The Highway CapacityManual Delay Formula forSignalized Intersections

BY RAHMI AKCELIK

The purpose of this article is to com-pare the 1985 U.S. Highway Capac-

ity Manual (HCM)’ delay for”mula forsignalized intersections with theAustralian’ and Canadian3 formulas andto present a generalized form that em-braces them all. The aim is to promoteinternational cooperation in this area ofresearch and development.

Compared with the other delay for-mulas, the HCM formula predicts higherdelays for oversaturated conditions, andthe differences between the predictionfrom the HCM formula and the otherformulas increase with increasing degreeof saturation. An alternative to theHCM delay formula, which matches theother formulas for oversaturated condi-tions, is given. The alternative formula,derived from the generalized formula,predicts delays that are very close tothose predicted by the original HCMformula for undersaturated conditions,and at the same time predicts delays thatare very close to the results from theAustralian and Canadian formulas foroversaturated conditions.

The HCM signalized intersectionchapter states that its delay formula“yields reasonable results for values of .xbetween 0.0 and 1.0. . . The equationmay be used with caution for values of xup to 1.2, but delay estimates for highervalues are not recommended.” Althoughtraffic engineers do not design for over-saturation, a delay formula that can beexpected to give reasonable results foroversaturated, as well as undersaturatedconditions, is preferred because the lim-itations of this type of formula are often

forgotten and the formula misused inpractice (e.g., in evaluating alternativedesigns or in stating benefits from im-provements to an existing oversaturatedintersection).

IThe HCM formulapredicts higherdelays foroversaturtedconditions.

The generalized formula could be cal-ibrated to develop a more suitable for-mula for U.S. conditions. It might alsobe possible to calibrate it for vehicle-ac-tuated and fixed-time signals separately.Various other issues related to this dis-cussion and briefly mentioned in this ar-ticle will be discussed elsewhere.

A Generalized FormulaThe HCM, Australian, and Canadianformulas for delay at traffic signals canbe generalized as the following two-termequation:

~ = 0.5C(1 –u)’

1–UX+ 900 TX”

[(x-1)

+ v J(x- 1~ + m(x-xo)/QT (1)

where:d = average overall delay (including

stop-start delays) in seconds pervehicle,

c = signal cycle time in seconds,u = g/c (ratio of effective green time

to cycle time),x = degree of saturation (ratio of ar-

—

;:m,n =X. =

x“ =where:

Sg =

a,b =

rival flow rate to capacity),flow period in hours,capacity in vehicles per hour,calibration parameters, andthe degree of saturation belowwhich the second term of the de-lay formula is zero. This can beexpressed asa + bsg (2)

capacity per cycle (s = satura-tion flow rate in vehicles per sec-ond and g = effective greentime in seconds), andcalibration parameters.

The two terms of the delay formulacan be referred to as the uniform delay(du) and the overflow (dO) terms:

d=du+ do. (3)

The overflow delay is called the incre-mental delay because of random arrivalsand individual cycle failures in theHCM. The usefulness of this concept isthat an overflow queue formulation canbe used as a common base for the for-mulas to predict delay, number of stops,and queue length as in the Australianmethod. 2“ The relation between the av-erage overflow queue (No) in vehiclesand the overflow delay (dO) in seconds is

3600 do = NOIQ (4)

where Q is the capacity in veh.lhr. In thissense, Equation 1 is based on a gener-

ITE JOURNAL . MARCH 1988023

alized overflow queue formula. More de-tailed discussion on the use of the over-flow queue concept to predict delay,number of stops, and queue length isgiven in the Appendix.

Various specific delay formulas can bederived from Equation 1 by setting thecalibration parameters n, m, a, and b inthe overflow delay term to appropriatevalues. Therefore, only the second termof Equation 1 will differ between alter-native models as no calibration parame-ters are considered in the first term (dJ.The values of the calibration parametersn, m, a, and b for the 1985 HCM ,’ 1981Australian,’ and 1984 Canadian,3models are given in Table 1. The HCMmodel differs from the other two modelswith the x’ factor (n = 2) whereas theAustralian model differs from the othertwo models with a nonzero XOparameter(X. = 0.67 + sg/600).

The exact form of the HCM formuladiffers from Equation 1 because a factorof 1/1.3 = 0.77 is applied to convert theoverall delay to stopped delay (i.e., itassumes that the stopped delay is always779%of the overall delay). Whilst the au-thor does not necessarily agree with thissimplifying assumption, it is outside thescope of this article.

To facilitate comparisons, T = 0.25hr., which is fixed in the HCM model,will be used for all models in the follow-ing numerical example although it is bestto leave this as a variable in the generalmodel. It will also be assumed, by wayof example, that c = 90 sec., g = 30sec., s = 1500 veh.lhr., and therefore glc = 1/3, Q = 500 veh.lhr., and sg =12.5 veh. The overflow delay, do, fromthe second term of Equation 1 for thisexample is given by

do = 225 X“ [(X– 1)

+ V(X - 1~ + m(x-xO)/125] (5)for x > x,, (O otherwise)

The results from Equation 5 for theHCM, Australian, and Canadian modelsare shown in Figure 1, and tabulated inTable 2. For this example, x. = 0.691 forthe Australian model and x. = Ofor theother two.

Time-Dependent DelayFormulationItis seen from Figure 1 that the Austra-lian and Canadian models produce ov-erflow delay curves that are asymptoticto a deterministic oversaturation delay

Table 1. Values of the Calibration Parameters in the HCM, Australian, and CanadianOverflow Deiay Formuias, Expressed in Terms of Equation 1

n m a b

HCM 2 4 0 0Australian o 12 0.67 1/600Canadian o 4 0 0TRANSW 8 –1 4 0 0Alternative

to HCM o 8 0.50 0

Tabie 2. Overflow Delays in 3econds per Vehicie from the HCM, Australian, andCanadian Delay Formulas (c = 90 sec., g = 30 sec., and Q = 500 veh./ho

x HCM Alternative” Australian Canadian Deterministicb

o0.40.60.80.90.951.01.11.21.4

0.00.41.98.4

17,626.640.285.4

155.5376.0

0.00.01.89.7

19.928.540.272.0

110.5195.0

0.00.00.05.5

16.525.938.872.4

112.1197.5

0.02.45.2

12.621.829.540.270.3

108,0191.8

—

—0.0

45.090.0

180.0

WVhererr =0, rn=8, c7=0.5, andb= O.bFrom Equation 60,

160

t _ USHCM

140

t— _ Australian

I .._-Canadian

120

t

—.— Deterministic

Overflowde~y,

(secon;s pervehicle)

20I/d““/

-.---:””-”--/”/---o~

0.2 0.4 0.6 0.8 1.0 1.2Degree of saturation, x

Figure 1. Overtlow delays predicted by the HCM,Australian,andCanadianformulas(c= 90 sec., g=30 sec., Q=500 veh.hw., and T= 0.25 hr.).

24. iTE JOURNAL . MARCH 1988

line (Akcelik”) given by

d = 1800T(x– 1) for .x=1 (6)

For the example above,

d = 450 (x–l). (6a)

This deterministic delay formula isbased on a single flow period of lengthT with constant flow and capacity andwith no initial overflow queue. The delayfrom Equations 6 or 6a includes delayduring T as well as delay after T (as ex-perienced by the vehicles that arrive dur-ing T but may depart after T). Subse-quently, the generalized formula,Equation 1, is based on the same simpli-fying assumption, which is used as analternative to the more complicated var-iable-demand analysis methods thattreat the peaking of arrival flows explic-itly.

The asymptotic curve is an importantcharacteristic of time-dependent delayformulation and was originally devel-oped by researchers at the U.K. Trans-port and Road Research Laboratory forthe TRANSYT programs. In fact, theCanadian delay formula is the same asthe formula given by Robertson.’ How-ever, TRANSYT Version 8 uses a differ-ent form of the function’ equivalent to~=–l, m=4, a=O, and b=Oin

Equation 1. Different forms of the func-tion were used in earlier versions ofTRANSYT.

Converting theoverall delay tospeed delay needsparticularattention beforeany calibrationeffort.

The time-dependent delay models arederived by converting a steady-state de-lay function, which is applicable to un-dersaturated conditions only, to anasymptotic time-dependent function,which becomes applicable to oversatur-ated conditions also. The steady-statefunction that corresponds to Equation 1when n = Ocan be expressed as

k(x –x<,)x/(l ‘X) (7)

where parameter k is related to param-eter m in Equation 1 by m = 8k. TheCanadian” formula corresponds toEquation 7 where k = 0.5 and X. = O,which is the random delay term of the

well-known Webster formula. 7The Aus-tralian formula’ corresponds to k = 1.5with a variable value of XO,which is anapproximation4 to Miller’s delayformula’. Unlike the Webster formula,Miller’s original equation was based onthe formulation of overflow queues. Thiswas extended to oversaturated condi-tions by the author using the TRRLtime-dependent delay method and intro-ducing a simplification to the originalMiller formula.4 The reader is also re-ferred to a paper by Hurdle,’ which dis-cusses the relationship between thesteady-state and time-dependent delayformulas.

As seen in Figure 1, the HCM formulaappears to produce a curve that does nothave the fundamental characteristic ofthe time-dependent delay formulation.For x above 1.0, it diverges from the de-terministic delay line and predicts verylarge delay values as indicated by theshaded area. This is due to the X2factor(n = 2 in Equation 1). It is not clear ifthe X2factor was introduced in an effortto calibrate the delay formula for under-saturated conditions, or for reasons re-lated to the use of a fixed value of T =15 minutes (peak flow period). A clari-fication of this issue would explain a fun-damental difference between the HCMformula and the other formulas.

An Alternative to the HCMFormulaA formula that gives delay values closeto the HCM formula for x values lessthan 1.0, but remains asymptotic to thedeterministic oversaturation line (Equa-tion 6), can be derived from Equation 1

by setting n = O (i.e., deleting the x’factor and choosing the appropriate val-ues of parameters m, a, and b). It hasbeen found by a best-fit analysis of theoverflow delay values from the HCM for-mula for x values in the range 0.6 to 1.0that the parameter values of m = 8 anda = 0.50 give a satisfactory solutionwhen parameter b is set to zero to obtaina formula with constant Xo. Using T =0.25 hr., this overflow delay formula isexpressed as

dO = 225 [(X– 1)

+ V(X- 1)2 + 32 [(x- O.5)/Q] (8)

for x >0.5 (zero otherwise).

This equation gives the same delay valueas the HCM equation for flow at capacity(x = 1.0). Interestingly, it correspondsto an earlier steady-state overflow delayformula given by Miller,’” which approx-imates to Equation 7 with k = 1.0 andx“ = 0.5. The overflow delay resultsfrom Equation 8 for Q = 500 veh./hr.are given in Table 2 as the alternativemodel.

For direct comparison with the fullHCM formula, the stopped delay for-mula obtained by replacing the secondterm of Equation 1 by Equation 8 andapplying the factor of 0.77 to both termsis as follows:

d, =0.385c(1 – u)’

1–UX+ 173 [(x– 1)

+ V(X - 1)2 + 32 (x- O.5)/Q (9)

The results from Equation 9 and theHCM formula for stopped delay for theabove example (c= 90 sec., g = 30 sec.,and Q = 500 veh./hr. ) are given in Table3. The values in Table 3 for x larger than

Table 3. Comparison of Sopped Deiavs from the HCM Formula and theRecommended Alternative Formulation Using Factor of 0.77 (c= 90 sec., g= 30 sec.,and Q= 500 veh./hr.)

HCM” Alternative’ Difference’

x (sec.] (8ec.] (sec.] Y. Differenced

0.2 16.5 16.5 0 0

0.4 18.0 17.8 – 0.2 –1,1

0.6 20.7 20.6 –0.1 – 0.5

0.8 27.2 28.5 +1.3 + 4.8

0.9 35.6 37.3 +1.7 + 4.8

0.95 43.0 44.5 +1.5 + 3.5

1.0 54.0 54.0 0 0

1.1 88.5 78.5 – 10.0 –11,3

1.2 442.7 108.1 – 34.6 – 24.2

1.3 216.9 140.0 -76.9 -35.5

1,4 312.3 173.0 -139.3 -44.6

‘%Yhere n=2, m=4, a=O, and/J=O.bWheren=O, m=8, a=0,5, and b=0.C(Alternative - HCM).’100 x (Alternative – HCM)/HCM.

ITE JOURNAL. MARCH 1988.25

1.0 have been calculated using the valueof the uniform delay at capacity (firstterm for x=1) as a fixed value (d= 23.1sec. ) for both models .Explanation ofthis method used by the SIDRA com-puter program’’-” can be found in anearlier article by the author’. The maxi-mum difference between the HCM for-mula and the alternative formula for xless than 1,0 is about 2 sec. (59%).

ConclusionEquation 9 should be considered as analternative to the HCM delay formula.This formula gives values close to the

HMC formula for degrees of saturationless than 1.0, and at the same time issimilar to the Australian, Canadian, andTRANSYT formulas in producing a de-lay curve asymptotic to the deterministicdelay line for degrees of saturationgreater than 1.0.

However, the calibration of the gen-eralized formula (Equation 1) directlyusing actual data for U.S. conditionsrather than the results from the originalHCM formula would, of course, be abetter way of developing an alternativeformula. It would also be useful to seeka value of X. dependent on capacity percycle (sg) in this process, because this

form of the model has certain advan-tages.

Furthermore, the question of convert-ing the overall delay to stopped delayneeds particular attention before anycalibration effort. Researchers couldalso try deriving separate formulas forfixed-time and vehicle-actuated signalsby calibrating the generalized Equation1 of this article (n = O recommended).Again, the important question of apply-ing the delay formulas for individuallanes as against lane groups has not beendiscussed here.” In this respect, any cal-ibration effort should take the particularmethod of application into account.

26. ITE JOURNAL . MARCH 1988

Acknowledgment

The author thanks the executive directorof the Australian Road Research Board,Dr. M. G. Lay, for permission to publishthis article.

References1. Transportation Research Board, NationalResearch Council. Highway Capacity Man-

ual. Special Report 209. Washington, DC:Transportation Research Board, 1985.2. Akcelik, R. Traffic Signals: Capacity andTiming Analysis. Research Report ARR No.123. Nunawading, Australia: AustralianRoad Research Board, 1981. (Third reprint1986).

3. Teply, S, (cd.). Canadian Capacity Guide

for Signalised Intersections. Edmonton, Al-berta, Canada: Institute of TransportationEngineers—District 7 and the University ofAlberta, 1984.4, Akcelik, R, Time-Dependent Expressions

for Delay, Stop Rate and Queue Length at

Traf/ic Signals. Internal Report AIR 367-1.Nunawading, Australia: Australian Road Re-search Board, 1980.5. Robertson, D. I. “Traffic Models and Op-timum Strategies of Control: A Review.” In:Proceedings of the International Symposium

on Traffic Control Systems, vol. 1. Universityof California, Berkeley, California: Instituteof Transportation Studies and the U.S. De-partment of Transportation, 1979, pp. 262-288.

6. Vincent, R, A.; Mitchell, A. 1.; and Rob-ertson, D. 1. User Guide to TRANSYT Ver-

sion 8. TRRL Laboratory Report LR 888.United Kingdom: Transportation Road Re-search Laboratory, 1980.7. Webster, F. Y Trafjc Signal Settings. RoadResearch Laboratory Technical Paper No. 39.London, England: HMSO, 1958.8. Miller, A. J, Australian Road Capacity

Guide—Provisional Introduction and Signal-

ised Intersections. Bulletin No. 4, 1968. (Re-printed as ARRB Research Report No. 79,1978. Replaced by ARR No. 123, 1981). Nun-awading, Australia: Australian Road Re-search Board, 1981.9. Hurdle, V F. Signalised Intersection DelayModels—A Primer for the Uninitiated. Trans-portation Research Record 971. Washington,DC: Transportation Research Board, 1984,pp. 96-104.10. Miller A. J. “Settings for Fixed-CycleTraffic Signals.” Operations Research Quar-terly 14(4): 373-386 (1963).11. Akcelik, R. “SIDRA-2 does it Lane byLane.” 1ss Proceedings of the 12th ARRBConference 12(4): 137-149(1984),12. Akcelik, R. Sidra Version 2.2 Input andOutput. Australian Road Research Board.TechnicalManual ATM No. 19, 1986.13. Akcelik, R. Introduction to SIDRA-2 forSignal Design: 13th ARRB Conference Work-shop Papers and Discussions. AustralianRoad Research Board. Research ReportARR No. 148, 1987.14. Akcelik, R. “Estimating the Capacity ofa Shared Lane.” Presented at the Third MiniEuro Conf. on Operations Research Methodsin Transport Planning and Traffic Control.Herceg Novi, Yugoslavia, June 1987.15. Powell, J.L. “Project evaluation at sig-nalised intersections using the 1985 HCM.”In Compendium of Technical Papers. Wash-ington, DC: Institute of Transportation” En-gineers, pp. 81-85, 1987. I

Rahmi Akcelik isa principal re-search scientist atthe AustralianRoad ResearchBoard. His currentwork is in the traf-

jic signal and fuel consumption areas ofresearch and development. He is a mem-ber of ITE Technical Committee 5B-27jLeft-Turn Lane Storage Length DesignCriteria. He graduated from IstanbulTechnical University, Turkey, in 1968 andreceived his Ph. D. degree in transporta-tion engineering from the University ofLeeds, England, in 1974. He is a Memberof ITE.

ITE JOURNAL . MARCH 1988.27