Propagation of on-ramp density perturbations on unidirectional two- and three-lane freeways

Tmrspn Res. Vol. 5, pp. 241-255. Pergamon Press 1971. Printed in Great Britain

PROPAGATION UNIDIRECTIONAL

OF

ON-RAMP

DENSITY

TWO- AND

PERTURBATIONS

THREE-LANE

ON

FREEWAYSt

P. K. MUNJAL and L. A. PIPES System Development Corporation, Santa Monica, California, U.S.A. (Received 15 March 1970) 1. INTRODUCTION causes a sudden increase in the traffic density in the right-hand lane of the highways in the region of the on-ramp, which in turn introduces more lane-changing maneuvers of cars in other lanes of relatively less density. As the distance from the on-ramp increases, the traffic density in the right-hand lane decreases, with relatively fewer lanechanging maneuvers. After a certain distance from the on-ramp entry point, the perturbation in the traffic density due to on-ramp flow will diminish significantly through the successive propagation of the perturbed traffic density into other lanes. Based upon the principles of expected value continuum models, this paper describes the propagation in space and time of such perturbed densities initiated by the on-ramp flow. This perturbed traffic density is defined as the difference in the actual instantaneous prevailing traffic density and the equilibrium traffic density which would prevail under steadystate traffic conditions. The present analysis involves setting up appropriate equations for the study of the effects of on-ramp flow on unidirectional two- and three-lane freeways. This setup is based upon the primary governing mechanisms, which contribute significantly to traffic behavior in the vicinity of an on-ramp. This paper also contains a brief discussion of the various assumptions of this theory which must be validated. THE ON-RAMP flow

2. EXPECTED VALUE EQUATION OF CONTINUITY FOR TRAFFIC FLOW Lighthill and Whitham (1955) introduced a continuum approach in describing the flow of traffic on long crowded highways. Richards (1956) has also considered a particular case This macroscopic treatment of of Lighthill and Whitham in his independent analysis. traffic replaces individual vehicles with a compressible traffic fluid that has a certain density k(x, t) (veh./mile) and a flow q(x, t) (veh./hr), where x is a linear coordinate measured in the direction of flow in miles and t is the time measured in hours. Pipes (1968) discussed the effects of diffusional and inertial phenomena in the analysis of continuum traffic models. Gazis et al. (1962) have applied the continuum theory in describing the inter-lane density oscillations of a multi-lane highway by neglecting the effect of longitudinal waves of traffic 7 Prepared for the U.S. Department of Transportation, Federal Highway Administration. Bureau of Public Roads under Contract No. FH-11-6623. The opinions, findings and conclusions expressed in this publication are those of the authors and not necessarily those of the Bureau of Public Roads. This publication results from work performed by System Development Corporation, prime contractor, and UCLA Institute of Transportation and Traffic Engineering, subcontractor, and has been approved by both parties. Presented to the Transportation Science Section of the Joint ORSA/AAS National Meeting, Denver, Colorado, 17 June 1969. 241 13

242

P. K. MUNJAL and L. A. PIPES

densities. Oliver and Lam (1965) have described the lane-changing behavior in the framework of a continuum model. The main advantage of suchexpected value traffic models is that they give useful averages of the overall statistical properties of traffic. However, some researchers favor the development of detailed stochastic traffic models and later find that no analytic solution exists, while others seek a detailed study of a particular traffic property without relating it to the overall statistical properties needed in the analysis of freeway design and control problems. The traffic flow resulting from the general setting of the continuum models behaves like real freeway traffic, but only up to a certain point. At any given time there are hundreds of different drivers on the road, in hundreds of different types of vehicles. The driver in each vehicle has his own rules and style of driving which may again change from one freeway to another. It is impossible to completely reflect this amount of diversity in any traffic model. However, it is important that the overall flow of vehicles in the model slows up about the right amount on a grade, and becomes appropriately congested moving past a high-volume, poorly designed on-ramp. It is in the analysis of just such average properties of traffic that the expected valSe continuum models prove useful tools. Lighthill and Whitham (1955) have shown that if traffic of density k(x, t) and rate of flow q(x, t) is traveling along a freeway, the following continuity equation must be satisfied,

Q&O

&k+D,q=

(2.1)

where Q(x, t) denotes the flow rate per mile entering the freeway and the operators Dt and D, denote the partial derivatives with respect to t and x, respectively. Q(x, t) is measured in veh./mile per hour. Equation (2.1) can also be derived on the basis of expected value concepts by using the speed distribution function (see, for example, Munjal, 1970). The speed distribution function f(x, v, t), for a given concentration, has been defined such that f(x, v, t)dxdv is the expected number of vehicles at time t whose actual speeds are between v and v+dv in the road interval between x and x + dx. The elements dx, dv must be very small. Thus,

s 0

mf(x, v, t) dv =

k(x, t)

(2.2)

where k(x, t) is the concentration at point x and time t. When the highway consists of two unidirectional lanes, then it is possible for an interchange of vehicles to occur between the two lanes and we may write two equations of continuity, one for each lane, of the form D,q,+

Q, = Q,

D,k, =

D,q,+D&,

(2.3) (2.4)

where Q, is the net flow per unit distance of vehicles entering or leaving lane 1; and Q, is the net flow per unit distance of vehicles entering or leaving lane 2, such that ki(x, t) =

“fi(x, v, t) dv

s

0

s

0

qi(x, t) =

mv&(x, v, t) dv

where i denotes the lane number. If there are no on- and off-ramps, leaving or entering the road, we must have

Qz=-Q,

(2.5)

(2.6) so that no vehicles are (2.7)

On-ramp

density perturbations 3. EXPECTED

on unidirectional

VALUE

two- and three-lane

SPEED-DENSITY

freeways

243

LAWS

A form of expected value speed-density law has been given by Pipes (1967). It gives the average speed fi for a given concentration k, in terms of the desired average speed d and jam traffic concentration kj, such that (3.1) where n is a real number whose magnitude is greater than zero. The desired average speed of the traffic is obtained from the free-speed or desired speed distribution functionfO(x, u, t), which is essentially the distribution for very dilute traffic concentration where all interactions between vehicles can be neglected. This desired speed distribution function fo(x, D,t) for all the drivers on the road is thus defined so thatfO(x, u, t) dx da is the expected number of vehicles at time t whose drivers have desired speeds between v and v+dv in the road interval between x and x + dx. Using equation (3.1), Fig. 1 shows the steady-state speed-density relationship for various values of the parameter n. For n = 1, (3.1) reduces to the straight line deduced by Greenshields (1935) as a result of some measurements. If n is less than one, we obtain concave

nzo

2,. FIG.

dimensionless

1. Speed-density

density

relation.

downward curves lying above the straight line and representing situations where vehicles do not slow down appreciably until the concentration becomes quite large. n> 1 gives concave upward curves lying below the straight line and represents situations where slowing down begins at low concentrations. Curves of this type seem to fit experimental curves obtained by Wardrop (1963). Figure 2 shows some data on the speed-density relationship obtained during constant flow periods for a two-lane freeway using the BPR Traffic Analyzer Data. It can be seen from this figure that our experimental data for the speeddensity relationship correspond roughly to n < 1. In order to estimate the equilibrium density kio in the ith lane in terms of the equilibrium (or average steady state) speeds fiSo,we shall assume that each lane satisfies the generalized speed density relation (3.1), so that we have Vi0

=

-0

Vi

(l_k2ki1n

(3.2)

P. K. MUNJAL and L. A. PWES

244

IO-

0

I 20

I 40

I 60

I 80 k,

I 100 density

I 120

I 140 in

I 160

I

1

‘:*J

180

200

220

240

cars/mile

FIG. 2. Partial relationship of the average speed and traffic density for the constant flow intervals for a two-lane freeway (BPR data).

Equation

(3.2) may be solved for ki, to obtain (3.3)

IfSi, andyto are the actual and desired speed distribution equilibrium conditions, we have

kio=

s0

mfi&)dl:

where all the speed distributions instantaneous speed distribution

functions

for the ith lane at

= are realized at sready state. It will be assumed now that the function f<(x, U, t) for the ith lane is given by (3.6)

.A(& u, t) = Lo(u) + Fi(x, u, r) where &(x, u, t) is the perturbation of the speed distribution equation (3.6) over all speeds gives

function

ki(X, t) = k, + &(x, t) where ki(x, t) is the traffic density in lane i; kio, the equilibrium equation &(x, t), the density perturbation in lane i. Integrating section gives q(t) = n, + N,(t)

in lane i. Integrating (3.7)

density in lane i; and (3.7) over a given road (3.8)

where ni(t) is the total number of vehicles in lane i; nio, the total number of vehicles in lane i at equilibrium; and N,(t), the perturbation of the number of vehicles in lane i from equilibrium. 4. DENSITY

PERTURBATION

BY ON-RAMP

3 shows a schematic diagram of an on-ramp configuration. If there is no onramp flow, it is assumed that the freeway will be in steady state. Now the incidence of onramp flow causes a sudden increase in the traffic density in the right-hand lane of the Figure

On-ramp

density perturbations

on unidirectional

two- and three-lane

freeways

245

highway which, in turn, induces a certain density perturbation about the original equilibrium density. The relationships between the equilibrium and perturbed densities have been given in equation (3.7). The net exchange in flow per mile between the various lanes was given in equations (2.3) and (2.4). We now postulate that Q&c, t), the net flow per unit distance of vehicles entering or leaving lane 1, is

Q&, f) = 4&k

t) - W, 0) -&w - M

= 4&c% 6 - mx, 01

(4.1)

where a is a positive constant with dimension time-‘. Equation (4.1) gives Qr(x, t) = 0 when k,(x, t) = k, and k,(x, t) = k,, and gives a positive flow when (k,(x, t) - k,(x, t)) exceeds (km-k,,,). This relation for Q, assumes that the rate of exchange of vehicles between two lanes is linearly proportional to the differences of the deviations of their densities from the equilibrium value. This type of lane-changing behavior is defined as the uniform lane-changing hypothesis. This hypothesis expresses simply a behavior in which drivers tend to leave a more crowded lane and enter a less crowded one without any particular preference for any lane. It is believed that this type of lane-changing hypothesis introduces only a first-order governing mechanism of traffic behavior. However, before we attempt to incorporate a more elaborate, complex lane-changing hypothesis, experimental data are necessary to validate the proposed uniform lane-changing hypothesis. Such an analysis will itself uncover more significant lane-changing features that may prove desirable in the development of a more realistic lane-changing rule.

Lane 2 Downstream side

Upstream --_----------------side

-$?yL=

FIG. 3. Schematic diagram of an on-ramp configuration freeway.

on a unidirectional

two-lane

Using the above lane-changing hypothesis given in equation (4.1) and the car conservation relationship given in equation (2.7), we get the continuity equation for a two-lane unidirectional highway as D, 41 + ot k, = a[(& - k3 - (k, - kr,,)I = a&-K,]

D, 42+ ot k, = a[(k, - kz) - (k, -

= a[& - K,]

(4.2)

kzo)l

(4.3) In order to determine the propagation of density perturbation, it is necessary to assume that the flow in each lane is a function of the density in that lane and of the position x along the highway; namely, qi=qi(ki,x), i= 1,2 (4.4) If the road is space homogeneous the flow will not depend upon the position, and equation (4.4) reduces to 42 = G,(k) (4.5)

P. K. MUNJAL

246

and

L.

A.

Equation (4.5) is regarded as the “equation of state” tiation with respect to x of equation (4.5) gives

Introducing

PIPES

of the traffic fluid.

Partial

differen-

the notation

dG,

C’=dk,’

dGa cz=z

(4.7)

i = 1,2

(4.8)

we get

D,q,

= Ci D,ki,

c1 and c2 are the velocities of wave propagation. Based upon the relationship of Fig. 2, we plotted a q-k diagram for the equilibrium flow as shown in Fig. 4. With the help of this diagram we have shown the wave propagation velocities for each lane that may be induced by small traffic perturbations. Figures 5 and 6 give the speed distribution functions for each lane for points A and E in the speed-density diagram. Lane

I

2

lane 2

2000 42 q, 1500

0

for

20

40

60

60 k.

FIG. 4.

y-k

diagram

for

lone

I

100 120 140 160 1130200 220

density

in cars /mile

the equilibrium (BPR

flow for data).

each

lane

of a two-lane

freeway

As seen in Fig. 4, wave speeds may take either positive or negative speeds depending If the wave speeds are positive, then the propaupon their position on the q-k diagram. gation due to the on-ramp density perturbations would take place on the downstream side of the on-ramp. However, if the wave speeds are negative, the propagation due to the onramp density perturbations would take place on the upstream side of the on-ramp. If we now substitute the relationship of equations (3.7) and (4.8) into (4.2) and (4.3) we get the equations Dt KI + c1 D, KI = a[K, - KJ (4.9) DtK,+c,

D,K,

= a[K,-K,]

(4.10)

Equations (4.9) and (4.10) are two coupled wave equations that serve to determine the nature of the density perturbations in the two lanes of the highway that are produced by

On-ramp

s

on unidirectional

two- and three-lane

241

freeways

0 120

.t

0.1 IO

5

0 I00

.-z % 0 L f .w

0090 0.080 0 070

m B ::

0.060

m z

0.050

G E Li c 2 0;

density perturbations

0 040 0 030 0 020 9 0 010

~--~

j

0

v.

FIG. 5. Desired

speed distribution

speed

in miles/

hr

functions for each lane of a two-lane (BPR data).

freeway

_..__ 0.120

-

z

O.llO-

II

O.lOO-

111 I I nl I III , “I ‘1

0 0900.0800.070

o,~60_

p -

1:

Lane Itpaint

El

I I L

,

Lane 2 (point

E)

Pl

0

Y. speed in mileslhr

FIG. 6. Actual

speed distribution

functions for each lane of a two-lane (BPR data).

freeway

248

P. K. MUNJAL and L. A. PIPES

the on-ramp disturbances from the state of equilibrium. If the density perturbations are small, we can take the values of wave speeds at equilibrium conditions and assume them to hold for the traffic densities under consideration, such that (4.11) 5.

PROPAGATION OF ON-RAMP DENSITY PERTURBATIONS ON A TWO-LANE UNIFORM FREEWAY

A freeway is termed uniform if the di- ki relationship for equilibrium conditions is the same for each lane and if the equilibrium densities are the same across lanes. This means for a given steady-state traffic flow, the relationships

4, = ho Cl

=

c2

(5.1)

6, = c2 41 = q2

hold for all concentrations.

I

In this case the equilibrium

equations

are (5.2)

(5.3) (klO)(n+ll?L)TL k, = i I The wave velocities

c1 and c2 given by equation

42

(5.4)

(4.11) would have the values (5.5)

In this uniform

case, equations

(4.9) and (4.10) take the form (since c1 = c2) D, Kl + CD, Kl = a[K, - KJ

(5.6)

Dt K, + CD, K2 = a[K, -K,]

(5.7)

To solve (5.6) and (5.7), we add and subtract

and introduce

Substituting

these equations

to get

Dt(Kl + K2) + c D,( Kl + K2) = 0

(5.8)

Dt(Kl - K2) + c D,C(Kl - K,) = 2a(K2 - Kl)

(5.9)

the variables Y,(x, t) = Kl + K2

(5.10)

Y&G t) = Kl-

(5.11)

K,

(5.10) and (5.11) into (5.8) and (5.9) gives D,y, + CD, y1 = 0 Dty2+cD,y2+2ay,

= 0

(5.12) (5.13)

On-ramp density perturbations

on unidirectional

two- and three-lane

In order to solve (5.12) and (5.13) the Laplace transforms CO exp(-sx)Y,(x,t)dx=z&,t), LYi(X,0 = s0 with respect to x will be introduced.

i= I,2

freeways

249

(5.14)

Then

LD,yi

=

SZ~-J'$(O, t), i =

I,2

(5.15)

Let Yi(O3 r, = YiOCf)

where YlO(0 = WA t) + &(Q t)9 y,(t) = &(O, 0 - &(O, 1)

(5.16)

Then the Laplace transform of (5.12) gives Dtzl + cszl = cylo(t)

If Dl denotes the operator for differentiation with respect to t we have

where {s+(DJc)}

is a differential operator.

Letting

gives z, -

hW

s+b,

Since L-1

-

l

s+b,

= exp(-b,x)

we have by the use of the symbolic form of Taylor’s theorem (5.17) Similarly, the Laplace transform of (5.13) gives Dt z, + csz, - cy&) + 2uzs = 0

or

Letting 2a Dt b, = T+F

we have z2 = Y&) s+b,

250

P. K. MUNJAL

and L. A. PIPES

Since

- l

L-1

= exp(-b,x)

s+b, we

have

Y&, t) = exp(- b2x)Y&) = exp

(5.18) To simulate the on-ramp input flow into lane 1, let the boundary conditions at x = 0 be G(O, 0 = k,(t),

GO, 0 = 0

where k,(t) = 0 for t < 0. This type of boundary condition requires a short acceleration lane. Then Yr(0,0 = k,(t), Yz(0,t) = 0 and (5.19) (5.20)

y,(x,t)=exp[$)k,(f-:)

Using the relationships (5.10), (5.1 l), (5.19) and (5.20) gives K1(x,t)=~[l+exp(~)]kO(t-:)

(5.21)

Ka(x,t)=i[l-exp(*)]k,,(f-:)

(5.22)

It can be seen from equations (5.21) and (5.22) that as x increases, the density perturbation in both lanes approaches k,(t - x/c)/2. Figure 7 gives the plot of &(x, t)/k,(t) vs. ax/c for the two-lane freeways, where the density perturbation caused by the on-ramp flow has a step function input, i.e. k,,(t) = 1,

= 0,

t>,O

otherwise

(5.23) 6. PROPAGATION OF ON-RAMP DENSITY PERTURBATIONS ON A THREE-LANE UNIFORM FREEWAY We shall now consider the case of a three-lane uniform freeway, i.e. one in which the i$ - k relationship for equilibrium conditions is the same for every lane. The corresponding basic equations of continuity for this case are

(Dl+cDJK,

= a(K,--K,)

(6.1)

(Dt+cDz)K,

= a(K,-K,)+a(K,-KJ

(6.2)

(D,+cD,)K,

= a(K,-K,)

(6.3)

On-ramp

density perturbations

;3oJopI

,

9

0.5

2

I

0

,

,

1.0

y,

FIG. 7. Relative propagation

on unidirectional

1.5

two- and three-lane

freeways

,

,

,

,

,

,

1

2.0

2.5

3.0

3.5

4.0

4.5

5.0

distance

from on-romp

(dimensionless)

of on-ramp density perturbations two-lane freeway.

where K,, K, and K, are the density perturbations density perturbation vector

251

in different lanes of a

in lanes 1, 2 and 3. Introducing the

(6.4)

and the matrix of coefficients of the density perturbations

A= [

1

-1

0

-1

2

-1

0

-1

1

I

(6.5)

equations (6.1)-(6.3) may be written in the following compact form (D,+cD,)K+aAK

= 0

(6.6)

A simple calculation shows that the model matrix M of A is

(6.7) and its inverse M-l

is

(6.8)

252

P. K.

The product

MUNJAL

and L. A.

of M-l AM then gives the spectral

PIPES

matrix

S

M-lAM=S=

The eigenvalues In equation

of A are therefore

(6.6) let us introduce

(6.9)

A, = 0, A, = 1 and A3 = 3. the normal

coordinates

y by the transformation

K=My

(6.10)

where

(6.11) to get (6.12) If we pre-multiply

(6.12) by M-l,

we have

(D,+~D,)y+d4-~.4My

Since M-l AM following form

is the diagonal

spectral

matrix

=

S, equation

(D,+cD,)y+aSy which is equivalent

0

(6.13) (6.13) may be written

= 0

in the (6.14)

to the three equations

(4+c~,)J5= 0 (4+c~z)Y,+w2 = 0 (D,+cD,)y,+3aJ$ Let us now introduce

the on-ramp

(6.15)

= 0

input flow into lane 1 by specifying &(O, t) = k,(t) K&O, t) = 0 &(O, t) = 0

If we pre-multiply

(6.10) by M-l,

(6.16) i

we obtain

(6.17)

If we substitute

the boundary

conditions

(6.16) with (6.17), we obtain

Y1(0, t) = W) YZ(O>t) = k#) Y3(0, t) = k,(t)

(6.18)

On-ramp

density perturbations

on unidirectional

two- and three-lane

freeways

253

The solutions of the differential equations (6.15) subject to the boundary conditions (6.18) are

y,(x,t)=exp($)k,(l-:)

ys(x,t)=exp(*)ko(l-:)

The various density perturbations equation (6.10) in the form

I

(6.19)

I

(6.20)

K,, K, and K3 may now be obtained by expanding

If (6.19) is substituted into (6.20), we get K,(x, t) =

k,,(t-:) [i+iexp

K&x, t) = &(t-$

6-i

K3(x, t> = ko(t-$

[i-iexp

(y)+aexp

(+)I

(6.21) (6.22)

exp (q)] (G+)kexp

(+)I

(6.23)

Using the above set of equations, Fig. 8 gives the propagation of on-ramp density perturbation &(x, t)/i&,(t) vs. ax/c in uniform three-lane freeways, where the density perturbation caused by the on-ramp flow has a step function input as given by equation (5.23).

OX ?,

FJG. 8. Relative propagation

distance

from on-romp

(dimensionless)

of on-ramp density perturbations three-way freeway.

in different lanes of a

254

P. K. MUNJAL and L. A. PIPES 7. MODEL

VALIDATION

Many elements of the model described in this paper require experimental validation. Some of these include: 1. For a long homogeneous roadway under “steady-state” conditions, does the functional relationship 4 =f@) hold ? 2. Does the above “steady-state” functional relationship hold for unsteady-state phenomena? 3. Is the above functional relationship the same for acceleration and deceleration situations? 4. Suppose q(x, t) is the local flow, i.e. the number of cars passing x per unit time at t, and k(x, t) is the local concentration, i.e. the number of cars per unit distance at the point x at time t. Is the roadway in “local equilibrium ?” That is, does the functional relationship between q and k for “steady-state” flow hold locally in the sense that 4(x, t) =

fW, t>>?

5.How many cars does one need to average over to get local averages ? 6. Is the speed-density relationship of equation (3.1) sufficiently accurate? Does it hold for unsteady-state phenomena? 7. Is the lane-changing rate expressed in equation (4.1) a reasonable representation of the lane-changing phenomena? The particular expression is ad hoc but appears reasonable. It should be noted that most of the questions raised above apply to the whole continuum approach in general.

CONCLUSIONS

Propagation of on-ramp density perturbations on undirectional two- and three-lane freeways has been discussed on the basis of a simple mathematical model. The basis of the model is the use of separate equations of continuity of an idealized “traffic fluid” under equilibrium conditions for each lane and a hypothesis for lane-interactions. This hypothesis assumes that the rate of exchange of vehicles between the two lanes is proportional to the differences of the deviations of their densities from their equilibrium values. Another assumption, which is made through defining the boundary conditions, is that the acceleration of lane is very short. Based upon the above assumptions, the relative propagation of perturbed densities in different lanes of a two- and three-lane freeway has been determined. In addition, the various model assumptions that require experimental validation were briefly discussed. Acknowledgements-The authors wish to express their indebtedness to their SDC co-workers Dr A. V. Gafarian and Mr. R. L. Lawrence for the many technical discussions on the subject of this paper. REFERENCES GAZIS D. C., HERMAN R. and WEISS G. H. (1962). Density oscillations between lanes of a multi-lane highway. Ops Res. 10,658-667. GREENSHIELDS B. D. (1935). A study of traffic capacity. Proc. Highwuy Res. Bd. 14, 468. LIGHTHILL M. J. and WHITHAMG. B. (1955). On kinematic waves-II: a theory of traffic flow on long crowded roads. Proc. R. Sot. A-229, 317-345.

On-ramp

density perturbations

on unidirectional

two- and three-lane

freeways

255

MUNJAL P. K. (1970). A simple off-ramp traffic model. Highway Res. Record No. 334,48-61. OLIVERR. M. and LAM T. (1967). Statistical experiments with a two-lane flow model. Vehicular Traffic Science, pp. 170-180. American Elsevier, New York. PIPES L. A. (1967). Car following models and the fundamental diagram of road traffic. Trunspn Res. 1, 21-29. PIPES L. A. (1968). Topics in the hydrodynamic theory of traffic flow. Transpn Res. 2, 143-149. bXIAItDS P. I. (1956). Shock waves on the highway. Ops Res. 4,42-51. WARDROPJ. G. (1963). Experimental speed-flow relations in a single lane. Proc. 2nd Znt. Symp. Theory of Traffic Flow, The Organization for Economic Co-operation and Development, London. pp. 104-119.

Note-A paper written by the same authors which gives the analysis of an n-lane case and effect of controlled ramD flow is scheduled for vublication in Tranmortation Science Vol. 5, No. 4.

I regret to announce the death on 16 January 1971 of my dear friend and colleague, Professor Louis A. Pipes. FRANK A. HAIGHT