assumed to be greenand continuous free flow permitted for an extended period. These

UNSTABLE-_",FLOW

I/

"

volume

VOLUME-

A speed/volume curve

°'"'0==-------------'

Figure 5

156 L. A. YARDENI

F ~ 1,

curves represent the road (or lane) flow capacity as it function ofspeed only. They may differ considerably in values represented,depending 011 road conditions (e.g., slipperiness, visibility). Thevolumes obtained from traffic counts on signalized arteries areobviously not for free flow. They are usually given for pulsed flow,or platoon flow, which results from signalfiow gating. The followinggating relationship can be assumed for an intersection:

Actual (pulsed) volume F ti. = green ra 10Free (capacity) volume

where F is a safety or inefficiency factor, which may be used tocompensate for acceleration and deceleration losses due to signal-ization and also for inaccuracy of the assumed speed/volume curve.F may also roughly compensate for the errors due to neglect ofintersection widths and related crossing times. Although it ispossible to develop theoretical expressions to account for theseeffects, it is felt that, in practice, it suffices to relate F inversely tocycle length. The speed/volume curve is employed to estimate freeflow volumes for given speeds.

Rewriting the above gating relationship, we have

Ct i L F _ o _ GiI~ff'iL· - ViL - c'

where, again,

ff'L = f(V L) = ff'L(V L),

as defined by the speed/volume curve.Now let Cti L be the required volume (flow demand) approach-

ing intersection I in direction L. Then the required free flowvolume is

(9)

(10)

The required equivalent free-flow volume for the artery is deter-mined by the bottleneck intersection, which is the one with a greenratio SmL and a demand Ct mL , which yield the largest free flowvolume:

ff'L = CtmL ·F.SmL

Checking against the speed/volume curve, we find the max-imum speed (if any) at which such equivalent free-flow volume isfeasible. If no such volume is feasible, it may be possible to increasethe green ratios sufficiently to allow the volume demand at a rea-sonable speed. This may impose impractical restriction of cross-street traffic flow. In such cases, the arterial flow demand cannotbe met and some compromise is required. In a properly controlledsystem, it may be desired to allow demands (entering traffic) onlyto the extent that normal flow (at above minimum speeds prior toflow breakdown) can be maintained.

ALGORITHMS FOR '['RAFFlC-SIGNAL CONTROL 157

Figure 6 shows a typical mid-green relationship for such trans-posed intersections. Again we transpose the base midpoint alongboth bands to the intersection under consideration, so that

p = c + tG b

yields

P - ti z = c + tGb - t i Z

and

P + til = C + tG b + til'

where

_ D,ti2 - V;

(11)

and

These two transposed points represent the band midpointsand as such may be slightly shifted as the band is being narrowedby red-interval interferences. The intersection's mid-green pointsare moved as close as possible to both band midpoints. Offset posi-tioning considers the unavoidable deviations .:lil and .:liZ • Thedesigner may select equal deviations or assign values inverselyrelated to traffic volumes. Generally, we assume some weightingfactor a, and (1 - Ct i ) for direction 1 and 2, respectively, whereo < a; < l.

Thus, we require:

Cti.:lil = (1 - Ct,).:li2'

But

.:lil = cPi + c + tG i - (P + til)

= cPi + !CG i - Gb) - til'

.:li2 = P - t i2 - (cPi + tG i )

= c - ti2 - t(G; - Gb) - cPi'

where cPi is the offset for intersection I.Substituting in Equation 11 and solving for cPi , we obtain

This formula provides a least deviations positioning for all inter-sections that transpose into the c zone (Figure 6), and it yields alead-trail interference configuration. This c zone is defined by

c2:::; T i < c.

For intersections that transpose into the c/2 zone, whereo :::; T; :::; c/2, a similar development yields

ALGORITHMS FOR TRAFFIC-SIGNAL CONTROL 159

curvilinearfit

networkcontrol

160

cPi = ti 1(X i + (C - ti 2)(1 - (Xi) - HGi - Go) + C(Xi

which provides for a trail-lead configuration.When extreme values of (X (close to zero or one) are used, it is

possible to interfere with one band while the other is given excessgreen time (outside of band). Such positioning is not permitted andthe algorithm overrides the design required by (X to take full advan-tage of the available green phase.

ExtensionsThe following extensions to the algorithms described may yieldsignificant results. These extensions may be incorporated in futureversions of the program.

A curvilinear fit for a set of system constants or speeds alongan artery may provide better results. Such a fit implies that, in-stead of the fixed speed along an arterial direction, speeds varyingfrom location to location would be imposed. In this case, two alter-native approaches are considered:

1. Obtain a second (or higher) order curvilinear fit, then approxi-mate this fit by connecting line segments, perferably improvingon the curvilinear fit.

2. Develop a simple linear fit for the first three intersections, andtest for goodness of fit; if good, include the fourth intersection,and test again; if not good, the first segment is composed ofthe first two intersections only. Continue until an appropriatenumber of segments for all intersections is developed.

In both cases, the smallest allowed cycle time that meets thevolume demand (via the speed/volume relation) is picked. Foreach segment and its systems constant K; , a speed is then calcu-lated for the condition

K; = ~ V;.

The actual speeds can be selected in relation to volume demandsin both directions. The offset design does not change, with theexception that each segment has its own base. Offsets can be easilytransformed to a common reference time at, say, the first intersec-tion. As long as each segment meets its volume requirements, itmay be desirable to obtain different bandwidths for differentsegments.

The previous modification leads to a further extension, i.e.,the adaptation of such a method to network control.

Consider a network element of 2 X 2 intersections. Althoughwe can design a progression around three legs of such an element,we are normally faced with the problem of nonclosure if only onespeed is considered. If we allow sufficient (but practical) speedchange around the element, closure can be obtained on some best-fit basis.

The extension to larger networks does not pose a conceptual

L. A. YARDENI

problem, although it may become quite complex computationally.Such a time/space approach to the network control problem maynot yield as good results as with an "instantaneous decision" con-trol system, but it could require considerably less equipment.

Computer programA computer program" was written in FORTRAN-language forthe IBM 1620, 7040, and 7090 systems. The input data acceptedand utilized by the program include:

• Speed/volume curve values• Intersection distances• Number of effective lanes• Cycle time range• Desired cycle time precision• Volumes approaching each intersection• Minimum pedestrian crossing times• Various program option controls, such as left-turn phase selec-

tion, double cycle control, etc.

Initially, green ratios are developed from volumes and min-imum pedestrian crossing times. Then a set of alternative systemconstants are developed. Next, minimum cycle time and maximumspeed are computed within the restrictions of given volume de-mands and speed/volume relation. Finally, offsets that providethrough-bands with maximum volumes are computed for all inter-sections.

Output is provided in terms of selected system cycle time,average speeds in both arterial directions, feasible system volumes,and intersection green splits and offsets. Time-space diagrams canbe machine-plotted if an IBM 1627 plotter is available.

The complete solution (with card-punched results) for a 15- to20-intersection arterial progression requires 20 to 30 minutes on a1620 MODEL I (40K and 1311 disk file) system. Larger problems canbe handled with an IBM 7000 series system in a few minutes.

CITED REFERENCE AND FOOTNOTES1. A similar version of this paper was presented at the Annual Meeting of

the Institute of Traffic Engineers, Miami Beach, Florida, November 1964.2. R. T. Underwood, "Some aspects of the theory of traffic How," Australian

Road Research, Ramsay, Ware Publishing Pty., Ltd., North Melbourne,N. 1., No.2, 35-47 (June 1962).

3. Two versions of the program, called "Vehicular Traffic Control Time/Space Diagram (Progression) Design Program," can be ordered throughany IBM branch Office. The 1620 version, PID 9.2.065, is included in theIBM 1620 General Program Library. The 7040/7090 version, PID 7040T4IBM0005, is part of the SHARE General Program Library.

ALGORITHMS FOR TRAFFIC-SIGNAL CONTROL 161