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U . S . A IR FORCEP R O J E C T R A N DR E SE A RC H M E M O R AN D U M

R M - 1 4 8 8

STUDIES IN THE ECONCI'nCS OF TRAliSPORTATION *Marlin BeckmannC. B. McGuireChristopher B. WinEten

12 May 1 9 5 5

A5signed to _

This is a working paper. It may be expanded, modified, or with-drawn at an} time. The views, conclusions, and recommendationsexpressed herein do not necessarily reflect the official views orpolicies of the United States Air Force.

------------~Q~nD~700 MAIN ST SANTA MONICA. CALIFOINIA ---* To be published by Yale University Pres6.~

Copyright, 1 9 5 5 .,;The' Cowles COmmiSSi0l1~...v


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CHAPTER 2

Demand . ' * ' " . . . . . . . 2.12.1. Decisions of R~)a.dUsers 2.22.2. Ccs ts . 2 .4

2.2.1-2.2.2. Comjos Lt i.on cf ~a.nsports.ti')n Cost.'ITansportati~n C o s t as a function of F l ( ) ' " 2.42 . 62.). 7ne Demand for Road Use . . 27

2.~.1.2.).2*2.).).The Cn:Jice:fRot:.tes . . . . Sh~rtest Rout~s ar.d T rip Costs ...The Induced De.I:!anl:'cr Use of a Read.

2 . 7.2.112.142.4. The Demand fc:'r Tranr;~'rtation . . 2.15

2.4.1-2.4.2 To e Dem and for Trips . . . . . . . . . . . . 2.16Demand Func t.f ons 9..nd Capac ity Funct:!.ons 2.20CHAPTER 3

Equl:ibrlum . . . 3 1Equilibrium in a Network . 3 3;,.1.1* F=--nm.:lation of the Eq,t:.ilibriUIt. CondI t.i ons ).1.2* Existence of Scluticns t - : . : the EquEibrium

C ~:md..itiODS ).1.3* Uniq'Uene.:;s of the Sc_,lt;ticns ...

. . . :3.538 3 10

3.2. E!'fects of Changes in Capacity and in Demand ..... ).14).2.1* tn Inequality on the Effect of Data ChanGes ,.1)

3.3. Stability ........... 3 1 9;.3.1. Adjustments of Road Users., .) .2 . N uo :: er ic a1~ y. am pl es Gf A . ? p r 0 a c n to Equil.ibriw:l . . ,.19 3 2 3

CH .AP l'ER 4

4 . 14.1. The Prvblern 4.1

4.1.1. The A.J..l;:.caticm cf Road Capacity 4.1


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4.1.2. T.-.eMea..r"J.ing cf Efficien.::y ...... 4.24.2. CC6~ Finir:lizatiaQ on Tlic' Roads ..... , .... 4.7

4.2.1. P;'f,CU 's Problem . . . .4 . 2 . 2 . P r : : ' v G . ' ; ; e C0S:' 8 J 1d S : : : c i a. : i Ccst 4.7. 4.10

4.?i. Effi::ient Transportation in tlet...cr-ks ... 4.1543.:* ~im.izatL:n Sub..;ect to I.ine3.T Ine,~t.;_:lli"ties asCon~:rain:6 . .4.16M i ni mi z .a ti on o f T r8 l1 5p ( )r ta t~ on Cost Su 't ject t o 6-

Fixed Program . . , ,Maximization of Consumers' Surplus .4.19.4.214.4. Effic.ency Tcl~s .. .4.23

4.4.1-4.4.?"*4 ,J+. 3.J : ; _ , 4.

Inteo-pret.stion:f t be E ff ic ic nc :.E~ui:1~ri'~ end E~flc:ency F101A Lmi ted. ':'')11 8~:::te..'ll The Value ~)f a R'Je..:!

C_nei tfons .'+.23.4.25.4.274.2;4.; . Tell R c~d3 R econs:derec 4.3C

4.. ).1* Ana l ys t a . . . .4.31

C DC .us icn; S'~mc Uns o.Lvco Problems . . . . . . . . .).1. . c . .. .: .

The')~'etical Capac ity FunctLons . . . .. .).1;'.lter:13.~ivE)::-t.i".:15 of Ce.pa.::ity f2,r FC:))l;:.ic Ana Lys Ls . .).4Commocity 'rTJ.r:f ,:)!'tiitiun _}:odels. _). ~-:>~-aa.T.i::: S-:.lEibr:i..r:::l M"dei.s . . . .;.5-? r 6 t : l e r r . , S c f t n e L : n L : R t _ ; _ n . . j . 1 0T~l_s 9. . . "10 !?in~1r:ce .......... ~;.~.:::

~ ; ; - . -. . J .. . .,I- I) j.

. . . . . . . '...;"PART II. A S '! '"u 'D Y O F R AILR OA D 'l'R A.U SP OR "!A T!M

CrihPT2JI :;

- Tra.r~3~ to :'!....~e and Inyentcr:. :03tS .').2_] .. ~l. . . . ......2 .. 1':'L'nsit T~::: : :e ~ . l r . r J_ the S ~ . C l C Y . . 'f Frei.::;.'1t Cars .04- i


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The :-,:;'vi::.;iun 0-" Scr"ti:1~ l,., 'c:-k ;:'et ...een Yards ........lO.~.: 0 . : : : : .: " C '

l~.': 'c.t1c~i,n . . . . .1 - . Pyb2.t~ : 2 . _ i t y Rel,rescr:":r.:1t::":-r i ""f' Trains

::'C.4'" Dc:~i"~:'~.;.m ,"f the ~\):.(. h.:l::'_i.J~,:r e > t .: 2 .t ;: :l . !":'alSe q',: eric ~ . . .::'C .;:.. Ext.er.s L.J!! t., a M,:ce Ger:e:r:l: T:yT'c .f Sequence . .

" -_ _ I . . . . . .

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A SiL.1l.1e Sche:....\lli:-..,:~ Pr ibLem C)r:si:iereQ Sclied~lle.5 of C~ven St:r~~.::tU!'c 7t.3.t ]I': r.i::li::eDe:,J .:.Y M_--st Economi ce I 3chedules.. . .Sc~e~L:i~gcn a ~~tworkw!th :~rcui~s.

rll:c-r:uls. tion

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12.4.1 2 . .. ..12.6.12.(.:. '?5'"

~;rp06e3 of the Study . . .1:;. ::..1.I::c :Jel'1omds (If e . Trar-.s;-'~rt" tior. ?r'.:;!'Q." J!1 t!:eFrc i..::;Lt ' : : : : e . l ' t~eet . . . . . .J2.1. 2. P . Me:,l..'l.s cf Eva Lua t.is.,: the ::Xi 5tir.~ Ca.r Sen i cePruc~dure5 . . . . . . . . .12.1.). Sti..'t;!.lit:y:;f t.he SL'rt-H;:;u::' R,~\.!ting PatterrisTne Data . . . . .Si.~I?luses anc Deficits ' , ' o f ' Etn:;-o'ty h:XCOT5. A Sir"c:li fkd Rai::. Net,,::')rk .Fi:l:...inc the Sh.::;rt-H3~J.l R:)'J_t~n;r, P~-tte:'ns Results ..........ConcLusicns With Recrect t'l St.:::J::Ei t.y (A;-:i-.end:'x) Fr. rraaL De:::;cripticIl ~f the Surp:uG-De~icit

Ccr; ..~\.:.-:'ati(")rl ? rl..ce-:...u :--e ... '" ... "" ..

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14::rClc l t>c tionby Tjallin[ C. K'Jopmans

The ex plo rat or y stu di es pr esen ted 1n this report are addressed toana'Iye t-s 1n various pr:)f es sioru; , i nc ludine ; econcat at.s, traffic e . . . " 1 d . r!J.ilroade n g f .n e e r 's m a n s .g e m e n t s c I e n t . t s t s , o p e r 'a t . L o n s r e s e a r c h e r - s a n d m a t . h e m a t . L c L a n s ,

who are interested in asse~s1nb the cap~b11it1es anC st~dJ~r~ t he effi ci entoreration of transportation systems. Studies relating t:i t....sys te. 'llS ,h1g h-way traffic and rellrc> ai! tr8 .n.6portatioD, are c; f! ered.

The tasks indicated assessini cap~b1lities and ap?raising efficiencyof oper at .L on - - are as vast end complicated as the tranapcr t.a tion s ys te msth~selves. The purpcse cf the present studies 1s to develop and 1llustratec~ rt ai n con cep ts; methods and models that may have usefulness as points ofc.epart'.1refSJr t he e xe cu ti on of these tasks. While the aim c:f these studiesis thus botb modest and provisional, they are cffered in tee hope of stimu-lating further factual, conceptual, mathenatical and computat10nal researchlntc the efficient utilization of transportation systema.

The method en:pl~yed consists in the c~nstruction of siI:l:;,le: :> de ls . ' !b evord .'model", freq-....ntly used io eneineering studies to mean a ;;h:;-Sical model,that is, an accuro.te copy of the system studied ( often with the scale alt er ed) ~1s here used in the more general meaning attached to it by pcysic1sts endsocial scientists alike. A mode~ here means a s~plified conceptual counter-part of the nystem studied. In such 8model the most i."1J .porta.ntariables ofthe system studied are enumerated and defined, end the relevant rela.tionships~et"'een them specified. V arls.bles an'::'e3.ationshi:; t:shus ex press the mostessentla:' aspects of the 3y\'>t~ in question bl..tleave out many ot her asp ect s,Be that an cpening weOt:e fo':" ana.Iys Ls may be prcNiied.


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This analJ~1s is often mathematical in character. H~weverl themathematical underpinning of tbe st~dies bere presented has been set of!io separate sections, marked by an asteriskC.). These sections can bepassed over by tbe reader interested mainly in their results, which arefully described io aections Without that symbol

The application.of the method of model construction to operating or

bUI!I i RIt is p ro blem s h as increased substantially in recent years. To namefirst a fev examples uorelated to transportation, we refer to two studieson inventory policy b y Arrow, Harris a nd M ar sch ak [ 1 9 5 1 ] and by Dvoretaky,Kiefer and Woltowitz [ 1 9 5 2 ] , io which models are designed to help balanceinventory costs against losses from stock deplet1on. Another family of modelsconstitute the general field of "linear p ro gr rum ni ng "; a t ech ni qu e to computeprograms tor the interdependent activities of a large organization. Besidesth e or igi nal pr esent at io ns by Dantz1g [1951a] (jointly with M. K. Wood),[195lb, c, and d] ve mention en expository discussion by Dorfman [1953] 1nterms ot a problem of automobile production, and an application to gasolinebleniins by Charnes, Cooper and Mellon [1952].

Si mi lar stu di es in terms ot a very simple linear model of transportationhave also been made. TVo mathematicians, F. L. Hitchcock [1941) 1n the UnitedStates and L. Kantoroviteh ( 1 9 4 2 1 1 1 1 0 R uSSia, and one economist (the authorof this introduction) independently cf e6Ch other formulated the problem ofmost econon:.ice.lex.cution of a given transportation program between a numberof locations in the case Where tbe per unit cost ot t ransp or t~ t1o n bet weeneach pair of location8 i8 1~ependent of the amount transported. This model

-belongs to the cla.s. of l"lir.ear pro gramming" models e.nd has served as one otj} Tois reference vas brought to our notice by M. M. Flood f l 9 5 4 ] .


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RJ,-li..335 - 1 2 - 5 5-3-the .tock ex~ples of this class. Dantzig [1951d] explored the computationalaspects of the above (linear) transportation problem, vhicb were carriedfurther end applied to a military tanker fleet problem b y Flood [ 1 9 5 3 ] , ( 1 9 5 4 ] .Economists studied various implications of this model, such as the relativecosts of alternative possible changes 1n the transportation program and therelationship of these cost ratios to freight rates for.med 1 n c om pe ti ti vemarkets (Koopmans [1947], Koopm.snsand Reiter [1951]) as vell a. s the relationof freight rates to interregIonal price differences and movements of goods(Eoke [ 1 9 5 1 ] , Se.uelsoD [ 1 9 5 2 ] , F o x [ 1 9 5 3 ] ) . AD extension of the model toa situation where places of origin and destination are continuously di.trib-uted in a plane W E : I .S given by Beckmann [1952]. AD important general resultof the studies mentioned i& that, under the circumstances of the simple modeldescribed, a competitive market solves tbe most economical routing problem asefficiently as & cectrally directed transportation organization could.

Fr~m the point of view of the efficient utiliz ation ot t he t ra ns po rtsystems of road and rail, the appl1cab111 ty ot the simple linear ("constantcost ") model 18 rather limited. It ignores al pbenomena of congestion, eitherat terminals, or !!!.~. Perhaps the most iI:lportantC8.8e were this assump-tion is approximately satisfied is that of ocean or lake 6hipp1ng betweenuncongested ports. The linear model further ignores 1nd1visib111ties, suchas arise from the bunching of a number of railroad cars into a traio tor whichOne indivisible engine provides traction. To extend the analysis of transportation systems by model construction, it will therefore be necessary to"forget" the linear model, s.nd . te.ke a fresh look at the technological and"organizational circumstances of various transportation systems. However,-before leaving the linear model, let us point out one practical applicationof impo:-tance for the indiv1d.ua.l business firm that bas plants in se.veral'


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locations. This application arises if, even though actual cost of trans-portatloD aD each route actually depends cn the amount transpcr ted, e publiccarrier makes the serv1ce 10 question aVailable at a const~t fre1;bt rate.The linear model then suffices to aho~ this firm bow to minimize its billtor -tranSport atioD. from plants to customers, even though the min.imum billedcost flow. of goods eo computed have DO necessary relation to the best utl1-1zation of the tr~portat1oD system

. . . *In our study of highway traffiC, the main emphasis is on the effects

of traffic conoestion. Coneestion phenomena have been Bubjected to mathematicals.nalysie in the theory of telepbone systerlS ey Erlang (1n Brockmeyer, et a1-[ 1 9 4 8 ] ) a n d Palm [ 1 9 4 3 ] , and more recently 1 n a more general a n a l y s i s o f q u e u e s( Kendall [1951] ) such as arise in many situations: people waiting for servicein a b~\, ships waiting for access to port or repair facilities, airplanescircling to land, pedestrians weiting for an opportunity to cross a street,etc.

the calculus of prc'babllity, ho ....he average wsiting time, and the extent offluctuation in indi vid.lalwa1ting times, depend on the opportunities forserviCing (number of servers, average of and fluctuations in service time),and. on the amount and irregularity of the inflow of cle.ilL.a.ntsor service'.In cr~pter 1 of th1s report, queue theory is applied to such traffic situationsas: the flow of cars throUgb an intersection, and tbe passing of slower carsby faster cars through the use of gaps occurring i n t he opposing traffiCstream. In addition, in Chapters 3 and 4, the main results of such studies


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ruF.-l4885 - 1 2 - 5 5- 5 -are incorpora.ted into the B...5Sum;,tiODS of a ( ncn-linear) model of high ...aytraffic on a road network.

It was said at ~he beginning ot this introduction that one important -objective to which our studies are ultimately directed 1s the determinationot the capabll1 ties of transportation sytitema -- in the present cue of aroad ~etvork. It will nov be clear that this cannot be expressed in a singlenumber like so many vehicle-miles per day. It h an essential aspecf of con-gestion phenomena that -- up to a limit beyond which overloading startemore "service" CM always be obtained at higher un! t cost. Hence J thecapability concept to be analyzed has more dimensions than a single number.Even tor one individual one-way road, it 15 represented by a curve, wh1chgives the relationship between the flow of traffiC through the road and thecost encountered on it, 1n terms of the money equivalent ot travel time, rueland depreciation, and possibly risk or other sacrifices measurable in moneyequivalente. The higher the flow, the greater the cost encountered. Thetime element alone, probably the most ~portant cost factor, ie represented1n the "capacity curve" of the traffic engineer. In this curve, flow is setoff against average speed, the reciprocal of tra.vel time. T'n18 curve and theconcepts associated with it, are presented and discussed in Chapter 1, whichie based on study of the relevant traffic engineering 11terature~ For 8roadnet~rk, the capabilities are expressed by the relat10nship between traff1cflows on all routes and the costs encountered on each as a result of thesefloW's.

A theory of highway traffic 6bould of course go beyond a descr~tion ofthe capabilities of a netvork to a study of hov these capabilities are utilized.This introduces the concept of demand tor transportation. The floW's of highwaytraffic are the result of a great many individual d eci si on s abo ut d est inat io ns,


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routes and prefe:-red speeds , Tnere is somewhat of an a.ne.loe:;yetveen highvaytre.!f1c equilibrium and. the models used by t he eco no mist to explain quant ftysold in a market bY,the interaction of demanders and supplier . The behaviorof demanders is summed up in a "deme..nd.urve" 1 which sta.tel the amounts takenoft the market at various alternative prices. Similarly, the supply curvestates amount. offered for the various possible prices. Actual price is thendetermined, by intersection of the curves, to be that price at which demandequals auppzy.

O tl the supply side the analogy is not a cLose one. Except tor toll road.,there is no party 1n the market selling access to roads for &. price .. Eacbpiece ot road not occupied or endangered by anotber vehicle 18 free tor useby whomever is near. However, tbere is a cOst of tr~spart& tion, incurred1nd1 vi d ually in t er ms a lr ea dy diecussed.. If we regard th1. cost as the "price"1n the transportation "market It, the .economist' 5 notion of a demand curve doesbecome applicable. For a 81ngle one -yay road 1t would state what flow oftraffic 1s demanded at any given cost encount-ered on that road. The higherthe cost, tbe smaller the flow demanded, otber things being equal. For aroad netvork, the demand function vould etate what flows are torthcocdng oneach route 1n response to given transportation coets along tbese routes.Equilibrium 1 s e st ab li sb ed if the floys on all roads arl&ing in response togiven costs are just of the magnitudes that produce these same coste.

The demand concept is developed in Cnapter 2. It is applied in Chapter3 to a study of traffic equilibrium on a highva,l' network. Tne stab111 ty of,this equilibrium is also discussed, ani same cbservations are made about theuse ot the analys1s 1n the prediction of traffic flows.

Mathematical tools used 1n this analysis (in particular Sections 3.1.2,3.1.3, and 3.2.1) may also bave an interest to the mathe~atical econ~mistapart from their present application to highway traffiC.


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R;,t-14835 - 1 2 - 5 5-8-e. s to himself) of hie choice. If there vere a. va.y to collect tolls fromthe users of congested road I, at rates that measure the cost to otherscaused by th e a ve ra ge road user, ~lle the revenue so collected is usedto lover gasoline taxes or otherwise benefit all road users, then a betteruse of the highway system would be obtained. Cha.pter 4 analyzes as a the-

_ o retica.l proposition how the amounts of such "efficiency toll rates" wouldbe determined. It also contains lome observations as to'how closely maximumefficiency can b e a pp ro ac hed by proper choice of the toll rates on roadsthat are at present toll roads, or by other ways of penaliziog addit100s tot raff ic co ng es tIo n.

In this diSCUSSion, tolls are looked upon, not al a means of financIngroad construction, but as a means of bringing about the best utilization ofthe highway network. This i8 in keeping with the growing acceptance amongmodern economists of the proposition that beat uae of facilities require.methoda of pricing the services of these facilIties that reflect the incre-mental cost attributable to each service demanded by aD individual user.Because of the non-linearity 1n the relatioD betveen emount of use and cost,sucb priCing does not necessarily produce revenues equal to the total cost,of opereiing and financ1ng the facility. This Bame principle has been appliedby William S. Vickrey [1952] in formulating proposals for fares 1n the NewYork subway which would diminish congestion by providing incentives fortraffic to shift froo; peak to off -pea..khours and to encourage fuller use ofoff-peak service by lover fares at these times when incremental costs arelow. It is also basic to cont~porary theory of electricity rates (see, e.g.h. S. Houthakker ( 1 9 5 1 ] ) .


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Rt.-14885 - 1 2 - 5 5-9-It will be clear that the8e considerations leave unansvered the q~estion

ot criteria t~r ex tensions o r improvement s ot the network to relieve coo-ge.tion. Some observations on the latter question that fl~' fram the presentanalysis are given in Cbapter 4,Section 4 . 4 . 6 , and a180 1n Chapter 5 , wh1chmakes brief mention of many u nso lved p ro blem s ot traffic theory and analysis.Hovever tbe main part of the present study concentrates on what the economistcalls ".hort run" problems. The road network, represented by & configurationof roads and a capacity curve for each road, i8 taken as given. Demand fortraffic on each route is represented by a fixed function of current cost;that is, if after considera.ble fluctua.tion cost were to return to its formerlevel on each road, then demand vould also return to its former level on eachroute. Thus the more gradual responses of demand to changes in coat that arise from relocation of residences} stores or plants are not taken intoaccount. It is believed, however, that the prese~t analYSis can be ulefulas a starting point in developing a theory of balanced extension of the hIgb-way network, concurrectly witb industrial expansion or relocation. The 1n-creased vulnerability of metropolitan areas under modern warfare adds a noteof urgency to the development of 8uch a theory, already highly de!irable beforethis complIcation arose

*D1fferences between our studies of railroad transportation and those of

highway traffiC reflect the different characteristics of the two transportationsystems. The tact that highway traffic results from the interdependent ehoicesof many decIsion makers, each with an objective of his ovo, gives to traffictheory a stroug soc1al science flavor. In a railroad system, operations are


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at least 1n principle centrally directec. 00 tbe o~her hand, the technicalaspects of railroad operation are R great deal more complicated than thoseof hlgbvB.y use. Hence, our exploratory study of models of certuo ra.ilroadoperations is sOQevhat closer to physics or engineering. In order to as.esland describe the cap&l:111ties of a railroad network, 1t is again neceuaryto conltruct stmple conceptual models of various parts of railroad operation.Our study, which makes a start with this task, eonfi~es it.elf almost entirelyto the supply side of railroad services. Ooe exception to this is the dis-cussion in Chapter 6, Section 6.1 of the value to the shipper of speedy trans-portatlon. Another exception is the discussion in Chapter 12 of best routingpatterns for ecpty cars, to which we return below.

It is probable that cong~tioo phenomena, which hold the center ofattention in our analye1s of bighway traffic, are 1m; > ortant also in railroadoperations. However, it hae appeared to t~e authors that other aspects ofrailroad operation require prior attention. Ooe of tbese arises from thetact that I n m os t c ir cu ms ta nc es it i s economjcal t o haul c a r s in trains rath~rthan Indlvidual.ly. This intruduces the problem ot " a c c U ! I I u l a t l o n delay", tbecar-time spent waiting f o r e no ug h t re. ffi c t o e c c u m u L a t . e 80 t h a t a. train can,economically be formed. Another 1s the protl~ ot classificatioD, that Is,the problec of sorting cars that arrive in 1nc~ng trains or are deliveredfrom loading tracks, so as to make up new trains that will take these carsto the next sorting point, closer to or at their destination. The problemis how to distribute this sorting work over classif1cation yards 1n a waythat min~1zes cost of claseification plus the money equivalent of accumulationdelay.

Basic concepts to make possible an accurate formulation and treatmentof problems of this kind are introduced in Chapter 7. In Chapter 8, tbe


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JUf , 14S55-12-55-11-o per at ion s of classi fi cat 10n yards are de8cribed, and simple approximativeformulae are proposed for tbe dependence of classification cost on toe cla.e-lflcat10n tuk performed. The.e formulae are used in Chapter 9 in a generaldiscussion of the distribution of classification work over l~rdS.

An earlier version of the material in this chapter and the ODe precedingit was presented to a meeting of the Railvay Syst~ and P ro ce du re s A ss OC ia ti on ,

held in Chicago on November 5, 1953, and was published as an appendix to thesiproceedings of that organization. 'The material i& bere reused and extendedwith the permission ot the R. S. P. A. Cbapters 10 and 11 are more detailedstudies of specific problems. Each of these chapters ignores ..mat the otherconcentrates 00. Chapter 10 1s devoted to the problem of distributing class-if1cation work between a bump yard and a flat yard located down the line fromthe hump yard, when problems of scheduling are ignored. In Ch~pter 11, clasl-ificatioD cost is 19noredl and instead the problem of scheduling trains betweenyards to minimize accumulation delay is discussed in detail for a Single-linerailroad, and in more general terms for mOre comp licated rai lro ad n etw ork s.

From this summary the reader will see that time and resources availablefor the study did nct permit us to construct a model that simultaneou~lyincorporates all the main aspects of railroad operations, on the basis ofwhich one couldl for instance, discuss the beat dovetailing of classification,scheduling and line-hauling operations. For that purpose, further "partial"models would be oeeded first, such as a model tor the study of track capacity.In ad ~i t1o n, co nsi der ab le co mp li ca ti on s of a purely mathematical characterwould arise in the attempt to put the partial models together into a modelthat would fully express the 1nterdepe~dence of the main elements of railroadoperation. It is felt. hcwever, that the first steps on the road to a moreiJ- Beckmann, K oopmc~s, M cGuire, and Winsten [ 1 9 5 3 ) .


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integrated model have beeu marie, ~~d that this in itself justifies thepUblication of these st~dies.

A more integrated model ....Uld have several uses. It w'mld help incomputing the capabilities of a g!ven r a i l r o a d network, and the rollingstock requirements of any tre.nsportat1 In prcgram t h a t is vithin tt.e capa-bilities o f the n e t.vo rk . I t would a l l ! : : ; a c i 2 . i t a t e e a t i : : I a t i c . g t h e "incremental"or "marginal" cost (i.e.) t h e cost i n c r e a s e ) occasioned by t he r en der in g ota n extra u n i t o f s e r v i c e .

Th e i mp or tan ce e, f tbe latter c:)n5icieraticmlies in tbe fact that freightrates are an important element i~ business decls~ons about industrial locationsand about rnudelJof transportation used. Only if t h e rete on each unit ofservice reflects its incremental cost can we expect that such decisions, takenby ~r0fit-seeklng entrepreneurs in response to freigtt rates and geographicalprice differences will lead to the mQst efficient utilization of the nat1aa's

It is true tha.t this vie....held by mQst eCQn:)!:listeJ..tas not beenes .urcee .accepted by reeulatory bodies as relevant to rate making. However, theecvnomist's ca.se is likely to rema.in just a nice peint of theory unless theoperations of railroads are analyzed t,: the extent nece saary to provide amethod of estimating the incremental ccs ts ':If trc..nsportation services renderec..

It may be useful here to recall the main properties of incremental costin the very simplest linear model of transportation reentioned at the beginningof th1s intrcauct1on. vf.ere b ot h co ng est io n ~d tile fact that care are l~.pedinto trains are ignored. In this modeI "efficiency freight rates ", that is,rates reflecting incremental cost, a re r el at iv el y low} per mile, 1n directionsin which empty cars pr')ceed regularly, and. relatively bigh in opposite direc-

:'-'." tiona. On r:.:.)uteso t t ravelled b;>t empty cars t he effi c: ;'en cy fr ei gh t r at esjf S~e, f o r instance, J . Dupuit [ l 8 4 4 ] , R . H ot elli ng [ 1 9 3 8 ] .


19/358

h av e i nt er me ti at e values, which can be ~eterzined 1n the 5~e calc~lationby ..'hicb the beet routing plan f::r empty cars is d.eterm:ne6..hL In any morerefined model, incremental cost freight rates are likely to exhibit thesemain features, with the effects of ccngestion and lumpiness 5upe~imposedas modifications. It is therefore worthwhile to examine the pattern ofempty car orig1n3tions and terminations associated with tbe movement ofgoods on U. S. railroads I the pattern of best r'out.Lngof empty cars betweenorigination and d est in at io n po in ts, and the stability or varlabili ty of thispattern between years. The examination of this questl~n in Chapter 12 revealsa substantial stability of best routing patterns of emrty boxcars -- and henceof incremental ecsts cf transportation of goods Shipped in tbese cars -- asbetween peace-time years, and renarkab1e changes connected with war-timemovements cf 6upplies to Pacific coast ports.

* *: *

This report has resulted from a research project carried out by theCovles Commission for Research in Economics under contract with the RANDCsrporeti0D. Tjalling C. Koopmana was the research leader of the project.The several authors came to this study witb different skills and backgrounds,and accordingly contributed.. i n different and c:)n:plemente.ryways to theircommon task. M ar ti n B eck nan n, a math~t1cal eco nom ist esp eci ally i nt er est edin linear programming and. economic activity analysis t cr.nt r-fbut.edmost of thecbapters of the highway traffic analysis, with the exception of Chapter 1 00capacity. Christopher WinsteD, mathematician and economist with a specialinterest in applications of probability calculus to industrial phenomena,~ See Koopmans and Reiter ( 1951), Dantzig (19)le).


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I'JI,-i4oo5 - l 2 - 5 5-14-contributed the analysis of queues reported 1n Charter 1, and. t he an aly si sof the division of serting work between yards. given io Chapter 10. C. B.McGuire, ecoo~ist, was assigned primary resp~oslbillty for the degree of

recorda. Most of the Chapters on railroad problems are the result of jointwork by McGuire and Winsten. Chapter 11 was contributed by Koopmans. Chapter12 was prepared by McGuire cn the basis of earlier work by Kirk Fox, MarcNerlcve J Rar1an Suits and Thomas A. Healy. The computations for a simpleexample of a highway netvork vere prepared, and the report thereon in Sec-tion ).3.2 written, by Goldman.

This dry enumeration of contributions does not indicate the extent 'tovilleh practically every chapter has been e.ffected by the thinking of allmembers of the group. A good deal of group discussion has been devoted inparticular to choice of concepts and mJdels cf railroad operations. In tbelater stage of preparation of manuscript, the group met 1n veekly sessionsfor critiCism and evaluation of successive drafts. Final editing of them an us cri pt wa s done by McGuire with the assistance of James M . Terrell.

Proper acknowledgment cannot be given in this space to all those whobelped the authors 1n their project. Especially deserving of mentionl how-ever, for the patient way they dealt with the authors' questions, are thefol1cwing men 1n tbe railroad industry: C. H. Bremhoret, L. H. Dyer, and


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S-J2-~~)-l~-

E. P. Stine of the Chicago, Bur L i.ngt.cn and Guincy; k'thur H . Gas s ofthe Car Servf.ce Division, Ass)ciatioo of Amer!can Ra1lroa6.s; C. E. McCartyand R. M. Zimmermann of Potorrc Yard: T. J . O'Cornell and ~. E. Bertrand ofthe Baltimore ~~ Chic; W. h' McClintic of the Pere ~:arquette Division ofthe Chesapeake and Ohio; E. E. Foulkes of the ~~ck Islani; and Val ~~cecf M; "dern RailrQa.is. For the frontis; ieee. an aerial photograph of Bensen-ville Yard. cn t he : :u tsk ir ts cr Chicago, we are indebted t:)the MihraukeeRead.

Discussions with vther m~bers of the Cow~ee C~issiJn research staftand with visiting scho rar-s have also been stimulating and helpful. Amongthese, we wish to mention in particula.r ....Feller of Pr:! .ncetonUniversity,H. S. nouthakker. fcrmerly of the Commission and n:-w of Stanf'crd University,D. G. Kendall cf Oxford Unl ver-c tty , Rar:-y r1arkowitz , Ge()rge Dant.z Lg , andT. E. P.o:r':-is cf the RA.!IDCcrporat.f on , and r1::'111am S. Vickrey of ColumbiaUtivers:ty. The le.st-mentioned has read the entire menuBcript and giventhe auth~rs the benefit ~f many detai~ed csmments. Of course responsibilityfor what is cffered ~est6 with the respective authors alone. If thesestudies stimulate otbers to impr~vlng on them, tbe auth0rB v111 feel thatthelr endeavc r nas been fully re....rc.ed.


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,,, ,. ,+ ~

__ ~--,;".- _ . : . ' . . . . . . ;-

PA.l1T I

1~ S?lIDY OF


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Cha.pter1R!1-14.885 - 1 2 - 5 5-1.1-

Road and I nt er s~t io n Cap ac it y

1.1 Introduction.In moat cues where an attempt bas been made t o mea. au .r et he cap aci ty

of 8road or inter.ection, this capacity baa been taken to be a number,repreient1ng the highest possible flow of traffic through the facil1tybeing studied. Accepting th1. v1ew tor a moment, let ua examine a particularc a. se 1n .cae detail and eee into what complica.tions we are led. Supposea cer ts. 1n u ns1E; lle. .lledut ersectio n is uaed by eutbound and northboundtratt1c only. What 1& 1ts c~ac1ty? Obvioualy no one number Will sufficeto describe th e capacity of the intersect10n tor nortbbound tloV5 alone,

,.. .,

Figure 1

for it 16 q~ite clear that the more eastbound traff1c there 15, the lesswill be the amount ot northbound traffic that can get through unleas thislatter flow in lame way dominate! the 1ntersection, and we rule this out


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R.:"~-14885 - l 2 - ; 5-1.2-

for the preleat. Perh~p6. hoveverl it 16 pos!ible to express the capacityin terms of a number, eay 1600 vehIcles per houri Which i8 not to be ex-eeeded by the sum of the two floys. If this is so, then this. capacitycan be expressed 1n the f'cllowine lent; thier but more 1nstr..I.Ct1ve way. '!becapa.elty 1e

o v eh ic le s p er h ou r n or th b~ ~ and 1 6 0 0 v e h i c l e s p e r hour eastboundor 100 " " " n 1500

1400. . " " n

or 200 n " to n . . " " II, .or 1 6 0 0 . . " H t. o n "

This set of combinations YTitten out 1n full says precisely the same thinge .a the sho rt er d efin In g sentence preceding it; lie can if \Ievish thereforealvayB describe such capacIties in terms of the various highest possiblecombinations of floys. Slnce tbe set of such combinations 1s rather tediousto write downl it 1s convenient to describe it graphically in the followingway.

c' ...,_.C I,d' -- ",- - - - - -II

D

cFigure 2


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R:>i-lt.;.3b5 - 1 2 - 5 5-1.3-

Tbe set of capacity flow ccxr.tins.tloIlfl Ct.D be re:presellteo. graph1caJ.ly inFigure 2 as all those points witb non-negative coordinates xe' ~ Whichlie on the dOYDvard sloping line whoseequation is xe + Xc 1000. Thewbole set ot combinations is therefore co:::1pletely


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Rl~-l4.&35 - 1 2 - 5 5-1.4-

combinationa, the total flow represented by A is less than the totalflow re:pre&ented by B. Total floy in these examples is no longer animportant element in ca.pacity cons 1d.erat1ons

Betore leaving thia stmple intersection exsmple it 1s worth pointingout that the curve, or set, we have been te.l..k.1ngbout repres.eots capacity1n t o e .ense that it forms a part of the boundary of all those poiots rep-resentlog poas ible flov combinations. Thus 1n Figure 2, the poiot C withcoordinates c and c' is a possible flow camr1natloD since the e~ ofc and c ' 1s leu than i soo, while tbe point D Vith cocrd1::J.!lel d andd' 1s an impossible flow combination since the aum ot d and d' isgreater than 1600. The possible combinations are those repreaented bypoints on or beloy the capacity curve; the impossible ones by pOints abovethe curve.

The reader may object at thie point that ell ot this elaboration haabeen unneces.ary, because for most of the important cases the capacity curveodoes 1n tact just happen to be a stra1&ht line sloped at 45 eo that capacityexpreued as 8 11m! atioD on total flow is rea.1ly all that isneeded. Itall of the interesting cases were 11ke the ones in Figure 2, the objectionwould indeed be veil taken. We intend to ahow however that as soon as thecapacity Dotion Is complicated someYhat to make it more useful fram aneconomie point of vlew1 the interpretation 88 8 eet of possibilities beeomesimperative. To make this clear, another _ imple and ratber artificial examplevill be described. So t e x ve have not conSidered the driving con~1t1onsWhich the traffic Viil meet, even when tbe flow. ere possIble ones. Onev~ of introducing the&e conditions into a definition of capacity 18 given1n the ijir.b ......y Cane,eity Mn.nual,!i where, very briefly:


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RJv;-l4()8')-I~'~)j-1. s -

basic _ca;..a:::1y is the maximua, fl:.' " \ o ' " uncer "must near Ly idealroad: ..ay and. t~affic conc itt cns ":

pOEsic} e ca;adu js the maximum f _ ; . . , ' ) w under "preve.iling r oacvay

'fract1cal C!o.;:,a;j,y is zne maximuz r : .. o . .. short of causing "unr'eas cn-

Thus f:.r each traffic ~~'nd1tl,:-n ~recifiec. there is a cr r-res ponctng maxtmumf l o " ' .

A_.crnat ive Ly caj.ac ity may be definec as a :-e:!..ati'n s~)""in8 how trafficconditiens depeno ::..nflo ... l"n15 is tc be ;::.referred if the w:.t:g'J.1ty ofterns like "unr-eas cnat :e " is t,:~be tc,roiieu. This _;.,rJce'.:mremeans that c:,t.....l or t.nree artitra.::-ily selected trs.ffic c~'n:;'::'i::ms are c.mcent.r-at.ed DC,but T6.:he:, the w h o' :" eran ~e c f !:raffic C'Dc.iti)ns is examined. Just as the

TIle word "de'. ay" in the a.escriptian vf the traffic conc itions thatdefine practica.l capacity cover-s what is }rolo..b:"_;y t~e most i.JI:porte.nt. elementin trs..ffic ccnd itIons . Crnc itions ar-e gcod if d,~le.y is smaLk ; they are badif delay is Lar ge . In the folbwing XEl.!!l; le. all,j_ in fa.ct in near-Ly all ofthe subs equer.t. discuss i::-n ir. this chapt.e r we sha.:'l Su~,::~o8e that "trafficconditions" are f'uLl.y ceser-feed by an as se s smerrt d' the delays that occur .

Let us iIlla[:ine a. certain vehicle ins}>ectl,'n s t.at t on at vua ch all vehicLestr&velinz a . particular r::,;a.dmust stop , Suppos-e tnat the L~s}ectI cn t.akesexac tLy t.....minutes f,_r each vehLcLe . that oz.1y cne vehicle can be inspe.ctedat a tir;le and that the arriva.: cf vehicl.es at the station is known to ber-andom, 1r.'hat is the capac Lcy uf the statiDn?


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?J/-14-825-l2-5~'-1.-

The exampl.e woulj be comp Let.e Ly ur trt.er-e stIr.g were it not, for the factthat a.6 f'lOVBbecomelarger, "tra.ffic coni! t1oW!", that 1s, delays, getprogressi vely trorse. Any particular car ha.e not only to wt the tw ominutes until its own inspection 1s completed~ but until the inspectionsof all cars w u t1ng whenit arrived are finished &8 well. The heavier thetraffic, the more cars it 1s likely to find waiting ahead ot it. If theaverage delay that ears suffer 1s plotted as a function of tlow, a curvelike . A B C D in 'igure 5 results. For very small rlm.'s tbe staUon vi11 seldanbe occupied and the aversee delay will be close tc two minutes. As flowapproaches ,0 vehicle; per bour, the stat10n vill usually have several vehieles waiting in line and the averaee delay i. 11~.ly to be large.

Averagedelay

d--------c

I' % E I.I Ib II2 A III

IIIIb' cl d' 30

F1g. .u-e 5

IIIIIIIIIIIIIIII40 Flow(vehicles/hour)


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/.

R:I-l4005 - 1 2 - 5 5-1.7--Proce~~1ng just as 10 the 1nterseetioo example ve can express the

capa.city as the set of combinations0 vehiclea per hour and 2 minutes average delay

or b' " . . endb . . . . . .or c' " , . " and c . . . . . .or d' " , . . . and d " " . .

etc.

Nov 1ma.gine another station at whlch tvo vehicles OWl be 1D.1pected11m'..l1Wleously, but ",here such iospection ta.kes exactly three minutes.The capacity curve f o r this station 15 IF in Figure 5. With flows cloaeto zero the average delay 1s three minutes} a little greater than for theotber station. At C where the tvo curves crOSB toe advantage of the firststat10n in terms at a ah or ter ser vi ce time i. exactly compensated for bythe second station's ability to deal with congestion. T"nu'J in a sense,for amell flovs the flr.t etatioD has the greater capacity; fer high flowsthe second stet10n h o. s t he g reat er C6Ilacity. Such a capacity ccmparison,would be arbitrary Inc.eed if it were ma:le on the basis of knowled.Ge ofdelays at tile tvo statiO!lS corresponding tc one particular flow, or on thebasis of knowledge of the floVB at the two stations corresponding to oneparticular value of average delay.

Note that, if we vere to plot a sim11ar function for the intersectionexatlple given above , then for each com'tination of flows ",l'Jichhe 1nter-Bec tLon could handle) w e would bave an average delay. The capac1 ty curveshown 1n the figure might nov be euppoced to separate the combinations offlows whieh bad acceptable delays fram those where the averaGe del~~ weretoo large to be acceptable. \.'benw e are con61iering tvo flovs in this ~.y1


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Rhi14b65 - 1 2 - 5 5l.8

ve could 'break this average delG.y ciown into tva puts J those experiencedby eacb of the tva Itresma. ~~t kind of tuoctione of the traffic den.iti.,t hese d elay s ar e depen! s especially OD the traffic rule5 governing the inter-section. Thi6 question will be discussed in Section 1 .2 be lo w. . .

With tbi8 introduction, it 18 time to enter into 8more detailed d1.-cuss Ion of the capacities of various types of intersections and roads.1.2 The Capacity ot Intersections.

Wehave sketched above the cot ion of cape.city we intend to use. Beforew e can develop quantitative expressioc! for this capacity notion, we musteo~1~r 10 more detail the rules governing traffic at intersections, andwe nruJt specify t he si mp li fi cat io ns in evit ably involved 10 constructing amodel o f traffic behavi~r. Some of the previous work DC intersection capacitywill be d16CUSSed, and a nev model will be given, which it Ls hoped, v111 beuseful fo r q ua nt it at iv e p re di ct io n in s~e situations.1.2.1 The Stop-Sign Intersection.

In a . simple type of such an intersection, a minor road crosses a mdJorroad. It 11assumed that trs.1'fic 1n the minor road m'J.stnot interfere withmajor-road tr~!1c. Henee a car reedy to ercss the major road must waituntil there 1s a sufficient gap io the major+road traffic. The burden' otJudSiog when a gap is sufficient lies on the m in er -r oad d ri ver . It 18poslible for theee J udgpents to vary considerably, botb between one drivera.nd tbe next, and bet10ieen d ifferent intersections. Driven tw.y vary beca.l;seof d if fe re nt d ee ;r ee s o f cautiousness, e.bility to pick up speed, and 60 on.Intersections var~: 1n y1aib111ty 1 width, and. many other factors. It isnecessary to consider vhich of tbese factors to bring into a q ua nt it at iv emodel, and jUBt how they should be brought 1n.


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R ! " - ~ . .d,S5 - 1 2 - 5 5-19-

1.2.2 Inter.~tion Cootrolled by Tr affic L igh tsTraffic l1ghts may have various rules of operation, and the delay

Vill depend on whIch of these 1s be1n.g used. '!'he most CCIIllinOD 1e the fixedrepeated cycle. With this rule a green interval of fixed. length ia fo11o ..8dby a red interval of fixed length (though the length of the red interval mayvery vell d,iffer from the length of the green one). 'lbe wole cycle 1s re-peated in5.erto.1el)'. There may be add! tional werning intervals between thegreen and the red intervaJ.s or betweo the red and. the green. There aresome pos.1ble variat!Jn8; for example1 lights can be made to ~~ange theircycle according to the c..eMIies of the traffic in the tvo ro&d.s. Sometypes of light. give priority to one of the roads, but we do not diseua.then here.

In botb the .stop-sign case and the traffic-light case , cars turninglett or right are en add1tlocal cause ot variation. Buppose for exampleIthat at IS stop 1160 eocar in the minor road. wi.bee to make a left turn intothe major road. It will have to vait for a gap in the major-road traffIc,and allo tor a. gap in the minor-road tre.tt1c i% 1 the oppoling .tream. Thusit will 00 the average bave to ~t longer than a car going straIght ahead,and 1t 1s likely to delay traffiC behind it longer. 'lhus the delay at the1ntersection may well depend 00 the ouQber of cars makln&lett turns, andon the densIty of traffIc 1n the opposing stream in this situatioo. Stm1larcone1derations apply to the traff1c-Ught cue. Tneee complications arementioned to show that quantitative statements about d~l~~ at intersectiona,and their relations to traffic flows vill oot be of completely general applica-tion, ~~ether they are derived from theory or f r om e x p er ~ eD t al observat1onof actusl road conddtions. Many Intersections maypresent specIal featureswhich call to~~odification of the reault


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5 - 1 2 - 5 5-1.10-

1.2.3. Related W~rk in tbe LiteTsture.Before we present our contribution to the theory of intersection

capacity, we will give some very brief reterencee to related work dee-cr1bed in the literature. The discus.ion bas no intention of being ex-h all' ti ve; i t 1s only meant to call attention to some attempts to developa quantitative theory of congestion at intersections.

In order to develop a theory of dela J w e bave to have someYay ofdelScr1bing tratt"1c tlev. Because of all the various unrelated factorsw hi ch d et er mi ne vbether there will be a car at a particular point at ap ar ti cu la r t im e, it w as s ug ges te d by ~ [ 1 9 3 6 ] that 8p ro ba bi li ty m od elmi&ht give a good tit to some classes of tra.tfic. Adoms suggested a s!J~e1a1type of prob&b111ty codel called the Po1B~on process 88 a possible des-cr1ptloo of the times at which cars PMS a particular point. Ris exper1meritalwork confirms that thil process gives an adequate deaer1ption for some road.,and. for some purposes , It has since been used in vork on traffic congestionby o th er w rit er .: G ar wo od, G reen ah 1el~ , R e t t , and Tanner, for example.

A theoretical t reat ment o f Itop-sign intersection de~~ 1s given 1npapers by Rs!'f (MarCh, 19511 and T'oner r 1952] Both use the same assumptionsaDd simplifications. It 1s IUppoSed that C~ in the minor road do notinterfere wi tb ee.ch otber. Tn15 assumption vill be e. valid one if tra.fficin the minor road 1a sparse. Tanner framee his discuBsion in terms o tpedestrians crossing a road; since pede6tri~~ c an u su all y crOBS 1n group.if necessary, all those waiting can cross almost s 600n as there 1s asufficient gap in the traffic. Thus it seems 8 good approx imation tosuppose that the lenGth ct time a pedestrian vti ha.ve to wait to crossdoes not depeod ~terially on hov many other pedestrian! are ueing the


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RM-l4885 - 1 2 - 5 5-1.11..eros.ins. In other yords, the assumption of non-interference seems &valid one in this case too.

It1s alao De~ee6a.ry to have acme more preeise detini tiOD of "sufficientgap". In the theories developed 10 far the DotioD of a tixe:l critical timegap is used. It 1. supposed tbat the dr1ver 10. the minor road. will onlycroes if there i8 no car 10. tbe major road due at the intersection in thenext v eecoo(ia, say. Under theu circumstances w 1s called the criticalgap.

Wecan reprelent the arrival. at the intersection o r car& 1n the majorroad as po1tlt. of' e . time aJds, 8.6 1n Figure 6.

- - - - - - ~ ~ - - - - - - - - - - ~ ~ ~ - - - - - - _ * ~ - - - - ~ ~ - - - - - -imeFigure 6

By us1ng the not1on of a fixed critical gap, we can, it \{e are given a plotof the traffic of thiB ecrt., say whether a car 1n the minor road can orcannot cross at any given time. It i8 na.tural to call the time intervalduring which a car in the minor road i! not able to cross a red interval andthe time 1ntervsJ. in which 1tcan cross a geen interval. ThusI if' we havea plot of arrivals in the major road, v1th our a!sumption of a fixed criticalga.p, w e can d1vide up the time into a sequence of ioterva.ls, alternating redend green. This procedure if!. 111Ultre.ted in Figure 7, where the care on theleft end of the time axis are later arrivals than those on the right.


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RY-ll.835-:!2-55-1.12-

J( x it X Arr! vaJ.s inMajor Road

1------. ,..--I ---1 - - . . . .

red green red geen red greeD redRed andGreenIntervals

rigure 7

Acar arrineg 1n the green interval will Dot bav-eto wa1t, but a cararriving in the red interval will have to wait juet until the end of thatinterval. Aa ve are 5'..lppos1n,sthere 11 DO other car 1n the minor road 1nfront of it to delay it, it Will be able to leave at the start ot the greeninterve.l.

To f1nd the average wait io the minor road, we bave to Itudy the die-tr1butiaa of the length. of red and green intervals. The papers by Re.:rr andTlaJll1erdo thi6 tor the case 10 wh1ch the arr1 vals 10 the major road. can bedescribed as a sequence formed. by a POi:!8oDprocess. Toe MButlptlon thatsufficiently large spaces occur between vehicles 1s the only one neeeS8aryfor the m:inor road.

Bot1ce that the reasoning in this caee can easily be ex~ended to thecase where the critical gap eccept.ed varies from car to car. If we knowthe relative rre~enc1es of the different critical gaps, we can find the


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fe'!-14885-12-55-1.13-

mean delay tor eacb one, L~ the~ aver&o~ the reeults us1llb these relativefrequencies as weights. In fact I Raff pre&ent&emIpir1ce.l evidence whichshows very clearly thst there is considerable variation in the gaps driver.consider lone; enough to alloy them to cross; so that 6 refinement ot thissort ~! be useful. In th~ ease of dense traffic in the minor road, Ybichw e treat later J there is no s'-1cheasy general1:tation to the case of 8. vari-able cr1tical gap J and we Il:iUsttor the time beins be content With the approx-imation that aupposes it to be constant.

Samemetbo~s of calculation tor the repeated-cycle tr~fIc-lIsbt caseare presented by Garwood [1940-l941} a n d . Ratt [19:)OJ. In these papers ana.ttempt is made to deal vith the phena:nenonof "sluggishness". If a lineof oere is wa.1t1ng when the light t.uroa green, 1tWill t.ake Bane time toclear. Bela,.. we give a ne...method ot des.11ngwith this case, which give.us a uaeful approximat.e formula wben traffic 1s sparse, and which cea beused to give re.~ltEvUCD traffic 16 heavy.1.2.4 A MJdel of I nt er se ct io n D el aj ~

TOe resulte we bave meot10ned above for the stop-sign intersectionwere restricted to tbe cue for lo'h1chtra.f't'1c in the minor road vas sparse.In this section ve g1ve a ~thod of extending th1e type at mod.el to the casewbere the tre.!tic in the minor road 11 dense, end CaTS axe delayed Dot onlyby the traf':fic in the major road , but by cars in front of tbem 1n the minorroad too. The type of model ve ceveLop turns out to be Uletul for the re-peat.ed cyc Le traffIc l1i;ht case also.

The basiC tact ~~ have tried to expreas in the model ie thnt a queueof cara waiting to cross a road cannot clear all a.t once, and the longerthe queue the lODGer 1t vill take to clear (the pheno.men~nalread.y described


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R1


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5 - 1 2 - 5 5-1.15-

tvc. Acar err1ved at the fiftb t!.De-po1nt, but not at the next three,-and 10 OIh

If ve represent red t1me-po1Iltl by the symbol r, 8..'1dGTee~t1me-point. by the symbol g, then a sequence of the rom:

r r r g g g g r r g

shov& the firat three t1me-polnte vere red, and they vere tollo,,'"e

Weare primarily intereSited 1n the dela.y whicb care surrer. T o f1ndthis delay w e must cal.ct.:J.ate&nather aequence: the ~ sequ.ence. Thequeue at any particular time ""Point we define, for our parposel t as thenumber of cars held over trClllthat t1lne-point to the oext. With tbe Implerule. ot traffic bebavior 1ven above, once we know the arr1val sequence,the red/ green lequence, and the number of cera wai till{; at the interllectionat a~ initial t1me-pciot ve can calculate the ~eue8 tor all later t1me-po1ntE. successively. We1llustrate such a. ca.1cu..lation below.

There 1&one more lequencs of interest: that of departures. Roydoe.the process of eros51ng the 1nter.ection alter the .pacing of cars? Thedeparture seq.;.ence too can be calcul..e.ted from the others.

Wecan nov 11lustrate these sequences. The t1.l 'rle -POints are nw:.beredt 1,2, 3.. and we will suppose that at the beginning of the period. noCs..T6 ''ere \I'dt11lg to croes.


38/358


39/358

P:l.-Jl585 - 1 2 - 5 5-1.11-

a possible flow of traffic hav1ng the spse1f1~ average density. Toobtain a different average density of tratt1c, a corre.ponding~ differentc la asl t1 ca t1o n o f tbe D~er8 1 to 100 would be ueed. This average density1n the aiDor road w e have repreBented. by a. It 16 defined as the ra.tioot the number of care wh1ch. arrive in a l OD g p er io d of time to the ma:x.1mumnumber whicb could arrive provIded the proper time ISpa.cine; between care 18ma1nta.ined. The correspoodlnc:; quanti ty tor the major road w e have called p.

'Wehave calculated the average delay for the case when ".tatbtlcalstability" has been reached. 'll1is a,verate 11 the same as the averfLt:,' over aperIod of time suffIc1ently long for the particular state of the intersectionat the beginning of our observation to beccae unimportant. However, tbecalculatIons lnvolve the 85sumption that conge.t1oD will not teed to 1ncreeaeIDdefInl tely. The probabil1 ty model w e have used has tbe property tbe.t this

, - . j event Will 1n fact a.lmost certainly DOt bappen provid.ed the average numberof green po1nta exceeds the average n;.maberof care arriving at the intersection in the minor road. If t hi s c on di ti on is satIsfied, the queue ofcars will certainly vanish again and a.ga.1o, even though at times 1tmaybuild up to a considerable length.

TOe critical gap also needs redefining for the purposes of our d1sc:etemooel. Wehave called 1t w , and defIned. it as the number of t1me-po1ntsbefore the arrival of a car in the major road vh1ch are blocked by tbe close-ness of that ear. For ex~ple, if v. 2} the tYD time p Oi nt s b ef or e thearr ;val of the car 1n the major road will be blocked, a n d . alBO the t1me-point at vh1cb i t act ually ar rives.

'7nc curves Ln Ficurec 8, 9, and lC p!'('=>cnt the.-rccul. t.s of t.he cr-i::ull3.Consfor this model. BOleed Oil the !!l.8.themntice.1ana1~'8is given 1n Section 1.5,tbey ehov the mean delay 111the minor re,ad $6 a function ot the traffic


40/358

?!-!-14385-12-55-1.18-

dens1ties 0 and p in the minor rca.d a . nd major road1 and the criticalgap v. lor a giveD value of p and w, as 0 increases, the mean delayincreases, until at a certaic value of a the cullgeation increases 1ndetin-Italy. T h e intersection is not :ahle to'accommodate a de~sity of trafficcorresponding to that value of 0or greater tor an 1ndefinite lengtb oftime.

It 1s important to note for purposes of analysis in other chapters,that t hese average delay functions possess a curvature p ro pert y calledZ)'coavext t;V' .

'!'be same type at 8llalysiB bas been Clade in Section 1.5 for the repeated-cycle tr&tt'1c -light case. The trat'fic light. are .uppoeed to be red tor rt1me-points, and green for S tlme-po1nts. The density of arrivals 1s rep-ruented by a as inthe stop--e1gn case. A s 10 that case the average va.1t

( '\ over ~ long time period. 11 calcu lated. A (3 ;a1ncon geston will certainly buildup indefinitely unles5 the Dumber of green points exceeds the n~er of arrivalsover ~ Longperiod. Thue the average delay 1e ce.lculated 00 the supposf tlonthat this condltioQ ia satisfied.

A mo6t striking by-product of our calculat10ns 1s the adeQ.u.a.eyf anapprox imation which has been used 1n the past for sparse traffic. The approx-!matioD 1s based on the 6uppoel tiOD that the queue of care has vanished atthe beginning at each red per-Lod, Though this vould seem a drastic MsumptioD,the results calcula ted by using it turn out to be Valid for nearly all tbeva.l.uesof a, r and B for which w e have carr fed through the ealeulaUooa.Toe stability condition) for queues not to grow Indetlnitelyt is a < _g_ ,r+ gand the approxim~tlon is valid even for a quite close to this limiting value

:.,.: definition of convexity 1e given 1n Section 3.U).


41/358

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42/358

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44/358

RMl4-.~5 - 1 2 - 5 5 1 . 2 ' 2 -1.3. 'me- Uniform~Speed Capacit;;." of a Road

In d1 ecussing road c~ ac1ty w e shall first examine a particular typeof capacity curve which will be euled the u n1t or m .p eed c ap aci ty curve.Amongour re&eons t o r giving same attention to this curve are1) Its pol bl e u se tu ln es , 10 the plano1ng of large directed traffic

movements sucb as &rmy convoys, urban evacuations, ete.2) Its very dist1nct difference t r a m , and relatioD to, the more com-

p11cated nottoD ot a capacity curve which 1s to tallow. This ditfer-ence 18 of lome interest 1n itaelf for 1t demonstrates the fact thatthe "capacity" or a facility must 'be def1ned with e. n eye tovard theuse to lJhicb the tac111ty 11 put.

Consider a lingle-lane road over which traffic moves in ODe direction.Aup'torm-soeed vill be laid to exist when every vehicle moves at the lamespeed. At e . given point on the road, the uniform-speed. capacity at a givenuniform speed Is defined as the ma.x . imum tratrlc tlow (mea.s'.u-ed1n vehiclesper bour) which can pus that poi.D.t at that speed under certain assumed.con-d1tions. Obviously flow vill depend on th~ ~1Itance-6p6c:1ne between vehicles,and v111 be at 8max~~ for any given speed when spacing is at &minimum,since floy equa.ls speed divided by spacing.

!~e a5surr,edeon~itione are those that affect the mintmumspacing betweenvehicles at each unifon:. speed. These include not onlJ' such things as thephysical characteristic. of the road beillG examined, but also Bane character~iat1c5 of the vehicle population being studied, ani in addition a stateceotot vnether the spaces betveen ~hiclee ,hall be determined by reference to8eGe standard of safety, or by obaerving t he & p3 .( :1 nga ct ua lly m ai nt ai ne d byd.rivers, or perhaps 1n sane other manner. The main problem involved indetermio1ng theie capacities comel tram the fact that by any realistiC


45/358

R!'1-1~865-12-55-1.23criterion, tbe spac1n.g_between vehicles mu. st in crease with Bpeed.. The formt h1 1 6 pa .c 1n g : f" 1. i. nc t1 oDakes vill .eriously affect the shape of the curverelating uniform-speed capacity and uniform $peed.

Four ways of determIning these capacItIes suggest themselves:1) Hypothetical .peeing functions can be constructed and eubat1tuted

in the relat10nflay speed.spacIng

to give the maximumflo\( at each un1form speed.2) Empirical spacing functions can be dete~ined by observation of

existing traff1c flows, and they can be substituted io the re-lat100 of method (1),

3) Emp1rical. estimates of uniform :",speeiapac 1 ties ean be attempted.by direct observation ot flows I by -paasiog a determi nati on atspacing.

4) Experiments can be performed.To our knowledge the experimental method, interesting as it iSt has

never been carried out. Each of the otbere will be described below.

1 . , . 1 . H ypothetical pacing ~~ctionsA minimum "ade" sp aci ng f un ct io n may be postulated on the basis of

information about tbe distances required to decelerate vehicles from ver10~speeds, and aleo on the bais of one's be11efs as to what con!t1tutes sarety.This he..sbeen the most COJZ: t : lOD approach.if Without muoh Justification it haausually been aasumed that safety requires a distance between vehicles at leastequal to that necessary to bring a vehicle to a full stoP.j} T he m at he ma ti ca l

E e r r e y a n a Berrey, ( 1 9 4 S ]Hess [1950]Extensive blblioge.phy in Hormann[February, 1942, p. 122].s J The most notable exception tc this etatament 18 the work of Greensh1eldaand weida [19)2, pp. 1 5 2 - 1 5 1 ~ 1 , where it is ar~~~i thct full-stop spacing 1s tooccnservat.I ve an interpretation of setety.


46/358

PJ '1 -14885 - 1 2 - 5 5-1.24-

torm typically given this spaciug function ~ been

(1.1) a + bu + cu2

Wherea average ler~th of vehicleb perception plus reaction timeC . . J . . . . .-2r where r 18 the maY.im~~ rate at decelerat13nu speed

All three values are affocted by the composttiOD of tra.ff1c (number ot trucks,age of drivers, etc.). Time of day n n e location affect b ; and. road surf'aee ,veatberJ and grade and curvatu:-e of the r:'.)a.d&:ttect c. Someof the mostcaref"u.l studies ot b have been ma:!eby D e SY1Va.,5..1.n d of c by Moyer~Various values usej in c0n8tr.J.ct1ng these spacing tunct1oll5 are 1ntereet1nglycompared by Ncnuann and waJ..:s.erJ J While mach disagreeoent is evident, the

a (15 feet)(1 second)1.3) b 2 . 7 8 x 10-4 hours

c 9.47 x 10-6 boux.2jmile (1.e., deceleration.21.6 tt./&ec.2 )

are fairly typical aDd give no idea of the ~itudeB iovolv&d. It will be

seen that the "safe" trailing d.ist&nce groWlS more and mere rapidly as speed.increases. The resulting uniform sp eed cap aci ty function Which can be written

(1.4) u 2a + bu + CUD e Sylva ( 1 9 3 7 ] .Moyer [1947).Ncnna.n..'1 end. Wcl.ker [1949, p. 120].


47/358


48/358

R'>I- l4&5-12~55-1.26-

ample reason to believe that drivers de Dot space th~elveG at distancesdhich are perfectly sa.te anyvay. While sco.e atter::pts have been made torefine this concept of safety 10 order to secure a better correspondencebetween theory and observation, tbeee etteopts are more properly classif1edunder the method to be deacribed next.

1. ;.2. Empirical Spacing !\L"lCtlo118An empirical spacing function can be found, and used. 1n the same way

as above to determine the un1fcrm-speed. cu.pe.c1y curve. One such procedure,reported 'by No:rm&nn, vas as tOllova.l, 1 / or 8,500 vehicles recorded at onelocation, about 2,000 were traveling at the s~ speed as the vehicles Justahead ot them. These were classified into speed groups. -rne nerl step vasto find for eecb speed. group the mean s;pacln.s between vehicles. To ensurehowever that the mean spsc1o.g thus derived vas an accurate estimate ot them.eanminim:JIil spacing tor that speed group, 1t vas neceseary to exclude tromthe sample &6 many as posElble of tho8e vehlcle8 ~hicb just hap~ened to bemov1ng at the same speed as preceding vehicles, ~thout in ~. vay be1ng1m~ded by these precedinG vebicle6. 10 order to perfonI! thi6 eliminationthe time spacing, that is, the epace in front of a vehicle divided by 1t aspeed, was ca.lculated tor all me:nbersof the sample.

: 6 1 Hormann (1939, pp.226227].)1/ Ibid.

"


49/358

5 - 1 2 - 5 5-1.27-

Number o~Vehicle.

figure 12. AP'req.leccy Distribution ot Time Spac1n.gaFor a Single Speed Group. '

The histogram ot Figure 12 gives a general indication of the relative tre-quenc1e, of varioue t1Qe spacings in a BinGle speed. group. Since, in thecase pictured, moat of the time spacllls are eoncentrated at the low values,and a very fev are to be f0und at l ev el s ~ater than to~r seconds, th@ in-velt1getors were prcb~ly justified 10 throwing out of the sample thosevehicles vith time spacing" greater than tour seconds , These, they assumed,were not trave11ng at m1nimu: epe.c1ng. !'ne same procedure vas tollo~ forall the speed groups , a n a . the meBll 9,.1stMce spacing ot the reduced semple toreach speed group v68 determ1ned.


50/358

IN-14885 - 1 2 - 5 5-1.28-

Spe.c1ng

r1~e 1;. Observed Minimum Spacings,

Figure 1; shows J for each .peed. group, this meanmin1llumspac10g 1 . .D.

teet of ell vehicles sp aced at leiS than four s8coDds. U81ng the data inFigure 1" a u n1 fo rm -s pe ed capacity cu rve w as cO nlt rd ct ed Whlch 1ndicatesfor each speed the maximumlow posl1ble it all traffic movedat just thatspeed . The resulting curve, OB 10 !'1gt.;.re11, differs from a ty;>1cal tbeore-tical curve, CA 'A 1n Figure 11, mainly in indicating greater flows at thehii;her epeeds. Notice also that the uniform speed at which thi curve attainsits maximUl:1flow is higher than before.

1.).3 Direct Observations of F low.The unlforr:l-speed capacity curve might be estimated directly by fitting

a curve to a set of maxil!rumsor observed fleve tor leveral dirt'erent uoiflrlllspeeds. The mnin d.lfficulty encountered here il S that very eeldom are thehigher u l ' l l f o r m speeds observed. When speeds are hlgh SCEe passing nearlyalways takes place. This difficulty points up one o f the rather a r t i f i c i a lfeatures ot a u n1 fo rm -s pe ed e ep ac fty function. No r-oadsof importance are


51/358

Jr:-l(%5-12-55-1.29-

one-lane road.} yet the ~~ction is de~ined with the latter as a baail. Amultiple-lane road is quite a different taing fram a Bum ot s in gle l ac es.The complications 1ntroduced by passing will be dealt With 10 Section 1.4.

One of the reescne tor t he va ri et y ot methode used to determine uniform-speed capacity f'unctiooa 18 that congestion often a.ppeared to be present whenflows were flU' sbort ot the z:w.;.1mal levels indicated by the pe.rticular curvebeing used. It w as then thought that 8more careful procedure would lead toa tu.nctioo without tb1a detect. But the trouble 10 me . . . " l y eases probe.bly w asthat an average rather than a uniform speed w aa be1ns used as the independentvariable 10 the capacity function. Tbe d1N'erence between the two CMllot beneglected, &I w e shall see.

Difficulties such as these make 1t impossible to use Q un1fo~-speedcapac1ty carve uone to explain speed and flow l1ml tat1oos. Teere are, hov-

". , ever, atill same reasons f or f in di ng realistic uniform-speed capacity curvestor typical roads or point.. Such 1nf'ome.tioD would be U8etul to an authority(such as the ar.my) which had complete control ot &tr&tt1c movement, 80 thatall speeds could be specif1ed. These fUnctioos also represent uleful pointaof reference in the .~e that they show the 1016 in terms of tlow when driversselect a varlety of speeds. In addition, they v111 be found useful below inthe d iscu ssio n o f capacity concepts which take into account the fact thatuniform speeds are nct ordinarily found in practice.

1.4 'file Free~Speed Capac1 ty or a RoadFor the reasoDs 1ndicated the uniform-speed ca.pacity curve was never

found to be very useful. People do not al ar1ve at the same speed ord1narily,and in those cases where they do same of the at least wish to go taster andWill pass the cars ahead of them a s soon as the oJ:'portunly arises. Th1s


52/358

5 - 1 2 - 5 5. -". .i .. j-.'-lea.d.s to a bunching of traffic which the uniform-speed cs;pac1ty curve doesnot take into account.

The capac1ty notion to be c..escribeGhere 111the very different enedeveloped by O. K. Normannand hie associate. in the B..trea'J,of Public Roads,

, . .. ,Iand 6.escr1bed. in the B1ghYaYCe:pe..eiy Manual.~ 'or a particular road &capacity curve 18 derive1 empirically Which relates ver~3C .peed and flow.Tbe curves whicb b.&vebeen publisbed sboy that flow increases only at theexpen5e ot' a marked reduction ot average .peed. Purther work 1s neeessarybefore this mechanismcan be rully underet::>Od.}ut it .eems plauaibl. thatat h1gher floV5 more pudng per car 1s required to maintain a given dis-tr1but1oD ot speeds, that lLmited opportunities tor passing prevent theaaintenance of the higher speea.., and that flOY rea.c.hes a maximumwen alcars are traveling at very close to the same speed, vhlcb then is necessarily

(' a relatively slov ODe.

1.4.1 The Free~Speed p1stribution Betore diEcus6ing this matter further, a word ahoul~ be said nbo~t the

way ~iver6r p r ~ t e r e n c e s affect the capacity relations w e are about to diseuse.N e a r l y e v e r - . i kind. o f e a p : J . . c 1y curve 1 . m . a { ; i n a . b l e d e p e n d . . 8 1n s o a e way OD . t h echaracteristics o f t h e v e b 1 c l e a n d d r i v e r population 1t p r e t e n d s t o describe.

Wbenthie popuJ.atioo changes J then the eapa.eity curve elso changes e.ndthe problem is to relate the capacity curve to tbe underlying vehicle popul.a-tion in some Simple way. 'rae unlfcnr.-speed capac1ty curve depended on themin1mum bet ....en-vehicle spacing required at d iN 'er en t speeds by dr1vers, andit prob&bly sufficed 1n tb,il case to def1ne "different veh1cle populat1oastlto mean "different spacing functions". While the populations Qe.y have differed

"


53/358

R K - _ j _ 4 - J 05-22-5)-1.3l-

1n other respects as veil, thelle were not importent so fe.r B.8 uniform-speedcap&eity curves were concerned.

For the tree-speed capacity curve the eharacter1stic ve have cho8en todescribe the vehicle population 1s (as the name of the curve 1~icate.) itlfree-speed distribution. A driver!s desired or ~ apeea OD a g1veD road 18that speed at which be chooses to travel vhen "aloo." on the road, that 1.,when the preceding car is 80 far ahead as to have no 1.J lt'lueneehatsoever ODthe speed of the car 1n question. Since the matter 1s discus,ad again inSecti on 2 .2 .2 we shall only say here that the tree speeds drivers selectprobably depend OIl the physical cbara.cteriaUe& of the road and on theirestimates ot the risk. and costs involved end the value they place on time.The "d16tribut1on" of tree-speeds over a given road tor e . .pec1t1ed groupot drivers is 8imply a summ ary deecr1ptioc of the tree speeds ot the group;it tells us what fraction of the driver. have free apeede between any twogiven limits. AgraphIc repre&enta.tioD or a tree -speed d18tribution 1s given1u Figure l4 where the ~otal area under the curve 1s equal to one and thefraction of' driverll With tree speeds between 30 mph and 40 mpb 1s represented.

III I!c I E Miles per

40 5 0 60 70 hourIIIA

20

. : : : - }/

Figlll'e14. Graphic: Repreeentation ot a Sp eed D istr ib ut io n


54/358

~-14885-12-55-1.32-

'b y the area AOOC, the fraction \fith t ree speeds between 40 mph and 50 IrLphby the area CDYE, etc. The reader wo is interested in tbl! actual free-speedd15tribu.tions found to prevail in certain 1neta.nees is referred to the H1~'

,Capacity Marr. l .6. l . Ui

To describe, as w e are doing, a vehicle population by its free-speeddistribution 1s very obViously an oversimplification; t o r cer tai nly t he reare many other real d1trereocea which may be qu1te icportant, such as theproportion of trucks 1n the population being considered, or the lengths oto pp os1n c- lan e g ap s d e: :: l8 ll dedo r p assi ng by the drivers. However the simpledescription v1l1 probably do for our purpose Whether or not more o f the characteristics that distinguish different groups of drivers ahould be used18 a que.tioD that can only be decided in practice. Once the reader 1& con-vinced of the usefulness of description by m ean s o f fr ee- speed d istr ibu tio na,it should not be difficult t o r him to imagine how oth er mo re cocpllcateddescription! might be handled.

1.4.2 Tbe Reali~ed Speed Pl~tributionWe sh&ll now att~t both to summarize and to enlarge somewhat upon the

work of Normann and Walker In ex amitl1.oghighWay c ap ac 1 ty in terms of speedreductions. It should be made clear tha.t while cuch ot what follows 1s baseddirectly on material in the H1~Vay Cgp~c1ty~4 P ual (end r el at ed p ub li ca t1 on iJ1 ,] )the autbors of tbat report are not to be b~ld responsible for possible misinter-pretations on our.par\, nc.r for additions ve have made with which they may notagree. ~oe deecription will refer to a tvo-lane roed which carrie8 traffic 1nboth directions. The free-speed distributions of the two streams of traff1c

N J r m a n n a n d W a l k e r ( 1 9 4 9 , p. ~ 2 , Fisure 6).N o r m s n n [ 1 9 3 9 ) , [ 1 9 4 1 ) , [ j ~ e , 1 9 4 2 ] .


55/358

RM-l4885-12-55-1.33-vill be s..ssumedknow, e.nd we lhe..li luppose that e . vehicle will Clove at 1tafree I~ed. whenever traffic cond1 t10Wl perm!t

Tbe distribution of desired speed8 will ordinarily not be the same asthe distribution or actual speed.. ror a fast car to pus Ii. slow car laDeempty 8pace in the opposing lane is necessary. If the density of vehicles(i.e., the number of vehicles per mile of road at any instant) 1n thisopposing lane is high, then luch .paces vill be seldom avaIlable, and thepasSIng maneuver will be delayed or prevented. As a result, the actual speeddistribution will difter trom the free-speed distribution, the lower speed.belng relatively more trequent in the former.

Another way of looking at this effect that beavy traftic haa on actualspeeds 11 to concentrate our attention for a moment on a one-mile 8tre\eh ofroad With unrestrIcted visIbility. SuPFoee that the southbound trartlc flow

.) 1s very low, so lOY in tact that tor northbound traff'ie the tree-speed dis.-tribut10ll 18 the same as the a.ctual._peed distribution. Row let us ukour-selve. how the total number of passing me.oeuvera per hour carried out by north-bound vehiclee on this onelmile etreteh varies with the level of northboundflow. Suppose narthbouod flow doubles. A 50 mph driver will now find itneceslary to complete tvice as mny p ass in g m an eu ve rs over the one~ilestretch as before, due to the doubling at the tlov of vehicles movinC at leB8th.en 50 mph. But since tbe flov of 50 npb vehicles has aleo doubled, thismeans the hourly 1lUDber of passing maneuver_ by 50 mph vehicles has increased.by a factor of four. The same bolda tor all .peeds, and it 1s only a step~~rther to the concluI1on that the total number ot passes per mile at roadper hour increases witb the 8q~are of flow it the free-speed d1Gtrlbutionremains unehanged.


56/358

R.Vo-14885 - 1 2 - 5 5-1.34-If now the assump tion o f oeg:igibl= 6outhbo~~d flow i~ dropped, it

bec~e6 clear that at same level of northbound flov, the number cf oorth-.bound pas. log man~~ver8 re~~1reQ to maintain tbe free-epeed distributionWill be more than the ga.ps 10 the opposing traffic strear!.pem.i t. As aresult, 60me vehicles will be ~ed and the realiz ed speed distributionwill no longer be the same aI t oe free-speed d ist ri bu ti on . I n v e s t i g a t 1 0 n shave revealed roughly the way speed distributions chan..:;evitb increases inflow tor some cases, and h a v e .hovn th a. t t he actu al t ot e. .ln um ber o f p assin gsper mi1e of roed. per hour incree.ees ....h flow up to a certain point I and tlieodecreases gradually down to zer-o as flow becomes IS O great that all vehiclesare forced to move at tbe l am e . p ee d. ~

The realiz ed speed distribut10n for northbound traffic is clearly 10-flueoced by t!le flows of traffic in ~ d1rections 0 0 the r o a d . It'nortb-bound flow i n c r e a s e s , maintaining the same 8~ed d i s t r i b u t i o n requires ~repasling o p p 8 r t u o 1 t i e l t han b efor e; it sou thb oun d flo w i ncr eases, few er passingopportuni ties present the:.selvel than before. Both phenoeena cause a change1n the r eali zed speed d h t : - l b . . . : . t l o n .Ilelee;a.nttheory of r oad cap aci ty 'WOUldbe o~e which described just t h l l relationship. Given th~ tvc flows and thetwo free-speed distributions we could determine tho two eorr~lponding reel1zedspeed. distributions. With th1a InfoJ"l't)3.tlonw e 'WOuldbe able to assess tbeeffect of c~ocestlon on vehicles of each spee1-class 1n the two traffic streams.However J this is obviously asking too much at least 1n our present stege ofdeveloping a body of information on the behavior ot traffic. A speed. ciistr1"button is a rather 1arr3.e8.Sae:obly of information. Weshall settle for a simpler,substitute measure of the degree to which speeds have been affected by flow Toe mee.sare we use is the mean real1z.ed speed. Normann'e group has in fact.;)/ Bormann [June 1942 J p. 70].. .'


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attempted more the.r. this by makine use to same extent of the eta.nd.ard.e-Wv1ation of realized speed, 10 ad!ition to mew speed.,. In subsequerrt dis-casaioo we ehall for the most part iGnore these more sophisticated descriptionsof the speed-flov mechanism_ It .h~uld of ccurse be berne in mind that hereis a poss ible weakness in our theory. Practical work mc.y vell incUcate thatmean speed is too gross a measure of the ecnaequencel of increased :r'lov, inwhich case it will be necessary to incorporat.e add1tional measures such asstandard deviation of mean epeedJ or one of the other. liuggestoo by NoI"!DB.l l l l .

1.4.; Me~~ Speeds ani the Free-Speed Capacity ~~ct1ooThe que.tion nov beccmes, how do northbound and 8outhbound mean speeds

depend on the two flows? When we move intbe direction of further e1mp11t1ca-tiOD we are faced with a situation much like alding together the two inter-section fl~ 1n tbe exrumplein Section 1.1. Ncrthbound mean speed dependson so ut bb ou nd aA vell as o:)rthbound flow; i8 itgoing too tar to suppose thatnorthb:)uod mean speed depenis only 00 the sum of the tw o flovs? Or, iOOeed,car. ve drop tbe final c~'l1catioD and sa,),tb~t speed a.veraged over bothDortb- ~ ~d s ou th bo un d traffic depen~8 only on the sum of the tva flows!

Of course these progTeui vely .1.r.:plerepresentations are at most approx-tmations and a final answer as to which of them 1s be!t for the purpose c~otbe given here. In practice the choice vill be d1ctated by tbe cost ot obtain-108 data, the reliabilIty of the resulting flev-speed relationships 1n makingpredictions, and tbe consequencos of errors vbeo the esttmated relationship'are applied to specific problems.

I~ any case, such quezt10ns ae these, important as they are, need notbe decid~ here. The sna.1ys1s tb.e.t follows mey be a.pplied to any of the' 1 ; / Normann [1939, p. 228}.


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R),!-14535~12-55- 1 3 6 -various repre5e~tationn of the !lev-speed relation. It is tc m3ke thislast point clear that w e ~h&8i~e these d!rferences.

In one of tb. earliest articles ulon the subject N~ma.nnchose one otthe more complicated relationships tor Doe particular rond a n e c y tbe m etb odof least squares fitted 8plane to a Bet of observations on northbound mean ..speed, northbound flow, end so uth bou nd flo w. Sinee the coefficient. heestimated are of intereet 'lie reproduce his result:

( 1 . 5 )

where u is in miles per hour and x and x in vehicles per hour. Thisn n Iplane, saovn graphlcal.ly in l'igure 15 1e DO exaz::.plef wha.t ve ahall call. a

Figure 15

ul Normann [1939, F1ry.re 11, p. 2291.


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free~speed capacity function or r w@re no conf'usLon 18 likely to result,simply a CaIlB.c1ty function (or plane or curve, depending on tbe nU!nber ofd im en si on s i nvo lved ). T he term ttte 8?eei in this use does not meM (ae intbe ease of the un1fcrm~speed capacity function) that the ~~ct1on appliesonly to vehiCle! traveling at free speedaj w e retain the term here ooly toemphasize tbe tact tbat the ~ctlon is based on a freespeed distribution.

More generally, a free-speed capacity function is the function relatingflays and m ean r eali zed . apeeda OD B particular road for a particular populationot vehicles. Other end simpler examples are presented in the R1Ghva y C ap ac! t yllil '"V..::.nual'.here mean speed at all traffic on the road is made to depend on the,total fiow on the road. Graphs of thelSe curvee (which In fa.et have usuallybeen 13 .B5umed to be str&J.ght l1nes) are quite a bIt euler to draw and tovtsu$l1ze, the number of dimenslons being lese. Figure 16 i & o ne ex am ple.For Teasooa to be discussed below, the curve has not been exten~ed all the

Flow

Figure 16. A n Example of a Free~Speed Capacity Function18/ Op. cit., FiE; Ul"e 5, p.examples are tc be found in: 31; CUrve 2 in Figure 7, p. 33. Other similarNcrmann [June 1942, Figure 6, p. 61; Figure 8 ,p.63No~enn [ 1 9 3 9 , Ficure 12, p. 229).Wardrop [Feb. a.'1!j. Mar. 1952, Fic;ure 1, p. 21 .G~~ville [ 1 9 4 9 , Firure 9 , p. 411.Forbes [1952, Yibures 5, 6, pp. 6 , 1 respectively].


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way to the flow b.Xis. N::>ticealso that in this example the c'U.rveis not& str alg bt lin e; w e have dra\nl it the ...ay itherely tc emph as1:l. et hetact that from a theoretical point of view there 1s no reason to believeit to be so. We shall, nevertheless, etton dray 1t as though it were, itonly for convenience. Moat Invest13stors have found that a straight lineprovides a fairly good fit to ob servat1one.

The ~ortant characteristic of all these curves; no mat ter \ o"hattb e1rdegree ot cCI%I1plicet1oD,1s that theJ' are negatively sloped; that 18, meanspeed decr-eaaes as tlo,",1ncreuea. It should be emph!Uized that, &}:oart tromsampling errors and errors of observation, o~ly points 2n the curve are possible\flo~-speed combinations t o r the vehicle populat1on with t he a ssu me d fr ee- sp eeddi6tribution. Points belov the curve, such as A 10 Figure 16, would implythat sa:ne drivers are not going as fast ae they both can and wish to, end theWidth and condition of the roed and the prelence ot t raffiC p revent pointeBuch as B trom occur~1ng.

1.4.4 Th e R elatio n Bet~een the Free-Speed Capacity Curve ~~ the Free-SpeedDistribution.

The position of a capacity curve at levels of flow approaching zero villvary wltb the vehicle population that the curve describes. When very Uttlecongestion 1s present nearly every vehicle will be able to travel at its freespeed. At zero flow therefore mean realized speed 1s equal to mean free speed.If for a certain r-oad it is known that Sunday drivers have a lover mean freespeed than veekday dri ve ra , then ve should expect the capac1 ty curves for thetwo days to d1ffer 1n a fashion like that indicated in 'igure 1 7 (assumingthe eurve5 to be of the same simple ~~ctional fcrm as the one in rl~~re 16).Here the point! A and B,


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1m-14885 - 1 2 - 5 5-139-

Flow

Mes . n Speed

T1eure 11. Capacity Cu..-veaCornsporA-ins to Different Free -Spee~Distributions

the mean speeds corresponding to &ero flo'W'l for the Sund.ay and.weekiay traffiereepect1vely, are allo equal to the respective aean tree spee~s of thoae twodays. The capacf ty curves both slope upward.and to the left from these pointe.

Tbe difference in capacities bet~een wide roads and narrow roads shouldahow up 1n these curves more a .e a difference io alope, thar. intercept. Supposean _improvement program 1. contemplated tor a cert.a.1n road vnot;e capacity isrepresented by the curve AS in Figure 18. If the proposed improvements are

Flow

A' ..

c . . . . . . . .

F1L'Ul"C18. The Effect of ROM ImFrovement 00 a Capacity Curve.


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e!f'ective in reducing congeat.aon, a new capaci t; y curve CB will result whichviti indicate tor each level of [lev a bigher mean speed then before. A s w eshould ex pect, the difference is greate.t where conGest1on 1s h1~~, that 1sat high levels of floli. When flowl approacb ser-o, m~an free .peetd preva.11stor both curves by the preceding arguments, and if ve have reason to be11evethat people's tree speeds Will Dot be altered by the road ~provement thantheee tvo pointe vill b. the same. The type of shift described seems to agreevery well vith ecoe of the r e s u . l t s of highway stuCies curled out by Ent;.l.e..nd t6R oad R esearch L a b o r a t o r y . 1 2 1

It 1 1 5 quite conceivable ot course that tree .peeds vill change with theimprovement. Wider roe:ie are very likely to ca.use an upward shirt 1n thewhole free -speed d.1stributioo, Vith 8 con sequ en t shi ft in the mean tree speedtrcm B to B in FiQJ.re 18. The capac! ty curve in this cue m1t;ht looksomething 11lte C'B', were the road. imJlrovement has ehanged 1) the slopeof the curve tbro1.l.ghts direct effect on, I5&Y,pes,we opportcnit1es, and2) the speed intercept at the curve through its 1ndirect effect on drivers'choices ae to apeeds.

Before an y further d18cU86ion of the influence of free speeds on theslope of capacity curves we must cover a point ve have so far glossed over.10 Figures 16, 171 and 18 the curves have not been extende~ all the way tothe flew axiB. It in F1gure 18 the dctted section AA' vere added to thecapac! ty curve ~, then the whole curve would seem to tell us that, tor tbeapp ro pr iate r oad and vehicle p~pulat1oc, mean speed will be zero if flow ri sest o the level A'. But it meen 5peed 11 z ero, then all speeds ar e zero and floycannot be positive. Clearly, the upward ex tens10n ot the curve ~~st end atsome pD1nt such 8B A.l,~I r h 2 1 ]~ Ward..rop] tYeeruary a . . ' 1 d Ms..rc , 19) t Figure ; p. 2


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Gene:-ally speaking, as nov Lncreaaes frOC! very 10\0' levels, mean speeddeclines and the apread of the realized speed dietrlbutioD grovs 6~ler an dsma.ller. Figure 19 portrays, in l8 very r ouGh way, this chenge 1n the realized~pe.d distribution. If flow continues to increase until the point is Ju.treached where every vehicle 1& traveling at the same speed, then the endpoint,referred to above, of the free-speed ca.pacity curve ,,1.11have been attained.TIle mean speed at tbill point will, under our as sumpt ions, be the free speedIot the Bloweat vehicle prelent.~I. It will not be less, tor the free-speedcapacity curve as we have defined it only describes speed reductions broughtabout by the absence of pasling opportunities. Any reduction 1n the speed ottbis slowest vehicle, which has no n e e d to pass, cannct therefore be e x p l a i n e dby the capacity curve we have been c.lIcus81nS' It if! tr.;,e th&

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