GAP ACCEPTANCE IN THE FREEWAY MERGING PROCESS

by

Donald R. Drew Head of Design and Traffic Department

and Principal Investigator

Lynn R. LaMotte Research Assistant

Joseph A. Wattleworth Assistant Research Engineer

and Co-Principal Investigator

and

Johann H. Buhr Research Assistant

REPORT 430-2

Volume 2 of the Final Report on "Gap Acceptance And Traffic Interaction in the Freeway Merging Process"

Sponsored by U.S. Bureau of Public Roads

Contract No. CPR-11-2842

with the

Texas Transportation Institute Texas A&M University College Station, Texas

ACKNOWLEDGMENTS

The success of this phase of the Project with the national scope of its field studies is due to a large extent on the cooperation of the Fed­eral Government, six State Governments, and many local govern­mental agencies. The following partial list of personnel, their staffs and agencies is offered:

California Division of Highways - Mr. James E. Wilson, Mr, Karl Moskowitz and Mr. Leonard Newman

Illinois Division of Highways - Mr, Charles H. McLean

Michigan State Highway Department - Mr. Joseph Marlowe

Missouri Highway Department - Mr. James Little and Mr. James Roberts

Texas Highway Department - Mr. Dale D. Marvel, Mr, John N. Lipscomb and William V. Ward

Detroit Department of Traffic - Mr. Alger F. Malo

New York City Department of Traffic - Mr. Edward Bonelli

New York Department of Public Works - Mr. N. C. Parsons

Long Island State Park Commission - Mr. Boyce

Deserving special mention is Mr. Joseph W. Hess, Acting Leader, Improved Utilization of High Speed Highways Task Group of the U.S. Bureau of Public Roads. Mr. Hess, who was instrumental in defining the scope and direction of the Project in the Project Prospectus, has helped greatly in making contacts with local agencies and selecting study sites.

The staff of the Texas Transportation Institute who worked in the collection, reduction, and presentation of the data are to be congratu­lated. Special recognition is due to Mr. Thomas G. Williams, Re­search Assistant, who assumed the responsibility for both the flight and filming aspects of the study procedure, and to Mr. Charles E. Wallace, Research Assistant, who coordinated the many activities in­herent in the presentation of this report,

11

GAP ACCEPTANCE IN THE FREEWAY MERGING PROCESS

ABSTRACT

This study is the first phase of a four-year program on freeway merging undertaken by the Bureau of Public Roads to (1) furnish more detailed information on the effect that geometric variables have on the merging of ramp traffic, (2) develop usable distributions of traffic variables for simulation programs, and (3) develop an optimum ramp metering and merging control system. The emphasis in this report is the collection and collation of gap acceptance characteristics.

The theoretical development of models and useful parameters for describing the merging process include (1) the derivation of the forms of the mean and variance of the delay to a ramp vehicle in position to merge and (2) the treatment of the variability of critical gaps and gap acceptance among drivers through the identification of the representa­tive forms for both critical gap distributions and gap acceptance func­tions.

Through the application of "individual record probit analyses;" simple, statistically significant relations between the percent gap acceptance and gap size is established. Using this approach, the char­acteristics of lags and gaps and single and multiple entry merges are compared, as well as fast to slow moving merging vehicles. The pro­bit analyses are generalized to establish a relationship between percent acceptance as the dependent variable and gap size and vehicular speed as the dependent variables.

The fact that 32 ramps-chosen to reflect diverse operating, geo­metric, geographic and environmental conditions-were continuously filmed at 5 frames per second for an average of an hour, and that enough data was collected to run 1344 usable gap acceptance regres­sions serve to demonstrate not only the vast quantity of data involved, but the nature of the characteristics now available to interested re­searchers.

iii

Contents of the Final Report on "Gap Acceptance and Traffic Interaction in the Freeway Merging Process"

Summary Report: "Gap Acceptance and Traffic Interaction in the Freeway Merging Process"

Report 430-1: "A Nationwide Study of Freeway Merging Operations"

Report 430-2: "Gap Acceptance in the Freeway Merging Process"

Report 430-3: "Operational Effects of Some Entrance Ramp Geometrics on Freeway Merging"

Report 430-4:

Report 430-5:

Report 430-6:

Report 430-7:

"The Determination of Merging Capacity and Its Application to Freeway Design and Control"

11 Traffic Interaction in the Freeway Merging Pro­cess"

"Digital Simulation of Freeway Merging Operation"

"Annotated Bibliography on Gap Acceptance and Its Applications 11

The op1n1ons, findings and conclusions expressed in these publications are those of the authors and not necessarily those of the Bureau of Public Roads o

INTRODUCTION

Scope of the Project

The subject of ramp vehicles merging into the freeway stream has deservedly been treated by a number of researchers. Most of the research has been devoted to empirical studies leading to design and operational procedures. Mathematical treatment of the merging maneuver has been attempted, too, with somewhat limited success be­cause of the complexity of the vehicle interactions. Computer tech­nologists have contributed several digital computer simulation pro­grams, but lack of detailed criteria on gap acceptance and merging logic has hampered progress.

In the summer of 1965, the U.S. Bureau of Public Roads under­took research to furnish detailed criteria on the merging of ramp vehicles into the freeway system. A contract, "Gap Acceptance and Traffic Interaction in the Freeway Merging Process, " was awarded to the Texas Transportation Institute. The general aim of this research is the conception of a relationship between the many variables asso­ciated with the interaction of vehicles traversing a ramp and merging onto a freeway so as to determine the effects of the following on merg­ing operation and level of service:

(1) Traffic characteristics such as gap availability, gap accep­tance, speed and volume;

(2) Ramp geometrics such as length, curvature, angle of con­vergence and grades, and acceleration lane geometrics such as length, shape, delineation and location of lateral obstruc­tions;

(3) System considerations such as interchange type, ramp con­figurations, frontage roads and upstream or downstream bottlenecks, and environmental elements such as metropolitan area size, location within the city, and lighting;

(4) Control devices such as freeway lane controls, yield or merge-ahead signs, traffic signal feeding the_ entrance ramp, and ramp metering stations.

The underlying purpose of this research is the application of the above information to the following:

1

(1) In design and operation--the furnishing of more-detailed in­formation on the effect that geometric variables and traffic characteristics have on merging traffic;

(2) In simulation- -the development of usable distributions of traf­fic variables for simulation programs.

In order to fulfill the broad project objectives, 32 ramp-freeway connections located in 8 metropolitan areas in 6 states from coast to coast and from border to border were chosen. These locations -specially chosen to give a complete range of geographic, environmental, geo:rnetric, and operating conditions-are summarized in Appendix A.

Specific Objectives

There are three purposes for conducting field studies of traffic characteristics in ramp-freeway merging areas. The first reason is for the eventual testing and refinement of models. The second purpose for collecting data is that at the present time only very limited data of this type are available. Gap acceptance data is a prime example of this. Much of the meager gap acceptance data available is very old data, based on a small sample size, or for a peculiar situation such as a left-hand ramp or stop sign control. These conditions severely limit the usefulness of these data. No data has been collected on the effect of ramp and acceleration lane geometrics on the·gap acceptance characteristics of vehicles on the ramp. A third very important ap­plication of the data on merging characteristics is for simulation in­puts.

Specific objectives of this phase of the project research are:

(1) Development of models and useful parameters for describing gap and lag acceptance and delay in the merging maneuver;

(2) Presentation of gap and lag acceptance characteristics ob­tained from studying some 32 ramps across the United States.

(3) Determination of any differences in gap acceptance charac­teristics for different points of entry along the acceleration lane.

(4) Delineation of the roles of absolute and relative speeds in the merging process and their effect on gap acceptance.

2

(5) Identification of gap acceptance characteristics of more than single ramp vehicles so as to determine the efficiency of platoon merging;

(6) Investigation of the effect of outside freeway lane volumes on gap acceptance.

3

THEORY

Definitions and Terminology

In actual practice, the entrance ramp-freeway connection may be regarded as a special case of the uncontrolled intersection. The three fundamental maneuvers performed by vehicles in the vicinity of this connection area may be identified as follows:

Lane Change. The transfe i: of a vehicle from one traffic lane to the next adjacent traffic lane.

Merging. The process by which vehicles in two separate streams moving in the same general direction combine or unite to form a single stream.

Weaving. The oblique crossing of one stream of traffic by another accomplished by the merging of the two streams into one and then the diverging of this common stream into separate streams again.

The freeway elements associated with the above maneuvers are:

Acceleration lane. An added width of pavement adjacent to the main roadway traffic lanes enabling vehicles entering the main road­way to adjust their speed to the speed of through traffic before mer g­ing.

Ramp. A connecting roadway between two intersecting or parallel roadways, one end of which joins in such a way as to produce a merg­ing maneuver.

Frontage road. A roadway paralleling a freeway so as to inter­cept traffic entering or leaving the facility and to furnish access to abut­ting property.

Basic to the description of traffic interaction in the merging pro­cess are the following variables:

Headway. The interval of time between successive vehicles mov­ing in the same lane measured from head to head as they pass a point on the road.

Space headway. The distance between successive vehicles moving in the same lane measured from head to head at a given instant in time.

4

Gap. A major stream headway that is evaluated by a minor stream vehicle desiring to either merge into or cross the major stream. The units may be those of either time or distance (space gap).

Lag. The interval of time between the arrival of a minor stream vehicle and the arrival of a major stream vehicle at a reference point or points in the vicinity of the area where the streams either cross or merge.

Space lag. The distance between a minor stream vehicle and a reference point where the minor stream crosses or joins a major stream, subtracted from the distance between a major stream ve­hicle and the same reference point, with both distances measured at a given instant in time.

Figure 1 is a time- space diagram prepared to illustrate the re­lationship between the geometric elements comprising the merging area and the movements of mainstream and ramp vehicles within the area. Distance is plotted as the ordinate, time is the abscissa, and the slope of the traces denote speed. The traces for freeway and ramp vehicles are identified in the figure. A vehicle is regarded as a ramp vehicle as long as it remains at least partially on the acceleration lane.

It is convenient to subdivide merges into their basic types and classifications, then to determine the kind and amount of information needed for intelligent analysis of the movements. Following is a list­ing of merge maneuvers, defined for use in this report: (see Figure 2)

Optional merge. The merging vehicle voluntarily moves from the acceleration lane into the outside traffic lane.

Confined merge. The merging vehicle is forced into the outside traffic lane by the presence of the end of the acceleration lane.

Ideal merge. The merging vehicle is able to enter the freeway stream without causing a freeway vehicle to reduce its speed or change lanes.

Forced merge. The ramp vehicle effects the merging maneuver into the freeway stream so that an oncoming freeway vehicle or ve­hicles must either slow down or change lanes.

Single entry merge. One ramp vehicle moves into a single free­way time gap.

5

i Ii 0 c:i: 0 0:::

LIJ (!)

~ z 0 0::: IJ...

SPACE DIAGRAM FOR TIME A-A

i LIJ (.)

z c:i: 1-(/)

0

;q!/j;/;;1mlmrll; OUTSIDE LANE == ON-RAMP

DELAY TO VEHICLE 2

A

TIME ---+-

TIME- SPACE RELATIONSHIPS OF FREEWAY MERGING MANEUVER FIGURE 1

I a la 10

I

I

I

I a

~

lo la I

I~

OPTIONAL MERGE

IDEAL MERGE

GAP MERGE

I c

la a I

~

la la

:~ SECT. A-A

MERGE

FORCED MERGE

LAG MERGE

TYPES OF FREEWAY MERGES

Figure 2

7

Multiple entry merge. Two or more ramp vehicles merge into a single freeway time gap.

From Figures 1 and 2 and the above classification, it is apparent that a merge must be qualified as being either single or multiple entry, optional or confined, ideal or forced, and gap or lag. For example, a given merge might be described as being a "single entry, optional, ideal gap merge. 11

Merging Parameters

In the previous section, a few traffic variables--headways, gaps and lags- -were identified. In addition to these variables, the speed of the major stream vehicles, speed of the merging vehicles, relative speed, major stream flow, and minor stream flow are additional variables that must be considered in any rational description of the merging process.

Whereas a variable is a quantity which can assume any value or number, a parameter is a term used in identifying a particular variable or constant other than the coordinate variables. For example, 11 vol­ume11 is a traffic variable and "capacity"--the maximum volume that a facility can accommodate- -is the corresponding traffic parameter. Some important figures of merit for describing the gap acceptance phenomenon in freeway merging are the critical gap, percent of ramp vehicles delayed, mean duration of static delay accepting a gap, mean length of queue, and total waiting time on the ramp.

Several "critical" values have been discussed in the literature. Greenshields 1 defined the "acceptable average-minimum time gap" as a gap accepted by half the drivers. Raff2 used a slightly different pa­rameter, the "critical lag. 11 The critical lag is the size lag which has the property that the number of accepted lags shorter than the critical lag is equal to the number of rejected lags longer than the critical lag. As such the Greenshields and Raff parameters are median values.

The principal use of gap acceptance parameters is to simplify the computation of the delay duration by permitting the assumption that all intervals shorter than the critical value (lag or gap) are rejected while all intervals longer are accepted. It has been suggested3, 4 that the mean of the critical gap distribution, not the median should be used in delay computations. Nevertheless, the "median critical gap" remains a practical parameter because, as shall be shown, it can be readily obtained graphically.

8

Delay Models

There have been a number of theoretical papers 5

' 6

dealing with the delay to a single waiting vehicle on the minor street of an uncon­trolled intersection, due to the traffic on the outside lane of an inter­secting highway. It can be shown that this theory is valid for a driver on a ramp waiting to join a stream of traffic.

Most discussions of this problem have assumed that the distribu­tion of main stream arrivals is Poisson, i.e., that the probability that a given gap is between t and t + dt seconds is given by an expres­sion of the form qe-qt where q is the flow. Raff2 considered the delay problem as related to vehicles whereas Tanner's analysis5 was spe­cifically applied to pedestrian delays. The pedestrians were assumed to arrive at random, and all waited for a critical time gap of T seconds in order to cross the highway. Although Mayne6 showed that one could obtain results for the Laplace transform of the delay duration for other than Poisson traffic on the main stream traffic, the only case con­sidered in detail was for this headway distribution. The derivation which follows assumes Erlang headways and as such is a generalization of the combinatorial techniques suggested by Raff2 , Tanners and Mayne6.

It may be assumed that a ramp driver waiting to merge measures each time gap, t, in the traffic on the outside lane of the freeway until he finds an acceptable gap, T, which he believes to be of sufficient length to permit his safe entry. If he accepts the first gap (t>T), his waiting time is zero. If he rejects the first gap (t<T), but accepts the second gap, his expected waiting time would be one gap interval. If it is assumed that the driver's gap acceptance does not change with time, then by induction, the individual waiting periods form a geometric dis -tribution and the probability, P(n), of any driver having a wait for n intervals each less than T seconds before merging is

n P(n) = p (1-p), n = 1, 2 .• ( 1)

where

ITO p = P(t< T) = f(t) dt (2)

and f(t) is the distribution of gaps in the major stream. It is conven­ient to define a normalized random variable w such that

w=t/T (3)

9

or t = Tw ( 4)

and dt = T dw (5)

Substituting (4) and (5) into (2) yields

p = P(wd) = f f(Tw) T dw. 0 .

(6)

The density of w may be defined as

g(w) = f( Tw) T (7)

which when inserted in (6) gives

P -- Jl g(w) dw (8) 0

If the distribution of gaps on an outside freeway lane of flow q can be described by the Erlang distribution,

a f(t) = (aq) ta-1 e -aqt

Y (a)

then it follows from (3) to (8) that

p = ( 0

a (aqT)

Y (a) a-1 -aqTw d

w e w .

Equation ( 10) is of the form of the incomplete gamma distribution which is of course equivalent to the cumulative Poisson:

, a-1

( 9)

(10)

-aqT i p = 1-e l (aqT) /i ! (11)

i=o and

a-1 -aqT i

1 - p = e l (aqT) Ii ! ( 12) i=o

Placing (11) and (12) in (1) gives the probability that the first n time-gaps between vehicles in the major stream are all less than the normalized critical gap, (w = 1) but that the n + 1th is greater than the nr.rmalized critical gap. This probability P(n) averaged over the distribution of gaps less than the normalized critical gap, g(w) (o<w <l), gives the distribution of waiting time or delay f(µ) to a ramp

10

vehicle in position to merge. In terms of the moment generating func­tions this may be expressed? as

00

M (8) = l µ

n=o

n P(n) M (8) ,

w ( 13)

where Mµ ( 8) is the m. g. £. of the distribution of delay to a ramp ve­hicle waiting for a gap greater than the normalized critical gap. From (1), (11) and (12), it is apparent that

00

M (8) = (1 - p) l [pM (8)]n µ w

( 14) n=o

1 - p =---~--

1 - p M (8) w

(15)

In order to use (15), Mw (8) must be found. It is apparent that g(w) ( o<w<l) is a conditional probability which by definition becomes

g(w) (o<w<l) = P(w<w I w<l)

( g(w) dw

= r g(w) dw

(16)

0

Since the denominator of (16) is given is (8), one obtains the density of gaps less than the normalized critical gap as

and the m. g. f. as

g(w) (o<wd) = g(w)/p

1

M (8) = ..!_ I w p 0

9w e g(w) dw

Changing the limits of summation and substituting for g(w) yields

( 1 7)

(18)

(aqT)a [Joo a-1 -w(aqT-8) Joo a-1 -w(aqT-8) J M ( 8) = ( ) w e dw - w e · dw

w PY a o 1

If a change of variable is made such that u = w(aqT- 8) in the first in­tegral and u + 1 = w in the second integral, then

(a q T ) a ( T ) - a r00

a - 1 - u d . - (a q T -8 ) J00

( u + 1 ) a - l M ( 8) = ( ) [ aq - 8 J u e u-e

w Py a o o -u(aqT-9) d ]

e u

11

Use of Newton's binomial identity,

a-1 (u+ l)a-1 = l

i=o

a-i-1 u

in the second integral, and noting that the first integral is a gamma function gives

a-1 oo a 'i' a-1 I a-i-1 -u(aqT-e)

M ( e) = .!_ ( l __ e ) -a _ (aqT) e - (aqT- e) l ( i ) 0

u e du. w p aqT Py( a)· i=o ( 19)

Since the second integral is a gamma function with parameters (a-i) and (aqT-e), the second term may be written as

or

a.,.l a 'i' y(a) Y(a-i)

(aqT) e -(aqT-e) .4 i!y(a-i) a-i py(a) · i~o (aqT-e)

a ( aqT ) aqT-e

-(aqT-e) e

p

a-1

l i=o

i (aqT-e)

• I 1 .

(20)

Substituting (20) for the second term in ( 19) and collecting terms, one obtains

M (e) w

-a a-1 i = 1 e [l-e -(aqT-e) 'i' (a~T1 -e)

p (1- aqT) l i.

i=o . ] (21)

Now it is possible to find the moments of the delay to a ramp ve­hicle in position to merge by evaluating the derivatives of (15),

and

_dM_µ_(_e_) = --"-p-'('--l-'-p::...;.) ___ dM_w_(e_)

d2

M ( e) µ

de [l-pM (e)]2 de w

p(l-p) [dM 2

(e)/d62

J 2(1-p) [pdM ('a)/daJ2

w w ----=---------- + ________ 3 __ _

de2 2 [1-pM (e)]

w [1-pM (e)]

w

(22)

' (23)

at e = 0, where the derivatives of M (e) in the expressions are given by w

12

dMn ( e ) a +n - l . ( ) ( T) a - (a q T - e) (a q T - 8) 'i' ( T e) i

w = y a+n aq e [ e - l aq - ] . (24)

Py(a) (aqT-e)(a+n) . i=o i!

Expansion of (22) and (23) leads to the following expressions for the mean and variance of the delay distribution:

aqT a 1 e - I (aqT)

. ' µ(w) =

i=o 1 •

a a-1 (aqT)i qT I

i I

i=o . ( 25)

a+l i

[eaqT _ \ (aqT)

(a+l) l ., 2

lo 2 i=o

a (w) a = -----a---1----i-- + µ (w) a

a( qT) 2 l (a~~) . 1 •

(26)

l=O

Converting from the normalized parameter w to the original variable t, the mean and variable for values a = l, 2, 3 and 4 become:

-1 qT µ(t) =q (e -1-qT)

l

e 2 qT -l-2qT-2(qT) 2

µ (t)2 = q(l+2qT)

( ) _ e

3qT -l-3qT-4. 5(qT)

2 -4. 5(qT)

3

µ t 3 - 2 q [1+3qT=4. 5(qT) ]

( ) _ e

4qT -l-4qT-8(qT)

2 -10. 7(qT)

3 -10. 7(qT)

4

µ t4- 2 3 q [ 1+4qT+8(qT) +10. 67(qT) ]

= 2qT 2 T qT l e - q e -

2 q

(2 7)

(28)

( 2 9)

(30)

( 31)

2 2e 4 qT _e 2 qT -2qTe 2 qT -8(qT) 2e 2 qT -l-4qT-2(qT) 2

(J (t)2 = 2 2 (32) 2q ( 1+2qT)

13

3qT 2 3 4 _ 4e -4-12qT-18(qT) -18(qT) -13. 5(qT) - 2 2 +

3q [ 1+3qT+4. 5(qT) ] ( 33)

= 5e 4

qT _5-20qT-40(qT)2

-53. 3(qT)3

-53. 3(qT)4

-42. 67(qT)5

2 . 2 3 4q [1+4qT+8(qT) +10. 67(qT) ]

2 + µ (t) 4

(34)

Equations 2 7-30 are plotted in Figure 3 and Equations 31-34 in Figure 4.

Critical Cap Distributions

The theoretical delay values of the previous section are based on a fixed critical gap for all drivers. A more realistic description of delays can be obtained by replacing the fixed critical gap with a distri­bution of critical gaps, f(T). 8 • 9 The forms for the mean and variance of the distribution of delay become

M(T) = J: µ (T) f(T) dT (35)

and

(36)

Assuming that the headway distribution on the freeway is negative exponential, for example, substitution of (27) in (35) yields

-1 M(T) = q .· J

oo

0

eqT f(T)dT- J~ 0

T f(T) dT-q-l J00

f(T)dT, 0

(3 7)

Realizing that the second term defines the mean of the critical gap distribution and the integral in the last term must equal unity gives

-1 Joo qT - -1 M(T) = q e f(T)dT - T - q

0

(38)

uome representative forms for critical gap distributions are shown in Figure 5. If one assumes that the critical gaps of the drivers are distributed uniformly between c and c 1 then

14

30

20

(/) c z 810 lJJ (/)

-II "

=!:-

~ 5 ...J lJJ c Cl> z ~ 3 lJJ :Ii

(/)

c z 0 ~10 (/)

"' II

" ::I.

~ 5 ...J lJJ c Cl> z Cl> 3 0: lJJ ::;:

µ. a= I

=q-I (eqT_qT-1)

1200 .333

FREEWAY VOLUME IN OUTSIDE LANE , q

3qT 3 2 µ. • e -4.5(qT) -4.5(qT) -3qT-I

a=3

I'--~~~~~~~~~~~~~~~~~~

600VPH 800 1000 .167VPS .250

1200 .333

1400 1600 .417

FREEWAY VOLUME IN OUTSIDE LANE, q

30

20 µ. =

a=2

(/) c z 810 lJJ (/)

C\I II

" :!: ~ 5 ...J lJJ c Cl> z c:o

3 0: lJJ :Ii

~O~O~V~P=H-"-!--~----,.,,-!,-~~~12~0~0~__,.~,c-~_..,-=--' .I 67VPS .333

FREEWAY VOLUME IN OUTSIDE LANE , q

4qT 4 3 2 fL = e -10.67(qT) -10.67(qT) -(8qT) -4qT-I a•4 (/)

c z q [ 10.67 (qT)

3 +8(qT)

2 +4qT +I]

0 ~IOt-----:--r.--~-----,-.~~~,-r--~-.-~~-+~­(/)

::I.

<t II

" ~ 5t--~~-i-~~----,..,.-~~t-~~-t---,"---+~-o ...J lJJ 0

Cl> z ~ 317-~~-j--f-~-+~~~t--r--~-t-~~-+~-o lJJ ::;:

IL-~~--'---L-~---'-~~~'---~~_,.L_~""'-----'-~~ 600VPH 800 1000 .167 .250

1200 .333

1400 1600 .417

FREEWAY VOLUME IN OUTSIDE LANE, q

MERGING DELAY IN TERMS OF THE FREEWAY FLOW q, CRITICAL GAP T, AND ERLANG CONSTANT, a.

FIGURE 3

0

"' a: <( :::J 0

"' "' 0 z 0 ()

"' "'

"'b

l:c _J

"' 0

~

"' ()

z <(

ii: :i!

0

"' a: <( :::J 0

"' "' 0 z 0 ()

"' "'

"'

200

"' " "

"' " "

100

50

30

10

IL-~~__i_.~ _ _j_ ___ L__ __ _L __ -+__J

600VPH 800 1000 1200 1400 1600 .167 VPS .250 .333 .417

100 2

<T o•3

FREEWAY FLOW IN OUTSIDE LANE, q

ff " 3~ 2 3 4 4e -4-12qT-18(qT) -18(qT) -13.5(qT)

3q 2

[1+3qT+4.5(qT)2

]

2 +/J. o=3

0

"' ~ 0

"' gi z

200 1------1--

100

§ 30f---F--+---i->~--+---l-----+-~

"'

"'~ ~ _J

"' 0

~

"' () z <(

ii: :i!

0

"' a: <( :::J 0

"' "' 0 z 0 ()

"' "'

"' " "

10

1'-----'---~<'----'----1----.....L---' 600 VPH 800 1000 .167 VPS .250

1200 .333

1400 1600 .417

FREEWAY FLOW IN OUTSIDE LANE, q

2aal 4qT v I ( 13 ~ 15 I 2 5e -5-20qT-40(qT) -53.3(qT) -53.3(qT) -42.67(qT) 2

""a=4= 2 [ 2 3 J + /J. a=4 4q 1+4qT+B(qT) +I0.67(qT)

IOOL--1--~-=--+--......,e----~--__,=------+----<

50

30

"' " " "! "'b

l:c _J

"' 0

~

"' () z ~ a: :i!

10 ~ _J

"' 10

0

~

"' () z ~ a: :i!

1.__ __ _,_ ___ J__..L:__-1-___ J_ __ _,___J 600VPH 800 1000 1200 1400 160D 1400 1600 .167VPS .250 .333 .417 .417

FREEWAY FLOW IN OUTSIDE LANE, q FREEWAY FLOW IN OUTSIDE LANE , q

VARIANCE IN DELAY IN TERMS OF THE FREEWAY FLOW q, CRITICAL GAP T, AND ERLANG CONSTANT, a.

FIGURE 4

16

,.... -....... -~ -w N c;; LL 0 Cl) a.. <{ (!)

...J <{ (.)

l­a: (.)

LL 0

>­(.)

z w ~ 0 w a:: LL

0

RECTANGULAR DISTRIBUTION

r (,)

I (,)

TRANSLATED NEGATIVE EXPONENTIAL

ERLANG DISTRIBUTION

LOG-NORMAL DISTRIBUTION

c

f(t) = _I_ e-(t-c)/(T-c) l-c ·

'f

f(t) =_I_ e-(ln t-iiit)/2s2

s"12TI

GAP SIZE, t 2 'f

REPRESENTATIVE FORMS FOR CRITICAL GAP DISTRIBUTION: Figure 5

l 7

. -1 ' f ( T ) = ( c1 - c )

The translated negative exponential distribution,

f(T) = (T-c)-1 e-(T-c)/(T-c)

(3 9}

(40)

has also been suggested 1 O as a gap distribution function, as has the Erlang distribution, 11

and

f(t) (a/ T)a

=----(a-1)

a-1 -aT/T T e ( 41)

Substitution of (39), (40) and (41) in (38) gives respectively:

M(T) -1 -2 qc qc - -1 = (c 1 -c) q [ e 1 -e ] -T - q (42)

M(T) = [q(l-qT+qc)]-l eqc -f -q-l (43)

M(T) ( 44)

which correspond to the distribution of delay for the first three criti­cal gap distributions in Figure 5.

The derivations of this section (Equations 35-44) are based on utilization of a distribution of critical gaps for all drivers, f(T). This is a concession to the obvious fact that not all drivers have the same critical gap. The difficulty lies, however, in measuring the critical gap for individual drivers in order to obtain a distribution of critical gaps. One technique 4 to obtain such a frequency distribu­tion4 samples only drivers who have rejected at least one gap before merging and is based on the reasonable assumption that the driver's critical gap must lie somewhere between the largest gap he rejected and the gap he finally accepted.

Although observations of a narrow range delimit a driver's gap more closely, by the very nature of the technique observations with wider ranges have greater influence on the shape of the density func­tion. Dawsonl2 suggests two weighting procedures to overcome this shortcoming which seems promising. Still, there exist some practical as w~ll as conceptual shortcomings in the technique which tended to rule it out as a basis of parameter determination for this project.

18

First, the technique only considers drivers who have rejected a gap; and secondly, the technique is obviously not applicable to lags.

Gap Acceptance Functions

In the derivations in the preceding sections based on the critical gap concept, it is assumed that the waiting driver evaluates each inter­car time gap, choosing to merge if the gap is greater than some pre­determined time gap T, or not, if the gap is less than T seconds. Herman and Weiss 13 suggest that a more realistic model would be to associate with each time gap, a gap acceptance probability P(T) such that the waiting driver crosses the highway with the probability P(T) when confronted with a gap of duration T. Some representative gap acceptance functions are illustrated in Figure 6. .

Using methods strongly dependent on renewal theory, Weiss,and Maradudinl4 formulate the merging delay problem in terms of an in­tegral equation instead of using the combinatorial reasoning ~pproach inherent in Equations 1 through 44. The expression for the mean of the delay distribution is:

where

M(T) = f" 0

1-P 0

TiJ! 0

(T)dT + -p-

p 0

P(T)f (T)dT 0

P = f'" P(T)f(T)dT 0

ijJ (T) = f (T). [ 1-P(T)) 0 0

ljJ (T) = f(T) [1- P(T) ]

TijJ(T)dT (45)

(46)

(4 7)

(48)

(49)

and f(T) is the distribution of gaps on the outside freeway lane and P(T) is the gap acceptance function. The zero subscripts refer to the gap availability distribution and gap acceptance function for the first gap.

W . 13

• 14

h 1 d h h f' d 'b e1ss as app ie is t eory to some speci ic istr1 utions.

19

-+--a.. ~

+-

a.. <( (!)

<(

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>-I-_J

CD <( CD 0 0:: a..

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.60

.40

.20 {

-o P(t} = (t-c}/(c1-c}

= I

,(t<c} , (c<t<c1}

, ( t>c1}

oi----.c.__----------------------1

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.60

.40

.20

0

.80

.60

.40

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0

.80

.60

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ERLANG FUNCTION (a= 1,2,···} NEG. EXPONENTIAL (a= I}

STEP FUNCTION (a= co}

( >={0,Ct<:> pt I ,(t>t}

LOG-NORMAL FUNCTION

c

8 II

c

-(t-c}/ct-c} 1-e 0

, ( t>C} , ( t<C}

P(t}= ~ ta-1 e-at/t dt :L\a it -(a-1)! o

_ 1 lt e-(ln t-fri1}/2s2

P(t}- s,J2.Tf t dt 0

t GAP SIZE, t

2t

REPRESENTATIVE FORMS FOR GAP ACCEPTANCE FUNCTIONS Figure 6

20

For example if the distribution of gaps on the outside lane of the free­way is taken as

-qT f(T) = f (T) = qe

0

and the gap acceptance function is

P(T) = P (T) = 1-e -;\(T-c) 0

(50)

(51)

one obtains for the mean delay to a ramp vehicle in position to merge

{ e q c - 1- q c + (q I ( q +;\ ) ) 2

( 1 +q c + c) {1- e - q c)}

+ (q/(q+;\)+qc] e -qc) (52)

It is interesting to note that Equation 27 is, as one would expect, a special case of both (44) and (52). Taking (44) first,

lim a+ oo

- -a (1- .9.!.)

a = eqT (5 3)

from the definition of "e ", the base of the natural system of logarithms. Since the variance of the Erlang distribution is zero for a = oo, this may be interpreted as the case for which the critical gap in (41) is constant and has the value T. Substitution of (53) in (44) gives {27).

Likewise in (52) since ;\=(T-c)- 1 (see Figure 6), if T=c we have a step function for the gap acceptance probability and (52) also reduces to (27).

The fact that the delay obtained using the fixed value is both a special case of the delay obtained using the critical gap distribution in (41) and a special case of the gap acceptance function in (51) tends to establish (1) that the probability of a driver accepting a gap of size T, P(T} in Figure 6, is the same as the probability of that driver having a critical gap less than T (see Figure 5),

P(T) = J: f(t) dT

21

(54)

and (2) that the mean not the median of both the critical gap distribu­tion or the gap acceptance function is the correct parameter for delay computations.

Theoretically then, the critical gap may be obtained from a gap acceptance function. From a practical point of view this is desirable because it is much easier to obtain data on the gap acceptance charac­teristics of drivers than it is to try to measure the critical gaps of drivers directly. Moreover4, it has been shown that the mean of the critical gap distribution and the mean of the gap acceptance function for a given ramp do in fact exhibit close agreement.

22

PROCEDURE

Data Collection

Since data on merging characteristics was to be collected at some 32 ramp locations from the far corners of the United States, there was a definite need for a standardized method of data collection in order to facilitate both the collection and analysis of data. The simultaneous collection of the many variables pertinent to the overall project objec­tives required the continuous viewing of an area of influence of about a quarter of a mile.

An aerial photographic technique was developed utilizing a 35 mm Automax data recording camera attached to a special tripod which was mounted in the rear of a Cessna P206 aircraft. The rear baggage com­partment door was fitted with a large plexiglass window. The aircraft was circled above each entrance ramp in a radius of.about 1I4 to 1 /2 mile during the peak traffic period and the traffic was filmed at 5 frames per second. Two 400 ft. magazines were used with the camera which allowed two filming periods of 22 minutes each with approxi­mately 2 minutes in between to switch magazines. The camera was fitted with a data chamber which included a clock, a frame counter, and a data slate where information about the study location could be recorded. The data chamber information was automatically recorded on one edge of the film.

Data Reduction

The frame number in which each vehicle in the outside lane of the freeway and in which each ramp vehicle crossed 200 ft. stations (marked on the shoulder by 1 by 6 ft. plastic stripes so as to be seen clearly in the films) was recorded. In this way, gaps could be deter­mined at any point within the 1I4 mile study area to an accuracy of 1I5 second.

A battery of programs was written. These include a time-space diagram of the merging area for the entire study period using the Cal­Comp plotter of the Texas A&M University Data Processing Center; plots of contour maps of such descriptive variables as speed, volume, density, acceleration noise, speed noise, energy, and shock wave speed; and summaries of volume-speed-density relations. These are discussed in detail in another report.

23

Pursuant to the objectives of this particular report, a computer program was written which identifies for each ramp vehicle, the re­jected (or accepted) lag, all rejected gaps, the gap finally accepted, the gap after the one accepted, the delay to the ramp vehicle, and the station of entry in the acceleration lane. A sample of this output is illustrated in Appendix B. AU lags and gaps are referenced to the ramp nose.

The Binomial Response

If an entrance ramp driver, selected at random from a population, is given an opportunity to merge, the probability he will accept is p; the probability to rejecting the opportunity is ( 1-p) or q. The oppor­tunity to merge here may be measured by the time gap, space gap, relative speed, or some combination of these or other factors. If two drivers are given the same opportunity, and if their reactions are completely independent, the probability that both accept is p 2 , and the probability tha.t both reject is q 2 ; the probability that only the first accepts is pq, and the probability that only the second accepts is qp. Thus the total probabilities of 2, 1, and 0 accepting are p 2 , 2pq and q2

respectively, the successive terms in the expansion of (p+q) 2 . In a similar manner it may be seen that if a group of n drivers is exposed to the same merging conditions, and all react independently, the prob­abilities of n, (n-1), (n-2), ... , 2, 1, 0 responding are the (n+l) terms in the binomial expansion (p+q)n. The probability of exactly x accep­tances is therfore

( 55)

Equation (55) of course represents the Binomial Distribution of probabilities. The average number of gaps accepted in n opportunities is np and the average number rejected is nq. The variance of the dis­tribution is npq.

It is apparent that the gap acceptance phenomenon involves a 11 stimulus 11 (available time gap) applied to a 11 subject 11 (the driver of a ramp vehicle). Variation of the stimulus is followed by a change in some measurable characteristic associated with the subject. This measurable characteristic- -the acceptance of a certain gap, in the case at hand--is referred to as the 11 response 11 of the subject.

Methods employed for the estimation of the nature of a process by means of the reaction that follows its application to living matter are

24

called "biological assays. 11 In its widest sense the term should be understood to mean the measurement of any stimulus {physical, chemi­cal, biological, physiological, or psychological) by means of the re­actions which it produces. One type of assay which has been found use­ful in many different disciplines is that dependent on the all or nothing response. The decision to accept or reject a gap is the type of response which permit no graduation and which can only be expressed as occurring or not occurring. The statistical treatment of this particular assay has been greatly facilitated by the development of probit analysis.

The Probit Method

Probit analysis is a well-established technique used widely in toxicology and bioassay work. Reference is made to two books by D. J. Finneyl5, 16 in which, in the first, the technique is described and a rather lengthy computational method set forth; in the second many examples covering a wide variety of cases are presented and the under­lying statistical theory presented in some detail.

Normally, in an experiment to which a probit analysis is applied, the magnitude of the stimulus is controlled by the experimenter. Thus, a number of subjects (usually insects) are administered a stimulus (usually a poison) to which they either do or do not exhibit a certain response (usually dying). Each subject is assumed to have a tolerance for the stimulus; if the stimulus is greater then the subject responds (insect dies), etc. Typically what is desired is a relationship express­ing the percent killed as a function of the dose. Since this is not a linear relationship the estimation of the equation of the curve and tests of significance are severely complicated. The transformation from percentages to "probits" forces the curve into a linear relationship.

In a study of lag and gap acceptances at stop-controlled intersec­tions, Solberg and Oppenlanderl7 showed that the probit of the percent accepting a time gap· is related to the logarithm of the time gap x by the equation

Y =a+ bx

By means of the probit transformation the study data were used to ob­tain an estimate of this equation. The parameters of the tolerance dis­tribution, mean and variance, were also determined. In particular the median gap and lag acceptance times were easily estimated from that value of x when Y=5 (percent acceptance is 50%).

25

Whereas, Solberg and Oppenlanderl 7 tabulated the data into groups at 1- second intervals in order to obtain an estimate of drivers accept­ing this interval in the time series, in this report the data were not grouped. Such groupings are usually reserved for experiments in which the magnitude of the stimulus is .controlled. In traffic studies such control is not possible though the magnitude of the stimulus (available gap size) may be measured and recorded along with the re­sponse (accepted or rejected). Finneyl6 terms such data "individual records." The probit method is just as applicable to individual records as to data from a controlled (grouped) experiment.

In the present study there were two practical considerations for not grouping the data. First, in the Solberg-Oppenlander data for in­tersections the effective range of accepted and rejected gaps was from about 2 to 13 seconds, however in the present investigation for many ramps studied the range was as low as from 1 to 3 seconds. Secondly, the wide geographic distribution of the thirty-ii ve study sites and the relative travel made it difficult to film for an adequate duration at each location in order to obtain representative samples. It was quickly as­certained that, for grouping, an interval of . 2 sec. would be needed to give enough groups and several hours of data would be needed at each ramp to obtain enough gap data for each . 2 sec. interval.

There are, of course, problems associated with the individual records approach used here. As Finney notes, the chi- squared sta­tistics computed in the statistical tests of significance are not as re­liable as with the grouped data. Another problem with the individual records is that the iterative solution technique employed may fail to converge in cases where the sample size is small or where the data are particularly irregular.

The development of the probit analysis for individual records is made in Appendix C.

26

RESULTS OF SINGLE VARIABLE ANALYSIS

Notation and Format

It is important to remember that what is desired is the form of the gap (or lag) acceptance function for the freeway merging maneuver. Many theoretical forms of this function were described in the section on "Theory" and are illustrated in Figure 6. Several researchers have shown that the log-normal function provides the best description of the gap acceptance phenomena. The purpose of the probit analysis is to transform this log-normal function into the linear form

Y =a+ bx (55)

where x = log t, t being the time interval (either lag or gap as the case may be) and Y is the probit of P, P being the probability of accepting a gap or lag oft {see Figure 6d).

If x and Y are plotted on a grid, Equation 55 has the form of a straight line. For convenience then, log-probability graph paper has been used throughout this section so that the relationship between t and P from Figure 6d may be viewed as a straight line. The abscissa of this graph paper (see Figure 7) is in the form of a logarithmic scale denoting the gap (or lag) interval t in seconds; the right ordinate is a probability scale denoting Percent Acceptance P and the left ordinate is a linear scale denoting the Pro bit of Acceptance Y. Thus the ab­scissa provides the transformation between x and t and the ordinate establishes the relationship between Y and P.

In the discussion which follows, a "zero" subscript will be used for lags and a "one" will be used in the subscript for gaps.

Gaps and Lags

In the past, some investigators have chosen to work with gaps 1

, some with lags 2, and some with both l 7 The choice of variable should not, of course, be arbitrary. Ramp drivers approaching the freeway merging area evaluate lags. In many cases, the driver evaluates the lag rather than the first gap since the lead vehicle on the freeway may have passed long before, the ramp driver was in position to merge.

The reason then for studying gaps as well as lags is a practical one. It is anticipated that one important application of this research will be in

27

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LAG AND GAP ACCEPTANCE REGRESSIONS FOR U.S. RAMPS FIGURE 7

the eventual development of a merging control system to help drivers execute this difficult maneuver. Presumably, such a system will be based on fitting merging ramp vehicles into openings in the freeway stream. This is most easily accomplished by a gap detector on the out­side lane of the freeway located far enough upstream from the merging area to allow a metered ramp vehicle to reach the merging area at the same time as an acceptable gap.

In Figure 7, the correspondence between the effects of lag size and gap size on acceptance may be seen. Illustrated are curves for 12 films for 7 ramps taken in San Francisco (designated SF in the Figure), Sacramento (SC), and Los Angeles (LA). The two non-parallel dashed lines (designated "unconstrained" in the lower right corner of the Figure) define the regression of acceptance on lags and acceptance on gaps. The two solid lines (designated "constrained" in the lower right corner of the Figure) illustrate the effect of forcing the lag and gap regressions to be parallel.

The results of the probit analyses for the San Francisco studies in which lags are considered the stimulus appear in Table 1. The x 2

(chi- squared) statistic with N0 - 2 degrees of freedom provides a test of the linearity of the transformed data. For all films except SF4- 1 the computed x2 statistics are small enough to be attributed to random variation. Examination of the data for SF4- 1 reveals that at several short periods during the filming, traffic was congested at the ramp. As a consequence, in several instances ramp drivers were lined up on the ramp and forced to reject large lags. Such occurrences contribute ~inordinately large amounts to the X 2 statistic.

All the regression coefficients estimated are of the same order of magnitude. It is not evident, however, that all the b 0 ' s are equal. As an approximate test of the equality of the b 0 's a x2 statistic with 7 degrees of freedom may be calculated as

2 (S wx y )

0 0

2 s wx 0

2 (I Swx y )

0 0

2 I Swx

0

= 27. 17

which is clearly significant, indicating that the b 0 ' s are not all equal. Similar tests may be made on the x2 for the equality of the b

01 s ob­

tained from two films on the same ramp. These tests indicate that the b 0 's do not differ between films from the same ramp. Taken together, these results indicate that the regression coefficient is constant for a given ramp and some range of traffic conditions but that the b

0' s may

29

TABLE 1

RESULTS OF PROBIT ANALYSES ON EIGHT FILMS

Stimulus: XO

Line Fitted: y = ao + boxo

Film: SFl-1 SFl-3 SF3-l SF3-2 SF4-l SF7-l SF8-l SF8-2

NO 196 197 291 311 328 164 159 179

Sw 76.949 78. 51 7 76. 961 116. 140 141. 424 41. 048 59.216 57.774

XO 0. 112 0.049 0.220 0.268 0.274 o. 105 o. 067 0.056 w - 0.762 0.686 0.846 o. 723 0.345 0.530 0. 708 0.461 0 Yo

2 Swx

0 15.586 14. 1 71 9.857 20.690 17.066 3.470 8.740 7. 175

Swx0y 0 21. 298 22.265 24.058 35. 1 78 39.035 11. 871 16.247 19.104 2

189.883 193.564 239.783 354.345 511. 978 182.492 161. 794 193.001 Swy0 2

160. 78 158.582 181.065 294.534 422.694 140.881 131.593 142. 136 x ao 0.609 0.608 0.309 0.267 -0.281 o. 172 0.585 0.312

bo 1. 366 1. 571 2.441 1. 700 2.287 3.421 1.859 2.663

m 0.358 0.410 0.747 0. 6 97 1. 327 0.891 0.485 0.764

ml 0. 150 0.218 0.519 0.451 1. 085 0.670 0.261 0.570

m o. 578 0.602 0. 962 0.941 1. 573 1. 100 o. 698 0.959 u

differ among ramps.

The results of probit analyses in which the first gap is considered the stimulus are shown in Table 2. The only large departure from linearity occurs in SF7- l (see Figure 7). Examination of the data shows that this heterogeneity is attributable to several short periods during which cars were lined up on the ramp. In several instances ramp drivers rejected quite large gaps. The contribution to the ?C2 from these instances is large. In each instance because the ramp driver was queued for the gap, ramp velocity was very slow compared to the speed at which freeway vehicles were traveling. As will be seen later adjust­ment for ramp velocity accounts for much of this apparent heterogeneity.

In general, the b1's are less than b 0 's. This is reasonable insofar as ramp drivers are le'ss sensitive to small differences in gap size than to small differences in lag size. However, it should be noted that inclusion of queued ramp vehicles in the analysis for gaps may cause the regression coefficient to be underestimated. A case in point is that of SF8- l in which the relatively small b1 may be accounted for in part by the fact that there were several occasions in which a large gap was available to several cars of which the last rejected the gap. This em­phasizes the importance of separating periods of stable flow from periods of congested flow in the analysis.

The last three rows of Tables 1 and 2 give the fifty percent point of the tolerance (median critical lag in Table 1 and median critical gap in Table 2), the lower 95% confidence limit (next to last row), and the upper 95% confidence limit (last row) for each of the San Francisco films. Again the results indicate differences among ramps but little difference, with the possible exception of SF8, between films of the same ramp. The 95% confidence limits for gap acceptance are illus -trated in Figure 8.

Since a lag is a fraction of a gap by definition, one might expect that for any response (percent acceptance) the gap size producing that response should be a constant factor times the corresponding lag size. That is the probit lines should be parallel. To check this and to esti­mate the ratio of lag size to gap size (relative potency in the probit jargon) probit analyses were performed in which the two lines were constrained to be parallel.

The results of probit analyses in which the lines for lags and gaps are constrained to be parallel appear in Table 3. x2 fLI ) has N

0 + N l - 3 degrees of freedom and x2 (PAR) for tests or parallelism of

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·,' I/

I :

---·-'--- ~ ·-· COLDWATER CANYON 17

,,.,,. ,,. ... /

v/ /

/ ... / /'

/ ,·'/ [,..

[,~ /v

/ ,, [)

I/

MILBRAE SF7-0I N=l64

I

I v

/ " .. v

"~)/ / ,. h~«,."- /' "o"" V I/

L.· / / qV

G~ I v /~

1./ ~/'---I-

/1%v

1/1§Q. IJ

" IJ . -·

COLDWATER CANYON

70

60

UJ u z .. :;: UJ

8 50 .. ,_

z 4 0 tJ 30

20

10

5

90

80

er "' a. a:

UJ u

10 z ~

60 UJ

8 50 .. ,_

z 40 UJ

u ffi

30 a.

a. 20

10

LAS-01 N=.118 IL~~ LA5-02

N=l69 5 .. I 1.52 340.5 I 1.5 2 3 4 TIME (SEC) TIME (SEC)

GAP ACCEPTANCE WITH CONFIDENCE LIMITS FIGURE 8

TABLE 2

RESULTS OF PROBIT ANALYSES ON EIGHT FILMS

Stimulus: xl

Line Fitted: y = al + blxl

Film: SFl-1 SFl-3 SF3-l SF3-2 SF4-l SF7-l SF8-l SF8-2

Nl 196' 197 291 311 328 164 159 179

Sw 80. 146 79.427 109.890 141.276 181. 660 75.080 76.572 86.609

~l 0.604 0.545 0.795 0.780 o. 731 0.652 0.667 o. 608 w - 0.749 0.665 o. 988 0.387 0.826 w Y1 o. 811 o. 747 0.579

2 Swx

1 6.057 5. 148 7. 149 14.659 16.~477. 6. 966 7.580 7.034

Swx1

y 1

11. 476 12.568 13.332 18.206 24. 15 9 10.508 4.988 12. 715 2

365.436 165. 145 195.213 Swy1

198.931 199.508 317.135 337.973 210. 780 2

x 177.188 168.826 292.273 315.362 330.014 194. 930 161.863 172.229

al -0. 396 -0.665 -0.495 -0.158 -0.684 -0.236 -0.387 -0,. 520

bl 1. 894 2.441 1. 865 1.242 1. 466 1. 509 0.658 1. 808

m 1. 618 1. 873 1.843 1. 340 2.927 1. 434 0.258 1. 939

ml o. 768 1. 222 0.793 0.434 1. 954 0.430 o. L 049

m 2.306 2. 398 2.742 2.234 3. 781 2.308 1. 279 2.665 u

the regression lines has one degree of freedom. The only significantly large x 2LIN is the one for SF?-1. This is probably due to the hetero­geneit~ of the gap data as discussed earlier. There are three signifi­cant x PAR statistics for parallelism. Each comes from one of the troublesome films discussed earlier. It is thought that inasmuch as the departure from the model in these three cases are explicable in terms of gross freeway conditions unaccounted for in the present analyses, there need be no serious doubt as to the validity of the model. The analyses of Table 3 indicate that it is not wholly unreasonable to assume that b 0 = b1 = b for each ramp, though the b's may differ among ramps. The "constrained" regression lines appear as the solid lines on the graphs in Figure 7.

In conjunction with checking the parallelism of the two pr obit lines, the primary purpose of the analyses in Table 3 is to estimate the "relative potencies" Ro. 1 of lags to gaps: that is, to estimate the effectiveness of lags relative to gaps in inducing drivers to merge. ~This is estimated as the antilogarithm of the horizontal distance be­tween the two parallel probit lines with R being greater than unity in the case lying for lags to the left of the pr obit line for gaps. Thus a value of R = 2 implies that in order to induce the same percentage of response as a particular lag, the gap must be twice the size of the lag. It is seen that the estimates of R in Table 3 the 3rd row from the bottom differ little among ramps indicating that the ratio of median critical lag to median critical gap is the same for all the San Francisco ramps.

Graphs of the lag and gap acceptance regressions for the remaining ramps, similar to Figure 7, appear in Appendix D. The graphs de­picting confidence limits on gap acceptance for the remaining ramps, similar to Figure 8, appear in Appendix E.

Multiple Entries

A multiple entry occurs when two or more drivers accept the same gap. Three probit lines may be determined considering ri the responses {Appendix C) and Xi the stimuli. The three lines are first fitted sepa­rately as:

Y. =a. + b. x. 1 1 1 1

i = l, 2, 3.

With these lines, it is possible to estimate for any given gap size the probc....:>ility that one, two or three cars will accept that available gap.

Tables 4, 5 and 6 show the results of analyses for multiple entries.

34

TABLE 3

RESULTS OF PROBIT ANALYSES ON EIGHT FILMS

Probit Lines for x0

and x1

Are Constrained to be Parallel:

y 0 = ao + bxo

y 1 = al + bxl

Film: SFl-1 SFl-3 SF3-l SF3-2 SF4-l SF7-l SF8-l SF8-2

NO+ Nl 392 394 582 622 656 328 318 358

l:Sw 157.644 159.234 187.849 258. 3 76 325.463 122.809 139. 584 145. 759

XO 0.093 0.025 0.236 0.285 0.283 o. 149 o. 117 0.073

xl o.629 0.573 o. 780 0.765 0. 719 0.620 o.625 0.590

Yo o. 736 0.647 0.881 0.750 0.364 o.651 o. 781 o. 496 .;.)

Jl - o. 790 o. 957 o. 789 0.366 0.674 o. 776 0.540 Y1 o. 722

l:Swx 2 21.452 18.682 18. 166 36.923 35.850 13.024 18.031 15.152

l:Swxy 32. 76 9 34.310 40. 113 56.015 68.866 29. 918 23. 975 32.693

L:Swy 2 384.251 388. 192 588.583 699.474 818.528 565.707 359. 750 393.329

2 333.048 322.534 498.631 612.642 680.223 487.053 321. 442 311. 280

X 1 in 2 1. 150 2.646 1. 378 1. 850 6.015 9. 932 6.428 2.610

X par

ao o. 595 0.601 0.359 0.317 -0. 180 0.309 0.626 0.328

b 1. 528 1. 837 2.208 1. 519 1. 921 2.297 1. 330 2. 290

R 3. 169 3. 213 3. 231 2.844 2. 722 2.889 3.249 3.144

Rl 1. 926 2. 133 2.366 1.940 2. 089 1. 999 1. 762 2.238

R 5. 144 4. 782 4.380 4. 147 3.547 4. 166 6.008 4.395 u

The significantly large x2 statistic for SF7- l and the small b1 for SF7- I and the small b 1 for SF8- I are explicable as before. For each ramp there is a tendency for the b 1 s to become successively larger: i.e., b1 < bz < b3 {see Figure 9). As the sensitivity of such platoons of cars to differences in gap size is affected by the decision of the last car in the platoon, this trend indicates the reasonable conclusion that the i + 1-th car in line for a given gap is more sensitive to differences in gap size than the i-th car. Furthermore, these analyses do not take account of ramp speed: it is expected that the i + I-th vehicle in line must slow down more than the i-th vehicle in line. Thus barring any interaction of ramp speed and gap size, the probit line for three or more cars per gap would be shifted to the right of the probit line for two or more, and the line for two or more to the right of one or more. This would accentuate the shifting of the lines due to the simple fact that it takes a bigger gap for two cars than for one and for three than for two.

The 50% point and its 95% confidence limits are estimated and ap­pear in the last three rows of Tables 4-6. In Table 4 the estimate of m for SF8- l is very small due to the position and_very small slope of the probit line. The explanation for this is as presented earlier. The upper confidence limit was too large to fit into the space provided for it.

In Table 5 the X 2 for linearity for SF 1-3 is significantly large. As in other such cases this statistic is inordinately inflated by contribu­tions from a few cases in which less than 2 cars rejected large gaps. The x2 statistic for SF7- l again indicates misbehavior, the reasons being discussed earlier. Of special noteworthiness is the fact that the estimates for SF8- l are much more reasonable than in Table 5. The elimination of the cases in which the gap was available to only one car evidently eliminates the heterogeneity in the data.

In Table 6 are the results of probit analyses for three or more cars per gap. The results for SFI-3 are typical of a situation encountered frequently when the gaps are not widely dispersed: the iterative pro­cedure goes through more iterations than usual before converging, the x2 statistic is very small and b is very large.

Table 7 and Figure 10 shows the results of probit analyses in which the probit lines for one, two and three cars per gap are constrained to be parallel under the assumption that single, double, and triple entries are equally sensitive to differences in gap size. The x2LIN for linearity has Ni + Nz + N3 -4 degrees of freedom and the x2PAR for parallelism has two degrees of freedom. The. x 2LIN statistic for SFl-1 is clearly

36

TABLE 4

RESULTS OF PROBIT ANALYSES ON EIGHT FILMS

Multiple Entries: One or More Cars per Gap

yl =al + blxl

Film: SFl-1 SFl-3 SF3-l SF3-2 SF4-l SF7-l SF8-l SF8-2

Nl 120 129 131 146 176 104 106 114

Sw1

52.986 50.771 52.742 78.079 101. 772 43.244 49 .. 389 47.045

xl 0.501 0.447 0.663 0.657 0.605 0.526 o. 569 0.473 (.V·

-.]

Y1 o. 696 0.573 0.782 0.586 o. 269 0.707 0.869 0.603

Swx1

2 2.687 2. 1 75 2.817 7.480 8.053 3.521 3.899 2.645

Swx1

y 1 5. 981 7. 276 6.979 7. 576 11.633 7.653 3.070 7.592

2 138. 611 129. 110 144.049 153.215 198.445 152.464 112. 043 128.414 Swy

1 2 126. 759 145.542 181.641 135.830 108.626 105.623 x 125.298 104.770

al -0.419 -0.922 -0.861 -0.080 -0.605 -0.437 0.421 -0.755

bl 2.226 3.346 2.478 1. 013 1. 445 2.174 o. 787 2.870

m 1.543 1. 886 2.226 1. 199 2.624 1.588 0.292 1. 832

ml 0.610 1. 327 1. 071 0.042 1. 486 o. 703 o. 1. 161

m 2. 1 77 2.312 3.056 2.371 3.594 2.289 2.351 u

TABLE 5

RESULTS OF PROBIT ANALYSES ON EIGHT FILMS

Multiple Entries: Two or More Cars per Gap

Film: SFl-1 SFl-3 SF3-l SF4-l SF7-l SF8-l SF8-2

NZ 63 71 89 107 139 64 53 70

Sw2

23.662 27.351 42.237 54. 298 6 9. 988 26. 360 29,025 32.147

X. o. 6 75 0 .. 601 0 .. 752 0. 760 0.707 0 .. 667 0.701 0 .. 677 >J l 0

-0" 104 ·-0. 064 0,287 0, 108 -0.253 0,, 118 ~0.034 -0, 190 Yz 2

Swx1

0,, 975 L 110 2 .. 240 3. 965 4.720 L 400 2.513 2 .. 031

Swx2

y2

4,. 394 4.833 7. 167 10.519 12. 093 5 .. 402 5. 108 6 .. 280 2

80.073 159.604 112. 572 160 .. 872 65.593 79., 628 Swy2

130,. 943 103.515 2

60 .. 271 138. 561 89,,641 103. 037 82.671 60.210 x 129 .. 889 55 .. 211

a2 -3 .. 145 ... 2. 683 -2., 120 ·-L 909 -2.065 -2.455 -1. 459 -2.285

b2 4 .. 507 4 .. 356 3. 199 2 .. 6.53 2.562 3 .. 857 2.033 3. 093

m 4. 987 4, 131 4 .. 600 5.244 6. 394 4.329 5.219 5.480

ml 4., 008 3., 341 3 .. 407 4. 016 5. 180 3 .. 302 3 .. 173 4. 213

m 6 .. 364 5. 186 5 .. 715 6.646 8 .. 419 5. 498 8. 986 7.643 u

Film: SFl-1

N3 51

Sw3

10.784

xl 0.883 w

-0.210 ·'° Y3 2

Swx1

0.297

Swxl y 3 1. 853 2

33.210 Swy3 2

21.650 x

a3 -5. 716

b3 6.235

m 8.255

ml 6. 498

m 11. 330 u

TABLE 6

RESULTS OF PROBIT ANALYSES ON EIGHT FILMS

Multiple Entries: Three or More Cars per Gap

y 3 = a3 + b3xl

SFl-3 SF3-l SF3-2 SF4-l SF7-l

52 72 90 120 40

3.533 25.608 36.746 31. 272 9. 2 71

0.868 0.833 . o. 876 o. 908 0.920

-0.163 0.057 -0.047 -0.704 -0.325

0.015 0.709 1. 842 1. 609 0.471

0.221 4.071 7. 126 6. llO 2.065

9. 391 85.416 98.910 95. 5ll 26.044

6.135 62.041 71. 343 72.309 16. 991

-13. 043 -4.722 -3.436 -4. 154 -4.354

14.836 5. 738 3.869 3. 798 4.382

7. 571 6.651 7. 731 12.410 9.854

4.446 5.584 6.309 9.886 6. 8 75

9. 094 7.851 9.561 18.454 18. 164

SF8-l SF8-2

37 53

12.213 10.483

0.869 o. 871

-0.665 -0.550

0.979 0.283

2.685 1. 561

32.461 28.526

25.098 19. 916

-3.095 -5.361

2.744 5.526

13.426 9.337

8.354 7.252

72. 728 17.364

6.5

~6.0 z ;'! :!; 5.5 8 <(

~ 5.0

1--~ 4.5 0: Q.

,.:4.0

3.5

6.5

~6.0 z ~ :!; 5.5 0

~ "- 5.0 0 I-­iii 4.5 0 0: Q.

,:4.0

3.5

6.5

~ 6.0 z ~ :!; 5.5

~ "- 5.0 0 I-­iii 4.5 0 0: Q.

,.:4.0

3.5

.....

0.5

~

I-'

/ I I/

I II II I

/ I .. / / I/

I/ I I ,1 I/ ,,. I , I

I / J -ASHBY

J SFl-01

I N1= N2=

' N3=

I ASHBY I/ SFl-03

N1=129 N2= 71

I J N0=52 --...,.........-·

1. J v I I

v; / I L_, / I' I

v; / v ~ I .. v / I I / II / I/ J I/ ....

... II I/ ,I

/ v ,. I/ I/ ~ II I

/ LAKE SHORE

/ SF4-0I N1•176

/ v N2=139 N3=120

I MIL BRAE I J SF7-0I

N1=104

I/ I N2=64 N3•40

-

I"'),. -/: ,,.v / ,,,1 V/ I I I/

.......... '/ •• II I II

./" i/ J I/ .... / I lo"

I/ '/ I ~

/ ,

.' / I

I/ I ~ / I/

I / SACRAMENTO I

I SCl-01

v I N1=86 Nz•61 N3=44

I If SACRAMENTO J SCl-02

v N1=85 N2=66

J N3=50

/; / I' ~

I ~I ~

/ / _,.., ·' I'

/ v i.-

/ / i PLEASANT HILL

I/ SF3-0I

I I N1=131 N2= 89 N3=72

J /

I

..... -; I I

..... ,~ .... / /

~ ..... I II

/. / I v I

I/ J l/ / BROADWAY

) SFB-01

/ I ~~~b~6 I N3=37 I J

I ,....,,...., ...

" .... ,...., .... -,-

J / ,'

I I~

COLDWATER CANYON I LA5-0I

I N1=84 N2•37 N3=28

v I

I

_ .... -

I

J I

I

I

/ II'

I J

I/

($

)' v

J J

I/ I

J

.,, ~ '/ 1 .....

I

I

I/ I

) _ .... -,,,..

95

90

UJ

80~ <(

70 li: UJ

60 8 50:

40~ 30 ii!

UJ 20 Q.

a: PLEASANT HILL ! ~;~i°4i2 i I N2•107 '

0

5

95

N3=90

I 1 II II I

I J

I I I

'/

BROADWAY SF8-02 N1 =114 N2•70 N3=53

lo~ i' --

°' I ~ ~<

" b':'/ '-.• ., ~

I I

~

i J

COLDWATER CANYON LA5-02 N1 =109 N2=61 N3=38

90 UJ

80~ <(

70 li: UJ

608

50~ 40~

0 30 0:

UJ 20 Q.

0:

10

5

95

90

UJ 80~

70~ Q.

60~ 50~ 40 !z

UJ 30i;!

UJ 20 Q.

0:

10

1.0 1.52 3 4 6 810 1520 400.5 1.0 1.52 3 4 6 810 1520 400.5 1.0 1.52 3 4 6 810 1520 4005 1.0 1.52 34 6 810 1520 5

40 TIME (SEC) TIME (SEC) TIME (SE'C) TIME {SEC)

GAP ACCEPTANCE FOR MULTIPLE VEHICLE MERGES FIGURE 9

6.5

~ 6.0 z ;'.'! ::; 5.5 u ~ u. 5.0 0 .... ~ 4.5 a: Q. ,:4.0

3.5

6.5

~ 6.0 z ~ It 5.5 u ~ u. 5.0 0 .... iii 4.5 0 a: Q. ,:4.0

3.5

6.5

~ 6.0 z ~ ::; 5.5

~ u. 5.0 0 .... iii 4.5 0 a: 0. ,:4.0

3.5

0.5

IJ

[..

ii

I ) 7 I I/

/ I/ ' 7 . I ,,

II . I I ,, I/ / II I/

I/ Jj I '

" 1 I IJ

/ / I j ,

7 ASHBY 7 7 SFl-01

7 J\11=120 N2=63 ) N3 =51

I / ASHBY 1/ I/ ! SFl-03

Nr=l29

J I N2= 71 N3= 52

,;"' II r, J / ,,

17 I/ I.' 7 ) .I 7

1.'' / .I v " j

l.1 I/ / / 11 17 L. I.I \I 11 7

/ 1. \,' I )

ll I/ ) \,

,' 17 L. ~) I ,, \I 7 1. LAKESHORE-

SF4-0I Nr=l76

/ / N2=139 N3 =120

MILBRAE I/ v SF7-0I

1. J I/ Nr•104

IJ N2=64 N3=40

., [..

ii f / // / .I / /

7

I 1/' 1, ,, :,·

J ~ i.· 17 I ,·

I l.1 /I.I

v 71J v J IJ

/ 1717 / / 11 .I I/I/ ·' v 1, \I

7 SACRAMENTO v SCl-01

I~ N1=86 N2=51 N3 =44

I/ I/ ·' SACRAMENTO / / SCl-02

I/ ,. Nr = 85 1, N2 =66

I N3=50

v /Ii LI ...

I / II v

I/ J

I II

j

I ) / I I / PLEASANT HILL

/ / ) SF3-0I

// Nr=l31 N2=89 N3 =72

ii .I ./

,., ,17

I/

./ 17

.I

I/ . /' [.. ·/

I.I 1...-_,. v .

I/ , I/ / "' BROADWAY'

J SF8-0I

/ Nr=I06 N2=53

, N3 =37

.1'

II v .I / ,

_,"· I/ 17 ,'

' IJ' .I v ,' l.1

II .I 7 1/ \/

J I/ COLDWATER CANYON .I / LA5-0I

v Nr= 84

II N2=37 N3 =28.

\,

17 v

,I

.

v / .I

I/ /

I 7

j

' 7

j

/ I/

)

v I/

' l/

,/ I/ / '

' I/

I,· v \/ / , /

\,

II I/ y

/ PLEASANT HILL SF3-02 Nr =146

. N2=107 N3•90

II v v I/ I I

' \I I/ J

II

I v

v I/ II

1.

BROADWAY SF8-02

, Nr=ll4 N2= 70 N~·-~3 _

.!; . &'1 ~ &-'

~ f-

er. "'' ~y

~ ~ v

II I/

Ii II

I/ COLDWATER CANVON LA5-02 Nr =109 N2=61 ! N3=38

95

90

"' BO~ <I

70 Ii: "' 60 u u

50 <I .... 40z

"' 30 ~ "' 20 0.

10

5

95

90

a:

"' 80~ <I

70 Ii: "' 6og

50~ 40/5

u 30 a:

"' 20Q.

10

5

95

90

0:

"' BO~

70~ 0.

60~

50~ 40~

"' 30~

"' 20 0.

0:

10

1.0 1.52 3 4 6 810 1520 400.5 1.0 1.52 3 4 6 BIO 1520 400.5 LO 1.52 3 4 6 810 1520 400.5 1.0 1.52 3 4 6 BIO 1520 TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC)

5 40

GAP ACCEPTANCE FOR MULTIPLE VEHICLE MERGES (CONSTRAINED OR PARALLEL) FIGURE 10

significant even though none of the X 2LIN statistics for the analyses on SF 1- 1 in Tables 4, 5 or 6 were significant. This phenomenon results from the different weights for each observation in the present analysis as compared to the analyses of Tables 4-6. Thus, if the b's are cal­culated separately using the weights of the present analysis, b 1 = 1. 811, b2 = 4. 278 and b3 = 4. 825 result.

Because of the underestimation of b1 in the present analysis and the large number of vehicles in the "one or more" group, the contri­bution to the X 2LIN of 278 from that group is great. The x 2PAR for SFl-1 is also significantly large though not extremely so. SF3- l, SF3-2, and SF4- l show significantly large X2PAR statistics. For each ramp, with the exception of SF7- l, the trend b1 < b2 < b3 is quite pro­nounced and the X2PAR statistics fairly large (all are significant at the 95% level). This fact indicates that the model considering the lines parallel is statistically inadequate. It may be concluded that double entries are more sensitive than singles, and triples than doubles, to differences in gap size. Even so, the analysis may be of use in assess­ing the sizes of equally effective gaps for two or more (R1, 2) and three or more (R 1 , 3) vehicles relative to one or more. These estimates and their 95% confidence limits are tabulated in Table 7, as are the esti­mated 50% gap points for one or more (m1). two or more (m2) and three or more (m3) vehicles. ·

Graphs of gap acceptance for multiple vehicle merges similar to Figures 9 and 10 appear in Appendix F for the unconstrained and Ap­pendix G for the constrained for the remaining ramps.

42

TABLE 7

RESULTS OF PROBIT ANALYSES ON EIGHT FILMS

Multiple Entries: Parallel Analyses

Film: SFl-1 SFl-3 SF3-l SF3-2 SF4-l SF7-l SF8-l SF8-2

Nl+N2+N3 234 252 292 343 435 208 196 237

Sw 92. 158 85.845 124. 218 177. 243 209.401 81. 677 93. 190 91. 207

(1) x1

0.468 0.432 0.636 o·. 614 0.594 o. 496 0.530 0.464

(2)-xl 0.674 0.601 o. 750 o. 758 o. 700 o. 660 0.697 o.679

(3)-yl

0.819 0.798 0.830 0,824 0.824 0.845 o. 795 0.812

yl o. 594 o. 516 o. 703 0.513 0.240 0.623 0.815 0.575

y2 -0. 108 -0,071 o. 281 o. 105 -0. 264 o. 113 -0.032 -0. 185

y3 -0. 498 -0.654 0.060 -0.189 -0. 932 -0. 577 -0.834 -0. 781

sw·.xl 2

4.412 3.476 6.024 14.645 15. 177 5.877 7. 926 5. 125

Swy1

2 398.657 305,305 3 91. 590 415. 154 487.346 418. 959 234. 981 251. 426

Sw• x1

y 15. 180 14.427 20. 213 3 0. 123 32.574 1 7. 184 12. 118 16. 4 76

21. x lll 338. 595 241. 512 317.583 337.517 407. 910 364.094 211.783 196. 815

2 7.834 3. 925 6. 1 79 X par 15.677 9.528 4. 61 7 4.671 1. 645

al -1. 016 -1.275 -1. 431 -0,750 -1. 034 -0.827 0,004 -0. 917

a2 -2.427 -2. 564 -2. 23 7 -1. 454 -1. 76 7 -1. 818 -1. 097 -2.368

a3 -3.314 -3. 966 -2. 725 -1. 883 -2.699 -3.047 -2.049 -3. 3 90

b 3.441 4. 150 3. 356 2.057 2. 146 2.924 1. 529 3.215

Rl, 2 2.571 2.045 1. 738 2. 199 2. 194 2. 183 5.253 2.828

Rl 1. 874 1. 576 1. 306 1. 492 1. 578 1. 503 2.538 2.031

R 3.801 2.774 2.395 3.443 3. 250 3.390 21. 184 4.288 u

Rl, 3 4.655 4.451 2.430 3.556 5. 967 5. 745 22.026 5.881

RL 3. 177 3.086 1. 800 2.335 3.848 3.425 7.474 3. 768

RU 7.660 7.019 3.461 6. 008 10. 709 11. 26 9 238. 072 10.685

ml 1. 973 2,028 2. 670 2. 316 3.033 1. 91 7 0.994 1. 928

m2 5.074 4. 148 4.641 5. 092 6.655 4. 186 5. 220 5.453

m3 9. 186 9.029 6.488 8.234 18.098 11. 016 21. 888 11. 341

43

EFFECT OF SPEEDS

The Ideal Merge

In the first part of this report, it was suggested that a merge must be qualified according to the terminology illustrated in Figure 2. Thus, the most desirable type of 11 gap 11 merge would be both 11 optional11 and 11 ideal 11 in that it would not be made because the ramp driver had run out of acceleration lane nor would it cause turbulence in the freeway stream. The requirements of such a merge serve to document the interaction of the basic traffic elements-the driver and the vehicle, and the basic traffic characteristics-headways and vehicular speeds.

If it is assumed that the average driver's normal acceleration of the merging vehicle may be represented by the following differential equationl8

du dt

= a - bu (56)

where u is the speed of the merging vehicle, t is time, and a and bare constants. If the merging vehicle is moving at a speed ur at the begin­ning of the merge, then the limits of integration for (56) are

1 -b r

u r

and the speed-time relationship is

ln (a - bu) -b

= J

t -bdu

dt a-bu

-bt

u

u r

0

= t

a - bu a - bu

r

= e

u = ~ ( 1 - e - bt) + u e - bt b r

(57)

Since u = dx/ dt, integration of (57) provides the equation of the time­space curve

44

a x--

b

u ( 1 - e -bt) + .2

b bt

(1 -- e ) (58)

Substitution of (57) in (56) gives the acceleration-time relationship for the merging maneuver:

du = (a - bu ) e -bt dt r

(5 9)

The units of the constants a and b - l are those of acceleration and time respectively, where a is the maximum acceleration and (a/b) is the free speed in the merging area. The forms of Equations 56 to 59 are illustrated in Figure 11.

The time-space relationship of Figure 11 has been reproduced in Figure 12 along with the procedure for determining the theoretical minimum ideal gap for merging. Such a gap is made up of three time intervals: (1) a safe time headway between the merging vehicle and the freeway vehicle ahead Tr• (2) the time lost accelerating during the merging maneuver, TL, (3) a safe time headway between the second freeway vehicle and the merging vehicle T f• The safe headway referred to is that headway between two vehicles in a lane which will allow the following vehicle to stop safely even if the vehicle in front makes an emergency stop. If reaction times T, braking capabilities and speeds u are assumed to be equal, then the safe headway is (L/u) + T where L is the length of the vehicle in front.

The time necessary for the merging ramp vehicle to accelerate from a speed ur at the beginning of the merge to attain the speed of the freeway traffic u may be obtained by solving Equation 57 for time.

1 b

(a - bu)

ln a - bu r

(6 O)

Subtracting the travel time T 1 to travel the same distance covered during merging, only at a constant speed u, from (60) gives the time lost during merging TL. The theoretical minimum ideal gap for merg­ing (T + Tf + T ) is

r L

u

u+ur (a/b) - u + 2T + -- +

bu bu a-bu

ln (a-bu ) r

(61)

in which Lf and Lr are the lengths of the freeway and ramp vehicles.

45

-Q w w a. U)

a/b

Ur

a-bur

-'O ..... ~

'O

..J w 0

~

TIME ,t

----------------------------------------------------, I I I

u• ~ (1-e-"l+u, ,-bf"\ !

------------------~--------------,

TIME,t

TIME,t

I I I I I I I I I

Ur a/b SPEED,u

iHEORETICAL SPEED-ACCELERATION RELATIONSHIP FOR A MERGING VEHICLE

Figure 11

46

SPACE DIAGRAM FOR TIME A-A

t w u z <( t­en 0

=:=:::::::::::::::::::::::: OUTSIDE LANE === ON-RAMP

Tr = SAFE TIME HEADWAY BETWEEN RAMP AND LEAD VEHICLES

T1 = TIME LOST ACCELERATING DURING MERGING

Tt = SAFE TIME HEADWAY BETWE­EN LAG AND RAMP VEHICLES

Tj = MINIMUM IDEAL GAP

A

VEHICLE 3

THEORETICAL MINIMUM IDEAL GAP FOR MERGING

Figure 12

It is apparent from (61) that a merging truck would require a larger gap than a merging passenger car by virtue of its longer length, Lr, and its reduced accelerating capabilities, a. Similarly, if the first of the two freeway vehicles is a truck, the merging vehicle requires a larger gap because of an increased Lf in (61).

. In order to determine representative values of minimum safe 1.aps, the parameters a and bin equation (61) must be estimated, Haight 9 suggests that some information on the parameters could be obtained from drag races which are now widely held. Knox20 used the average result of road tests obtaining values of a = 4. 8 mph/ sec. and a/b = 80 mph for Australian conditions.

Speed of the Merging Vehicle

It is apparent from the previous section that the ramp drivers' problem in executing the merging maneuver is more than one of simply evaluating successive time headways in the freeway traffic stream un­til he finds a large enough gap. For example, Equation 61 shows thatthe ideal gap is based on the ramp driver's reaction time T; the vehicle characteristics Lf, Lr, a, and b; and the speeds of the mainstream ve­hicles and ramp vehicle u and ur. The influence of speeds on merging should come as no surprise; we know that, for example, under con­ditions of forced flow on the freeway that comparatively large time headways are rejected. Theoretically, infinitely large time headways would necessarily be rejected as the concentration and movement ceased.

In delineating the effect of speeds on merging, consider a very simple model consisting of two vehicles A and B separated by a space headway s traveling at a constant velocity u on the outside lane of the freeway, and a vehicle C traveling at speed ur on a corner intersecting the freeway at an angle o . It follows that the gap between A and B is

t = s/u (62)

However, since the driver in vehicle C is moving he views the time gap between A and B as

s t' =----­

u-urcos o (6 3)

wher~ urcos o is vehicle C's speed component along the freeway. Solv­ing for s in (63) and substituting in (62) yields

48

u-u cos 6 t = r ) t'

u (64)

If t' > T' where T' is a constant, the gap will be accepted. Therefore, the minimum acceptable gap T,

u-u cos 6 r

T = (-· ----) T 1

u (65)

is clearly a function of the speed of the freeway traffic u and the speed of the ramp vehicle ur.

The effect of ramp geometry on merging operation (the subject of a companion report21) is indicated in Equation 65. The equation sug­gests that the difference between traffic operation at an ordinary inter­section and that at a freeway ramp terminal is primarily in the angle at which entering traffic and through traffic converge. For an inter­section ( 6=90°) one obtains T=T 1 from (65); for a lane change (6 =P 0

)

u-u T = (--r) T1

u ( 66)

Whereas, for an intersection, gap acceptance is independent of the speed of the merging vehicle, the speed of a vehicle merging from an entrance ramp has a profound effect on gap acceptance. This is illus­trated in the graphs in Figure 13 for the California ramps. The dashed line pertains to ramp vehicles with speeds at the beginning of the merge greater than 20 mph; the solid line to ramp vehicles with speeds at the beginning of the merge less than 20 mph. For most of the California ramps there were virtually no slow moving ramp vehicles hence only the > 20 mph line appears. However, for the Ashby, Pleasant Hill and Sacramento ramps the effect of the speed of the merging vehicle ur in accordance with the model exemplified by Equation 65 is evidenced. For the higher speed a much smaller gap was needed.

Graphs of gap acceptance for fast and slow merging vehicles for the remaining ramps studied appear in Appendix H.

Angular Velocity Model

Several researchers have hypothesized the use of angular velocity as the basis for the gap acceptance decision. The advantage of this

49

U1 0

w <> z

--,_

6.5 -

6.0 -----

,'

/ /

/ /

~f 'j

II II

LA) /V

v , / ·/

!.- ....

,,,. L,... i

I t,..fl I

1, I Jtf I I I

)" / .. /

95

90

1.11,

80 ... <>

~ 5.5 Q. w

--I/

/ v

) / / _,.' / 1 :/ A

I i I /~ ... //

70 :i Ii:

60 ~ 8 ~ 5.0 0 ,_ ~ a: 4.5 0.

,:

... <> z

4.0

3.5

6.5

6.0

~ 5.5 0. ... 8 <I ... 5.0 0

!:: 8 g: 4.5

,:

4.0

3.5

- I ~

:/ ---- I --

~v ------------ .... -------: ------ --· --: ---

/ I

v I

I

I I ': ASHBY

SFl-01 Nu•l47 Ni.•49

........

,.,-l.--..... -

i !

I ! I I l I !

I ! I I r 1· 1

' BROADWAY I SFB-01 : N,,•149

'-·· 2 3 4 TIME (SEC)

" ·' I' / I , ,

/ I

I I --

GAP ACCEPTANCE

ASHBY I SFl-03 11'4.=150

NL=47

BROADWAY SFB-02 Nu• 177

/ ,, ,j

)

PLEASANT HILL ) . SF3-02

I Nu•285 Ni.•26 ·J

I

l j

'' -·- -/"

,./'

,./

/

/ v ~

SACRAMENTO SCl-02 Nu• 195

-

v /

~/

·--- --~-1...--

PLEASANT HILL SF3-0I Nu•291

;~ y _,.//

~~. I iJOC/ v ~o. p / ~

v I J

<i7 ~ '<i

i) ~ SACRAMENTO

/ SACl-01 N,,•174 NL• 16 ., I 1 r r

I/ //

,/ v '

--··-' LAKESHORE

SF4-0I

I Nu•318

'

.... v ,,/

/' /

/

- COLDWATER CANYON LA5-0I Nu• 118

- '" "

/ v

<> 50 <I

!z 40 ...

<.> a:

30 ~ c{

20

-MIL8RA£ l 1 SF7-01 I 0

Nu•l64 1

i 5

l-----+--+--+-+-+-+-195

/ ,. 90

/ -~·

1-----t---t--o~"+-+-+-1--l80 / ...

/ <> 1----/-j/-,,'-+-+-+-+-1--l70 ~

1---/-_,r-i--+-+-+-+-1--l160 8 ~/-r~-+-~+--+--l-+-1-1!50 ~

!z t-----+--_,___.._..--1-......_..40w <> a:

l-----t--+--+-+-i--l-130 ~

0.

l-----t--+--+-+-l--l-120

- COLDWIQ'ER CANYON IQ LA5-02 Nu•l69

5 6 7 8 I 2 3 4 5 678 I 2 3 4 5 678 I 2 3456781 2 3 45678 TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC)

FOR FAST B SLOW MERGING VEHICLES FIGURE 13

parameter is it takes into consideration the distance of the approaching freeway vehicle and its speed of approach.

The angular velocity model for gap acceptance is an application of the calculus of related rates. Referring to the moving coordinate system in Figure 14, we can write the equations for the relative speed of the freeway vehicle and merging vehicle and their space lag:

dx dt

= u - u r

-x = w cot e

(6 7)

(68)

Differentiating (68} with ~espect to time, equating it to (67) and solving for the angular velocity 8 yields

e = ~ (u - u ) 2 r (69)

s

The basic aspect of the theory is that a ramp driver rejects a lag if he detects an angular velocity but accepts the lag if he cannot per­ceive any motion. Michaels and Weingarten2 2 utilize this criterion to develop the equation for the minimum acceptable gap T.

T = s/u 1/2

w (u - u ) . 2 r

(70) eu

Although the direct application of equation (70) to determine gap acceptance times depends on the evaluation of drivers' threshold of angular velocity, the role of speed in gap acceptance is again evidenced. The relative speed (u - ur) appearing in the numerator of {70) suggests that the lower the relative speed the smaller the gap needed. The speed of the mainstream traffic u in the denominator indicates an inverse re­lationship. For low speed freeway traffic, a larger gap is needed. Moreover, the fact that the latter variable is raised to a higher power than relative speed suggests it might have more effect on the gap acceptance performance.

Before the effect of speeds on gap acceptance is explored further, some aspects of a two variable probit analysis need to be discussed.

51

\JI N

1 1 t c <t 0 0:

UJ (!)

~ z 0 0: u..

t )( .

UJ (.)

z ~

A :::::::::::::::::::::::::::: OUTSIDE LANE

===ON-RAMP

--------------~--o+-~~~..fi4-~~--/4;;11L-~~~~%i<---~----,F.*il-~~~~~­

>­<t

== UJ UJ 0: u..

SPACE DIAGRAM FOR TIME A-A

c

TIME,t ---

THEORETICAL MINIMUM ACCEPTABLE GAP FOR MERGING Figure 14

Results of Two Variable Probit Analysis

It has been demonstrated in the single variable probit analysis how the acceptance curve (percent acceptance as a function of the single variable gap or lag size) may be transformed into a straight line. Just as a regression analysis may involve more than one independent variable, a probit analysis may be generalized to include more than one independent variable. Based on the theory of the preceding sec­tions the two obvious independent variables are gap (or lag) and some speed parameter (u, u , or u-u ). Figure 15 illustrates a hypothetical

r r gap acceptance surface for a freeway speed u = 50 mph. A vertical plane parallel to the T axis yields a gap acceptance curve ·Of the form shown Figure 6d and discussed in the single variable analysis. It should be apparent that the equation of the surface in Figure 15 would be difficult to write-the value of the parameters of the equation from the row data even more difficult to estimate.

A probit analysis for each ramp is presented in which a combination of x

0 and ur is considered the stimulus and r the response. In each

analysis there is obtained an estimate of a probit plane defined by Y = a + b1x0 + bz Ur· Thus for any combination of x~ 0 and ur, it is possible to estimate Y and the corresponding percentage of all :ramp drivers who would accept that combination. Or, going in reverse, it is possible to estimate for any given percentage acceptance the corres­ponding combinations of x 0 and ur producing that percentage. The ef­fect of the two variables probit analysis is to change an acceptance plane such as shown in Figure 16. Again, Figures 15 and 16 are meant to be illustrative rather than conclusive, the actual independent vari­ables used being the lag X: 0 and the speed of the ramp vehicle ur.

Table 8 shows the results of analyses in which x and ur together constitute the stimulus variable. The x2 statistic with N-3 degrees of freedom provides a test of linearity. The only significant x2 is that for SF4- l; again, the congested traffic conditions probably caused the apparent heterogeneity. Ramp SF7 shows the greatest effect of ramp speed on lag acceptance; this result supports the arguments proffered earlier explaining the apparent heterogeneity of data for this ramp. The last two rows of Table 8 give estimates of the 50% points for lags when Ur = Ur and Ur = o.

The results of the analysis summarized in Table 8 are shown in Figure 1 7 for the California ramps. The three lines on each graph depict from left to right a speed at the start of the merge of 50 mph, the mean speed for the study period and . O mph. Similar graphs for the remaining ramps appear in Appendix I.

53

GAP ACCEPTANCE SURFACE FOR FREEWAY SPEED. U= 50 MPH

Figure 15

54

ACCEPTANCE PLANE Figure 16

6.5

~6.0 z ~ ~ 5.5 '-' !:! u. 5.0 0 t-

~ 4.5

Q.

,.:4.0

3.5

6.5

~6.0 z ;: ~ 5.5 '-' !:! u. 5.0 0 t­iii 45 0 a: Q.

,.:4.0

3.5

6.5

~ 6.0 z ;: ~ 5.5

~ u. 5.0 0

t­iii 4.5 0 a: Q.

,.:4.0

3.5

,, // I/ /

I/

I/ ;

I." v /

/

/

--t.r - ,, -~ '.I

v ,· I/

I / , ,

; I/ I / I./

,' I; ,. "'

/

,, , If

J

I/

~~ 1,

, COLDWATER CANYON LA5-0I N=llB

I /

v I

ASHBY SFl·OI ,_ N=l96

+-j -t-- i.----1-I ,,1

1' v

- L__ - .

LAKESHORE SF4-0I N•328

I/

J ~

I/ I

~

/ ,,/ / v

I; v ;

/ l/i i

"' / I

v v /

)

v" II

ASHBY SFl-03 N=197

I- 1--+-I

~

~ I I

I I I

' I I II I/ I .....

I II' Ii

Ii I i " MILBRAE

SF7-0I N=164

ii I/ , v

I If

v I 11 -~

/ IJ J II

I/

I.'

v II'

COLDWATER CANYON LA5-02

J N=l69

I )' /I I

, ,, ,.

v /

I 7

/ ;

I I;~

I." I/

v 1.

, 1_, PLEASANT HILL

I/ SF3-0I N=291

v,... v ; L'

,I v I/

·' v , 11

11 V )' 11" /

1.

.. -- -BROADWAY SF8-0I N=l59

_I--~ ~

/v v '-1--1-

..... I/ I, '-· ~ v / -Ii 1--~ 1--.- ~

/ ./ - i- 1--1-v ,,1

·' 1. , / IJ ~

L

SACRAMENTO SCl-01 N=l90

I ·" ;

I/ / /

/

v / I/'

7 /

/

/

/ r7

J

j

17

'I IJ

' 7

I

J};I/ ~ .~°)

+ . -- L- :..._

I/ I/ '.!l •J

~-0 ~.

/

1/

/

! / _,· _,;

)'

~~§~6~NT HILL I N•311 i

If j

v v

I/

- . - -BROADWAY SF8-02 N•l79

J If

" 1--~ 1--~

___ ..____,______ --·

SACRAMENTO SCl-02 N=l95

95

90

10

5

95

90

10

5

95

90

w 80~

70;: Q.

60~ so!:! 40 !z

w 30~

w 20 Q.

cL

10

0.5 LO 152 34 6810 1520 400.5 1.01.52 34 6810 1520 4005 1.0 1.52 34 6810 1520 4005 l.O 152 34 6810 1520 5

40 TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC)

EFFECT OF SPEED OF RAMP VEHICLE ON LAG ACCEPTANCE FIGURE 17

The two variable probit analysis was repeated using the relative speed of the merging vehicle with respect to the freeway lag vehicle,

The results for the California ramps, shown in Figure 18, verify the models suggested by Equation 65 and 70. Similar graphs for the re­maining ramps appear in Appendix J.

The coefficients for all two variable probit analyses are summa­rized in Appendix K.

57

TABLE 8

RESULTS OF PROBIT ANALYSES ON EIGHT FILMS

Two-Dose Analyses: Y = a+ b 1x 0 + b 2 V r

Film: SFl-1 SFl-3 SF3-l SF3-2 SF4-l SF7-l SF8-l SF8-2

N 196 197 291 311 328 164 159 179

Sw 70.991 74.667 68. 148 111.185 139. 179 31. 596 58.869 53.082 -XO o. 098 0.048 0.203 0.270 0.275 o. 097 0.064 0.056

v 26. 110 26.382 37.184 34.210 30.343 37.753 31.205 32.708 r y o. 711 o. 672 o. 722 o. 723 0.348 0.378 0.702 0.436

2 13.879 13.070 19.504 2.529 6.250 Swx 8. 919 17. 184 8. 728

0 2 SwV 7550.095 7262. 986 2713.574 10108. 283 4280.659 767. 460 3072. 899 1512.316

\J1 r 00 Swx V -114.085 -77.616 -63. 768 -35.274 -8.788 -27.331 -29.845 -17.034 0 r

Swx0

y 17,444 19. 731 19.455 32. 110 38.655 7.225 16. 178 16.220

SwV y 123.334 106.030 56. 143 238.825 133.231 -3.576 -13. 511 68.524

Swy l

180.575 182.233 245.760 349. 495 505. 946 105. 377 159. 074 182.510 2

147.891 x 145. 119 186.437 287.857 413. 515 72. 984 128.497 131.747

a -0. 498 -0.275 -3. 06 3 -0.747 -1. 36 2 -6.019 o. 142 -2.236

bl 1. 588 1. 705 2.800 1. 700 2.268 4.562 1. 902 2.805

b2 0.040 0.033 0.086 0.030 0.036 o. 158 0.014 o. 077

V =V :m 0.447 0.450 0.882 r r 0.699 1. 323 1. 032 0.495 o. 795

V =O:m 2.058 1. 451 12.419 2. 751 3. 985 20.875 0.842 6.270 r

6.5

~6.0 z ;'! :::; 5.5 u ~ u. 5.0 0 f-

~ 4.5

"' Q.

,:4.Q

6.5

~6.0 z ~ :::; 5.5 u ~ u. 5.0 0 f­iii 4.5 0

"' Q.

,:4.0

3.5

6,5

~ 6.0 z ~ :::; 5.5 u ~ u. 5.0 0 f­iii 4.5 0

"' a. ,: 4.0

3.5

II

I,

,

0.5

\,

\,

1,

II II

" II

II

~

1,., I/ "' 17 ·' ,I

7 r7 / v --:;; v

/ • I/ II • ,

/

," /7 ....

v / /

~' ASHBY SFl-01 N=l96 /''

7 LI II ' I r7 /

' 17 I/

v I I I 17

I I 17 I/ I/

J v

LI ,, ' l,..._.L....-... ··--

/ LAKE SHORE SF4-0I N=32B

I J ; / I

7 I

/

I I IJ I I ..

I/ ~ r7 J

I I/ /

~ J

I

11 COLDWATER CANYON LA5-0I N=llB

v·

I/ I/ ;

I/ -· I/ 11 ,. / 17 I/

I, , LI LI

I

I

ASHBY SFl-03 N=l97 /

J / 1 I

--, 7 , I/

1,11

I 1,

; 7 7

1 MILBRAE SF7-0I N=l64 /

IJ IJ

,,, II

// I/

11r:. I/

E~'s~'!YrER CANYON N=169

I [J l7 /! ,•

I/ I.I

' II

,

/

PLEAS ANT HILL SF3-0I N=291

/ 17 7

1.· ' 17

\,

11· II

I

/ -BROADWAY SFB-01 N=l59

/ il h I

'/ ,,. I

SACRAMENTO SCl-01 N=l90

i=i=r:i:i:r:=+:=+==i=::i:::::g::++1i:i:=::::i=:i=:i;::i95

,:'I/ ." l-++Hl+--+~1hr,.+v->'f-~f-t-t++---11-i./~t---1so

i-+++++-~11-h~/+-+-+-H-+++--<+--+---+~ao~ 1/I/ v ~

l-+++J;f---};'-+-+-+-+-H-l'f+--+--+---+-170~

1-+.l'++-lli.<--/l+-+--+-+-+ll-,lq-+++--+--+---+-160 8 ~1"+-b.f+lf--l+-+--+-++-lf-t-t-t+-~f---+-150~

,. f-hv.f-++-H-+-+-.-l.',,-+-+-+-++++---+--+-+-140 ~

l-+++-l+--+-+r-1-e-++++f.l--+--+--+~30 ~ Iv" OJ

l-+++-1+-44-1-e-++++f.l--+--+--+~20 a.

v _, a:

PLEASANT HILL 10 SF3-02 N=311 5

95

/ I/ i l-+++H~"-+.-J*--+-++-+-++++---+-}-l---+-190 OJ

l-+-HIH--lll/'-+-+-+-+-++-+++-~l~/l---+-180~ l-+~~+-H-+1+-+--+-++-+-++++-~1-+--+---+-170t

11 f7 v OJ ~]l-+++!,f-+-+--+-++-+-+++lo<--+--+l--+-1608

J<.+.+.J-~--l--l-+-+-+-+-1-J.J+--+-+--ll--50~

l-+~~ll-+l-l+-+--+-+-+-+~++-l--l--ll--+-140~

l--l<<H-++---+--+-+-+-+-ll-l-+++--+--+---+-130~

~l'+++-H-l+-+--+-+~1'4-l-+++--+--+---+-120~

~++l+---+-+-~v4-+-+-+++r BROADWAY 10 SF8-02 ' N=179 5

J

I ~::~ENTO

110

N=l45 5

a'.'

LO 152 3 4 6 810 1520 "00.5 1.0 1.52 3 4 6 810 1520 4005 1.0 L5 2 3 4 6 810 1520 400.5 1.0 1.52 3 4 6 810 1520 40 TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC)

EFFECT OF RELATIVE SPEED ON LAG ACCEPTANCE FIGURE 18

SUMMARY

Theory

The announced objectives of this phase of the project research listed in the introduction will be discussed in conjunction with the findings.

The first objective of this particular study was the development of models and useful parameters for describing the freeway merging pro­cess. This theoretical development began with the derivation of the forms for the mean and variance of the delay to a ramp vehicle in po­sition to merge in terms of the ramp driver's mean critical gap, free­way flow, and freeway gap availability expressed in terms of the appro­priate Erlang distribution (Equations 2 7-34), This, a generalization of previous delay models, does not appear in the literature, The import­ance of this delay parameter lies in its applicability as a means to accurately forecast the capacity and operating conditions which can be expected for individual ramp-freeway junction designs, This applica­tion is the subject of a future report,

The remainder of the theoretical development concerns the treat­ment of the variability of critical gaps and gap acceptance among drivers. The significance of this portion of the investigation lies in the identification and application of the representative forms for both critical gap distributions and gap acceptance functions.

Characteristics

The emphasis in this report, consistent with specific objectives 2-5 in the introduction, is the collection and collation of gap acceptance characteristics, The fact that 32 ramps -chosen to reflect diverse operating, geometric, geographic and environmental conditions (See Appendix A) - were continuously filmed at 5 frames per second for an average of an hour, and that enough data was collected to run 1344 usable gap acceptance regressions serve to demonstrate not only the vast quantity of data involved, but the nature of the characteristics now available to interested researchers.

It is thought that the application of the individual record probit analysis has provided simple, statistically significant linear relations betw~en gap acceptance and gap size. The idea of constraining pertinent regression lines to achieve parallelism affords the means of quantita~ tively comparing lags to gaps, single entry to multiple entry merges

60

and fast to slow merging vehicles.

Specific objective (3) in the introduction needs some explanation. It is well known that the driver of a merging vehicle possesses a certain ability to adjust his merging environment. By speeding up or slowing down, he may adjust his position relative to that of a sequence of free­way vehicles, thereby enhancing his opportunity to merge. Indeed, in the computer output exemplified in Appendix B an appreciable percentage of drivers accepted a gap before the first available gap (denoted by a minus sign before the number of the gap accepted) simply by overtaking it. To account for this "dynamic" aspect of merging, lag acceptance characteristics at points along the acceleration lane were measured and summarized in the conventional graphical format used throughout this report. However, be cause of its applicability, these graphs are to be included in another project report establishing the effects of ramp and acceleration lane geometrics on merging operation.

Figure 13 and Appendix H suggested that not only might the percent acceptance be related to the logarithm of the time gap for fixed freeway and ramp vehicle speeds, but that this linearity might extend to the logarithm of these speeds for a fixed critical gap, Therefore, as a re­presentation of the effect of time and speed on gap acceptance, a surface similar to the one shown in Figure 15 seemed appropriate. To fit such a surface (with the desired log normal curve in the percent acceptance­time plane) to real data, and then test it statistically, a two variable probit analysis was used. Under this transformation, the surface of Figure 15 becomes the plane of Figure 16. This model for the percent acceptance-time -speed data proved statistically reliable tending to establish that the logarithms of the critical gap and some function of speed (relative or absolute) required to give any percent acceptance are linearly related.

Application

The effect of outside freeway lane volumes on gap acceptance has not ~een discussed. In a study of lag acceptance at urban intersection, Raff wrote "it is safe to say that the main-street volume does not haz3 an appreciable effect in the _critical lag". Similarly for ramps, Wohl attributed a slightly smaller percent acceptance at higher freeway volumes to sampling rather than a difference in driver behavior.

The results of this investigation tend to support the conclusions of Raff and Wohl. The graphs for individual films taken at the same loca­tion illustrate the consistancy of the gap and lag acceptance curves

61

(See Figure 7 and Appendix D). However, since gap acceptance is a function of speed and speed is a function of volume, a relationship does exist, Because of the form of the speed-volume relationship (the classic parabola showing two speeds for any given volume), it is evident that the volume-gap acceptance relationship is a very complex one. It would suffice to say that gap acceptance is independent of volume for a given level of service (free, stable, unstable, or forced flow) and that changes in gap acceptance between different levels of service may be predicted from differences in speed rather than the service volumes. From a practical point of view, since a much larger gap is needed for a freeway volume of 1800 vph under a forced flow condition (20 mph) than under a stable flow condition (40 mph), positive means should be taken to pre­vent the freeway from becoming congested.

An important application of the results of this report lies in the field of simulation. Computer technologists have been quite active with several digital computer programs developed for simulating freeway and interchange operation, As a design and operational tool, present programs are not sufficiently validated to warrant confidence in their ability to predict behavior or needs at freeway interchanges, While development of simulator hardware and programs has proven feasible, lack of detailed criteria on traffic stream interaction has hampered progress. While some simplifications are necessary in simulation models, simplification necessitated because of lack of knowledge of the pertinent variables reduces the model's realism. Over-simplification of the merging logic is a case in point. For example, distributions for merging gap acceptance under different-geometric, traffic, and envi­ronmental conditions have been lacking.

The results of the probit analyses serve to present the pertinent gap acceptance variables in the form of distributions (the log-normal) and equations (the linear probit equation), making them extremely usable as inputs to digital simulation models. A sub-routine is being written to perform this.

As mentioned before, this Project is but the first phase of a four­year research program the Bureau of Public Roads is undertaking to ( 1) furnish more detailed information on the effect that geometric vari­ables have on the merging of ramp traffic, (2) develop usable distribu,.. tions of traffic variables for simulation programs, and (3) develop an optimum ramp metering and merging control system. The material presented in this report should aid materially in fulfilling these ultimate objec':ives.

The importance of preventing freeway congestion was alluded to

62

earlier in this Summary with reference to the volume-gap acceptance relationship. Of course, the concept of ramp control as a means of deferring congestion is now a reality. It only remains for these ramp control systems to be generalized to include merging control. The design of such a merging congrol system will depend on the proper interpretation of merging gap acceptance characteristics.

63

REFERENCES

1. Greenshields, B. D., Shapiro, D., and Erickson, E. L., "Traffic Performance at Urban Street Intersections", Technical Report No. 1, Bureau of Highway Traffic, Yale University, 1947.

2. Raff, M. S., and Hart, J. W., "A Volume Warrant for Urban Stop Signs, "The Eno Foundation for Highway Traffic Control, Sauga­tuck, Conn., 1950,

3. Blunden, W.R., Clissold, C. M., and Fisher, R. B., "Distribution of Acceptance Gaps for Crossing and Turning Maneuvers," Austra­lian Road Research Board Proceedings, Vol. 1, Part 1, 1962.

4. Drew, D.R., "Gap Acceptance Characteristics for Ramp-Freeway Surveillance and Control," Presented at the Annual Meeting of the Highway Research Board, Washington, D. C., January, 1966.

5. Tanner, J.C., "The Delay to Pedestrians Crossing a Road," Biometrika, 38:3 and 4, December, 1951.

6. Mayne, A. J., "Some Further Results in the Theory of Pedestrians and Road Traffic," Biometrika 41, 375, 1954.

7. Feller, W. , An Introduction to Probability Theory and its Appli­cations, Vol. 1, Wiley and Sons, New York, 1950,

8. Garwood, F., "An Application of the Theory of Probability to the Operation of Vehicular-Controlled Traffic Signals", J. Roy. Stat. Soc. 7, 65, 1940.

9. Tanner, J.C., "A Theoretical Analysis of Delays at an Uncon­trolled Intersection", Biometrika, Vol. 49, Nos. 1 and 2, pp. 163-170, 1962.

10. Evans, D. H., and Herman, R., "The Highway Merging and Queue­ing Problem", Oper. Res. 12, 1964.

11. Major, N. G. and Buckley, D. J., "Entry to a Traffic Stream," Australian Road Research Board Proceedings, Vol. 1, Part 1, pp. 206-228, 1962.

12. Dawson, R., Comments on "Gap Acceptance Characteristics for Ramp-Freeway Surveillance and Control"., Presented at 1965 Meeting of the Board.

64

13. Herman, R., and Weiss, G., "Comments on the Highway-Cross­ing Problem", Oper.Res., 9:6, 828-840, Nov. -Dec., 1961.

14. Weiss, G. , and Maradudin, A. , "Some Problems in Traffic Delay", Oper. Res. 10:1, 74-104, Jan. -Feb., 1962.

15. Finney, D. J., Probit Analys'is, Cambridge University Press, 1947.

16. Finney, D. J., Statistical Method in Biological Assay, Hafner Publishing Company, New York, N. Y., 1964.

17. Solberg, Per and Oppenlander, J.C., "Lag and Gap Acceptances at Stop-Controlled Intersections, "Highway Research Record No. 118, 1965.

18. Mats on, T. M. , Smith, W. S. , and Hurd, F. W. , Traffic Engineer­ing, McGraw-Hill Book Company, New York, N. Y., 1955.

19. Haight, F. A., Bisbee, E. R., and Wojcik, D., "Some Mathemat­ical Aspects of the Problem of Merging", Highway Research Board, Bulletin 356.

20. Knox, D. W., "Merging and Weaving Operations in Traffic", Australian Road Research, Vol. 2, No. 2, December, 1964.

21. Wattleworth, J.A., Buhr, J.H., Drew, D.R., and Gerig, F.A., "Operational Effects of Some Entrance Ramp Geometrics on Freeway Merging, 11 T. T. I. 5430-3, Texas A & M University.

22. Michaels, R. M., and Weingarten, H., "Driver Judgment in Gap Acceptance, 11 Unpublished, April, 1965.

23. Wohl, M., and Brand, D. , "Applications of Simulation to Highway Design", Dept. of Civil Engineering, M. I. T. , Cambridge, Mass., 1963.

65

APPENDICES

APPENDIX A

Summary of Geometrics of Ramp Study Locations

Length and Angle Angle Shape of of of

Ramp Accel. Lane Converg. Converg. Grade Frwy. Ramp Ramp Curb Length

Comments Location at Ramp 2 off at Direct- Type Curva- Offset of

Name P=Parallel Nose Pvmt. Nose ti on ture at Ramp T=Taper Edge Nose

Houston Weslayan 12°15 1 Area to SW Fry 610'-P 12°15• level East diamond tangent 2' 103 1 4-lane

Buff Spdy 12°

0 4-lane, frwy curve

to SW Fr_y 620'-P 12 level West diamond tangent 2' 160' 10 right

Dumble to Frwy. Grade past Gulf Fry 240'-T 70 70 +o. 5% East diamond tangent 2' 170' ramp +5. 0%

Wayside to 10°

0 Gulf Fry 680 1-P 10 +0.4% East diamond tangent 2' 380'

Cullen to East diamond tangent 2' 180' Gulf Fry 340 1-T 10° 100

Broad St.to Gulf Fry 335'-T 14° 14° level East diamond tangent 2' 225 1

Griggs to Gulf Fry West diamond

Moss rose to Gulf Fry West diamond

Chicago Pulaski Rd Area Entrance to

Ramp grade is SW Expy 1500' -T 20 11-0 -0. 80% East diamond tangent 10' 900 1

-4. 7%

Kastner Ave Ramp grade is Ent. to slight -2. 5%, Frwy Eisenhower 712'-T 60 6 0 +l. ~% East diamond right 2' 350' curve slight right

Independence Ave to Eisen-bower Expy 738'-T 6 0 6 0 -0.4% West diamond tangent 2' 432'

Dempster to Edens outer slight Expy 785'-P 110 110 level North connector right 2' 785'

Peterson Ent. to outer slight Edens Expy 468'-P 110 110 level North connector right 2' 1020'

Los Angeles Coldwater Area Canyon to

Venture 970'-P 6 0 60 -1.0% West diamond tangent 2' 600 1

San Fran- Monument Rural, 2-lane, cisco Area Jun. Ent.to 0

slight Frwy curve Inst. 680 600'-T 4 40 +0.4% South diamond right 2' 800' slight right

Ashby Ave. to Eastshore

7°30' trumpet

Frwy 1120 1 ..-T 20 level South directional tangent 10' 500' 4-lane

Lake shore to Mac-Arthor 700 1-T 5

0 +2.5% South diamond tangent 10' 450 1 4-lane

Milbrae from to cloverleaf See

Bayshore 675 1 -T · 30 20 level South C-D road comment 10' 350'

Broadway 2-lane to slight Frwy curve Bay shore 690 1 -P 40 40 -0. 2% South diamond right 2' 500' slight right

67

APPENDIX A (continued)

Length and Angle Angle Shape of of of

Ramp Accel. Lane Converg, Converg. Grade Frwy. Ramp Ramp Curb Length

Comments Location at Ramp 2 off at Direct- Type Curva- Offset of

Name P=Parallel Nose Pvmt, Nose ti on ture at Ramp T=Taper Edge Nose

Sacramento Watt Ave. 2-lane frwy, Area to

40 40 slight curve slight

I 80 690 1-P -0. 2% South diamond right 2' 500 1 right

Detroit Chene to Area Edsel Fd. Ramp grade

Expy 693 1-T 6°15' 6°15 1 -0. 4% East diamond tangent 2' 450 1 is -4.1%

Linwood Ent. to Ramp grade John Lodge 600 1-P 80 80 +o. 5% North diamond tangent 2' 550 1 is -3. 5%

Warren to Southfield Freeway grade Expy, 6601-T 3°30 1 3°30 1 -0.4% South diamond tangent 2' 50' is+3,0%

Gratiot to Edsel Fd. Ramp grade Expy 500 1 -T 7°15 1 1°15 1 +o. 4% East diamond tangent 2' 600 1 is -0. 70

New York Jewel (69) City Area to Grand

50 Ramp grade

Central 500 1 -P 50 level South diamond tangent 12' 520 1 is -7. 0%

Rockaway Ramp grade Ent, to Van Wyek 400 1 -T 20 20 -0. 5% North diamond tangent 2' 380 1 is -5. lo/o

Long Is, Ramp grade is Exp, to outer slight -5, Oo/o, Frwy curve Cross I. 1200'-P lo lo +0.8% North connector left 2' 1200 1 slight left

Commun-ity Dr, to slight Frwy. curve L.I. Exp, 700 1-P 30 30 -2. 0% East diamond left 2' 450 1 slight left

Jerico (Rt. 25) on L. outer slight I. Exp. 1200 1-P lo lo -1.0% East connector right 2' 660 1

Brush Hollow Rd. to North-ern St, Pky East

Broadway Ramp grade is N.B. to N. outer slight -4. 4%. Frwy St. Pky. 340 1-P 8t

0 8.!.o -1. Oo/o East connector right 2' 600 1 curve slight right '

St. Louis Brentwood Area to Daniel +0.5% outer slight Ramp grade is

Boone Exp. 593 1-T 14t0 14±0 +0.43% West connector right 2' 600 1 +3. 8%, 2-lane

68

Ramp Veh. No.

2

4

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

Type

2

Arrival Time

0. 4

0.4

0.5

0. 5

0.6

0.7

0.7

0.8

0.8

1. 6

1. 6

1. 6

1. 7

1. 7

·l. 9

2. 0

2. 1

2. 1

2. 1

2.2

2.3

2. 3

2.3

2. 3

2.4

2.4

3. 1

3, 1

3. 1

3.2

Appendix B

Sample Computer Output for Lag and Gap Analyses

Lag Speed Entry Ramp Speed At Nose At Nose Station

Gap No. Lag Gap 1 Gap 2 Gap 3 Gap 4

43.0

38.3

43.0

45.9

45. 9

43.0

40.5

31. 3

32.8

43.0

43.0

43.0

45.9

49. 2

45.9

40.5

40. 5

38.3

43.0

38. 3

38.3

38. 3

40.5

36.2

34.4

36.2

49.2

49. 2

49.2

43.0

62.6

62.6

62.6

62.6

49. 2

49.2

40.5

40. 5

40.5

53.0

53.0

53.0

53.0

53.0

53.0

57.4

62.6

43.0

43.0

49.2

49.2

49."2

49.2

49. 2

49.2

49.2

49.2

49. 2

49. 2

49.2

8

4

8

8

4

2

8

4

8

6

2

2

6

4

4

4

4

4

4

4

4

4

4

4

4

4

6

4

2

6

2

2

2

2

2

2

2

3.2

1. 2

14.5

10. 3

5.9

3.2

0.2

3.6

2.4

12. 3

9. 5

8.3

5.9

1. 8

5.0

2.0

0.4

5.3

3.2

9.9

8.9

7.3

5.3

4.2

3.2

0. 2

7.7

5. 5

4.0

0.

7. 3

7. 3

17.4

17.4

. 17. 4

17.4

17.4

4.0

4.0

12. 7

12. 7

12.7

12. 7

12. 7

5.9

2.8

2.8

5.9

5.9

13. 1

13. I

13. I

13. I

13. I

13. l

13. I

8.7

8. 7

8.7

8.7

17.4

17.4

0.

0.

3. 2

3. 2

3. 2

0.

0.

0.

0.

0.

0.

0.

0.

0.

1. 0

0.

0.

0.

0.

0.

0.

0.

0.

4.0

0.

0.

0.

3.6

0.

0.

0.

0.

0 .

0.

4. 0

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

Appendix C

DEVELOPMENT OF PROBIT ANALYSIS FOR INDIVIDUAL RECORDS

The development in this section follows closely a similar develop­ment in Finney. An adaptation for individual records is made of his development.

Consider the discrete probability density function Pr(r} = Pr Ql-r, r = 0, 1, defined for each number x, where

(1) P = Pr(t <x} = L f(e1,e

2,t}dt

and f(81

, 8 2

, x} is a probability density function. k of x's and- r's is obtained:

A sample of size

(x1

, r1

}, (x2

, r2

}, ... , (xk, rk}. The likelihood of r1

, r2

, ... , rk is

proportional to eL, where

(2) L = k l r .

i i=l

k

ln P. +· l 1

i=l (1-r.}

1 ln Q.

1

The maximum likelihood estimates of e1

, e2

are solutions of

and

oL ( 4} """36"2 = 0 .

Substituting for L, these equations become

(5)

and

70

(6) aL r-P aP a 8

2 = l PQ Cl 8

2 = O.

Let t 1

, t2

be first approximations to the M. L. E. 's of 81

, 82

. By

the Taylor-Maclaurin expansion,

(7) aL aL

I a

2L

I + 08 a

2L = 0 = -- + 081 a81 d 81 a8 2 2 a8

1a8

2 tl,t2 1 tl,t2 tl,t2

similarly,

As in Finney (3), replace

their expectations by putting r = P after differentiation. Substituting for L, adjustments 38

1, 38

2 to t

1,t

2 are solutions of the equations

(9)

( 10)

2 o8 I-1- ( aP)

1 PQ 38

o8 I-1 1 PQ

1

aP a8

1

= l r-P aP PQ 38

1

a P aP

=Ir-P aP PQ 382

A new pair of adjustments is found by recalculating (9) and ( 10) using t

1 + 08

1, t

2 + 08

2 in place of t

1, t

2. The iteration continues until

the adjustments become as small as desired. The variances and

71

covariances of the estimates are obtained as the elements of the in­verse of the matrix of coefficients of 08

1, 08

2 in (7) and (8), all terms

being evaluated at the values t1

+ 081

, t2

+ 082

obtained in the last

iteration. No conditions for convergence are mentioned by Finney (4) and it appears difficult to obtain general conditions of this kind. Though modifications may be attempted along the lines described by Hartley (5), this pro~lem is not investigated here.

( 11)

In this thesis f is assumed to be such that

P= 1

crffn

2 (t- µ)

exp[- ] dt 2 cr2

Letting a - - .l:!.. and i3 =land Y=a.+13x, it is seen that cr cr

1 Jy 2 (12) P = - exp(-u )du

l21f _oo

Y is called the probit of P.

( 13)

and

( 14)

ap 1 1 2 Let z = -- = -- exp(- -

2Y ). Then

3Y rz;

aP -- = z

dCl

aP d i3 = zx .

Let a, b be first approximations to the M. L. E. 's of a , 13; then ad­justments o a, ob to a, b are solutions of

2 2 2 \ z \ z \ z r- P

( 15) o a l PQ + ob l PQ x = l PQ (-z - )

2 2 2 ( 6) \ z ob \ z 2 \ z (r- p ) 1 o a l PQ x + l PQ x + l PQ PQ x

Let 2

z w=--

PQ so that the equations become

72

r-P ( 17) oa E w t ob E wx = E w(--)

z

2 r-P (18) &a Ewx. t ob E wx. = E w(--)x

z

Let x EWX -

= ~ and a' = a + bx . Ew

- r-P ( 19) o a E w + bx E w = E w(--)

z

Then equation (17) becomes

but oa I = oa + X ob, SO the equation reduces tO

r-P (20) oa I E w = L: w(--)

z

Equation (18) becomes

2 r-P (21) ob E wx. = E w(-)x - oa L: wx.

z

or,

r-P -= L: w(-)x - (oa' - xob) L: wx

z 2

= E w(r-P) (x-x) + ob(E wx.) z L: w

(22) - 2 - r- P

ob E w(x-x) = L: w(x-x) (--) . z

r-P Let y = Y + -- ; y is called the working probit. Then (20) becomes

z

(23) oa' L: w = E w(y-Y)

=Ewy-a' L:w

from which

..J,.:!!Y. -(24) a I t oa I = = Y , L: w

Proceeding similarly with (22),

(25) ob Ew(x-x)2

=L: w(x-x) [y-a'-b(x-x)]

so that

73

(26) {b +ob) E w(x-x)2

= E w(x-x) {y-y)

and hence

(27) b +ob = Ew(x-x) {y-y)

- 2 E w(x-x)

Thus improved estimates of a , S are obtained as the coefficients of a weighted linear regression of y on x. With the improved esti­mates another iteration is carried out, and so on until the desired pre­cision is obtained.

Throughout this thesis, r is defined by r = 1 if a gap (or lag) is accepted, r = 0 if a gap (or lag) is rejected. The x-variable is taken as log (lag) or log (gap) or, in the analyses of Table 4, as a linear combination of log (lag) and ramp velocity.

2 There are a number of x tests monitoring certain features of the

fitted model.

In cases where one probit line Y = a+ Sx is estimated, a x 2

statistic with k- 2 degrees of freedom provides a test of linearity:

(28) 2 - 2

X = E w(y-y) -- - 2

[E w(x-x) {y-y)] - 2

E w(x-x)

If two pro bit lines are estimated and constrained to be parallel, a x 2

with k1

+ k2

- 3 degrees of freedom for linearity and a x2 with 1 degree of freedom for parallelism may be computed; if j lines are estimated and constrained to be parallel, the linearity x2 has ~ ki - j - 1 degrees

1

of freedom and the parallelism x2 has j-1 degrees of freedom.

If two parallel probit lines are estimated as Y 1

= a1

+ bx1

and

Y 2

= a2

+ bx2

and x1

and x2

are measured on the log(stimulus) scale,

the relative potency R1

, 2

of stimulus one to stimulus two is estimated

as the antilogarithm of the horizontal distance between the two lines, or as lorn, where

Y2 - Y1 (29) m = x - x - ----

2 1 b

As a result of Fieller's Theorem, confidence limits on RI> 2

may be

assessed as 10m1, lOmu, where

74

2 where g =

2 t

2 and t is the normal deviate for the desired signifi-

b l: Sx r;:ance level. Tne summations are over sets one and two. The notation

k

Sw = L' i=l

w., l

2 k Sx = l

i=l

- 2 w. (x.-x) ,

l l

k Sxy = l w.(x.-x)(y.-y),

. 1 l l l i=

used in this thesis where convenient and unambiguous.

etc., are

When one probit line Y = a + bx is estimated, the estimate of µ

is m '='- t ; the 50% point of the tolerance distribution is then estimated 1 m1 m

as m = lOm . Confidence limits on mare estimates as 10 , 10 u, where m

1, mu are the roots of the quadratic

- 2 (31) Y

2 = t

2V(Y) = t

2 [-1- + (x-x) ] in x. Similarly, if Sw Sx2

Y =a+ b1x

1 + b

2x

2 is estimated, the 50% point for x

1 at a given x

2 is

obtained as -a - b

2x

2 (32) m =---­

bl

and confidence limits on m are obtained as the roots of the quadratic

where V .. is the i, j-th element of lJ

Sx1 2

Sx1

x2

(34) v = Sx

1x

2 Sx

2 2

75

APPENDIX D

Lag and Gap Acceptance for U.S. Ramps

Figs. 19-23

(See also Figure 7)

--

6.5 -----

6.0 ,_

UJ (.) z

----·-~ 5.5

Q_

UJ (.) 0

----

~ 5.0 ,_

0 --1- --iii

~4.5 ,-Q_ -

-,: : ,_ 4.0 ----

3.5 ·-

UJ (.)

z

6.5

6.0

~ 5.5 Q_ UJ

~ u... 5.0 0

!::

"'

-

----------------------·-

J , )

I/ )

J

/ II / j

II

j J

I/ - -;7 -- I

~ 4.5 Q_

,:

·=11 I !/ = I ,~ - I

4.0

= I Ii/ I I 3.5

I/ I

I/ I

I J

I

~~L j

ll GRIGGS GRG-OA N=310

I

1//

Jr~ I/

~J ?ff1.

ti-I

I/

WAYSIDE WAY-14 N=217

I/

/J !#/ I~' y

I

I '/

j

;f j 'I 111 I / I

I //

I/ I I GRIGGS

GRl-04 I N=40

I

It I/

i/11 /1 ,

II $,I/ , //

1[/ J

I/ .. 1, ~

1 ;J

_!/ J /I /,

v? I I

// TELEPHONE TEL-13 N= 141

/ I I I

,, I/ ' II j '/ VII

I: ti J/ ,,_

g ;;rl ~

I I II /

/ I / I I/ ,

t! " I / l/11 / II

h v~ ~~ v w ~}' f.1 /! /V

I/ I J~ ;,[,.· / /! ll,1 iv

I/// / If v !' I I/ I [/

I// v v v

I GRIGGS

I/ GRIG-05 N=79 // WAYSIDE

WAY-18 N=l94 /l WAYSIDE

WAY-17 N=229

I 7 /I

I I I

/I //

J

~~

,bl'

I ~.J "~

~ '-¥

I/ I

v J ,Ji

~~; '""'<:) ~'"~' ~"'"" ari-~ 'c,~ "'" -{;/ 1..)'

"'"" I / I/ / / )"

,I / 1l //

//

/~ // TELEPHONE

TEL-14 N=78

I v / /

I

I v I v I/ TELEPHONE

TEL-12 N=l25

~ /, 7

&t/ 4-"' o-1' vJ /.~

/Q;-~~ v:Jf;' c"· CULLEN

"" ·-ii· CUL-03 N=l49

I I

v /

I I/ I I I I I I

2 3456781 2 3 4 5 6 78 I 2 3456781 2 3456781 2 3456781 TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC)

)/"

)./ -~v "

~~ 17

/

v WAYSIDE WAY-20 N=/83

.~

I ~

JJ / IJ

J ".v / II

J

/ /

v I

v CULLEN CUL-01 N=l39 I I

95

90

80 UJ (.)

70 ~ Q_

60 ~ (.)

50 "" 1-z

40 ~ a:

30 ~

20

I 0

5

9 5

90

80 UJ (.)

70 ~ Q_

60 UJ (.) 0

50 "" 1-

40 ~ u a:

30 ~

20

I 0

5

2 3 4 5678 TIME (SEC)

LAG AND GAP ACCEPTANCE REGRESSIONS FOR U.S. RAMPS FIGURE 19

DUMBLE DUM-27

_ ~~~-l-~-l-----+-~N·~l~72;::.._,.~

w (.) z

--,_ 6.5 ----·-6.0

----,_ ;:: 5.5 a. w (.) (.) <t

----·-LL 5.0 0 -,_ iii 0

---,_ -a: 4.5

a. --.,: -·-4.0 ·--

)

I I /;;

·:V 3.5

-

,, I/ 'I,

1v I/

11 J

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APPENDIX E

Gap Acceptance with Confidence Limits

Figs. 24-28

(See also Figure 8)

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GAP ACCEPTANCE WITH CONFIDENCE LIMITS FIGURE 28

APPENDIX F

Gap Acceptance for Multiple Vehicle Merges

Figs. 29-33

(See also Figure 9)

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GAP ACCEPTANCE FOR MULTIPLE VEHICLE MERGES FIGURE 29

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TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC)

GAP ACCEPTANCE FOR MULTIPLE VEHICLE MERGES FIGURE 30

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GAP ACCEPTANCE FOR MULTIPLE VEHICLE MERGES FIGURE 33

APPENDIX G

Gap Acceptance for Mult. Veh. Merges (Constrained)

Figs. 34-38

(See also Figure 10)

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GAP ACCEPTANCE FOR MULTIPLE VEHICLE MERGES (CONSTRAINED) FIGURE 34

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N1 =113 N2=102 N•=IOO

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u W--++l-l--1--1--1--1-j.-l-!-f+I---+--+--+- ~-i-.+-++1--4--4---+-+-l-IH-1-++-l--1--+-~0~

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LO 1.5 2 3 4 6 810 1520 400.5 1.0 1.5 2 3 4 6 810 1520 4005 1.0 1.5 2 3 4 6 810 1520 400.5 10 1.52 3 4 6 810 1520 40 TIME (SEC) TIME (SEC) TIME (SEC)' TIME (SEC)

GAP ACCEPTANCE FOR MULTIPLE VEHICLE MERGES (CONSTRAINED) FIGURE 35

6.5

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NY RAMP

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TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC)

GAP ACCEPTANCE FOR MULTIPLE VEHICLE MERGES (CONSTRAINED OR PARALLEL) FIGURE 36

6.S

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t:±:Jt±±±::::±::±::±::±=t:H:tlt::±±:=l=:S LO 152 3 4 6 BIO IS20 400.5 1.0 152 3 4 6 BIO IS20 4005 1.0 152 3 4 6 610 IS20 4005 1.0 152 3 4 6 BIO IS20 40

TIME .(SEC) TIME _(SEC) TIME (SEC) TIME (SEC)

GAP ACCEPTANCE FOR MULTIPLE VEHICLE MERGES (CONSTRAINED)

Figure 37

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TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC)

GAP ACCEPTANCE FOR MULTIPLE VEHICLE MERGES (CONSTRAINED OR PARALLEL) FIGURE 38

APPENDIX H

Gap Acceptance for Fast and Slow Merging Vehicles

Figs. 39-43

{See also Figure 13)

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GAP ACCEPTANCE FOR FAST 8 SLOW MERGING VEHICLES FIGURE 39

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TIME (SEC) 2 3 45678

TIME (SEC) TIME (SEC)

GAP ACCEPTANCE FOR FAST a SLOW MERGING VEHICLES FIGURE 40

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GAPACCEPTANCE FOR FAST a SLOW MERGING VEHICLES FIGURE 41

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r DET-02 NL• I Nu= 168

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· N0=93 I I I I I

DETROIT RAMP 3 DET-05 NL=O N0•79

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I ~ST LOUIS RAMP I I STL-01 NL=21 :Nu=53

&~-ST LOUIS RAMP I

I STL-02 NL=32

cNu =54

I ST. LOUIS RAMP I , STL-03 NL=24 Nu=59

2 3456781 TIME (SEC)

2 3 4 5 678 I TIME (SEC)

2 3456781 TIME (SEC)

2 3 4 5678 1 TIME (SEC)

2 3456781 TIME (SEC)

,,,,/

........ [/

I/

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. Nu=l39

I

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95

90

80 w 0 z

70 <I

li: 60 ~

0 50 <I

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40 ~ Ir

30 ~ a:

20

10

5

95

90

80 w 0

70 ~ a.

60 ~ 0

50 <I 1-z

40 ~ Ir

30 ~ a:

20

I 0

5 2 3 4 5 678 TIME (SEC)

GAP ACCEPTANCE FOR FAST a SLOW MERGING VEHICLE FIGURE 42

....... 0 \.}1

UJ u z

6.5

6.0

i! 5.5 a_ UJ u u

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t:: CD

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I

2 3456781 2 3456781 2 3 4 5 678 I 2 3456781 2 3 4 TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC)

GAP ACCEPTANCE FOR FAST a SLOW MERGING FIGURE 43

I

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I 11 I

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11

I

95

90

80 UJ u z

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--t--+---+-+-+-+-150 "' I ';;; ---+- 40 t!

--i--. 30 ~ -1---1---i-l-l 20

VEHICLES

APPENDIX I

Effect of Speed of Ramp Vehicle on Lag Acceptance

Figs. 44-48

(See also Figure 17)

65

~6.0 z

ii: w 5.5 (.) (.)

"" u. 5.0 0 .... ~ 4.5 a: a. ,:4.0

3.5

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1.0 152 3 4 6 810 1520 400.5 1.0 1.52 3 4 6 810 1520 400.5 1.0 1.52 3 4 6 810 1520 400.5 1.0 152 34 6 810 1520 TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC)

EFFECT OF SPEED OF RAMP VEHICLE ON LAG ACCEPTANCE FIGURE 44

5

95

90

w sou z 70;:

a. 60~

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l-+++++--+-+-+-+--+-H-+++--+--+--+--f5 0.5 1.0 152 3 4 ~ ~:o 1520 400.5 1.0 1.52 3 4 6 810 1520 400.5 1.0 1.52 3 4 6 810 1520 400.5 1.0 1.52 3 4 6 810 1520 40

TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC)

EFFECT OF SPEED OF RAMP VEHICLE ON LAG ACCEPTANCE FIGURE 45

6.5

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TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC) 40

EFFECT OF SPEED OF RAMP VEHICLE ON LAG ACCEPTANCE FIGURE 46

6.5

~6.0 z ~ f::i 5.5 8 .. "- 5.0 0

t-

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DETROiT-RAMP 4 DET-06 N•177

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TIME (SEC)

EFFECT OF SPEED OF RAMP VEHICLE ON LAG ACCEPTANCE FIGURE 47

6.5

~ 6.0 z ;! :!; 5.5

8 <(

u. 5.0 0

I­m 4.5 0 a: 0.. ,:4.0

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40 TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC)

EFFECT OF SPEED OF RAMP VEHICLE ON LAG ACCEPTANCE FIGURE 48

APPENDIX J

Effect of Relative Speed on Lag Acceptance

Figs. 49-53

(See also Figure 18)

6.5

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5 40

EFFECT OF RELATIVE SPEED ON LAG ACCEPTANCE FIGURE 49

6 .5:

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t-++t-t+--+-+-1--1-+-++-++t--+--+--+-ilO

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FFR=FF=+=F=:r=;FF:i=MFR=::::+=F::::+=l95

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TL

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t-+++-H--+-+-l--f-+++-++t---+-+--+-110

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TIME (SEC) TIME (SEC) TIME (SEC) TIME (SEC)

EFFECT OF RELATIVE SPEED ON LAG ACCEPTANCE FIGURE 50

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EFFECT OF RELATIVE SPEED ON LAG ACCEPTANCE FIGURE 51

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EFFECT OF RELATIVE SPEED ON LAG ACCEPTANCE FIGURE 52

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. 5 40

EFFECT OF RELATIVE VELOCITY ON LAG ACCEPTANCE FIGURE 53

Appendix K

Two Variable Probit Analysis

Film. Y = a + b 1x

0 + b 2ur Y = a+ b 1x

0 + b 2(u-ur)

a bl b2 a bl b2

SFl-01 -.4980 1.5884 0.0403 . 9188 1. 9214 -.0762 SFl-03 -.2754 1. 7046 0.0328 . 9199 1. 85 7 3 -.0534 SF3-0l -3.0631 2.7997 o. 0865 . 6906 2.7379 -. 0473 SF3-02 -.7469 1. 6998 0.0296 .5516 1. 8814 -.0356 SF4-01 -1.3617 2.2679 0.0358 .5187 2. 6698 -.0634 SF7-0l -6.0194 4.5615 0. 1578 2. 1908 6. 9433 -.2150 SF8-0l . 1425 1. 9017 0.0141 . 9044 2.2878 -.0584 SF8-02 -2.2362 2.8048 0.0769 1.0966 3.5646 -.0895 SCl-01 -.4930 2. 1646 0.0269 . 5018 2. 1165 -.0104 SCl-02 -1.8632 2. 9888 0.0460 . 4558 3. 1224 -.0480 LA5-0l -5.0140 3.5081 0. 1459 1. 6998 3.9525 -. 1156 LA5-02 -3.4966 2.9272 0. 1152 1. 2628 3. 2628 -.0574

HOU-OA -2. 5172 3.8960 0.0291 -1.7480 3.8350 -.0121 HOU-04 -1. 6584 4. 1702 0.0740 1. 0411 4.5001 -.0658 HOU-05 -1.7693 5. 9848 0.0837 2. 6992 7.7737 - . 1424 HOU-17 -2.2084 2. 7380 0.0593 -.5281 2. 6269 -.0312 HOU-18 -3. 1773 3.2002 0.0829 . 0814 3.2205 -.0652 HOU-19 -4. 1001 4. 6828 0.0980 -.6606 4. 1865 -.0701 HOU-20 -1. 2376 1. 5209 0.0308 - . 4235 1.7022 -.0476 HOU-12 -3.2555 4.4234 0.0785 -. 3329 4. 3281 -.0777 HOU-13 - 1. 0407 2.4375 0.0150 -.0824 2.8808 -.0613 HOU-14 -3.6268 3.3956 0.0939 -.3690 3. 1932 - . 0425 HOU-01 -3.4118 3. 1470 0.0855 . 2461 3.4107 -.0788 HOU-02 -5.0356 5.5036 0. 1372 0. 6875 4. 6041 - . 1071 HOU-03 -0.8871 1. 4108 0.0392 -.3870 1. 5 23 3 -.0838 HOU-09 04.2856 6.0221 0.0616 -1. 8523 5.8918 -. 0536 HOU-10 -8.7483 8. 8717 0. 1148 -3. 6133 7.4841 - . 07 81 HOU-11 -4.3051 5. 3352 0.0993 -.7902 4.5805 -. 0559 HOU-26 -3.5475 3. 6235 0.0952 -.4536 3. 1656 -.0436 HOU-27 -1. 9309 3. 1140 0.0537 0. 1234 2.9775 -.0461 HOU-21 -4.0786 3.4434 0. 1097 0. 1448 3.5484 - . 0931

118

Appendix K (continued)

Y =a+ b1x + b

2u Y =a+ b 1x

0 + b2(u-ur)

Film · o r

a bl b2 a bl b2

NY-03 . 2908 2.2206 0.0073 0.6135 2. 2459 -.0095 NY-04 -.9302 1. 8 7 88 0.0366 0.9542 2.0853 -.0559 N¥-05 -3.0698 2.5662 0.0759 0. 6334 2.8705 -.0952 NY-13 -5.4945 2.7491 0. 1536 o. 1757 2.2674 -.0353 NY-14 -3.7123 2.4832 0. 1047 0. 6608 2. 7856 -.0842 NY-01 -5.0739 2.8477 0. 1326 0.8473 2. 6781 -.0775 NY-02 -3.7311 3.0947 0.0959 0.6204 3.3697 -.0794 NY-09 -4.0017 2.8350 0. 1720 0. 6242 2.5667 - . 1027 NY-10 -3. 1057 0.7868 0.0333 -2.3648 0.7622 0.0269 NY-11 - .5586 1. 20 68 0.0518 0. 1324 1. 035 6 -.0205 NY-12 -2. 6550 2.7851 o. 1095 - . 0006 2.3734 - . 1028 NY-15 -1.1326 1. 6969 0.0502 - . 0073 1. 8356 -.0264 NY-16 -2.4107 0.9418 0. 1078 - .. 3009 0.9545 -. 0823 NY-06 -5.0497 3.2566 0. 1227 0.8371 3. 2290 -. 0908 NY-07 -5.0480 2.9605 0.1191 0.7503 3.0895 - . 1062 NY-08 -2.8156 3.8514 0.0743 0.3690 3. 7919 -. 0658

CHI-01 - . 2166 0.9694 0.0325 0.4744 1. 1114 - . 0546 CHI-02 -1. 0907 1. 4976 0.0755 0.3899 1.3035 -.0080 CHI-03 -2.0343 1. 8949 0. 1112 0.5965 1. 4226 - . 0123 CHI-04 - . 5848 1. 1745 0.0487 0.3977 1. 1063 -.0158 CHI-05 -4.4316 3.2024 o. 1258 0.4122 3.4376 - . 0595 CHI-06 -2. 9024 3.2979 0.0757 0.2702 3. 6044 - . 0565 CHI-07 -5. 2561 2.8767 0. 1468 2. 0841 4. 1313 - . 17 38 CHI-08 -6. 7314 4. 6188 0. 1970 0.7676 4.3682 -.0927 CHI-10 -5.9050 5.4241 0. 1569 0.8505 5.5845 - . 1200

DET-01 -5.2606 3. 1123 o. 1645 1. 0642 2.5874 -.0659 DET-02 -1. 2709 1. 550 0.0606 0.7891 1. 6788 -.0387 DET-04 0.8886 1. 6084 -.0074 0. 6698 1. 6900 -. 0333 DET-05 -4.5808 2.2830 0. 1210 1. 8385 3.7743 - . 1606 DET-06 - 1. 0907 2.2521 0.0659 1. 6773 2. 3232 - . 0611 DET-07 - . 1705 1. 0526 0.0422 0.3989 1. 0485 -.0412 DET-08 - . 0526 0.7780 0.0502 0. 3785 0.7481 -. 0454

•STL-01 -2.0329 1. 5039 0. 1094 0.0039 1. 4178 -. 0566 STL-02 o. 2657 1. 3859 0.0021 0. 1304 1. 4753 -. 0292 STL-03 -1.0301 1. 834 7 0.0746 0.5797 1. 7079 -. 0443

119