NATIONAL BUREAU OF STANDARDS REPORT

10111

ANALYSIS OF A CAPACITY CONCEPT

FOR

RUNWAY AND FINAL-APPROACH PATH AIRSPACE

Inter-Agency Agreement DoT FA69-WAI-166

FAA Project Number 187-601-01R

SRDS Report Number RD-69-47

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT

4314403 November 1969

NBS REPORT

10111

ANALYSIS OF A CAPACITY CONCEPT

FOR

RUNWAY AND FINAL -APPROACH PATH AIRSPACE

Inter-Agency Agreement DoT FA69-WAI-166

FAA Project Number 187-601-01R

SRDS Report Number RD-69-47

Technical Report

to the

Federal Aviation Administration

Systems Analysis Division

Systems Research and Development Service

by

Technical Analysis Division and

Applied Mathematics Division

IMPORTANT NOTICE NATIONAL BUREAU OE S for use within the Government

and review. For this reason, whole or in part, is not autt Bureau of Standards, Washin

the Report has been specitica

ress accounting documents intended

is subjected to additional evaluation

re listing of this Report, either in

Standards and Technology (NIST) Government agency for which on October 9, 2015.

'I copies for its own use.

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
«

e.

. ^ :
ABSTRACT

A "maximum throughput rate" concept, for the capacity of a runway

and its associated final -approach path airspace, is developed as a

possible alternative for some purposes to the present concept embodied in

the Airport Capacity Handbook which is based on a "tolerable average

delay" level. The new concept is shown to be representable by a single

mathematical formula. In the context of a stream of IFR landings, it

is shown to have other properties useful in an operational setting, in

particular to have potential value in connection with cost-effectiveness

analyses of proposed changes in ATC equipment or procedures. Illustrative

nunerical calculaticns and parametric sensitivity analyses are included.

A comparison with the ideas and numerical values in the Handbook is

carried out. Suggestions for natural extensions of the present brief

study are formulated. Technical appendices include peripheral studies

of capacity- increasing techniques for deviating from a first-come

first-served treatment of arriving aircraft.

TABLE OF CONTENTS

Page

1. INTRODUCTION AND SUMMARY 1

2. THE CAPACITY CONCEPT 7

2.1 PRELIMINARIES 7

2.2 INFORMAL DESCRIPTION 9

2.3 SYNOPSIS OF RESULTS 16

2.4 MATHEMATICAL FORMULATION AND ANALYSIS 22

3. APPLICATION TO IFR LANDINGS 33

3.1 PRELIMINARIES 33

3.2 TREATMEHT OF TIEUP TIMES 35

3.3 TREATMENT OF BUFFER TIMES 40

3.4 SYNTHESIS AND CONSENTS 42

4. NUMERICAL ILLUSTRATIONS 46

4.1 DATA SETS EMPLOYED 46

4.2 DISCUSSION OF RESULTS 52

4.3 COMPARISON WITH SIMULATION OUTPUTS 61

5. RELATION TO "HANDBOOK" VALUES 66

5.1 PRELIMINARIES 66

5.2 a basis for "MODIFIED HANDBOOK" VALUES 71

5.3 COMPARISON WITH HA^0fflOOK CONCEPT 79

6. POSSIBLE NEXT STEPS 87

7. REFERENCES 90
TABLE OF CONTENTS (CONT.)

Page

APPENDIX A; PROOF OF EQUATIOI (2.6) . 92

APPENDIX B: JUSTIFICATIOJ OF EQUATION (2.7) 96

APPENDIX C: ON SEQUENCING OF ACCEPTANCES 102

APPENDIX D: MORE ON SEQUENCING 105

APPENDIX E: A DELAY CONCEPT 117

APPENDIX F: A SUPPORTING SIMULATION 126

APPENDIX G: EXTENSION TO A DUAL-USE RUNWAY 135

Figure 2.3.1: DECOMPOSITION OF INTERTOUOIDOWN TIME 19

Figure 2.4.1: SEQUENCE OF INTERSERVICE INTERVALS 22

Figure 3.3.1: TRIANGULAR DISTRIBUTION ON [-R,R] 41

Figure 3.3.2: UNIFORM DISTRIBUTION ON [-R,R] 41

Figure 4.2.1: SENSITIVITY TO CHANGES IN FINAL APPROACH DISTANCE . . 54

Figure 4.2.2: CAPACITY AS A FUNCTION OF POPULATION MIX 56

Figure 4.2.3: SENSITIVITY TO CHANGES IN R 58

Figure 4.2.4: EFFECTS OF R ON EFFICIENCY MEASURE 59

Figure 4.3.1: ANALYTICAL VS. SIMULATION RESULTS (UNIFORM DISTRIBUTION) 63

Figure 4.3.2: ANALYTICAL VS. SIMULATION RESULTS (TRIANGULAR DISTRIBUTION) 64

Figure F.l: SOME SIMULATION OUTPUTS 127

Figure F.2: LISTING OF SIMULATION PROGRAM 133
TABLE OF CONTENTS (CONT.)

Page

TABLE 4.1.1: FINAL -APPROACH PATH LENGTHS AND MINIMUM

SEPARATION DISTANCES 47

TABLE 4.1.2: DISTRIBUTIONS OF ERROR IN INTERTOUCHDOWN TIME 47

TABLE 4.1.3: THRESHOLD PROBABILITIES OF VIOLATION 47

TABLE 4.1.4: MIXES OF AIRCRAFT TYPES 47

TABLE 4.1.5: FINAL -APPROACH SPEED AND RUNWAY OCCUPANCY TIME

TABLE 5.2.1: ESTIMATES OF AVERAGE HOLDING TIMES BY AIRCRAFT

TYPE 74

TABLE 5.2.2: PREDICTED VS. EXTRACTED HANDBOOK VALUES .... 75

TABLE 5.3.1: COMPARISON WITH VALUES FROM HANDBOOK 82

TABLE 5.3.2: COMPARISON WITH MODIFIED HANDBOOK VALUES USING

INTERTOUCHDOWN TIMES 82

TABLE 5.3.3: COMPARISON WITH MODIFIED HANDBOOK VALUES USING

RUNWAY OCCUPANCY TIMES 82

TABLE D.l: PENALTY COSTS FOR SEQUENCING EXAMPLE 110

TABLE D.2: FRACTIONAL PENALTY -REDUCING SOLUTION 110

TABLE F.l: SIMULATION APPROXIMATIONS OF K=c^/^ 129
-riji'

’!*> S
1. INTRODUCTION AND SUMMARY

The currently employed concept, for measuring the "capacity" of an

airport's runway system and the associated final -approach airspace, is

that set forth in the Airport Capacity Handbook [1]^^^ and based on a

number of mathematical and empirical studies carried out by the Airborne

Instruments Laboratory (e.g. [2], [3]). In connection with the FAA's

( 2 ) Continuing Study of Air Traffic Control System Capacity and Demand ^

, it

appeared advisable to examine and re-evaluate the assumptions on which

this concept was based, and as appropriate to initiate the exploration

of alternative concepts.

The present report docui'.ents one such technical exploration, carried

r 3') out at the National Bureau of Standards by members of the Bureau's

Technical Analysis Division (Simulation Group) and Applied Mathematics

Division (Operations Research Section) . Our assignment had as its

theoretical core the task of developing, and bringing to a point permitting

numerical application, a mathematical model of a "capacity" concept with

the following features:

(a) Arrivals represented by perturbations to a deterministic process,

rather than highly random (Poisson distributed) inputs.

(b) Known individual service times, rather than a constant service

time

.

^'^^Numbers in square braces indicate references in Section 7 of this

report

.

( 2 ) FAA Subprogram 187-601, assigned to the Systems Arialysis Division,

Systems Research and Development Service.

(3) Under Inter-Agency Agreement DoT FA69 WAI-166, PR No. WA5I-9-0629.

- 1

-
'iDest effort’* basis.

(' 4') The study was undertaken on a short-term'- ^

This had several implications, recognized in advance by the FM and the

study team, for the course of the work. One pertained to technical scope;

the new capacity concept was to be developed in detail only for the case

of a single runway, a "pure" situation in which attention could focus on

the concept's essential features without distracting complications.

Other implications were that only readily available data could be employed

(i.e., no data-gathering or data-assimilation tasks were included), and

that orientation for the study's staff would necessarily be quite limited.

Under these circumstances the study was of course critically dependent on

close cooperative liaison with the FAA, and we would be remiss in failing

to acknowledge here both the constant helpfulness and the substantive

contributions of our project monitor at the FAA, Mr. S. P. E. Price.

The body of the report is organized as follows. Section 2 describes

the underlying ideas of the new concept, in the general context of

"capacity of a service facility serving several customer types." The

essential technical difficulty stems from the fact that the allowable

interval, betv/een providing service for the first of two consecutive

customers and providing service for the second, depends upon the types

of both. This complication is overcome by drawing on some relatively

recent developments in the mathematical theory of random processes. The

result is a simple closed-form formula for the facility's capacity, in

74)

'^Roughly tw; calendar months.

Prior to operational use of the concept, its development would of

course have to be extended to encompass interactions between neighboring

or intersecting runways.

^^^E.g., the required distance separation between consecutive aircraft in

the final approach of IFR landing translates into a time separation which

depends on the approach speeds of both A/C.

- 2

-
terms of (^) the relative proportions of the different customer types in

the mix of customers to be served, (^) for each pair of customer types,

the nominal desired interval between provision of service to two consecutive customers of these respective types, and ( iii) the precision

with which such desired intervals can be achieved. See eq. (2.17), p.31.

This approach appears applicable, as it stands, to situations in

which a runway is being used either for arrivals (landings) only, or for

departures (takeoffs) only. In Section 3, the application process is

illustrated by carrying it out in detail for a stream of IFR landings.

(71

The resulting specialized formulas^ permit one to study the effects, on

capacity, of technological or operational in^rovements which for example

might permit reductions in runway occupancy times or minimum separation

criteria (e.g., the traditional "3-mile rule"), or might achieve higher

precision in the measurement and control processes involved in the

landing operation. The capability for such analyses seems quite

important, if a capacity concept is to be readily usable for costeffectiveness analyses of possible new equipment or procedures.

Section 4 continues the illustrative treatment of IFR landings,

presenting numerical values of the new capacities calculated for a

number of sets of input data supplied by the FM. These sets exhibit

variations in

(a) the mix of aircraft types in the arriving stream,

(b) the length of the final approach path.

See eqs. (3.25-26), p. 42.

- 3 -
(c) the prescribed minimum separation distance,

(d) the "just tolerated" probability of violating the separation

criterion, and

(e) the probability distribution of deviations from a desired

inter- touchdown time interval.

The results are displayed in several graphical formats to help

communicate the capacity measure’s sensitivity to the various parameters

involved.

In Section 5, some of these capacity values are compared with the

values given in the Airport Capacity Handbook for the corresponding

cases. Our information, on the full set of assumptions governing the

results in the Handbook , is not sufficiently explicit to leave us entirely

confident of the strict comparability of the two sets of values; on the

other hand, since the Handbook is intended to be employed "in the field"

without further explajiatory material, acceptance of its values for

use in the comparison seemed appropriate. In addition, "modified Handbook

values" intended to be better -suited for the comparison were derived.

The calculated new capacity values were found to differ systematically

and significantly from the Handbook values (both modified and

unmodified) .

The present document of course constitutes only an initial exploration

of the new capacity concept. Section 6 sketches some of the next steps

that would be involved in bringing the concept closer to operational

status. Such further development may well be useful to the FAA in

connection with its Continuing Study of Air Traffic Control System

Capacity and Demand.

- 4-
Apart from a bibliography, the remainder of the report consists of

technical appendices. Some of these (A and B) contain detailed mathematical

justifications of formulas given in the main text. One (Appendix F)

describes a fast- time simulation performed to check the adequacy of an

approximation entering the analytical treatment. Two (C and D) contain

mathematical studies, stimulated by but somewhat peripheral to the main

theme of the project; they deal with possible strategies for replacing

a first-come- first-served policy toward arrivals with some sort of

"optimal sequencing."

The two appendices not mentioned yet contain discussions of topics

quite germane to the central purposes of the present study. One (Appendix

G) deals with the extension of the capacity concept to the case of a

runway used for both takeoffs and landings. Such an extension is evidently

of considerable practical and theoretical inportance; only the beginnings

of a treatment could be developed during the time available, and this

topic probably tops the list of significant next steps in pursuing the

present line of research.

The other appendix (E) takes up the development of a delay concept

to be associated with the new capacity concept . What seems of special

interest here is that the attempt led (indeed, drove) us to take into

account the "capacity" of the next higher layer in the Air Traffic Control

System. This association, of a quality-of- service measure (delay) at

one level of the ATC system with a capacity measure one level higher,

is suggestive of possible similar relationships at other elements and

levels of the system.

- 5-
We conclude this introduction by listing the members of the project

staff

:

Simulation Group (Technical Analysis Division)

:

M. J. Aronoff (Project Manager)

M. D. Maltese

J. T. McQueen

W. Steele

M. Wanger (Consultant)

Operations Research Section (Applied Math Division)

:

A. J. Goldman

W. A. Horn

J. Levy

M. H. Pearl

( 8) Almost all participated on a part-time basis.

- 6-
2. THE CAPACITY CONCEPT

2 . 1 Preliminaries

Before the description of the new capacity concept is begun,

some preliminary remarks by way of orientation may be helpful.

The first point to be addressed is the whole idea of "alternative

capacity concepts." For contrast, consider the length of a rod. One

can propose different methods and instruments for measuring this length,

and discuss the relative precisions and accuracies of each, but such

discussions refer to alternative approaches to the measurement of a welldefined physical property of the rod , not to "alternative concepts"

of length! hTiy then is it sensible to consider alternative concepts for

(' 2

') the capacity of a runway

The answer, of course, is that "capacity" is not a "well-defined

physical property" of the runway. A particular capacity concept is an

attempt to capture the essentials of a particular intuitive notion, in

a quantitative form suitable for certain uses. Studies which begin with

different intuitive notions, or are directed toward different goals, may

well find different capacity concepts appropriate. (The situation is

analogous to that in engineering economics, where the appropriate

definition of "plant capacity" can be strongly dependent on the use to

Assuming specified environmental conditions, and that the context is

not a level of detail at which surface irregularities and the like

become significant.

( 2 ) Throughout the report, we shall use "runway" as an. abbreviation for

the awkwardly long phrase "runway and its final -approach path airspace."

- 7-
which such "capacity numbers" are to be put.)

This leads to the second point to be stressed here. The capacity

concept embodied in the Airport Capacity Handbook expresses the following

intuitive notion:

maximum traffic rate which can be accommodated without

average delay reaching an unacceptable level .

On the other hand, the capacity concept to be described below is based

on a quite different intuitive notion^ of the "maximum throughput rate"

variety; it is not directly associated with any quality- of- service

f3) indicator such as average delay ^

, the study of such indicators (in

particular, the evaluation of their levels as more or less acceptable)

being regarded as admitting and deserving attention somewhat apart

from "capacity" considerations. We must therefore explicitly ask the

reader, vfio through familiarity with or continuing use of the Handbook

has developed an automatic association of "capacity" with "delay," to

suspend this association of ideas in examining the new capacity model.

( 3 ) A tentative treatment of a "delay" concept compatible with the new

capacity concept is given in Appendix E.

- 8

-
2 . 2 Informal Description

The new capacity concept will now be described, in general

terms relating to a facility which serves a stream of customers of

(3a) several types . Adopting this initial generality of language has

two advantages. First, it promotes the recognition of possibly useful

analogies between airport runways and other specific examples of service

facilities; these may for example provide clues toward helpful literature

references. Second, it places in sharper focus those more special features

which characterize the airport runway situation.

As noted earlier, the intuitive notion underlying the capacity

concept is of a "maximum throughput" type. We therefore list two factors

which are major influences on a facility’s throughput rate:

(a) Customer availability . Since customers cannot be served before

they arrive, the throughput rate would be degraded by idle intervals

in which no customers are at hand.

(b) Tieup times . Very roughly, this refers to how long a customer

"ties up" the facility, i.e., how fast the facility can service

that customer and get on to the next one. Clearly, a reduction

in such times (due, say, to Improved equipment or operational

procedures) will tend to increase the facility's throughput rate.

Before leaving this subsection, we shall have to come back for a

more searching examination of this "tieup time" notion. But for the

moment, let us return to the "maximum throughput rate" notion and ask

with respect to which of the two preceding factors the "maximization"

( 33.") ^ -^Unless otherwise specified, customers are always assumed served

in order of arrival.

- 9-
should take place. This depends upon the anticipated uses of the

capacity concept. The concept under study here is intended to prove

useful in cost-effectiveness analyses and comparisons of possible changes

in equipment or procedures, changes which might reduce tieup times.

Thus the concept should not involve some prior optimization with respect

to tieup times, for this would "wash out" the capability to use the

concept in comparing approaches to the improvement of this factor.

On the other hand, the factor of customer availability is a natural

candidate for the "maximization." While idle periods at the facility

affect its actual throughput rate in a given situation, they are irrelevent to its potential for throughput under heavier workloads, and hence

to its "capacity" as normally conceived. The "maximum" aspect of our

concept will therefore be expressed in the following assumption of

continuous demand :

As each customer's service is completed, another customer

is at hand .

With this point clarified, the following informal description of

the capacity concept can be given:

the average mean throughput rate over a prolonged period

of continuous demand .

Here "throughput" has the familiar meaning of "number of customers

served," but other terms in the description require explanation. First,

^^^This is consistent with the treatment in [4] (p . 67), in which "capacity'

is considered only for situations satisfying the analog of the continuous demand assumption.

- 10 -
a "prolonged" period is one which is long relative to the maximum tieup

time for a single customer. If the capacity concept is to be useful,

it should be numerically insensitive to the exact duration of the period,

and the range of "prolonged periods" should include durations which are

typical rather than unreasonably long relative to peak load conditions

at the facllity^^\

Second , although the terms "average and "mean" are generally used

interchangeably, their employment above involves a deliberate distinction.

The mean throughput rate of the facility during a period of duration T

is the number of customers served during that period, divided by T. There

is some inexactness here, as to whether or not to count in customers whose

service is in progress at the beginning or end of the period. This,

however, involves an uncertainty of at most 1 or 2 customers in the count,

an uncertainty which is negligible since the period in question is a

prolonged one.

For the applications of interest here, the tieup time for a customer

can depend on his type , and perhaps on the type of the next customer as

well. Thus the mean throughput rate during a period of time will depend

on the exact sequence of types exhibited by the customers served during

that period. Clearly a capacity concept based on such detailed specifications, of a sequence of types almost certain to change from day to

day, would be of little practical use. What is done, therefore, is (in

effect) to average the mean throughput rate over all possible sequences

(^^That is, we would not want a runway capacity concept which applied only

to periods of several hours ' continuous demand

.

- 11 -
of those customer types liable to use the facility during the period;

the "weights" (probabilities) involved in the averaging operation are

of course based on the relative proportions of the various customer types

in the "mix" which the facility serves. This explains the usage of

"average." Of course the possibility of frequent large deviations from

the average must be looked into.

Because of the assumption of continuous demand, the facility’s

capacity (as described above) is limited only by the tieup times associated

with the successive customers it serves. This "tieup time" notion, which

is so critical to the capacity concept, must therefore be examined with

some care; this will occupy the next part of our informal discussion.

One initial point is that a customer can tie up the facility for

longer than the "processing time" physically required to provide his

service. For example, some sort of clean-up may be required before

another customer can be served. On a production line, a "setup time" after,

and in addition to, the processing time may be needed if the next customer

(production batch) is of a different type from the current one. This is

a good example for present purposes, since it illustrates how a customer's

tieup time might depend in part on the type of the next customer. At any

rate, the principal idea is that tieup times will in general be longer

than processing times; rather, they correspond to what are called "holding

times" in the queuing literature stimulated by communications-network

considerations

.

- 12 -
Another aspect of tieup times is that they may not be determined

solely by physical laws (in the simplest case, that two bodies can't

occupy the same space at the same time), but also by man-made regulations.

Safety considerations, for example, may lead to rules inhibiting the next

customer in approaching the facility while the current customer is being

served, or even from having followed the current customer too closely

toward the facility. For the runway case, this regulatory element figures

very strongly. The distinction is important, because it is physically

possible for constraints expressing man-made rules (rather than physical

laws) to be violated ; such violations may typically be inadvertent,

infrequent, and slight, but they can in principle occur.

The notion of tieup time will now be made more concrete, but at the

cost of introducing two more ideas. One is that of an identification

point in the use of the facility by a customer; this is merely some clearcut stage in the service process which can be defined in a uniform way

for all customers. It might for example be defined to be the start of

processing or the end (for landing aircraft, these correspond respectively

to initiation of final approach, and to turnoff from the runway) . We

have chosen it in later sections to be a clearly distinguishable intermediate point (namely, touchdown)

.

The second new idea is that of the interservice interval associated

with a customer. This is simply the time interval between the moment

at which the customer's identification point occurs, and the corresponding

moment for the next customer. The length (duration) of this interval will

- 13 -
be called the customer’s interservice time . The "identification point"

idea was introduced just in order to provide definite moments between

which to measure the interservice time.

We can now define

( 6 ) tieup time = required minimum value of interservice time.

The word "required" is intended to recall the largely "regulatory"

nature of the tieup time, and the fact that violations of the requirement

are physically possible. Because of this possibility, it is necessary

to distinguish clearly among

tieup time

(actual) interservice time

desired interservice time.

Ideally, the second and third of these would coincide with the first;

the capacity resulting in this case might be called the ideal capacity

of the facility. In practice, however, an interservice interval cannot

be made to assume a desired (aimed-at) duration exactly. Deviations

from perfect accuracy may be due to any or all of a variety of reasons:

inevitable residues of imprecision in knowledge as to when the current

service period began, as to the exact duration between start-of-service

and identification point, etc.; inability to bring the new customer into

"servicing position" at precisely an intended moment; and so forth.

^^^Thus, tieup time (more precisely, the associated interval)

corresponds to "service time" as treated in [4], p. 24.

- 14 -
As a consequence of this inevitably less-than-perfect accuracy,

choosing the desired interservice time to coincide with the tieup time

may be too risky, creating an unacceptably high probability that the

actual interservice time will turn out to be less than the tieup time --

i.e., that a violation will occur. We shall therefore assume that the

desired interservice time which the facility’s operators aim at achieving,

CJ') is systematically somewhat larger than the required minimum level

(the tieup time) . The increment will be called the buffer time ; it is

a ’’safety ftiargin” added to the tieup time in order to reduce the probability

of violation. With this addition, a violation will only occur when the

random error in achieving the desired interservice time is so large

(negative) as to more than offset the buffer time; such extreme values

of the random error should be relatively rare.

The ratio of the ’’actual” capacity (which involves the buffer times)

to the ideal capacity defined above, provides one measure of the facility’s

(7a) efficiency . This measure of course has only a limited scope of

relevance; it does not pertain to possible improvements (in equipment,

procedures, or whatever) or possible regulation changes which might reduce

tieup times, but only to the effect on capacity of the imperfections in

measurement and control which make the buffer times necessary.

(7) This idea is taken from [5]. There is no implication that the increase

need be officially sanctioned, or even deliberately set and consciously

recognized by the operators themselves; see however [8], p. 29, para. 1.

(7a) ^ "^This has little relation to the usage of the same term in [4] (pp.

154-155); the latter in fact comes close to our concept of ’’capacity.”

- 15 -
2.3 Syncpsis of Results

The preceding informal discussion can be summarized as follows:

First , the facility's "capacity" under the proposed concept is

defined as its average mean throughput rate over a prolonged interval

of continuous demand. Here the "average" is in effect one over all

possible sequences of customer types which might arise in the stream

being served, the necessary "weights" or probabilities for the average

being derived from the proportions of different customer types among the

population of facility users. The term "continuous demand" refers to

a heavy-traffic condition (the only situation in which a capacity concept

is of much interest) , represented by the assunptioh that at the coir^letion

of each service there is another customer at hand to be served.

Second , throughput rate (and hence capacity) is limited by two sets

of factors:

(a) One set consists of tieup times , which measure how long each

customer has to "tie up" the facility. These depend on the quality of

the technology and operational procedures enployed, which determines how

rapidly a customer can be served and how much time need be spent on other

necessary activities (cleanup, setup, etc.). They also depend on

regulations which prescribe minimum separations between consecutive customers;

since such regulations are not physical laws, their occasional violation

is physically possible. Specifically, tieup times enter as minimum

required interservice -interval durations.

- 16 -
Cb) The second set of limiting factors consists of buffer times . They

arise because the servlee process cannot avoid some degree of random

error in seeking to achieve a specified interservice time. Specifically,

the buffer times are increments vdiich are added to the tieup times to

obtain desired (aimed-at) interservice time spacings large enough that

atten^Dts to achieve them, despite random errors, have only a very

low probability of violating the tieup time regulations. Thus the

necessary sizes of the buffers depend on how tightly the distribution

of errors is "packed” around zero, i.e., on the precision available in

the landing and landing- control process.

In the next subsection 2.4, a single mathematical formula to express

the capacity concept will be derived. Then, in Section 3, this foimula

will be specialized to the case of a stream of IFR landings at a runway.

The resulting specialized formula is the basis for the numerical work

reported in Sections 4 and 5.

The derivations of formulas, in 2.4 and Section 3, necessarily

involve material which is relatively detailed 'and technical. Some readers

may prefer to emit this material, going at once to the more concrete

illustrations in Section 4. In the rest of the present subsection, we

therefore provide a synopsis of the results of subsection 2.4 and

Section 3, to provide a direct bridge to Section 4. The results are

of course stated without proof, and no mention is made of the various

alternatives and generalizations taken up in the detailed text.

- 17 -
The situation under consideration is a continuous -demand stream of

IFR landings at a single runway. Interservice times are measured, for

definiteness, between successive touchdowns.

Suppose first, for sinplicity, that only two aircraft types are

involved: A and B. These types are present, in the mix of aircraft

using the runway, in certain relative proportions p^ and

pg

(positive

numbers summing to 1) . Let

r^ = tieup time (required minimum interservice time^ for a type

A aircraft if followed by a type B aircraft;

^BA’ ^AA ^BB

defined analogously. Also, let

b^ = buffer time for a type A aircraft if followed by a type B

aircraft.

bg^, b^ and bgg are defined analogously.

Consider a touchdown chosen at random. The probability that it is

an "AB- touchdown”, i.e., involves a type A aircraft followed by a type

(7cl B aircraft, is given^ ^ by the product P^Pg- In this case the (actual)

intertouchdown time, denoted I^, is equal to the desired (aimed-at) intertouchdown time plus a random (unbiased) error term, e. Furthermore, as

described earlier, the desired intertouchdown time is the sum of tieup

time and buffer time. Therefore

^AB

" ^AB ^AB

See figure 2.3.1, in which the random error e is assumed to follow a

symmetric triangular distribution ranging from (-R) to (+R) , where R is

a measure of control precision.

^Minimum allowable time implied by separation regulations and physical laws,

(7c) Recall -he first-come -first-served assumption.

-18-
Figure 2.5.1 : Decomposition of Intertouchdown Time (see Explanation)

first touchdown

i

desired relative timing of next touchdown

/ V

tieup —r /^bufA

fer

k M L n— N

X

Explanation: Dotted "curve” shows probability distribution of the random

error in achieving desired intertouchdown time. Area both under this

curve and preceding end of tieup interval (see shaded triangle) gives

probability of violation; it would be increased to 0.5 if there were

no buffer time, i.e., if the dotted distribution were shifted to be

centered at the end of the tieup interval.

The average value of is therefore

•^Wav " ^AB ^AB*

2 Similarly, AA - touchdowns occur with probability == (p^) >

and give rise to an average intertouchdown time of

•^Wav "

’"aA ^AA*

- 19 -
with like results for the other two cases (BA- touchdowns and BBtouchdowns) . The average length of a random touchdown is therefore

given by the expression

Pa^B ^^AB ^AB^ Pa^A (^AA ^AA^

PgPA ^^bA ^BA^ PbPb (’^bB

''' ^BB^’

obtained by summing the average touchdown times for the four cases,

each "weighted" by the probability of that case.

It is shown in subsection 2.4 that the capacity is given by the

reciprocal of the average length of a random touchdown, i.e. the reciprocal

of the expression above. For more than two aircraft types, this formula

generalizes to

= 1/Z. . (r.. + b..)

where the sum is over all ordered pairs (i,j) of aircraft types.

Now it is necessary to specify how the numerical values of tieup

times (r's) and buffer times (b’s) are to be determined. For an ABtouchdown, these would be r.„ and b.„.

The tieup time r^ is found as the larger of two quantities,

corresponding to the two regulations to be observed. One of these

quantities is the typical runway occupancy time for type A aircraft;

its presence corresponds to the requirement that the next aircraft

(here, type B) not touch down until the current one (here, type A) has

cleared the runway. The second quantity is a straightforward translation,

into a separation of touchdown times , of the minimum distance separation

- 20 -
Imposed along the final approach path. This translation of course

involves the final approach speeds of the two aircraft (with due attention to whether the gap between them is opening or closing during the

approach), the size (S) of the required distance separation — e.g.,

3 n. mi. —

,

and the length (L) of the final approach path.

The buffer time b^ is chosen, as described earlier, to reduce the

probability of violation to a realistically "endurable" threshold level.

This permits its calculation (details in Section 3) in terms of (i) the

threshold level (p^) and (li) the probability distribution of the random

error in attaining a desired intertouchdown interval. For our illustrative

work, this distribution was taken to range from (-R) to (+R), and to be

either triangular (as in Figure 2.3.1) or uniform; thus it is characterized

by the single datum R.

This completes the synopsis. For the next subsection, we revert to

the "service facility" level of generality.

- 21 -
2.4 Mathematical Formulation and Analysis

To begin the mathematical formulation of the material described

informally in subsection 2.2, we consider a stream of customers^ ^

Cl,

for the facility. For simplicity, the moment at which the interservice

interval of begins is taken as the origin of the time axis, t = 0. Let

II,

designate the corresponding sequence of lengths of (actual) interservice

intervals, e.g., the interservice period of begins at time t =

For any interval of time [0,T] starting at t = 0, let

N(T) = number of interservice intervals in the period [0,T], so that

N(T)/T = mean throughput rate in [0,T]

.

N(T) is explicitly defined by the conditions

II + I2 + ... + ^ T < + . . . + + 1^(1)+!' ( 2 . 1 )

> pi >2 3

V h h ^N(T) ^N(T)+1

T

Figure 2.4.1: Sequence of Interservice Intervals

(7d) The symbol ''C” without subscript is reserved to mean "capacity.

- 22 -
We can delimit the mean throughput if we have estimates of the

largest and smallest interservice intervals that would ever occur under

continuous demand, i.e., quantities I and I . such that ^ max min

I . < I < I min — n — max

(all n)

.

For then it follows readily from (2.1) that

(l/I ) - (1/T) < N(T)/T < (l/I . ); (2.2)

for large T, the term (1/T) on the left can be ignored, leading to

(l/I ) < N(T)/T < (l/I . ). max — — min

Now the interservice interval I^ has been assumed to depend on

what types of customers and sequence of successive

types is to be regarded as randomized rather than specified. Thus each

I^ is a random variable, and so N(T) is a random variable as well, the

same holding true for the mean throughput rate N(T)/T. Furthermore,

our informal description of the capacity concept identified it as the

average value of this last-mentioned random variable, for prolonged

intervals [0,T]. Thus, using the customary symbol "E” for the operation

of taking the average (or "expected") value of a random variable, we have

Capacity = E [N(T)/T] (large T) . (2.3)

Since E [N(T)/T] is not perfectly constant for large T, we formulate

(2.3) more precisely as

Capacity = lim.^,

oo ^ [N(T)/T]. (2.4)

- 23 -
Once the intuitive ideas behind the capacity concept have been

made precise enough to admit explicit mathematical representation, the

questions to be resolved involve problems in mathematical analysis

rather than formulation. Specifically:

QUESTION 1. Is there a usefully simple formula for evaluating capacity,

in terms of the limit in equation (2.4)?

QUESTION 2. Is the convergence to the limit, as T increases, sufficiently

rapid that the limit is an adequate approximation to the result for

large but finite T in the practical — interest range of durations?

(Cf. footnote 5 of this chapter.)

QUESTION 3. For large T, is the variance of the random variable N(T)/T

so small as to indicate that the random variable is adequately represented

by its average value?

We proceed next to the analysis of these questions. The mix of

customer types can be represented by a set of numbers

p^ = relative proportion of type i customers in the mix.

These p^'s are non-negative and sum to 1; we can interpret them as

probabilities

p. = Prob {C is of type i} .

This interpretation requires that the type of each successive customer

be regarded as arising "at random" independent of the types of all

(81 previous customers, e.g., no deliberate "sequencing by type" is involved.^ ^

(81 ^ ^Some topics related to sequencing are taken up in Appendices C and D.

- 24 -
Suppose first, for simplicity, that the probability distribution

function of 1^ depends only on the type of customer C^. Let

= average value of random variable 1^, if is of type i;

these averages, "conditional" on the type of C^, can be combined to yield

^i^i^i

~ (unconditional) average value of 1^. (2.5)

The random process is one which starts afresh ("renews itself") with the

end of each interservice period; after the n-th period the type of the

next customer ^^ises at random in accordance with the probabilities

p. , and then the value of I , arises at random in accordance with the

probability distribution function for associated with whichever

customer type turns up. Such a process is called a renewal process , and

there is a fairly extensive technical literature on the subject. In particular, it is known ([6], p. 359) that for large T, the random variable

N(T) is approximately normally distributed with mean T/y where y is

2 3 2 here the quantity in (2.5), and with variance T o /y where a is the

(unconditional) common variance of the random variables I^.

(9) From this result it follows, first, that

linvj,^ ^

E [N(T)/T] = 1/E.p.T.. (2.6)

This gives an affirmative answer to Question 1, assuming that the T^’s

are readily calculable. Second, it follows that the variance of N(T)/T

_2

(T times that of N(T)) is for large T approximately proportional to 1/T;

since 1/T tends to zero rapidly for large T, an affirmative answer to

(9)

In Appendix A we give a proof of the next formula which is relatively

elementary, involving only Laplace Transform techniques.

- 25 -
Question 3 is also at hand. That the answer to the remaining Question 2

is also favorable, follows from a more general result to be cited later.

Thus, if the length of the n-th interservice interval could be

assumed to depend only on the type of the n-th customer C^, then our

analysis could be regarded as successfully concluded, and it would be

time to move from the present level of generality to more specific and

runway-oriented applications. But in fact, for the type of application

motivating this study, the probability distribution of 1^ really does

depend on the type of the following customer as well as that of

C , and so we are not yet done.

When this point in the project was reached, some thought was given

to the idea of trying an approximation in which is split into two

parts; a first one whose probability distribution depended only on

' s type, and a second one depending only on C^^^'s type. Then the

first part of and the second part of

^ together form an interval

depending only on C^'s type, and so the previous results on renewal

processes would be applicable. This idea might have worked out well;

it is noted here for its intrinsic interest and because it may suggest

analogies useful in other situations. However, it was laid aside when

the more compelling approach described next was uncovered.

- 26 -
Let us say that the facility is in "state i" if the current

interservice interval involves a customer (C^) who is of type i.

Then the time history of the facility is a random process involving

a sequence of transitions from state to state (or sometimes from a

state to the same state), these transitions occurring when a new interservice

interval begins. The probability distribution of the time interval

spent in a state, before the next transition, depends on the pair of

states between which the transition occurs.

Such a random process is known as a semi -Markov or Markov

renewal process. These processes, considerably more general than

renewal processes, were introduced more recently and have been

studied less extensively. By suitably assembling and specializing

results in the literature on this subject we arrive at the following

conclusions.

First , with the notation

T^j = average value of if is of type i and is of type j ^

we have

Capacity = lintp_^ E [N(T)/T]

= 1/E . p. p. T. .

,

1 j ij ’

(2.7)

( 10 ) Details are given in Appendix B.

- 27 -
where the summation is over all ordered pairs (i,j) of customer types.

This formula, which yields an affirmative solution to Question 1 if

the T^j's are readily calculable from known quantities, can be regarded

as the climax of the more theoretical part of the study. Second , the

error in approximating E [N(T)/T] for large T by the limit in (2.7),

is essentially proportional to 1/T, so that an affirmative answer to

Question 2 is indicated^^^^ . The coefficient of proportionality is also

available, as an explicit combination of complicated quantities whose

simplification we lacked time to attempt.

( 12 ) As for Question 3, in the time available we were unable to extract

from the literature (or to devise ourselves) an analytical proof that

the variance of E [N(T)/T] tends rapidly to zero as T increases, although

it seemed likely that this variance tends to zero roughly proportionally

to 1/T. We were therefore obliged to check Question 3 by a Monte Carlo

fast-time simulation for the type of application described in Sections 3

and 4. The results were positive; they are reported in subsection 4.3.

The considerations described at the end of subsection 2.2 will

now be brought into play, in order to reach a more specific form of

equation (2.7). Suppose customer is of type i, and is of

type j . Let

d. . = desired interservice time for C , 13 - n’

if (C^, are of types (i,j).

( 2 . 8 )

( 11 ) This implies the same result for the renewal-process case discussed

earlier.

^^^^For a later bulletin, see the portion of Appendix B following equation

(B.IS).

- 28 -
Then the actual interservice time, the random variable I , is

’ n

given by

In + en

(2.-9)

where

e^ = random error in achieving a desired interservice interval.

We now make the further assumption that

E(eJ = 0, ( 2 . 10 )

i.e. that there is no systematic bias in the actual interservice

intervals as compared with the desired ones. The justification is

that such a systematic bias, if it existed, would be detected by

the facility's management and cancelled by appropriately altering the

desired (almed-at) interval lengths. From eqs. (2.9) and (2.10) it

follows that

Efy = d...

But E(lj^)» under the stated assumptions on the types of and

Ties

( 2 . 11 )

is what was defined earlier as T^^ . Thus eq. (2.7) becomes

Capacity = l/l^. p. p^

d.^

- 29 -
Recall that the tieup time for C was defined as the smallest size n

that I should have, according to regulations. With the notation

n

r^^. = tieup time for if (C^, respective types (i,j),

the probability of violation of the tieup time constraint is given by

p^ = Prob < r^^. , given that (C^, are of types (i,j)} .

Using equation (2.9), and omitting the "given that" clause for brevity,

we have

p

= Prob {d. . + e < r. .} ^v ij n ij

or equivalently

p = Prob {e < r. . - d. .} ^v n ij ij ( 2 . 12 )

The random error e^ has some probability distribution, which might

conceivably depend on the types of and We therefore introduce

the cumulative distribution functions

Fij(x) = Prob {e^ < x, given that (C^^, are of types (i,j)>. (2.13)

Thus (2.12) takes the form

p

= F. . (r . . - d. .) . ^v ij " ij ij"

We now adopt the viewpoint that p^ is not a derived quantity, but

(2.14)

rather a ( small) realistically endurable probability of violation. The

equation (2.14) can be solved^^^^ for d^^. ; if F^j'^(p^) denotes the

smallest value of x for which (^) = then

(13) Assuming is continuous.

- 30 -
-1

(2.15)

Since is small, ^ (p^) will be negative, and the quantity

-1

. = - F. 4 "

(p ) ij ij (2.16)

represents the "buffer" or "safety margin" mentioned earlier (see pp.17,19).

The consequence dj^j = + bj^j of (2.15-16), on substitution in (2.11), yields

Capacity = 1/Z^^ p^ p^ (r^^ + b^^), (2.17)

where r^^ and b^^ are defined below (2.11) and in (2.16) respectively.

[Of course, more sophisticated models for treating the random error

can be introduced if they seem appropriate. For example, suppose that

e^^ might be better represented as a random multiple (sometimes negative)

of the desired interservice time to which it is the random error. Denoting

the multiplier by y, and its cumulative distribution function by G, we have

p^ = Prob { dij + < r^j >

= Prob { d^j (1 + y) < r^^ }

= Prob { y < (tji^j/dj^j) - 1 }

= G -!)•

Solving for d^^ yields

"ij

= ^

rather than (2.15), and thus

^ij d^j - r^j. = [- (p.^^)] rij/[l + G

^ (p^)] (2.18)

rather than (2.16).]

(14) This is intended as a hypothetical illustration.

- 31 -
This completes the theoretical analysis. The results appear clearly

applicable to a runway serving either a stream of landing aircraft, or

a stream of departing aircraft. To bear out this contention, the

preceding material is applied in some detail to the "IFR landings" case

in the following sections of this report.

Further investigation would be required to extend this material to

interacting runways, or to a runway serving both arrivals and departures.

The latter topic is taken up in a preliminary way in Appendix G.

- 32 -
3. APPLICATION TO IFR LANDINGS

3.1 Preliminaries

In the previous section, a capacity concept for a class of

facilities serving a multi-type stream of customers was introduced

and analyzed. A relatively simple formula for it was derived, namely

Capacity = p^Pj (r^^ + ( 3 . 1 )

where the summation is over all ordered pairs (i,j) of customer types,

and

P£ = relative proportion of customer "mix" which is of type i,

r^j = minimum time which should elapse between providing service

to a customer of type i, and providing service to the next

customer if of type j,

b^j = "buffer" time added to r^^ as a safety margin, because an

intended interservice spacing cannot be achieved with perfect

precision; it reduces the probability of violating the

• constraint expressed by to a prescribed level p^.

In the present section, our aim is to specialize equation (3.1),

and hence the capacity concept, to the case of a runway handling a

stream of IFR landings. This specialized formula provides the basis

for the specific numerical illustrations presented in Section 4.

The "facility" .in question is clearly the runway together with its

final-approach path airspace. The "customers" are evidently the aircraft

in the landing stream. Several definitions of "service" might be possible

- 33 -
the choice among them is immaterial so long as the chosen one is used

in a careful and consistent way. Here we shall regard the interservice

period of an aircraft as beginning with its touchdown, and ending with

the touchdown of the next aircraft.

The "input data" p^ in the capacity formula clearly represent the

mix of different aircraft types in the arriving scheme. It remains to

derive appropriate formulas for the other quantities appearing in

equation (3.1), namely the "tieup times" r^^ and the "buffer tiimes'

b... These two tasks are carried out in subsections 3.2 and 3.3

ij

respectively. The results are assembled, together with some comments,

in the concluding subsection 3.4.

( 1 ) Our treatment of these topics is generally similar to that in [5],

though differing in several specifics.

- 34 -
3 . 2 Treatment of Tleup Times

For IFR landings on a runway, the minimum period which should

elapse between the touchdown of one aircraft and that of the next, is

governed by two considerations. First , the second aircraft of the pair

should not touch down until the service of the first is complete. With

the notations

T’ = time of touchdown for first A/C of pair,

T'' = time of touchdown for second A/C of pair,

OT’ = runway occupancy time for first A/C of pair,

this condition reads

Second , for IFR landings there is a minimum distance separation

constraint on the two aircraft as they share the final approach path

(e.g., the "3-mile rule"). This requirement is expressed by a datum

S = prescribed minlmtim distance separation between two A/C on final

and translates (as will be seen below) into a time separation

SEP = minimum time separation, corresponding to S, between touchdowns

(3.2)

approach path (3.3)

for the two A/C. (3.4)

Thus, in addition to (3.3), we have

T’ ' - T' > SEP. (3.5)

Combining (3.2) and (3.5) yields the result:

tieup time for first A/C in pair = max (OT*, SEP). (3.6)

- 35 -
To make equation (3.6) useful, the translation from the distance

separation S to the time separation SEP must be carried out explicitly.

Transition from a distance to a time duration of course requires some

sort of velocity data; in the present instance these are

v' = final approach speed of first A/C,

(3.7)

v' ' = final approach speed of second A/C.

The same idealization as in [5] will be adopted, namely that these speeds

can be treated as constant over the final approach path; a more delicate

analysis could of course employ variable velocity- time profiles, if they

could be satisfactorily specified. In addition to v' and v’’, there is

an additional element of the situation's geometry which must be known;

this is

L = length of final approach path. (3.8)

There are two cases to be considered:

CASE 1. Suppose v' ^ v' '

.

Then the second aircraft is overtaking

the first, so that the gap between them is closing as both proceed down

the final approach path. The aircraft are therefore closest, together

at the last moment at which they share the approach path, namely at the

time (T') when the first member of the pair touches down. If the distance

separation criterion (S) is satisfied at this moment, it will also be

satisfied at all prior times when both aircraft are on the final approach

path

.

- 36 -
With the temporary notation

X

= distance between the A/C when the first one touches down,

we see that the second aircraft will touch down x/v'

later than the

first, i.e.

T” - T* = x/v".

The distance separation condition, which reads x 2. therefore

equivalent to

T' ' - T’ ^S/v’ '

.

In other words

SEP = S/v' ' (if v' < v") . (3.9)

CASE 2. Now suppose v* > v''. Then the gap between the two

aircraft is opening as they share the final approach path. The aircraft

are therefore closest together at the moment when they first share the

path, namely at the moment when the second aircraft begins its final

( 2 ) approach (passes the outer marker). Suppose this occurs at time t,

and now adopt the temporary notation

X

= distance between the two A/C at time t.

Then the second aircraft, before touching down, must cover the full

length (L) of the final approach path; it will therefore touch down at

time

T" = t + (L/v"). (3.10)

( 2 ) An alternative is noted below.

- 37 -
The first aircraft has only the distance L - x along the approach path

to cover; it will therefore touch down at time

T’ = t + (L - x)/v'.

From (3.10) and (3.11), we have

T” - T' = (x/V) + L (1/v” - 1/v').

The distance separation criterion, which reads x ^ S, is therefore

equivalent to

T” - r > (S/v') + L (1/v” - 1/v’).

In other words.

SEP = (S/v') + L (1/v” - 1/v') (if v' > v”). (3.12)

Finally, combining (3.6), (3.9) and (3.11) yields, for the tieup

time of the first aircraft of the pair.

tieup

in CASE 1,

time = max

and

(or, S/v”) (v' < v' ') (3.13)

tieup time = max (or , S/v' + L. d/v' ' - 1/v')) (v' > v”) (3.14)

in CASE 2.

Two technical points about the preceding analysis should be noted

before going on. First , it has of course been assumed that S < L, i.e.

that it is allowable for two aircraft simultaneously to be in the final

- 38 -
approach phase of their landings. Second , a different approach to

CASE 2 (opening gap) is adopted in [5]

.

The viewpoint there is that

as the two aircraft approach the outer marker, they would be altitude

separated, with the second (slower) one higher. The distance separation

criterion is then imposed when the first aircraft begins its final

approach, on the assumption that by then the second aircraft has already

begun its approach from the higher altitude. This situation of course

admits an analysis similar to the one given above, though resulting in

a slightly different formula for the tieup time. Our information on this

point is that the postulated altitude separation is not so universal a

practice as to command inclusion in the present illustrative application

of the capacity concept.

- 39 -
3.3 Treatment of Buffer Times (3)

Recall from equations (2.9), (2.15) and (2.16) that the actual

interval between the touchdowns of the two successive aircraft discussed

above is given by

(tieup time) + b + e

where b is the buffer time we wish to represent, and e is a random error

in achieving a desired time separation of the two touchdowns. The error

e was defined so as to have average value 0, and the desired buffer time

was calculable from (2.16) as

b = - (p^) (3.15)

where F is the cumulative distribution function of e, and p.^ is the

probability of violation.

Without an explicit investigation of error sources and their

statistical variation, nothing definitive can be said about the proper

choice of the distribution function F. Some considerations on this topic

appear in [5], [7] and [8]. In particular ([8], p.30), error patterns

that appeared to be normally distributed have been observed. For the

present illustrative purposes, we shall work with two mathematically

simple distributions which represent sharply different assumptions about

the tendency of errors to cluster about the zero (no-error) point. Each

of these distributions depends on a single parameter, which also serves as

a measure of precision for the landing and landing-control process

.

'^Based on (2.16) rather than the alternative (2.18).

- 40 -
The probability density (or frequency) functions for these two

distributions are shown in Figures 3.3.1 and 3.3.2 (whose vertical scales

differ). The first is the triangular distribution over the finite

interval [-R, R] , where R measures the spread of errors. This is an

analytically more tractable substitute for the normal distribution,

and also lacks the sometimes embarrassing infinite "tails" of the latter.

The second is the uniform distribution over the same interval [-R, R]

;

it is much more pessimistic, allowing for no peaking at 0 whatever.

For these two distributions, under the natural assumption p^ < 1/2,

equation (3.15) will be shown to yield

1/2

b = R [1 - (2p^) ] (triangular), (3.16)

b = R [1 - 2p^] (uniform). (3.17)

Figure 3.3.1 : Triangular

Distribution

on [-Rj R]

.

Figure 3.3.2 : Uniform

Distribution

on [-R, R]

.

- 41 -
f

I

\

I,

•f

!j

>i

1
The details for Figure 3.3.2 will be given first. The area of the

rectangle, like the area under any probability density curve, must be

unity. Since the rectangle’s base has length 2R, its height must be

1/2R. Thus the equation of the probability density function is

f(x) = 1/2R (-R <x <R) ,

and so that of the cumulative density function is

FW =

/ ^

f (e) de = (x + R)/2R (-R ^ x ^

R) .

The equation F(x) = p^ therefore leads to

X = (P^) = R •

so that

b = -F‘^ (p^) = R [l-2p^]

,

corroborating (3 . 16)

.

For Figure 3.3.1, the area must again be unity; since the base of the

triangle is 2R, its altitude is 1/R. Thus the left-hand leg of the

triangle joins the points (-R,0) and (0,1/R); its equation (that of the

probability density function) is therefore

f (x) = (1/R) (x + R) (-R < X < R)

,

so that the cumulative density function is

F(x) = f(e) de = (x + R)^/2R^ (-R ^ x ^R)

.

llie equation F(x) = p^ leads to

X = F'^(p^) = R [(2p^) 1/^-1],

SO that

b = = R [l-.(2p^)^^^l,

corroborating (3.17).

-41A-
3.4 Synthesis and Comments

It is relatively routine to assemble the results of the last two

subsections. For this purpose, let

OT^ = runway occupancy time for type i A/C,

= final approach speed for type i A/C.

Then (3.13) and (3.14) yield

r- . = max {OT. , S/v.}

13 13

1/v.)}

(3.18)

(3.19)

(v. < V.)

,

^ “ 3 (3.20)

A >

•H (3.21)

(triangular)

,

(3.22)

(uniform)

.

(3.23)

r. . = max {OT., S/v. + L (1/v.

13 1* ' 1

^

' j

Next, by (3.16) and (3.17),

b. . = R [1 - b..=R[l-2p]

When the last four formulas are substituted into equation (3.1),

the result is somewhat sinplified because b^^ as given above is not

dependent on the pair (i,j). This result, with notation

C = capacity,

takes the form

1/ 2 . C = 1/{Z. .p.p.r.. + R [1 - (2pp^'"])

for the triangular error distribution, and

C = 1/(Z. .p.p.r,. . R [1 - 2p^]}

(3.24)

(3.25)

(3.26)

for the unifom distribution, with r,^. given by (3.20) and (3.21).

- 42 -
Numerical calculations based on these formulas will be presented

in the next section. However, a few comments about the formulas seem

in order at this point.

First , the foimilas are sufficiently simple to permit easy derivation

of equations for sensitivity coefficients, i.e., partial derivatives with

respect to each of the quantities entering the capacity measure. (These

derivatives exist except in the unlikely coincidence of a tie between

the quantities competing for the maximum in (3.20) and (3.21),) Thus

much of the appropriate sensitivity analysis , with respect to changes

in (for example)

the separation distance (S)

the violation probability (p^)

the control -precision measure (R)

the runway occupancy times (OT^)

the mix of aircraft types (p^) can be carried out analytically, rather than purely numerically as in

Section 4.

Second , these formulas (3.25) and (3.26) are based on the mathematical

model of random errors leading to eq. (2.16). If for exan^le the

approach leading to (2.18) were used, the result would be

C = [1 + G

-1

Cp ) 1 /2 . .p.p . r. . (3.27)

In this case, the efficiency of the landing process, as measured by the

ratio of its capacities with and without the need for buffer times,

- 43 -
(3.28)

turns out to depend on alone.

(Pv)

* 1-

Third, three generalizations can be handled with essentially no

extra effort, except that for the first two will really depend on

the pair (i,j) so that (3.1) rather than (3.25 - 26) must be used: We

can replace the error range R by a set of R^^'s; this may be advantageous

in working toward a more sophisticated error model. We can generalize

the threshold violation probability p^ to (p^)j^j's; this is desirable

since in principle one might prefer a "safety policy" sensitive to

considerations (such as number of passengers) which may typically differ

among aircraft types. Similarly, the minimum desired separation distance

S can be generalized to a set of

This leads naturally to the fourth comment. In the previous

development, nothing specific had to be said about what classification

of the arriving aircraft into "types" was to be employed. Implicit in

equations (3.19) and (3.20), however, is a first restriction on which

classification schemes are admissible. (5) The requirement for OT^ and v^

^^^Incidentally, the treatment of S and p^ as independent parameters seems sanewhat odd; one would expect both to be related to some "safety

criterion", perhaps with a possibility of tradeoffs.

^^^It was for this reason that the notations OT.

,

v.

,

t. . i’ i’ 13

avoided in subsections 3.2 and 3.3.

b.

.

were

ij

- 44 -
to be meaningful, clearly, is that runway occupancy times (OT)

and final approach speeds (v) for the aircraft of any one type be

adequately representable by single nominal numerical values. This

phrase "adequately representable" is not as burdensome as it might

appear; for example it does not require that OT and v be essentially

constant within each type, a condition that would lead to an undesirable

proliferation of types. Rather, it requires that a reasonably representative

nominal value for each r- • + b. . be assignable.

In the present version, given by eqs. (3.25) and (3.26), this

stipulation restricts only the r^^’s. A glance at the formulas (3.20-21)

for these tieup times reveals that some degree of sophistication, in finding

nominal values for each class, may be called for. For example the v’s

appear only in the form 1/v, so that averaging of 1/v’s rather than of

v’s is indicated. Moreover, even the use of average OT’s and 1/v’s

is not rigorous, since these quantities enter the tieup time via a

nonlinear mathematical operator ("max") . This point cannot be pursued

further here, but seems to merit further analysis; we note only (a) that

the general difficulty in question seems to be equally pertinent to the

capacity concept embodied in the Handbook, and (b) that an aircraft

classification scheme developed for other purposes may not be especially

suitable for "capacity" analysis.

average of 1/v is not in general the reciprocal of the average

of v; the bias is systematic, in the direction [l/v]^^g ^ ^/^avg'
4. NUMERICAL ILLUSTRATIONS

4.1 Data Sets Employed

In Section 3, the general capacity concept fomulated in Section 2

was applied to the case of a single runway serving a stream of IFR landings.

In particular, the general formula (2.17) derived for this concept was

specialized (in a way based on further "modeling" assumptions) to a specific

version appropriate to this application, and given in equations (3.25) and

(3.26).

To obtain concrete illustrative numerical results for examination,

this formula was exercised to calculate "capacity" values for a number of

sets of input data provided mainly by the FAA. These results are discussed

in subsection 4.2, where they are displayed in several graphical formats

to help communicate the capacity measure's sensitivity to the various

parameters involved. In subsection 4.3 some of the results are compared

with the outputs of a fast-time Monte Carlo simulation, and excellent

agreement is found. A further comparison, this one with capacities given

in or derived from the Airport Capacity Handbook [I]

,

is carried out in

Section 5.

The remainder of this subsection is devoted to specifying tiie data

sets employed in calculating the capacity values. We will list the

quantities which enter the calculations, and present the numerical values

assigned to each.

First, there are two geometrical quantities

^^^We emphasize "illustrative," because an emphasis on data considerations

quite beyond this study's scope would be needed to produce numbers

sufficiently reliable for operational use.

- 46 -
Table 4.1.1: Final -Approach Path Lengths (L) and Minimum Separation

Distances (S) , in N. Mi.

L :

S :

Table 4.1.2 : Distributions over [-R, R] of Error in Intertouchdown Time.

Form

R(sec.)

triangular (symm.), uniform

15, 20, 25, 30, 35, 40, 45

Table 4.1.3: Threshold Probabilities of Violation

Pv 0.01 , 0.05

Table 4.1.4: Mixes of Aircraft Types

Mix

Ed Ee

I (Fast) 0.6 0.2 0.2 0.0 0.0

II(Med.) 0.2 0.4 0.2 0.2 0.0

Ill (Slow) 0.0 0.3 0.3 0.2 0.2

Table 4.1.5 : Final -Approach Speed (v, in knots) and Runway Occupancy

Time (OT, in sec.) by Aircraft Type.

A B C D E

V : 165 150 135 120 105

OT : 59 52 45 38 31

- 47 -
L = l^ength of final approach path (n. mi.),

S = required minimum distance separation along final approach

patn (n. mi . )

.

For these, the value -sets given in Table 4.1.1 were presented by the FAA

as covering the range of interest. Since L and S are to be varied

independently, the Table represents 9 combinations in all; the "nominal"

combination among these is taken to be

L=8n. mi. ; S=3n. mi.

The next two factors to be considered describe the probability distribution of the random error in achieving a desired intertouchdown

intem^al. This distribution is assumed to be symmetric and centered at

0, and to depend on a single parameter determining its "spread." The

two factors are

R = half-range of distribution (i.e., the errors range over the

interval [-R, R] )

,

F = (functional) form of distribution.

A traditional mathematical representation of random errors is that

of the normal (Gaussian) distribution; it is used for example in ,5], and

some empirical backup for this choice exists ([8], p. 30). On the other

hand, that distribution is a little awkward to work with analytically,

and its infinite "tails" are not really appropriate. We therefore

employed, as one functional form, the more tractable triangular distribution,

which has a satisfactory resemblance to the nomal for our present

illustrative purposes. As a deliberately pessimistic alternative, the

uniform distribution was used. (These distributions are sketched in

i'Lg’ires 3.3.1 and 3.3.2.)

- 48 -
Data to guide the choice of R-values were not initially at hand.

However our project monitor (Mr. S. P. E. Price of the FAA) obtained,

from one of the authors of [8], information showing deviations of between

(-8) sec. and roughly 52 sec. between scheduled and actual inter-aircraft

arrival intervals. Writing this as 22;^30 sec., we regard the 22 seconds

as a "buffer time" rather than a systematic unintended bias; this interpretation is supported by [8], p. 29, para. 1. Thus we are led to take

R = 30 sec.

as nominal value.

This however involves considerable uncertainty. Some intervals

in [8] are described (p. 28) as recorded "at the runway boundary"; the

seriousness of the difference between errors in such intervals and those

in intertouchdown times is not clear to us. The "mean errors" reported

([8], p. 29) are much smaller than 22 sec., apparently inconsistent with

our use of a symmetric error distribution. Using the observed range of

a random variable as an estimate of its theoretical range is of course

not sound practice, but time did not permit setting up a proper

estimation based on the distribution of observed errors . We tried to

allow in our calculations for some of these uncertainties, by permitting

R to vary quite widely as shown in Table 4.1.2. This is not really

adequate, however; quite clearly a more intensive effort to determine

suitable error distributions would be an important part of further work

along the lines of this report.

- 49 -
The next parameter to be considered is the threshold probability

(p ) of a violation. Values of 1% and 5%, which appeared in [5], were V

approved for use here. The value 0% of course suggests itself as of

special interest; however, because the error distributions we employ

are relatively "short-tailed," the results for 0% would not differ greatly

from those for 1 %

.

The three mixes of aircraft types shown in Table 4.1.4 were provided

by the FAA. Ranging from "fast" to "slow" in the average final -approach

speed involved, they represent a progression from a mix representative of

a large airport to one more typical of a smaller facility. The classification

into "types" is that employed in the Airport Capacity Handbook. In the

Table, p^ denotes the proportion of type i aircraft in the mix.

Counting up possible combinations of data at this point, we find 108

for each of the 7 R-values. This had two implications for our work. First ,

although the formulas are simple enough that hand computation for a small

number of cases is perfectly feasible, the number of cases involved here

is large enough to call for a computer program. Accordingly, a straightforward FORTRAN code was written and employed for the calculations.

Second , further proliferation of cases would be discouraged; the results

could be computed quickly enough, but time for their thoughtful interpretation and meaningful summarization (in this report) was lacking.

This decision affected our treatment of the remaining two sets of

input data, namely the typical final approach speeds (v^) and runway

occupancy times (OT^) for the various aircraft types. The particular

- 50 -
numerical values given in Table 4.1.5 were provided by the F// These

are only ’'representative" sets of values, however, and so sensitivity

analyses with respect to changes in them should be made. Such analyses

were not performed, for the reason just given, and so remain as unfinished

business for any possible continuation of this work.

- 51 -
4.2 Discussion of Results

In this subsection we present and discuss three groupings of calculated

capacities, and one grouping of values of the "efficiency" ineasure defined

at the end of subsection 2.2. There are three types of comments to be made

concerning these results. One type is sinply confirmatory and qualitative ;

it involves checking that as some parameter varies, the calculated values of

capacity or efficiency do indeed change in the direction indicated by commonsense. The second type examines these changes quantitatively , pointing out

the sensitivity of the calculated quantity to changes in the various input

data. The third type notes, in those cases where common-sense fails to

indicate a "proper" direction of change, which direction actually occurred.

All three types arise be lew.

( 2 ) We begin with Figure4.2.1^ ^

, in which R=30 sec., and the population

mix (II) is held fixed. There are a number of observations to be made:

(a) For the triangular distribution (believed more realistic

than the uniform), and the nominal value S=3 n. mi., the

capacity (C) varies from 32.5 to 36.5 aircraft/hr. For

population mixes I and III, the corresponding ranges are

35.3-38.7 and 29.0-32.5 aircraft/hr. These results appear

quite coirpatible with the range of values (31.0-37.0) reported

in [8], p. 22. Thus our procedure appears to be producing

numbers "in the right ball-park."

( 2 )To avoid misleading first inpressions, note that the ordinate scale of this

and other figures does not start at 0

.

- 52 -
(b) C is systematically higher for the triangular error distribution

than for the uniform distribution. This is as it should be, the

latter assumption not crediting the landing process with any

ability to make small errors more frequent than larger ones.

The variation amounts to 5% or more in many cases.

(c) Since capacity can presumably be increased at the cost of

greater indifference to the regulations , it is not surprising

that C increases with p^. This is more pronounced for the

triangular distribution than for the uniform. The difference

can be traced back to equations (3.16-17): the formula for

C contains p^ in the form (2p^) in the triangular case and

(2p^) in the uniform case, and for small p^ the first form

varies much more rapidly than does the second.

(d) As would be expected, capacity increases when the minimum distance separation (S) on the final -approach path can be reduced.

The sensitivity of C to S is quite marked; e.g. for L=8 n. mi.,

p^=0.01 and the triangular error distribution we have

C = 27.1, 33.0, 41.2 aircraft/hr.

corresponding respectively to

S = 4, 3, 2 n. mi

(e) C is consistently a decreasing function of the length L of

the final- approach path. This too is as it should be: a

C3)

Our model inposes no penalty to throughput rate for a violation; ....incorporating

such a feature might be desirable.

- 33 -
Figure

4.2.1

-

Sensitivity

to

Changes

in

Final

Approach

Distance

o

Pm

ISo

M

H

1:3 CQM

pc5 H

M

Q

o m o LO

sj- cn cn cN jnoH/3jHaoaTV ux

-54-

00

VD

00

v£)
longer path permits an "opening gap" between two successive

aircraft to open even wider, with consequent loss in throughput rate, whereas the "continuous demand" assumption assures

that "closing gaps" will finally close to a separation S

regardless of L, so that they are not shortened further by an

increase in L. What is somewhat surprising is the nearly

linear variation of C with L; it was not evident that our data

combinations would be on a linear portion of the generally nonlinear C vs . L curves

.

We turn now to Figure 4.2.2, designed to display the influence of

population mix on capacity. Clearly C decreases, as one moves from the "fast"

mix through the"medium" one to the "slow." Now aircraft making faster

approaches should indeed be able to achieve more landings per unit time —

the minimum distance separation translates into a smaller required time

separation — unless the constraint associated with runway occupancy prevents

realizing the potentially greater throughput from the faster mix. The

seriousness of this last possibility is accentuated by the fact that the

fast-approach types have longer runway occupancy times; see Table 4.1.5.

Thus there are two opposing tendencies involved in hav C will vary with

population mix, and without calculation it was not clear which would win

out. From the direction of variation shown in the Figure, it follows that

tieup times are in fact (for our data) generally determined by the final

approach separation criterion rather than by runway occupancy considerations.

This observation will gain in significance when a conparison with Handbook

values is made in Section 5; see there the discussion of Table 5.3.1.

- 55 -
Figure

4.2.2

-

Capacity

as

a

Function

of

Population

Mix

t-j o

CQ

c/1

LO r-i LO rH > O O O O

I • • • •

13O

M

H

pqM

PiH

COM

Q

Pi<

O O

3 3

Pi Pi H H

§§O O

P-l M M

gg

anoH/ 3 jEaoaxv ux X^xobcIfo

X

•H

- 56 -

Population
The observation also provides an exceptionally sharp example of how such a

model can prove useful in cost-effectiveness analyses: under the stated

circumstances, investments (in equipment, R. and D. or whatever) whose

principal aim is to reduce runway occupancy times are clearly ill-advised

if increasing capacity is the major goal!

Figure 4.2.3 was prepared to show how capacity (C) depends on the

measure (R) of error spread. The variation of C-values is considerable,

roughly 20% for Mix II for example, so that pinning down realistic values

of R more tightly is indeed important. The functional dependence of C

upon R can be seen from equations (3.25-26) on p.42 to have the form

C = 1/(A+BR)

where A and B are positive constants (relative to R) with B < 1, but near 1 (3a) when p^ is small. It follows^ ^ that as R is reduced, successive reductions

have increasing marginal benefit (in upping the value of C) . This contrasts

with the more common and less promising ’’ decreasing marginal benefits”

situation. Thus, purely from the ’’effectiveness” side of the cost-effectiveness

ledger, attempts aimed at reducing R in order to increase C seem to merit a

tentative label of "high-payoff.” (This of course says nothing of technical

feasibility and cost considerations.)

For R=0, C assumes the ’’ideal capacity” value corresponding to

perfectly precise control of intertouchdown times. Figure 4.2.4 repeats the

information of the previous figure in a ’’normalized” form, using as ordinate

not C but rather the efficiency measure

E = C/C. , ' T ideal

^^^^Since 3C/9R = -B/(A+BR)^ which increases in absolute magnitude as R decreases.

- 57 -
Figure

4«2.3

-

Sensitivity

to

Changes

in

Pi

LO

<r

o

cn

ocn

Pi

mCvJ

o

CNl

anoK/^jBaoaxv ut A:ijoede3

58

in

Seconds
TRIANGULAR

DISTRIBUTION

R

in

Seconds
suggested in subsection 2.2. This measure increases to the value 1 as R is

reduced to 0. As the Figure shows, the rate of increase in E per unit decrease

in R is considerable throughout at least the lower half of the R-range considered, again illustrating the potential from such reductions.

On comparing Figures 4.2.3 and 4.2.4, it is interesting to note that

the rank order of population mixes reverses when E rather than C is the

criterion. This must mean that for slower mixes, buffer times are less

significant relative to tieup times than for faster mixes. Thus average

tieup times must be less for slower mixes; since runway occupancy times are

higher , we conclude as before that "final- approach separation" rather than

"runway occupancy" must be dominant in determining tieup times.

Various other arrangements of our results could of course also have

been plotted and discussed, but it is hoped that those given above suffice

to illustrate the kinds of analysis that are possible.

- 60 -
4 . 3 Comparison with Simulation Outputs

Our capacity concept was defined, informally in subsection 2.2 and more precisely in subsection 2.4, as the average mean throughput rate of the facility

for a prolonged period of continuous demand. If the concept is to be useful

relative to a peak load interval at a runway during which one can meaningfully

speak of a single papulation mix, then a period of at most one or two hours

had better be sufficiently "prolonged" for the concept to apply. Moreover,

the concept is defined as an "average"; its potential usefulness is therefore

much enhanced if one can shew that the associated random variable has a small

variance, thus guaranteeing (e.g., by Tchebychev's Inequality; see [6], Vol. I,

p. 183) that large fluctuations from the average are very rare.

These points were explicitly addressed in subsection 2.4. Let T be the

duration of the "prolonged period," and N(T) — a random variable — the

number of customers served (aircraft landed) during that period. The "average"

in question is that of the random variable N(T)/T. Now it is shavn in the

second part of Appendix B that for large T, the variance of N(T)/T is

approximately proportional to 1/T, and so declines rapidly toward 0. It is also

shown that the average, which depends on T, differs from its limit as T

by a quantity which is approximately proportional to 1/T for large T, and so

also declines rapidly to 0, inplying that the average will soon (as T increases)

"settle down" to be virtually independent of T.

These arguments appear to provide assurance on the two points raised

above, and such arguments have generally proved to be reliable guides in

applying similar mathematical concepts to normal real-world situations. Havever,

- 61 -
two slim possibilities for trouble still exist. One of them can be identified

by the double appearance of the phrase "for large T" in the last paragraph; it

could conceivably be the case that 1-2 hours is not "large" enough for the clause

follwing the phrase to apply. The second danger is associated with the double

appearance of the phrase "proportional to 1/T"; we did not have time to try

to estimate the associated constants of proportionality, and just possibly they

might be so large that their quotient by a T of 1 or 2 hours would not be so

very small at all.

To check on these undesirable possibilities , a fast-time Monte Carlo

simulation of the IFR landing stream was designed, programmed, and run. For

each of several of the data combinations presented in subsection 4.1, a san^le ,

of 100 simulated 2-hour periods was generated. For the first

half-hour

j hour

hour-and-a half

2 hours

of this 2-hour period, we recorded the associated sample average (to conpare

with the limit as T of the theoretical average) , and the sairple standard

deviation.

The results shown in Figures 4.3.1 and 4.3.2 (for R=30 sec.) are quite

representative. The two curves delimit a band, of width (* 1 sample standard

( 4)And on whether the variance did appear to be converging to 0; this was not

yet known at the time. Incidentally, the bounds on N(T)/T provided by equation

(2.2) were calculated in our numerical work, and in general did not confine N(T)/T

so tightly as to rule out significant fluctuations from the average.

- 62 -
Figure

4.

3.1-

Analytical

vs.

Simulation

Results

n3

CJ (U H 3

ctj

TIME

IN

HOURS
Figure

4.3.2

-

Analytical

vs.

Simulation

Result

cOa 0) •H 3

CO

TRIANGULAR

DISTRIBUTION
deviation), around the capacity value calculated from the recommended formula.

We see that for the triangular -distribution case, as early as T=0.5 hr., the

sample average differs from the theoretical value by less than 0.5 aircraft/hr.,

while the sample standard deviation is only about 1.4 aircraft/hr. and drops

below 1.0 aircraft/hr. at T=1 hr. For the uniform distribution, the standard

deviations are just a bit larger, while the settling-down of the average is

even more rapid.

This establishes that the conceivable difficulties mentioned above should

not in fact arise. Further specifics are given in Appendix F. Note, incidentally,

that the sample averages all lie on the same side of the theoretical value; this

is consistent with the fact that the constant B, in eq. (B.14) , has a definite

sign.
S. RELATION TO "HANDBOOK” VALUES

5.1 Preliminaries

One of our specific assignments was to comnare the new capacity

concept with that embodied in the Airport Capacity Handbook [1].^ ^

So far as underlying ideas are concerned, a basic difference

between the two concepts is that pointed out in subsection 2.1: the

Handbook *s "capacities" represent maxim.um traffic rates which can be

handled without average delay exceeding a prescribed level, whereas

the present concept is based on a "maximum throughput rate" idea not

explicitly limited by delay considerations. On these grounds alone,

the present concept should assign a higher capacity number to a

given situation than would the Handbook .

At a more technical level, the Handbook ' s results are based on

a mathematical model which assumes a Poisson distribution of

"customer" arrival times. We understand that this assumption has

been questioned as being too pessimistic, since arrivals for the

facility in question (here, a runway and its final-approach path

airspace) have been subjected to previous control actions tending to

regularize their flow; thus a higher-order Erlang distribution has been

suggested as a possibly more appropriate mathematical representation

than the Poisson. At any rate, any really probabilistic treatment

of arrival times admits the possibility of occasional idle periods

for the facility, whereas our concept is defined in terms of a

This reference is the first edition of the Handbook ; the

second edition, dated 6/69, became available too late for use in

this study.

- 66 -
''continuous demand" scenario which rules out such periods as not

properly relevant to a "capacity" measure. This difference, too,

will tend to make our values higher than the Handbook's.

We turn now to a numerical comparison of the present concept's

capacity numbers with those given by the Handbook, for the case of

a single runway handling a stream of IFR landings. Our capacity

numbers are taken from the material in Section 4, using the data

( 2 ) and parameter values given in subsection 4.1. How the comparable

"Handbook values" were obtained will be discussed here and in subsection 5.2, before the comparative numbers are presented and

discussed in subsection 5.3.

The need to discuss how "Handbook values" for the comparison

were determined may at first seem absurd; one would think the

only possible answer is "we looked them up in the Handbook .

Moreover, the look-up procedure for IFR landings at a runivay is

quite simple:

(a) First one consults Figure 16-1 Cp-lb-2 of the Handbook.) .

This gives a curve for average delay (which we v^:ill denote D)

versus A , the arrival rate parameter in the underlying Poissondistribution model. The Handbook recommends D=4 min. as the

appropriate "tolerable level", and has A =21.7 landings/ hr.

explicitly marked on its Figure 16-1 as the corresponding nominal value of ''capacity.''

( 2 ) The half-range R, of the random error in achieving a desired intertouchdown spacing, was held fixed at a "nominal"

level R=30 sec.

- 67 -
Cb) Figure 16-1, however, is based on some single nominal

mix of aircraft tyoes among the users of the facility. The remainder of the Handbook ' s Chapter XVI, Figure 16-? through 16-10,

gives multiplicative correction factors to be applied to the basic

X =21.7 to obtain the capacity for the ''actual" population mix

at hand.

This straightforward look-up process was in fact employed

to obtain on_e of the sets of "Handbook values" which appears in

the comparison of subsection 5.3. However, we had two misgivings

about it. The less important of the two is that the procedure is

somewhat laborious and liable to human error: if a larger set

of comparisons were required in the future, a computer program

based on the mathematical models underlying the Handbook for at

least its Chapter XVT) would clearly be more suitable than a

visual-manual lookup from graphs.

The second misgiving is more serious. For a proper comparison of the Handbook 's values with ours, the two concepts should

be applied (as nearly as possible) to the same sets of cases . However, the Handbook ' s material permits variation only of the mix

of aircraft types and not of a number of ouantities used in our

f3)

concept which appear to be relevant:

(3) He would jpjt expect the parameters and R of our mathematical model to be directly relevant for the Handbook's capacity

concept. Insensitivity to these factors, as well as the four

listed above, appears to reduce the usefulness of the Handbook's

capacity information for cost-effectiveness analyses at a higher

level than airport design.

- 68 -
length (L) of final-apnroach oath

minimum distance separation (S) required along path

final approach speeds (v^) by A/C type

runway occupancy times (OTj^) by A/C type.

Presumably, nominal values of these quantities arc imnlicit in the

Handbook ' s contents, but these values might not He compatible with

those (in subsection 4.1) used in evaluating this report's capacity

concept. In particular, the data used in Section 4 are based on

experience more recent than the period (1963-65) mentioned in the

(4) title of the relevant Handbook chapter.

Thus, to ensure proper comparability capability we again needed

to know the mathematical models underlying the Handbook ' s graphs,

so that our data could be entered into these models to determine

what capacities the Handbook * s concept would yield for the same

cases evaluated in our Section 4. At this point, we cannot avoid

some critical comments concerning the documentation (at least, that

available to us) of the methodology underlying the Handbook . This

docimientation was found to be essentially impenetrable, despite the

high motivation and technical competence of the staff members

examining it. Some sections were more yielding than others, but

it was simply not possible for us to extract a clear checkable

statement of the mathematical models and calculations used in

(5)

obtaining the Handbook 's curves ^ even for the especially simple

^^^On the other band, Chanter XVII of the Handbook takes up IFR

operations for the period 1970-1975; possible use of this chapter

is taken up in subsection 5.3.

understand that others have had the same experience.

- 69 -
case of a single runway handling a stream of IFR landings. (Such documentation

does not even permit the reader, with a reasonable effort, to satisfy himself

as to the correctness of the reported work. More satisfactory supporting

documents may in fact exist, but the time -frame of our study precluded an

intensive search.)

We are therefore unable to be fully certain that any of the 'liandbook

values" enployed in subsection 5.3 are simultaneously (^) truly comparable

with those given by the present concept and ( ii) true representatives of

what the Handbook’s methods would produce. As one set of "Handbook values"

in the conparison, we employ capacity values extracted from Chapter XVI by

look-up; their use seems acceptable since the Handbook is after all intended

to be employed "as is" without supplementary explanatory material. In

addition, some detective work was performed to "estimate" the model used

in the Handbook '

Chapter XVI; this work, which provides a basis for

calculating "modified Handbook values" possibly better-suited for the canparisons to be made, is reported in subsection 5.2. Some readers may prefer

to skip over this next subsection, which is relatively technical in content.

- 70 -
5.2 A Basis for "Modified Handbook" Values

The text of a I965 professional-society presentation by an

FAA staff member describes ([l5J,p.l) the underlying model, for arrivals

only at a single runway, as a "simple Poisson-fed queue with first-come

first-served discipline". The term "simple" in this queuing context

has two possible Interpretations as regards the probabilistic distribution of holding times: either constant (= l/,j), or exponential

(with average = l/^)* The formulas for average delay in these two

cases are standard ones (C9]>P«3^7):

D = X/2M(m-X) (constant). (5.1)

D = X//z(/U-X) (exponential)

.

(5.2)

Although Figure 1 of [15] contains (5»l) rather than (5.2), implying

that constant rather than exponential holding times were assumed, both

possibilities were maintained through this phase of the analj^sis.

(5a) The capacity concept of the Handbook requires setting'^ '

D = D =4 min.

o

and solving for the corresponding value of \ (the capacity).

(5a) That is, will stand for h minutes (expressed in appropriate

units) in all the formulas to follow.

- 71 -
This yields

X = 2i^ / (l + 2/^ D^) (constant), (5«3)

X = / (l + /i D^) (exponential). (5*^)

For present purposes, however. It Is more relevant to solve for the

average holding time l/yL In equations (5«l) and (5»2) with D = :

l/p = [ + (2 Dq A)]^/^ - (constant) (5-5)

1/M = [( A)^ + ( A (exponential) (5.6)

In using the model for predicting capacity (X) by (5*3)

or (5*^)> a definite value of jx must be employed. The Handbook

values of X depend on the mix of alrcral“t types (the p^); since

Is constant, fx must depend on the mix. We now assvime that In the

method underlying the Handbook, this dependence Is of the form

1/m = (1/M^) (5-T)

where model parameter representing the average holding

time for type 1 aircraft; thus l//i is a general average holding

time

- 72 -
If the 1/y^ were known, then (5.7) together with (5.3) or (5.4)

represents the conjectured Handbook model employed in its Chapter XVI. It

only remains to estimate the 1/y^, which are initially unknown to us.

For this purpose, we can regard (5.7) as giving 1/y as a linear form

in the variables p^, with unknown coefficients 1/y^ which are to be estimated.

Such estimation becomes a standard least-squares linear regression problem

(for which computer codes are readily available), given the appropriate

data consisting of an adequate number of p^-combinations (aircraft-type

mixes) together with the value of 1/y corresponding to each. The value

of 1/y corresponding to a given population-mix can in turn be found by

first using look-up in the Handbook ' s Chapter XVI to determine a value

of capacity (A), and then determining 1/y from (5.5) or (5.6).

This procedure was followed, using a set of 15 different population

mixes. The resulting estimates are shown in Table 5.2.1; the closeness

of the statistic^^^^ to 1, and the smallness of the estimate standard

deviations (about 1% of the estimates) , provide some assurance that an

approach like that sketched above really does underlie the Handbook * s

"IFR landings, single runway" curves. In addition, eq. (5.7) is consistent

with the use of averaging procedures elsewhere in the Handbook , e.g.

in calculating nominal "runway ratings" ([1], p. C-2, right-hand column).

Note that neither column in Thble 5.2.1 has its entries arranged in

order of size; we do not know whether or not this is cause for disturbance.

^^^^Here and on the next page only . R denotes a multiple correlation

coefficient (a measure of goodness-of-fit of a proposed formula to

a set of data)

;

this should not be confused with the usage of "R"

in the bulk of the report as a measure of spread in intertouchdown

time errors.

- 73 -
As a further check, the estimated values of the 1/y^'s were used

together with equations (5.7) and (5.3-4) to "predict" the Handbook

values of capacity for three additional mixes of aircraft types (not

among the 15 mixes used in the estimation). Tkble 5.2.2 shows that the

predicted values are in fairly good agreement with

Table 5.2.1: Estimated Values of Average Holding

Times (1/y^) by Aircraft iype.(*)

((*)

**)

Constant Exponential^ ^

1/Pa

• 171.5 145.3

1/Ub : 112.9 98.8

l/^C • 115.9 101.3

^^^D+E: 134.3 116.0

(*) Each column based on 15 observations.

^ In seconds per A/C. R = 0.9818.

fifif if'\ 'y ^ -^In seconds per A/C. R = 0.9829.

- 74 -
Table 3«2.2 : Predicted vs. Extracted Veilues of

(a)

Handbook Capacities for Three Mixes

Mix Pm-E Const

.

„ (aaa) Extracted

. t 0.6 0.2 0.2 0.0 18.4 18.5 18.2

II 0.2 O.k 0.2 0.2 22.0 20.0 21.7

III 0.0 0.3 0.3 0.4 23.9 23.4 23.4

those extracted from the Handbook * s curves.

In connection with this predictive process, it is convenient

to rewrite equations (5*3) and (5»^) in terms of l//i (the quantity

given by (5.7)) rather than /i . The new versions are

X * 2 / (l/|i)C2 + 1/m] (constant), (5*8)

X = / (1/m)C 5^ + 1/m] (exponential). (5-9)

The ccwitents of this subsection so far, can be sunanarized as

follows: We have made a plausible case that the mathematicsG. formulas

(a) In A/C per hour.

(aa) Predicted vadues assuming constant holding-time model,

(aaa) Predicted values using exponential holding- time model.

- 75 -
underlying Chapter XVI of the Handhooh are equivalent to the procedure

of calculating a mix-dependent 1/m from eq. (5*T)^ and then computing

the capacity (X) from ( 5 « 8 ) or ( 5 » 9 )« Since eq. ( 5 »l) rather than

(5.2) appears in [15], it is probably (5*8) rather than (5.9) which

should be used. Moreover, the numerical values of the l//i^'s used.

for the 1963-65 curves in Chapter XVI sho\ald be approximately those

in Table 5 . 2 . 1 .

It follows that "modified Handbook values" of X, probably

more suitable for conqjarlson vith numerical values of this report's

cai>aclty concept than are those extracted from the Handbook ' s

Chapter XVI, can be obtained by using the procedure described in

the last paragraph. However, the l/ft^'s of Table 5 * 2.1 sho\ild be

replaced with values of average holding time (by aircraft type) consistent

with our data in Section k. This will be tried in the next subsection.

The remainder of the present subsection consists of two ccraments

of a more technical nature, which some readers may prefer to omit.

The first of these concerns the approach (involving linear least-squares

regression) used above to estimate values of the from the curves

in the Handbook * s Clutptar XVI. This approach was such that l//t played

the role of "dependent variable" in the regression. Since we eire

"fitting" a model to be used in calculating value* of X , it would

be theoretically preferable to perform parameter estimation with X

- 76 -
serving as dependent variable. The appropriate functional form is

that obtained by substituting equation (5 *7) into (5*8) and (5 •9)*

Note that the resultant functional form is nonlinear (specifically,

reciprocal quadratic) in the parameters 1/a*^ to be estlmatedj thus

the theoretically preferable procedure requires use of a nonlinear

regression algorithm (e.g., [l6]). Such an algorithm was therefore

applied, and proved to yield only negligible changes from the estimates

found by linear regression.

The second technical comment concerns the use (reported in

[ 15]) of equation (5«l)# apparently in conjunction with eqviation

(5»7); in producing the Handbook *s curves in Chapter XVI, Unfortunately

this procedure is simply not correct; eq. (5*l) applies only to constant

holding times, whereas 1/m as given by (5*7) is an average holding

time representing a probabilistic mlxttire of constant holding times,

but not Itself a constant holding time. Equation (5*l) should be

replaced by ([9J^P»3^5)!

U = [X /2iX (/i-X)} {l + sV] ( 5 . 10 )

2 where s is the variance of holding time, given (when constant holding

times l/Mj^ nne assumed for each aircraft type) by

- 77 -
Pi ^ • (5.11)

•“ 2 2 Thus (5*8) should be corrected by replacing with (l s ).

We have not attempted to investigate the numerical seriousness of

this flav.

- 78 -
5.3 Con^arison with Handbook Concept

With the preceding explanations and preparations conplete, we can

begin the actual comparison of "capacity values" (presented in Section

4) for this report’s concept with corresponding values of the concept

embodied in the Handbook . Since the two concepts are based on different

intuitive notions of "capacity", the meaningfulness of such numerical

comparisons should not be over-estimated.

Four sets of comparisons will be made. These represent four

different possibilities for obtaining 'Handbook values" properly

comparable with those obtained for the present concept. The first two

sets involve values directly extracted from the Handbook ; the third

and fourth involve "modified Handbook values" based on the ideas in

subsection 5.2.

The first and second sets of comparisons are presented in Table

5.3.1. Two columns of values are given for the capacity concept presented

in this report; they correspond to the assumptions of a triangular

or (more pessimistically) a uniform distribution for the random error

in achieving a desired intertouchdown interval. Each entry in these

two columns consists of (ij a range of values, followed by ( ii) a single

number. The range of values arises because the formula for this report’s

concept depends explicitly on a number of factors not employed in

extracting "values from the Handbook ; thus a single "Handbook case" cor-

- 79 -
responds to a whole family of the cases considered in Section 4, and

hence to a range of capacity-values . The single number in the entry

is one extracted (for definiteness) from this range, and corresponds

to the specific conditions

L = 8 n . mi .

,

S = 3 n. mi.

,

p

= 0.01.

The third column in Table 5.3.1 gives capacity values taken from

Chapter XVI of the HandbookAs noted earlier, these refer to the time

period 1963-65, which casts doubt on their comparibility with the material

in the preceding columns. Therefore, in the fourth column of Table

5-3-1, we present the analogous values extracted from the Handbook's

Chapter XVII, which deals with IFR operations in the period 1970-75.

These numbers are presented subject to the following reservations:

(a) They are based on the Airborne Instrument Laboratory's projections

(specified at least in part in [2]) on the effects by 1970 of improved

traffic-control techniques and equipment. These projections were made

no later than 1963 (the date of [1] and [2]), and so represent a forecast

roughly 7 years into the future. They may therefore be proving considerably

over- optimistic or over-pessimistic,^^^ and no doubt have been updated.

^^^This is not a criticism, but merely an acknowledgment of the difficulties

attending medium -range and long-range forecasting.

- 80 -
(b) Chapter XVII deals mainly with "mixed" IFR operations, i.e.,

both landings and takeoffs. While "landings only" is certainly a

special case of this, it is an extreme case, and the curves in Chapter

XVII may be based on approximations whose quality deteriorates for

extreme cases.

(c) From the instructions for Chapter XVII (see Example 1, p. 17-1

of [1]), it appears that this chapter's "capacity value" for a single

runway serving a stream of landings is obtained either from Figure

17-6, or from the smaller of the readings from Figures 17-1 and 17-6.

The uncertainty on this point causes no trouble here, since for the

particular mixes under examination both alternatives give the same result.

What 33 disturbing is that the capacity values in each of these two

figures depends only upon the proportion of Class A aircraft in the

mix. In addition, the "bending backwards" of the curve in Figure 17-6

causes concern as to vdiether the methodology is consistent with that

eiiqjloyed in Chapter XVI.

Examination of Table 5.3.1 reveals two salient points. First,

the values for the proposed concept are systematically and distinctly

greater than those taken from Chapter XVI of the Handbook . (The

same is not true for the values from Chapter XVII, but for the reasons

given above not much significance is attached to this.)

Second , the values of the proposed capacity concept diminish as

one goes from faster to slower mixes, whereas the Handbook values increase

.

- 81 -
Table 5.3.1: Conparison with Values Extracted from

Handbook. (A/C per hour)

MIX PROPOSED CONCEPT HANDBOOK EXTRACTS

Triang. Uniform Chap. XVI Chap. XVII

I (Fast) 29.0-46.2 (35.9) 28.1-42.7 (34.7) 18.2 36.5

II (Med.

)

26.5-45.4 (33.0) 25.8-41.9 (31.9) 21.7 38.0

III (Slow) 24.0-42.3 (30.0) 23.4-39.3 (29.1) 23.4 39.2

Table 5.3.2: Comparison of Proposed- Concept Values (C)

with Modified Handbook Values (A) using

Intertouchdown Times as Holding Times. (A/C

per hour)

C: 20 25 30 35 40 45 50 55 60

X (const.): 11.5 19.2 24.0 28.8 33.7 38.6 43.5 48.4 53.3

X (exp . ): 11.4 15.6 20.0 24.5 29.1 33.8

! 38.5 43.2 48.0

Table 5.3.3: Conparison with Modified Handbook Values

using Runway Occupancy Times as Holding

Times. (A/C per hour)

MIX PROPOSED CONCEPT MODIFIED HANDBOOK

Triang. Uniform Const. Exp.

I (Fast) 29.0-46.2 (35.9) 28.1-42.7 (34.7) 62.9 57.4

II (Med .

)

26.5-45.4 (33.0) 25.8-41.9 (31.9) 66.4 60.7

III (Slow) 24.0-42.3 (30.0) 23.4-39.3 (29.1) 77.0 71.2

- 82 -
This divergence can be explained in part by the following line of reasoning

On the one hand, in passing from aircraft types with higher final

appiK)ach speeds to those making slower approaches, runway occupancy times

typically diminish (see Table 4.1.5), and this tends to increase capacity

for the slower mixes. On the other hand, diminished speeds will increase

the time separation into which the minimum distance-separation requirement translates, and this tends to decrease capacity. Which of these two

opposing tendencies wins out is largely determined by the relative frequencies with which runway occupancy and distance separation (respectively)

determine tleup time; see equations (3.13) and (3.14). This last sentence

is couched in the language of the present report; in connection with the

Handbook, we must instead speak less concretely of the relative influences

of runway occupancy times and of distance separation criteria on the

"holding time" concept entered into the underlying queuing model.

Examination of intermediate results, in the calculations reported in

Section 4, reveals that tleup time was determined by "separation" rather

than "runway occupancy" except for sane cases with S = 2. 'Thus the decrease in our capacity values, for slower mixes of aircraft types, is

explained. Evidently runway occupancy times play a more influential role

in the "holding times" entering the Handbook’s mathanatlcal model.

It is interesting that the discrepancy probably derives fran the

"holding tine" concept in the work supporting the Handbook . The treatment

of holding times could after all be varied while retaining the essence

(7)

In the queuing-literature sense of tying up a facility, not the ATC

sense of "holding."

(8)

An alternative possibility, that the Handbook assumes a smaller required

distance separation, is ruled out by p. 3~9 of [2].

- 83 -
of the delay-based capacity concept, and so possibly the divergence

could be removed. It might also be removed by suitable changes in

the calculated values of the proposed concept, due say to variations

from the (nondefinitive) data on runway occupancy times and final

approach speeds used in Section 4. But the decreasing trend down

each of the first two columns of Table 5* 3*1 seems too clear-cut

to be reversed by reasonable perturbations of the input data.

We turn now to comparisons with "modified Handbook values."

In subsection 5*2, evidence was assenbled that such values couTd

be produced by one of the fommlas

^ = 2 Tyy(l/y)[2 + 1/y] (constant), (5.12)

A = D^/(l/y) [D^ + 1/y] (exponential), (5.13)

where the parenthetical term refers to the assumed distribution of

holding times in the underlying queuing model, = 4 min., and 1/y

is a mix-dependent average holding time given by

1/y = E. p. (1/y^ (5.14)

with 1/y^ the average holding time for type i aircraft.

A set of modified Handbook (capacity) values is obtained by

assigning appropriate values to the 1/y^. Without a clear understanding

of the holding- time concept

- 84 -
used In the Handbook’s implicit mathematical model, one cannot be certain

how such values should be assigned. However, two alternatives will be

examined. One of them should produce results quite properly comparable

with those of Section 4 for the proposed capacity concept, but at the

probable cost of doing real violence to the "holding time" idea embodied

in the Handbook’s Chapter XVI. The second is believed to hew more closely

to the ideas underlying the Handbook, but its results are less comparable

with those of Section 4.

Specifically, the first idea is to equate 1/y^ to the average intertouchdown time for type i aircraft (i.e., between the touchdown of such an

aircraft and that of the next aircraft in the stream) . In the context of

Sections 2 and 3, this becomes

l/bi = Zj Pj (rij + b^j),

and so equation (5.14) yields

I/U ' Pi Pj + b^^).

Now comparison with equation (2.17) yields

C = y (5.15)

where C is the capacity concept proposed in this report. Thus equations

(5.12) and (5.13) see the more convenient forms (5.3) and (5.4) become

X = 2 Dq C^/(l + 2 Dq C) (constant), (5.16)

X = D C^/(l + D C) (exponential). (5.17)

0 o

- 85 -
This gives an explicit algebraic conversion from values (C) , of the

capacity concept used in this report, to "associated" modified Handbook (X)

values. With the temporary notation

A = 2 in (5.16); A = D in (5.17),

o 0 the preceding relations become

X/C = AC/(1 + AC) = 1 - 1/(1 + AC). (5.18)

From this it is apparent that X < C, and that X/C Increases from 0 toward

1 as C increases from 0. In particular, X increases with C so that the

type of discrepancy noted in connection with Table 5.3.1 cannot occur;

this is not surprising since the "holding time" concept underlying X has

been bodily replaced by that used in our calculation of C-values . A

numerical comparison is given in Table 5.3.2; the second row is more likely

to be relevant than the third.

A second alternative is to equate the 1/y^ *s to the runway occupancy

times OT^ . In view of the above discussion of Table 5.3.1, this was

thought to reflect more closely the holding-time concept used in deriving

the Handbook’s curves. The resulting comparison, using the OT^-values in

Table 4.1.5, is given in Table 5.3.3. We see that the "modified Handbook

values" obtained look considerably too high. Thus the holding times used

in the Handbook probably include more than just runway occupancy.

- 86 -
6. POSSIBLE NEXT STEPS

The present report of course constitutes only an initial exploration

of the new capacity concept. In this Section, we sketch some of the next

steps that would be involved in investigating the concept more fully and

bringing it closer to operational status. Such further development may

well be useful to the FAA in connection with its Continuing Study of Air

Traffic Control System Capacity and Demand.

The steps to be described fall into three groups . One pertains to

extending the scope of the concept. The second involves more intensive

application and exploitation of the theoretical progress already made than

was possible within the present study’s time span. The third concerns

attempts to derive sharper infoimation on certain technical points arising

during the study.

In the first and most ambitious category, five tasks suggest

themselves:

(a) Extend the concept to deal with a runway serving both arrivals

and departures.

(b) Extend the concept to deal with interactions between runways

.

(c) Extend the concept to a ”weighted capacity" notion, in which

"throughput" is measured in a way sensitive (for example) to the number

of passengers served and not merely to the number of aircraft.

(d) ^fc)dify the mathematical model to incorporate scrnie penalty in

capacity associated with violations.

(e) Continue study of an associated delay concept (see Appendix E)

.

- 87 -
These five tasks are too much to tackle together in a small-scale

effort. Item (a) seems to deserve top priority, and progress on it might

well have spillover usefulness to the subsequent study of (b) . We believe

that task (c) requires only modest additional theoretical development

beyond vdiat has already been done, but are not quite so confident

concerning (d) . On balance, it would seem that (a) and (c) represent a

reasonable pair of targets for the "extension” portion of a follow-on

study; further work on (e) also seems called for.

There are several possibilities in the "further applications"

category:

(a) Perform sensitivity analyses with respect to variations in the

runway occupancy times and final approach speeds attributed to various

aircraft types.

(b) Examine the effects of introducing violation - probability thresholds

and distance separation rules which depend on the types of aircraft involved.

(c) Derive analytical formulas for sensitivity coefficients.

(d) Examine alternative and better -founded models for the distribution

of errors in achieving desired intertouchdown times.

(e) Apply the methods to a stream of IFR takeoffs, and perhaps to

VFR operations as well.

(f) Redo the comparison with the Handbook * s concept, using the

second edition of the Handbook .

Of these tasks, only (d) appears to be of fundamental significance,

and only (d) and (e) promise any real difficulty. The remaining items

- 88 -
should in principle require only relatively minor efforts, so that those

of them considered to hold sufficient interest might as well be undertaken.

We have identified three tasks of the "derive sharper information"

variety. The first two are aimed at reducing the incentive for auxiliary

simulations like that reported in subsection 4.3:

(a) Recall that the capacity measure, over a prolonged period of

duration T, differs from its limit as T by a term essentially proportional to 1/T. Analytically determine or estimate the constant of proportionality.

(b) Recall that the mean throughput rate, over a prolonged period

of duration T, has a variance essentially proportional to 1/T. Analytically

determine or estimate the constant of proportionality.

(c) Investigate the effects of the fact that approach speeds and

runway occupancy times vary within an aircraft "type."

In addition to the three categories of tasks listed above, we mention

also the possible investigation of capacity enhancement through, sequencing

of customers in other than "first-come-first-served" order. Such study

might begin along the lines of Appendices C and D, e.g. by investigating

what capacity increases are theoretically possible using the idea proposed

in Appendix C.

- 89 -
7. REFERENCES ( 1 )

L. Airport Capacity Handbook. Airborne Instruments Laboratory, 6/63.

* 2. Operational Evaluation of Airport Runway Design and Capacity. Airborne

Instruments Laboratory, 1/63. E. N. Hooten et al.

* 3. Airport Runway and Taxiway Design. Airborne Instruments Laboratory,

7/60. M. A. Warskow et al.

4. Staff Study: Air Traffic Control System Capacity and Demand (Report #1

of a Continuing Study) . FaA Systems Analysis Division, Eys terns Research and Development Service, 9/68 Working Draft. S. P. E. Price.

5. (Untitled manuscript.) MITRE, 6/9/69 Draft. R. M. Harris.

6. Introduction to Probability Theory and its Applications, Vol. II.

J. Wiley and Sons, 1966. W. Feller.

*

7. Probabilities of Missed Approach for Random vs. Controlled Arrivals.

Control Technology, Inc. , 4/63. R. V. Morse and T. W. Connolly.

8. Development and Application of a Terminal Spacing System. FAA (NAFEC)

Report NA-67-3, 12/67. D. A. Martin and F. M. Willett, Jr.

9. Mathematical Methods of Operations Research . J. Wiley and Sons, 1959.

T. L. Saaty.

10. The Laplace Transform . Princeton U. Press, 1946. D. V. Widder.

11. Finite Markov Chains . Van Nostrand, 1960. J. Kemeny and J. Snell.

12. Mathematical Theory of Reliability . J. Wiley and Sons, 1965. R. E.

Barlow and F. Proschan.

13. Introduction to Linear Programming . McGraw-Hill, 1960. W. W. Garvin.

14. Limiting Covariances in Markov Renewal Processes. Berkeley Operations

Research Center Report, 7/22/64. W. S^. Jewell.

15 . An Airport Capacity Handbook: Results of a FAA Research Program in

Mathematical Models and Techniques for Determining Airport Capacity . Paper D3b at 27th National Meeting, Operations Research Society of

America, 5/7/65. 0. Shapiro.

16. An Algorithm for Leastr Squares Estimation of Nonlinear Parameters.

J. Soc. Ind. Appl. Math. 11 (1963). pp. 431-441. D. W. Marquardt.

( 1 ) Titles followed by an asterisk indicate contractor reports to the FAA.

- 90 -
17 . In^strial Schelling . Prentice-Hall, 1963. J. F. Muth and G. L. Thon^son (ed. )

.

18. Theory of Scheduling . Addison-Wes ley, 1967. R. W. Conway, W. L. Maxwell

and L. W. Niiller.

19. Theory of Stochastic Processes. J. Wiley and Sons, 1965. D. R. Cox

and H. D. Miller.

20. Stochastic Service Systems . J. Wiley and Sons, 1962. J. Riordan.

21. Some Vistas of Modem Mathematics . U.of Kentucky Press, 1968.

R. Bellman.

22. Theory of Linear Economic Models . McGraw-Hill, 1960. D. Gale.

23. Priority Rules for Runway Use . Operations Research 12 (1964), pp. 941-

950. G. Pestalozzi.

24. Some Problems in Traffic Delay . Operations Research (1962), pp. 74-104.

G. H. Weiss and A. A. Maradudin.

25. Distribution of Gaps and Blocks in a Traffic Stream. Operations

Research 10 (1962), pp. 197-217. R. M. Oliver.

26. Waiting for a Gap in Traffic. Berkeley Operations Research Center

Report, 6/15/61. W. S. Jewell.

27. Aiiport Runway Capacity and Delays , I . Berkeley Operations Research

Center Report, 8/28/62. R. M. Oliver.

- 91 -
APPENDIX A: PROOF OF EQUATION {2.6'f

In this appendix we give a relatively elementary proof (using only

Laplace transfoms) of equation (2.6) of the main text^^^. The situation

being analyzed involves a sequence 1^,1^,... of random interservice times.

The probability distribution of each I^ is assumed (a deliberate over- simplification) to depend only on the type of customer , but this

type is in tirrn picked from a probability distribution (given by the

p, ’s) which is the same f#r all n . Thus, if I has cumulative

i n

distribution f\mction H. when C is of type i , then its unconditional

probability distribution is H = particular its (unconditional)

average, which here will be denoted ^ , is given by equation (2.5) of

the text.

The formula to be proved is

lim^ ^E[N(T)/T] = 1/^ , (A.l)

where the random variable N(T) , the number of customers served during

the period [0,T] , is defined by

^1 **• h{ T) ••• Vt) ^ Vt)+i •

By the definition of "average value" for a discrete random variable N(T) ,

E[N(T)] = m Prob[N(T) = m]

= 1^=0 ^ [Pt‘ob{N(T) > m] - Rrob{N(T) > m+l}]

(l) The treatment is formal, omitting some technical details required

for mathematical rigor.

- 92 -
" ^=o “ ^o^{N(T) > m}

- 1^=0 C(m+1)-1] ProbfN(T) > m+1}

= m Prob fN(T) > m}

- (m+l) Prob{N(T) > m+l}

+ ProbfN(T) > m+l} ,

Replacing the "diaininy variable" m in the first sum by m+l shows

that the second sum cancels the first except for the latter's term

for m = 0 ^ which is zero anyway. Maicing ^he same replacement in

the third term, we are left with

E[N(T)] = Prob{N(T) > m] . (A. 3)

Now by (a. 2), N(t) > m holds if and only if

I + . . . + I < T . (a. 4) 1 m —

Since are independent random variables, with cumulative

1 m

distribution function H , the distribution function of the sum

I + ... + I is the m-fold convolution H of H . Thus (A. 3) becomes

1 m

EfN(T)] = x^^^(h'^)(T) . (A. 5)

- 93 -
We nov take Laplace transforms of both sides of eq^uatlon (A. 5)*

Let E(s) and H(s) denote the Laplace transforms of E[N(T)] and

H(T) respectively. Since the Laplace transform converts convolutions

into products, the transform of function (H^)(T)' is the power

[H(s)]°^ . Hence

E(s) =

2^,3^ [1(8)]“ .

( 2 ) The right-hand side is a geometric series^ which can be summed explicitly

yielding

E(s) = H(s)/[1-H(s>] . (A.6)

Next, expand H(s) in a power series about s=0:

H(s) = H(0) + sH'(o) + o(s) (A.7)

where "o(s)” here represents terms which vanish at least as fast as

2

s as s - 0 . Since

H(s) = J** exp(-sT)dH(T) , o

we have

H(0) = dH(T) = 1

o

(because H is a cumulative distribution function), and

so that

dH/ds = - T exp(-sT)dH(T)

o

H’(0) = - T dH(T) = - ^

. o

C2) Convergent for > o, since the exponential is < 1 in the integral

expression for H(s) given below.

(31 Converger-ce is needed only near s = 0.
Substituting these results into (A. 6) yields (4)

E(s) = [l-fiB+o(s)]/[^s+o(s)]

= (l/^is) + o(s) .

(A.8)

Nov the "Tauberian theory" of the Laplace transform penoits one to

infer the limiting behavior as T - ® of the original function, here

ErH(T)] , from the limiting behavior as s -* 0 of its Laplace transform,

here E(s). Since E(s) behaves like l/|is as s - 0 , it follows^^^

that E[N(T)] behaves like T/j^ as T -• <» . From this the desired

result (A.I) follows.

The preceding material has been presented to make it conveniently

available to interested readers who have not previously encountered ±t;

it is not an original contribution.

^"^^Here different appearances of '’o(s)" can refer to different functions of

s with the property described on p.94.

•^^^See [10], p.l92.

- 95 -
APPENDIX B: JUSTIFICATION OF EQUATION (2.7f

In this appendix we provide background material leading up to

equation (2.7) of the main text. Recall that the runway is considered

to be in "state i" during the interservice interval of a customer of type i.

L et

p^j

= Prob{C^_|_^ is of type J , given that (B.l)

is of type i] .

Then the sequence of types of arriving customers constitutes a type of

random process known as a Markov chain, with (p ) as its table of

J transition probabilities . A process such as those to be considered

here admits a unique set of positive numbers x. such that

= X ^^i%J -J

Si^i = 1 >

(all J) , (B.2)

and these numbers have a number of properties^^^ permitting one to regard

x^ as the long-term probability of being in state i .

Our situation has still more mathematical structure, however. This

involves the sequence of interservice times Ij^. If customers C and Cn+1

are of respective types i and j , then is a random variable

whose cumulative distribution function will be denoted H ^ and whose

average value will be denoted T

ij

It will be convenient to set

which can be interpreted as the average interservice time for a customer

of type i.

‘^^See [11].

- 96 -
Our concern is vith the rajadom vajriable N(t) . To study it, it

is convenient to introduce the "initial conditions"

a^ = Proh{C^ is of type i] (B.4)

and the auxiliary random variables

N^j(T) = number of type j customers served during

given that is of type i .

Then the average number of customers of all types served during [0,T] , given that was of type i , is

E[r/ij(T)3 = i:jE[n.j(t)] .

Averaging over the possible types of , we find that the (unconditional)

average value of the total number of customers served during [0,T1, i.e.

of N(T) , is

(B.5)

To apply this, we first obtain a fonnula for E[N^j(t)] . Consider,

as a random variable, the time from the initiation of service to a type

customer ^ the next moment at which service to a type J customer is

rial started. ^ Let 2... denote the average value of this random variable.

( 2 ) ( 2 ) and its second moment. Both 1 . . and appear in the formula

13 ij 33

for EfN^^CT)], and so we must be able to evaluate them.

It can be shown ([12], pp. 132-3, eqs. (2.7-8)) that

^3 3 "

(la)Other customers of types different from j may have intervened.

- 97 -
vhere the x's and T 's are those of (B.2) and (B.3)^ and that

iC

+ T (B.7)

If the x's and T 's are knovn, then the I can be found from

Jo

(B.6); for each j , (B-7) gives a set of linear equations which can

in general be solved for the / with i^j . ^d

was the average value of a random variable with distribution

,( 2 ) ^i

Then ( [12],p.l3i^, eq.(2.10)).

denote the second moment of such a random variable.

Finally, the formula for E[N (T)] is given, in terms of i

( and the X. .'s, by^ ' ([12], p.l37^ Theorem 2.10)

-^d

(B.8)

(2)

JJ

EfN.j(T)] = T/ijj . alf- . 0 ( 1 ) (B.9)

where "o(l)" denotes a function of T which tends to 0 as T - ® . With the abbreviation

B. . ij

2

JJ

^ (B.IO)

this yields

E[N^j(T)/T] = 1/ijj + B^j/T + o(l/T) (B.ll)

(2) The formula in [12] contains an extra term allowing for certain

periodic phenomena not relevant here.

- 98 -
where "o(i/t)" denotes a function of T which tends to 0 , as T -* 00 ^ so fast that o(i/t)/(1/T) -> 0 .

Substitution into (B.5)^ and division by T , leads to

E[N(T)/T] =

Now set

B V. .a. B.

1 iJ (B.13)

use the fact ~ (B*2) and (B.6) to substitute for

1. in (b. 12). The result is

E[N(T)/T] = + B/T + o(l/T) . (b.14)

This is the main result. It asserts that

- »

ErH(T)/T] = 1/r^Xj^T^ , (B.l?)

and that for large T , since B/T » o(i/T), the error in replacing

E[N(T)/T] by the limit is essentially B/T . Using (B.3)^ we can rewrite (B.I5 ) as

- »

e[h(t)/t] = i/r^jX^p^jT.

j

. (B.16)

In the case of concern to us, the probability that a given customer

is of type J is assumed to depend only on the general population mix,

not on the type of the previous customer. Mathematically, we have P/s

J

- 99 -
rather than p^^'s. By direct substitution into equations (B.2), we find that in this case the x's are given by the p's themselves.

Thus (B.16) becomes

^ „

Eri»(T)/T] = l/r^jP.PjT^j , (B.17)

reproducing eq.(2.7) of the main text.

In this case we also have = Pj^ ^ so that (B.I3) becomes

Using (B.IO) and the fact that = 1 , we obtain

(o') o (B.18)

Since all \ possible to simplify (b. 18) considerably

further using (b. 6-8), but there has not been time to pursue this

possibility for getting sharper information on the error term In (B.14).

So far the discussion has dealt with the behavior of average value

E[N(T)/T] for large T . We are also concerned, however, with the behavior

of the variance Var[N(T)/T] of this random variable; see "QUESTION 3"

in subsection 2.4. For this purpose, set

N^(T) = SjNij(T) (B.I9 )

and observe that

Var[N(T)] = r^a^Var[R^(T) ]

. (B.20)

- 100 -
By (B.19),

Var[MjT)l = (t) , N^^(T)] . (B.21)

(q) We employ the results of* [l^]^p.9* These results are given

explicitly only for the case corresponding to the presence of just tvo

customer types, but it is stated in [l4] that the more general case displays the same behavior. Specifically, there exist constants c such that

- 0= = =jk- (B.22)

It follows from (b. 21) that

»

fVar[H^(T)]/T) = = 0 . (B.23)

From (B.20) and the fact r^^a^ = 1 , we have

lim^ {Var[N(T)]/T} = c . (B.24)

Since Var[N(T)/T] = (l/T^)Var [N(T) ] , we see that for large T , Var[N(T)/T] behaves like c/T . Further study may permit estimation

of c .

(3) We are grateful to Professor R. Fyke (University of Washington)

for directing us to this unpublished report.

- 101 -
APPENDIX C. ON SEQUENCING OF ACCEPTANCES

The biilk of Appendix B dealt with a situation more general

than that treated In the main text. Specifically, the probability

distribution of the "next customer type" was permitted to depend on the current customer type, rather than only the customer mix. Thus we had

p^j

= Erob Is of type j , given that (C.l)

C^ Is of type 1} .

The resultant formula for capacity (equation (B.I7 )) was found to be

C 1/lj^jXiPijTij (C.2)

where the positive x^ give the unique solution of the system

= Xj (all j) ,

^^1 = 1 *

(C.3)

Only after this derivation. In Appendix B , was the specialization

p . . = p. made. Its Implication Is that no attempt at sequencing customers

Ij J of different types Is made. With appropriate changes In technology and

operating procedures, however, such sequencing might be worth consideration.

We wish therefore to present one formulation (but not a solutlon|) of a type of optimal sequencing problem which may merit study In such a context.

The "Independent" variables of this optimization are the of

(c.l), which represent the control policy. The auxiliary variables are

the x^'s. The objective Is to maximize the capacity C as given by

(C.2), or equivalently, to

- 102 -
minimize . (c.k)

The variables must satisfy three sets of constraints. One

consists of the relations (C.3)^ which determine the x's in terms

of the p^j's. A second set consists of

Plj>0 (all 1,J), (C.5)

^ (C-6)

= Pj (all J). (C.7)

expressing the requirement that the p. .'s do constitute a set of

1 <3 transition probabilities" compatible with the given population mix

prescribed by the p 's. The third set, which cannot be written in

J closed form, restrain the to be such as to admit a unique

and positive solution (x's) to the system (C.3) of linear equations.

It is apparent, by comparison of (C.3) and (C.?), that all x^=p^

so that (C.4) becomes

minimize T^jP. j (p^T. ^ ) . (C.8)

Thus, ^ the above-mentioned third set of constraints could be ignored, we would have a linear programming problem of a special form called the

generalized transportation problem, which has known solution methods

more computationally effective than those for linear programs in general^^^

However, it is not apparent that every (or even any) optima.! solution

of the generalized transportation problem will satisfy the third set of

constraints, so that further investigation is needed. Moreover, a full

mathematical formulation of the sequencing problem should include some representation of the costs of sequencing control (both monetary, and

in delays to some customers) as well as its benefits (in increasing capacity).

(l) See Chapter 10 of [13]*

- 103 -
ADDENDUM: The uniquely reversible change of variables, from

^ij ^ij

~ ^i^ij * transforms the generalized transportation problem

into

minimize Z- . T. . y.

11 11 ^11

subject to

y. . > 0

^11 - (all i,j).

z . y. . = p.

1 ^11 ^1 (all i).

z . y. . = p 1 ^11

3 (all j).

The non-zero coefficients in the constraints are now all unity, i.e.,

the problem has been transformed into an "ordinary” transportation problem

for which still simpler and more readily available solution algorithms

exist. Here plays the role of a "unit cost" per pair of consecutive

landings consisting of a type i aircraft followed by one of type j; y.

is the relative frequency of such landing-pairs among al

consecutive

landing-pairs

- 104 -
APPENDIX D: MORE ON SEQUENCING

At present, landings at runways are typically handled on a firstcome-f irst-served basis. Technological advances, however, may permit

enhancing the capacity of this and other ATC System levels by adopting

some more sophisticated "sequencing” procedure.” This notion receives

considerable attention in [4], with its explicit recognition of "selection

algorithms" (p.64) and its careful distinction (pp. 68-69) between (i)

"maximum capacity" and (ii) the smaller "normative capacity" achievable

2 / by a first-come-first-served policy.”

The mathematical-logical justification for useful sequencing procedures

are likely to resemble or be based upon material in the standard references

on this field, e.g. [17] and [18]. Appendix C’s content, while closely

related to the body of the report, stands somewhat apart from the class of

problems typically considered in this body of technical literature. The

material that follows is much more in the tradition of "sequencing" research,

but makes relatively little contact with the balance of our text despite

having been directly stimulated by the present study. This is quite

understandable, since our report’s capacity concept as presently developed

is a "normative capacity" (see the last paragraph) rather than one defined

so as to allow for the potential benefits of deliberate deviations from

"first-come-first-served .

—^Conversely, the desire to achieve such a capability may influence the

planning of technological R. and D.

2 / —In this connection, we may note the (non-technical) lead paper in [17],

describing why operating staff are often practically unaware that

sequencing problems are significant in their activities.

- 105 -
We will be concerned with two of the many possible mathematical

formulations of the following probJ.em: If a facility can serve only

one customer at a time, in what order should it accept the customers

presented to it? The word "should” will be Interpreted here as referring

to some "penalty function" to be minimized. This function is assumed

to be the sum of penalties (possibly zero) relating to individual customers

and the times at which they are served.

An important question in the formulation of such mathematical models

is the nature of the information available to the "sequencer" how

much is known about the customers, and when is this knowledge available?

We shall deal here only with the simplest and most favorable case: before

the first arrival of a customer, all relevant information about all

customers (arrival time, length of service required, penalty-related data)

is assumed known, exactly rather than probabilistically. The natural

directions for generalization are obvious.

In the first formulation, time is treated as divided into discrete

equal- length periods. The model involves a set of decision variables.

X = 1 if customer n begins service in time period t,

nt

X =0 otherwise,

nt

There are three sets of constraints. The first ,

= 0 or 1 (all n, t) , (D.l)

expresses the dichotomous nature of each decision. The second ,

Z X =1 (all n) (D.2)

t nt

- 106 -
6xpress6s the fact that each customer (n) must begin receiving service

in one and only one of the time periods

.

The third set of constraints involves a set of problem data, namely

d(n) = required duration of service to customer n.

Consider any time period t. If customer n begins service in time period

s, this will tie up the facility during period t if and only if s _< t <

s + d(n) - 1, or equivalently

max (1, t + 1 - d(n)) = r(n,t) < s < t.

Thus the requirement that the facility can be used by at most one customer

at a time during any period (t) takes the form

2 X <1. (D.3) n s=r(n,tj ns —

Since we are interested in "good" sequencing of customers, the

mathematical model is naturally an optimizing one. Specifically, we seek

to

minimize (D.4)

subject to the constraints (D.l), (D.2), (D.3) given above. In the

situations we have in mind, for fixed n the "penalty" coefficient c^^ is 0

for small values of t (customer n is being served early enough), but then

1 increases with t (lateness penalties). This is however not essential; if

for example e(n) is the earliest possible time at which customer n can be

available for service, then this condition can be expressed by

Cj^t ^ ^ e(n),

3 / —For some of what follows, d(n) can be generalized to d(n,t), l.e. to

service durations which depend on when service begins. Each d(n) is

assumed integral.

- 107 -
though an attractive alternative is an extra constraint

^t < e(n)

X

nt

0 (all n)

What we have now is an integer linear programming problem (ILP) in

the constrained variables Its mathematical structure is sufficiently

special that it might admit an especially efficient solution method

(further study of this possibility is indicated), but for the moment the

only apparent solution methods are those used for ILP’s in general. For

problems involving moderate to large numbers of variables and/or constraints,

these methods are somewhat unpredictable in their ability to reach a

solution in a reasonable amount of computer time. If we deal either with

numerous customers (many n-values), or with a fine subdivision of the

time scale (many t-values), we will in fact have a large problem. Therefore

some experimentation, to see how the methods behave for this particular

class of problems, is called for.

This difficulty would not arise if we had a "continuous" rather than

an "integer" linear programming problem, since available solution methods

for the former category can handle quite large problems without strain.

Some important classes of ILP's have the property that their continuous

analogs possess optimal solutions in which all variables have integer

values. The pleasant consequence is that the more reliable solution

methods for the continuous case can be used to solve such ILP's. We

therefore Investigated whether the problems at hand might fall into this

category. Unfortunately the answer turned out to be negative , as is shown

4/ by the counter-example given in Tables E.l - E.2 and discussed below.”

4/ ~ Simpler counter-examples might exist.

- 108 -
The example involves 4 customers and 10 time periods. Table E.l

the associated penalty coefficients 5 is the entry in row

n and column t. Customer n is assumed to require n time periods of

service, i.e. d(n) = n.

We first show that the minimum value for the ILP is >_ 2. Note that

there are 10 time periods in which to accomplish d(n) = 10 units of

service; hence the facility can never be idle. To achieve a cost < 2,

at most one of the following four statements can be violated:

(a) Customer 1 begins service before period 10, hence ends it

before period 10 .

(b) Customer 2 begins service before period 6 , hence ends it before

period 7.

(c) Customer 3 begins service before period 7, hence ends it before

period 9

.

(d) Customer 4 begins service before period 6 , hence ends it before

period 9

.

If all but possibly (a) are true, at most customer I's single unit of

service is left to complete with periods 9 and 10 still left, leading to

an idle period. If all but possibly (b) are true, then customers 3 and 4

have received their 7 units of service during periods 1-8; if customer 1

used the remaining one of these 8 periods (it might have been in period

1 or 4 or 5 or 8 ) then customer 2 must begin in period 9, with a cost of

at least 7, while if customer 2 started service in period 8 (the only

other possibility), the cost is at least 4.

- 109 -
Table D. 1 : Penalty Costs (c^^) for Sequencing Example

t1 2 3 4 5 6 7 8 9 10

n = 1 0 0 0 0 0 0 0 0 0 1

2 0 0 0 0 0 1 2 4 7 11

3 0 0 0 0 0 0 1 2 4 7

4 0 0 0 0 0 1 2 4 7 11

Table p.2 : Fractional Solution Reducing Total Penalty to 5/3

n = 1 0 0 0 0 0 0 0 0 1/3 2/3

2 1/3 0 1/3 0 1/3 0 0 0 0 0

3 1/3 0 0 0 0 0 1/3 1/3 0 0

4 1/3 0 0 1/3 1/3 0 0 0 0 0

- 110 -
Now suppose all but possibly (c) are true. Then all customers

except the third have completed service before period 10 , hence

customer 3 receives service during this last period (since the facility

cannot be idle), hence customer 3 began service in period 8 yielding a

cost of at least 2. Similarly, if all but possibly (d) are true, then

period 10 finds customer 4 still receiving service which must have begun

in period 7, yielding a cost of at least 2.

The minimum value for the ILP is actually 2, as is seen for example

by setting

X

31 = X2 ^ = =1 (x

^ = 0 otherwise)

.

To prove the point, it now suffices to exhibit a (necessarily fractional)

solution to the continuous analog which achieves a value < 2. The

continuous analog is obtained by replacing (D.l) with 0 _< x^^ ^ 1, and a

fractional solution to it which yields a total penalty 5/3 is given in

Table E.2.

We turn now to the second formulation. It involves much more restrictive

assumptions, but yields quite specific instructions as to an optimal

sequencing policy. The restrictive assumptions are that all customers

have the same priority (i.e., penalty) function, differing only in timing,

and that all customers require the same duration of service. Further

research, to see what conclusions can be derived when these conditions are

weakened, seems worthwhile.

Specifically, we have a known availability time a^ for each customer

n, and also a non-decreasing continuous function P with P(0) = 0. The

- 111 -
penalty associated with serving customer n at time t is taken as

Pn (t) = P (t - a^).

Let t be the (variable) time at which customer n is served, and let t n

-> be the vector with n-th component t^. Then the problem is to choose t

so as to minimize the total penalty

S(t) = E p (t ) = E P(t - a ), (D.5) n n n n n n

subject to three sets of constraints. The first set,

t > a (all n) , (D. 6) n — n

simply asserts that a customer cannot be served before he is available.

The second asserts that the facility can serve at most one customer at a

time; in terms of the common duration (d) of each customer’s service,

this reads

1 t - t 1 > d (m 4 n) • (D.7)

' m n —

The third set expresses the requirement that all customers receive and

complete service during a specified time interval, say [0, T]

0<t <t +d<T (all n). (D.8) — n n —

Deferring the question of whether constraints (D.6) through (D.8)

are consistent , we assume consistency and discuss the problem of finding

an optimal (S-minimizing) service schedule. The solution will depend

on a further assumption, namely that the function P is convex (i.e., is

never under-estimated by linear interpolation). This has the significance

that increasing delay carries non-decreasing marginal penalties.

Let N be the number of customers. It is convenient to index them in

order of availability, so that

- 112 -
(D.9)

In terms of this ordering, we claim that the following simple rules

yield an optimal policy:

Rule 1 : At time a^, serve customer 1.

Rule 2 : At the conclusion of a service, if there are customers

available, serve the one with lowest index. If there are none, but

not all customers are yet served, serve the first one to become

available (breaking ties, say, by giving preference to lower indices).

This "f irst-come-f irst-served" policy serves customer 1 at time

* t* = a^ , and the other customers at times t determined recursively by

i i n

t* , = max {a t* + d}. (D.IO) n+1 n+1 n

To prove that it is optimal, we begin with any optimal schedule t^, and

show that t can be transformed into the schedule t* given by the two rules

in a finite sequence of steps none of which increases the value of the

function S to be minimized. Each step operates on a "current schedule" s

(t initially, and then the output of the preceding step) to produce a new

schedule which will obviously satisfy all the constraints (D.6-8) because

the current schedule did.

The steps are of two types. The first type simply "tightens up"

schedule eliminating slack periods but not changing the order in which

customers are served. In terms of the current schedule s with its

components s , this step can be formally described as follows:

- 113 -
Find the smallest s such that s > a and either (a) s = min s , n n n n m m

or (b) s > min* {s + d} where the "min*” is a minimum taken over

n m

all m such that smm< s . In situation (a) , change s to a . In n n situation (b) , change s to min* {s + d}. n m

When no such s^ exists, so that no further application of a Type 1

step is possible, the schedule has been "tightened up" as much as possible

subject to the current order of service. In applying the second kind

of step, the current schedule s is assumed to have been already tightened

by Type 1 steps.

A Type 2 step rectifies a situation in which service is begun to a

customer other than the one with smallest a^ among those available at the

time. The formal description is as follows:

Find the smallest s such that s > s for some m < n. With such

n m n

->•1 an m, form a new schedule s from s by interchanging components s^ and s^.

Note that m < n implies a < a < s , so that in the new schedule

m — n — n customer m is not served before he is available. To show that the

objective function is not increased, we calculate

AS = S (s) - S(s^)

= [p-nn(s„) + mm(s^)] - [P„nm(s^) + P^mn(s„)]

= [P (s - a ) + P (s - a )] - [P (s - a ) + P (s - a )] . n n m m m n n m

Setting

X = s a n n

y

= s s > 0

m n

z = a a > 0 n m — y

- 114 -
we have

AS = [P (x) + P (x + y + z)] - [P (x + y) +P (x + z)]

.

(D.ll)

Since function P is never under-estimated by linear interpolation,

P (x + y) = P ( 2 . X + y . (x + y + z)

)

y+z y+z ^ z P (x) + y P (x + y + z)

,

y+z y+z

and similarly

P (x + z) y P (x) + z P (x + y + z)

.

y+z y+z

Adding the last two inequalities gives

P (x + y) + P (x + z) _< P (x) + P (x + y + z) ,

which by comparison with (D.ll) shows that AS ^ 0 as desired.

The over-all process of passing from t to t* can now be described

as follows: Apply Type 1 steps as long as possible, then a Type 2 step,

then Type 1 steps as long as possible, etc. The process must terminate,

since each sequence of consecutive Type 1 steps contains no more than

N steps, while each Type 2 step brings the order in which customers

are served closer (i.e., fewer "inversions") to the order {1, 2, . . . ,N}.

This order is clearly achieved by the last Type 2 step. Note that the

process is a conceptual one used only for proof purposes; for calculating

the optimal schedule one sets t* = a^ and employs the recursive

formula (D.IO).

It only remains to consider whether the constraints (D.6-8) are

consistent. If they are satisfied by at least one t ( optimal or not),

the above construction will lead from t to t*, which must also satisfy

- 115 -
the constraints; conversely the constraints are surely consistent if

t* satisfies them. Thus consistency can be checked by computing t*

as described above, and seeing whether it obeys the constraints; this

is simply a question of whether t* + d _< T.

In conclusion, we emphasize that the two specific mathematical

models presented above are illustrative and exploratory; the directions

in which they might have to be modified and generalized to be useful

in an ATC setting are not yet known to us.

- 116 -
APPENDIX E : A DELAY CONCEPT

As emphasized in subsection 2.2, this report’s capacity concept

differs in a basic way from that in the Airport Capacity Handbook , in

that it is not bound up with some notion of a tolerable level of "delay."

We can be somewhat more explicit. In a textbook-type queuing situation,

there is no real problem in defining "delay." A customer arrives and is

served at definite times (in general, known only probabilistically in advance);

his delay is the duration of the period between these two events.

The "continuous demand" assumption of Section 2, however, only requires

that a customer be on hand whenever the facility is no longer tied up by

the previous customer. Nothing is said about how much earlier that "onhand" customer was in fact available to be served had no others been

ahead of him. One of the essential ingredients for discussing "delay,"

namely a specification of when customers arrive , is therefore missing.

This separation of capacity from delay has its advantages. It

permits capacity-focused studies to proceed without bogging down in

questions as to what delay levels are "acceptable." Even more significantly,

it avoids the complications stemming from the fact that the ATC System

and its elements are far from "textbook" in their complexity, and do not

provide clear and compellingly "right" ways to define "delay." These

complications are communicated very vividly in [4]

;

some of them can be

roughly described as follows:

(a) It does not take much sophistication to suggest

that delay, as a "quality of service" concept, might well be

measured not from the moment of a customer's "arrival" (which

might by chance be early), but rather from the moment of some

- 117 -
scheduled or anticipated initiation of service. How should

these be defined in an ATC setting? Flight plans are altered

en route ; should "scheduled or anticipated" refer to the

expectations at takeoff, or after the last fight-plan updating

prior to reaching the terminal area, or at some intermediate time?

(b) During its flight, an aircraft is served by a number of elements

of the ATC System. These may be regarded as service facilities,

but in some cases they can best "serve" a customer by increasing

his "local" delay, rather than by passing him as promptly as

possible into a soon-to-dissipate high-congestion situation

further "down stream. How should a delay concept be formulated

so as to reflect such a multi-facility situation, with its overtones of potential conflict of interest among facilities in case

each is rated solely by the delays which customers incur at its

hands?

(c) It is an economic truism that in situations of competition for

a "scarce resource" (e.g. airspace, or prime treatment from the

ATC System), better treatment for some of the competing entities

normally implies worse tceatment for others. Thus for example

"total system-wide delay" is not in itself adequate as a

measure of performance; a proper analysis should also consider

the equitabllity of the way in which this total is distributed

among various "customers." How should a delay concept be defined

so as to capture this aspect of the situation?

That is, performing some portion of a flight somewhat less quickly than

possible may be preferable if it avoids a severe slowdown later on.

- 118 -
We have just given good reasons why the absence of delay considerations

from our capacity concept might be considered cause for rejoicing. Still,

"delay" is so important in the ATC context as a measure of service quality,

as to compel at least one exploration of what type of demand concept might

reasonably be matched with the capacity concept developed in the main text.

The setting is that of Sections 3-5, namely a single runway (plus its

final^approach path airspace) serving a stream of landings. As noted

earlier, a treatment of delay will require some representation of the times

of customers* "arrivals." Arrivals where ? We will make two attempts at

answering this question, find that both run afoul of our need to preserve

mathematical simplicity in this initial exploration of delay concepts,

and then settle for a compromise.

(a) From the view of actual or imagined risk it is at least plausible

that "total time airborne" is what's to be minimized for a

given flight. This suggests that a customer's "arrival," the

moment from which his "delay" is measured in the standard queuing

language, should be regarded as occurring as early as possible

in the flight, say as soon as "enroute" status is reached. The

trouble with this approach becomes apparent if we consider two

flights departing for New York, one from Los Angeles at 10 a.m.

and the second from Boston at 10:30. The Los Angeles flight

would be counted as the first of the two to "arrive" at New York

for service, though of course the second would typically be served

first at New York. Thus the "far-out arrivals" approach would

lead to severe violation of the assumption of a "first-comef irs t-served" service policy, and the mathematical analysis would

become much more difficult than is tolerable at this stage of

our study. -119-
(b) Since our capacity concept is firmly associated with the runway

and its immediate vicinity, there is a natural inclination

(opposite to that in (a)) to define "arrival" so that the same

will be true of the associated delay concept. A natural "close-in"

definition would be that of arrival in the holding pattern.

The flow of entries to the holding pattern, however, may be too

well regulated by previous ATC controls to permit description

as "Poisson arrivals" and this assumption that the Poisson

distribution can be used is also important to the simplicity of

the analysis.

In what follows, therefore, we shall regard "arrival" as occurring

at a suitable point intermediate between the two extremes described above:

sufficiently "far out" from the runway that the aircrafts' diverse origins

and takeoff times combine to yield a pattern of arrival times that is

acceptably Poisson-like, yet sufficiently "close in" that the first-comefirst-served assumption is a reasonable approximation. Perhaps the

"terminal boundary" will do. In a longer-term study with less stress on

initial mathematical simplicity, one of the two extremes might possibly

prove preferable to this compromise.

We next consider how to express, for present purposes, the assumption of

continuous demand used in the capacity concept. Its statement there

(subsection 2.2) was that at the end of each service period, another

customer is invariably at hand. This condition cannot be achieved in a

setting of Poisson arrivals; to retain the simplicity of the Poisson

distribution, some other mathematical representation of the same intuitive

- 120 -
1

idea is needed. For this purpose, consider the following random

variable: The time, from a completion of a service period only to find

no "next customer" at hand, to the next occurrence of such a situation.

Our modified statement of the "continuous demand" assumption is that the

r lal

average value of this random variable be infinite ^ ^

.

The mathematical formulation of this requirement proceeds as follows.

We are dealing with a Poisson-fed queue at a facility operating under a

first-come-first-served policy. Let X denote the average arrival rate.

C2) For simplicity, it will be assumed that the "service time" for an aircraft

has a probability distribution depending on the type of that aircraft only

(and not on the type of the next aircraft) . If 1/y^ denotes the average

service time for type i aircraft, and p^ is the proportion of type i

aircraft in the customer "mix," then

1/y = p^ (1/y^) (E-1)

is the mean service time. The requirement stated at the end of the

last paragraph can be expressed (see [19], pp. 115-117) as X = y.

Unfortunately, the delay concept under these conditions is quite degenerate:

since average delay is given ([9], p. 345) by

D - {X/2y(y - X)} (1 + s^y^) (E.2)

where s is the variance of service time, setting X = y leads to an infinite

average delay.

Roughly speaking, instead of assuming such situations cannot happen

at all, we assume that on the average their occurrences are "infinitely

far apart;’’ this seems to express the same intuitive idea.

Here "service time" has the same meaning as did "holding time" in

Section 5.

- 121 -
This outcome is neither physically reasonable (in the "runway"

context) nor useful for our alms. We are therefore led to reexamine

the physical reasonableness of the assumptions underlying the queuing

model. The weak point appears to be a hypothesis so customary that it

was not even stated explicitly before: that there is no bound to the

length of the queue.

In fact, there must be a finite practical limit to what the airspace

and ATC capabilities of the teminal area can handle. We interpret this

as a finite value of

h = maximum queue length

= maximum number of A/C which can be held in the terminal area.

When the queue is "full" (has h members), new arrivals are abstractly

considered to be "turned away"; we will not try here to think through

3 / how best to interpret this phrase in terms of diversion to other

terminals and/or restrictions on original schedules. What seems worth

noting , for its possible analogy-value in other studies of capacity and

delay in an ATC setting, is the way in which the study of "delay" at

one level (the runway and final-approach airspace) of the ATC System

hierarchy has driven us to bring in the "capacity" at the next higher

level (the terminal area)

.

With a finite maximum queue length (h) , the average delay is at most

h times the average service-time, and so is certainly finite. However,

17

In the runway situation it typically represents a balking by the

customer at an unacceptable wait, not a rejection by the facility.

- 122 -
the simple condition A = y no longer expresses exactly the previous

formulation of the condition of continuous demand. To analyze the

situation further, let p = A/y and let I be the long-term probability

that the "facility" is idle. In terms of h, p, we have

I = (1 - p)/(l - (E.2)

for the simple case of exponentially distributed service times ([20],

p.73, formula for p^ with n = 0) . As p increases to 1 (i.e., A increases

to I approaches the nonzero limit 1/ (h+1) .

In general, one would expect that making I 0 (to express "continuous

demand") would require taking p

-> «> (i.e., A

->- «>, for fixed y). And one

would further expect, in this case, that those customers who do join the

queue almost invariably find h-1 others ahead of them and thus must suffer

an average delay

D = (h - l)/y . (E.3)

This can be verified for exponential service times (and probably more

generally, though we have not yet tried to do so), using the results on

p.73 of [20]: The limiting probability that an arriving customer finds

the queue less than full and so enters it ((1-pj^) in the notation of the

reference), is

h h+1

(1 - p )/(l - p ) (E.4)

and the mean delay for all customers (assuming 0 delay for one not

4 /, joining the queue“ ) is

—^This does not really jibe with the language on p.73 of [20], but appears

to check with the mathematics; time was lacking to resolve this point

definitively.

- 123 -
I [1 - h p (E.5)

h-1 h 2 + (h - 1) p ]/(y p) (1 - p) .

The average delay for those customers who ^ join the queue is then

obtained as the quotient of (E.5) by (E.4), with equation (E.2) used

to substitute for I:

D = (1/y) p [1 - h p h+1

+ (h - 1) p^]/(l - p) (1 - p^). (E.6)

It can be shown that the limit of this quotient as p is indeed given

If the sole source of an aircraft’s "delay", once within the terminal

area, is that associated with queuing-up for use of the (sole) finalapproach airspace and runway, then our capacity measure C = y (see eq . (5.15))

so that equation (E.3) gives the compact expression

for average delay as a quotient of capacities at two levels of the ATC

system. It remains to see how generally this holds.

Much of the previous material has involved an attempt to force the

delay concept to include the "continuous demand" assumption already present

in the capacity concept. The necessity for such an inclusion seems less

evident for delay than for capacity, however. It may be best simply to

accept D as a function of p (i.e. of arrival rate X, for fixed y) , to be

calculated using equation (E.6). Interest would naturally concentrate on

situations with X > y. The relationship with our capacity concept would

enter via C = y, or some more general 1/y = (1/C) + <5 where 6 is an average

delay within the terminal area due to factors other than queuing for the

runway.

by (E.3).

D = (h - 1)/C (E.7)

- 124 -
Further study of these ideas would include developing the analog

of eq. (E.6) for more general service-time distributions, in particular

those implicit in the main text. Our study of this whole topic has

been so brief, however, that further thought may well reveal a superior

conceptual approach.

- 125 -
APPENDIX F: A SUPPDRTING SIMULATION

In Appendix B a somewhat complicated analysis was given of the

behavior, for large T, of the average and variance of the mean throughput

rate of a facility over a time period [0,T]. The "facility” of interest

here consists of a runway (and its final-approach path airspace) serving

a stream of IFR landings. As a check on the analysis, and (more

significantly) to discover what constitutes "large T' to which the

analytical results apply, a model to simulate the runway over a period

of time was desired.

A fast-time, Monte-Carlo simulation was developed, coded in R3 RTRAN

(a sample listing follows), and operated for selected data combinations

from subsection 4.1. In particular, each run was made for a specific

population mix, error distribution type and range of distribution.

Within each run the probability of violation (p^) , minimum separation

distance between aircraft (S) , and final approach distance (L) were

allowed to take on all values mentioned in Tables 4.1.1 and 4.1.3.

For each combination of p , S and L, 100 simulations were made of

a two hour period, with "the number of aircraft serviced so far" recorded

after each half hour. This yielded four arrays, with 100 sample points

in each. For each array, the mean and standard deviation (a) of the

sample points were calculated and printed. (Illustrative outputs for one

set of cases are given in Figure F.l.)

- 126 -
o

o

c

o

c

Figure F ! : Some Simulation Outputs

THE population MIX USED FOR THIS RUN WAS .fS-A .2-B .2-C .0-D .0-E

THE VALUE OF R USED IN THIS RUI'J WAS 30. SECONDS WITH A TRI ANGULAR DISTI

L S PM MEAN STD. MEAN STD. MEAN STD. MEAN STD.

DEV. DEV. DEV. DEV.

. 5 .5 1.0 1.0 1.5 1.5 2.0 2.0

b. di. • . u5 46.22 1.61 46.21 1.05 46.18 .82 46.19 . 66

b. 2. .U1 43.25 1.54 43.31 .97 43.33 .72 43.32 .54

b • 3. • u s 38.72 1.31 38.71 .87 38.70 .71 38.68 .58

b • 3. .01 36 . 64 1.27 36.53 .92 36.56 .72 36.60 .55

b • A. • 05 31.21 1.05 31.20 .75 31.20 . 56 31.21 .47

b. H, .01 29.97 1.00 29.90 .69 29.67 .53 29.87 .45

b • 2. . 0 5 45.23 1.63 45.23 1.13 45.21 .83 45.21 .71

a. 2. .01 42.49 1.41 42.46 .91 42.47 .77 42.45 . 66

t> • 3. . U 5 37 .75 1.39 37 . 57 .90 37.71 .77 37.87 .68

a. o • .01 35 . 62 1.40 35.68 .95 35.89 .72 35.91 . 56

a • 4. .05 30 . 73 1.13 30.75 .72 30.70 .53 30.73 .46

ti* 4. .01 29.13 1.10 29.16 .78 29.36 .67 29.40 .48

.0. 2 • .05 44.22 1.50 44.20 1.13 44.12 .87 44.14 .69

.0 • 2. .01 41.52 1.51 41.50 1.04 41.46 .75 41,52 .64

.0. ^ • .05 37.22 1.63 37.03 1.06 37.08 .90 37.07 .73

,0. 3, .01 35.18 1.44 35.20 .94 35.27 .73 35.22 .62

.0. 4. .05 30.32 1.16 30.31 .77 30.25 .64 30.21 .51

,u. 4. .01 29. 06 1.17 28.94 .78 28.99 .63 28.91 .48

- 127 -
I

The resulting sample means were used to verify that the analytical

results are indeed applicable to relatively short time periods; this is

discussed in subsection 4.3.

The resulting values of a were also used to see whether the time

periods in question were long enough for the theoretical result

c T, (F.l)

expressed in eq. (B.24), to take hold.

apply, then the quantity

/ 1/2

K = a/T

If this relationship did indeed

should be approximately constant (equal to c ) . Values of K for

T = 0.5, 1.0, 1.5 and 2.0 hrs . were calculated for several sets of the

simulation outputs in Figure F.l, and are shown in Table F.l. In that

1/2 Table, K(T) designates the value of a/T for the simulated time period

[0,T].

- 128-
TABLE F.l Simulation Approximations of K = c 1/2

L

h.

DISTN. K(.5) K(l) K(1.5) K(2)

8 3 .05 UNIFORM .42 1.19 1.15 1.07

8 3 .05 TRIANGULAR .35 .90 .95 .96

10 4 .01 UNIFORM .31 00 .93 .89

10 4 .01 TRIANGULAR .29 .78 .77 .68

6 3 .05 UNIFORM .41 1.15 1.05 1.08

6 3 .05 TRIANGULAR .33 .87 .87 .82

- 129 -
These results indicate that the value of K at 0.5 hrs. is quite

misleading as to its limiting value for large T; moreover the "approach

to the limit" is not evident for T = 1.5 hrs. It may occur by T = 2 hrs.,

but this cannot be ascertained without calculations for more values of T

between 1.5 and 2 hrs. Unfortunately time did not permit either such

intermediate calculations, or going on to higher values of T.

To avoid possible misunderstanding, we note that these findings do

not limit the usefulness of the analytical results; as noted in subsection

4.3, a is "small enough" for T = 0.5 hrs., and certainly for T = 1 hr.

The findings merely illuminate the theoretical question of the range of

validity for equation (F.l).

We conclude this Appendix with a listing (Figure F.2) of the simulation

program. This is preceded by a glossary of the symbols appearing in the

listing. There are some divergences from the notation in the main text,

namely

0T->R0T, L^X

S -> Y, r ^ SERV.

The listing is given for the triangular distribution case only.

- 130 -
Glossary of Simulation Terms

V(I) = Final approach velocity of i-th A/C, defined in knots but

internally converted to N.M./sec.

ROT (I) - Runway occupancy time of i-th A/C, in sec.

X(I) = Length of i-th chosen value of final approach path, in N.M.

Y(I) = Length of ith chosen value of minimum separation distance, in N.M.

SERV(I,J) = Tieup time for A/C of type i followed by A/C of type j , in sec

PV(I) = Probability of violation, with PV(1) = .05 and PV(2) = .01.

PMIX(I) = Percentage of the population mix of type i.

RANDNO(ARG^, ARG^) = A function generating a uniformly distributed pseudo

random number between 0 and .9999, using the power

residual method: RANDNO = Mod^^^^^ (3ARG-j^ + ARG2

)

BUFFER(I) = Added separation time corresponding to PV(I) in sec.

R = Half-range of the error distribution, in sec.

ISEEDl, ISEED2 = Seeds of random number generator.

TIMINT(I) = Endpoint of 1-th time period for which samples are to be taken

in sec. (1/2, 1, 1 1/2, 2 hrs.)

QTYLND(I,J) = Number of A/C which have landed before the end of the 1-th

period, in the j-th sample.

AVF(I) = Mean of 100 samples for i-th time period, in A/C per hour.

STDDEV(I) = Standard Deviation of 100 samples for i-th time period, in

A/C per hour.

IPER = Index of current period.

- 131 -
ITER = Index of current iteration.

ENDTIM = Endpoint of current time period, in sec.

NOLAND = Number of landings finished at current clock time.

TIME = Current clock time.

VAR = A triangularly distributed random variable with mean 0 and range 2R.

ACTIM = Time required to land current A/C.

- 132 -
Figure F .

2

: Listing of Simulation Program

DIMENSION V(5)tR0T(5).X(3)»Y(3) f°V(2)fSERV(5#5)»PMlX(5) »BUFFER( 2 )

1

»OTYLND( 4, 100) . A VG (4 ) t ST DDEV ( 4 > * TI MI NT ( 4 )t NUM3RS <100

)

DATA V/ 165. #150 135, fl 20 . 105,

/

1 tROT/59. #52. #45. #38. .3 1. / #R/30-/ . TI MI NT / 0 . 5 # 1 . 0 » 1 . 5 # 2 . 0

/

2 #PV/0, 05 #0.01/ . X/6. 0*8.0# 10 .0/ # Y /2 .0 . 3 . 0 #4 .0 /

3 #PMI X/0. 6,0.2# 0. 2. 0. 0# 0. 0/ # IS EE Dl /3 11 3/ # IS EE D 2 /2 15 7/

C THIS FUNCTION GIVES A RANDOM NUMBER BETWEEN 0 AND 1 C USING THE POWER RESIDUAL METHOD

RANDNO( I# J) ::H00 (3 *I *0 #1 0000 )/ 10000.

C WRITE (6 #780 )PMIX

780 FORMATtM THE POPULATION MIX USED FOR THIS RUN WAS •#F3.1**-A*#

1 F3. 1# *-8» #F3.1# #F3. 1# *-D* #F3. 1# *-E* / ) D0TT^275

1 PMIX(I)-PMIX(I-1) PMIX( I)

D02I=1#5

2 V (I )=V( I) /360a.Q

8UFFER( 1) -R*< 1 .0-SQRT <2 .0*PV( 1) n

BUFFERC 2) =R*(1 .0-SQRT (2 .0*PV( 2) ))

D0500IX=1 #3

D0500IY-1 #3

ITEST-0

OOlOIrl ,5

DOlOjrl #5

IF(V(I).GT.V(J)»S-X<IX)/V<J)-(X(IX)-Y(IY)J/V(I)

IF<V(I).LE.V(J))S-Y(IY)/V(J)

10 SERV( I# J) =AMAX1(R0T (I ) # S)

IF( ITEST.EQ.l ) WRITE< 6# 78 1) < t SERV (I #J )# Jr 1# 5) #I =1 #5 » 781 FORMAT! //5F10.3 ) D0450 IPVrl#2

D0440 ITER=1#1Q0

IPERrl

ENDTIMr 1800.

40 TIMErO.O

NOLANDrO

JTYPErO

51 RNONUMrRANDNO (ISEED 1# ISEED2 ) ISEEDlrRNDNUM*10000

ITYPErJTYPE

JTYPErO

50 JTYPErJTYPE+1

I IFiRNDNUM.GT.PMIX (JTY PE )) GO TO 50

IF< ITYPE.EQ.QJG0T05

1

52 RNONUMrRANDNO CISEEDl# ISEED2 }

I ISEED1=RNDNUM*10000

RNDNM2rRAN0N0 (ISEED 1# ISEED2

>

ISEED1=RNDNM2*10QOO

VAR=(RNDNUM4RN0NM2-1. )*R

ACTIMErSERV ( I TYPE #J TY PE » BUFF ER (I PV » VA

R

T IME=TIME ACTIME

IF< TIME.GE.ENDTIM JG0T06 0

NOLANDrNOLAND^l

60T051

I 60 OTYLNDf IPER #ITER) rNOL AND+ <EN0TIM^ ACTIME-T IME) /ACTIME

- 133 -
NOL ANO^NOLAND+1

IP£R=IPER+1

ENDTIM-ENDT IM+18QQ.

IF( IPER.LE.4)G0T051

440 CONTINUE

IF( ITEST. EQ.l > WRITE( 6f 78 2) GTYLNO

782 F0RMAT(4F20.3

)

D0448IPER-1 »4

SUM^O.O

00441 Iz 1, 100

441 SUM^SUM+OTYLND ( IPER *I

)

A VG (IPER) rsuM/lQQ

SUM ir 0.

D0442I=lf 100

442 SUMl-SUMl ( AVG { IPER )-0T YLND (I DERt I) )**2

SiGSQDr SUMl /lOO

443 S TOPE y{ IPER )- SORT (S IGSQD) _

DO 449 IPER=1*4~

AVG (IPER) :=A VG (IPER) /TIMINK IPER ) 4 49 S TO DE V( IPER ) = S TDD EV (IPER) /TIM INK IPER ) WRITE (6*7 89 )X(IX) *Y(IY) *PV(IPV) tBUFFER(IPV) * (TIMINT(I)*AVG(I)*

1

STDDEV (I )*I=1*4)

789 FORMAT(///* FOR THE CASE OF X-**F3.0** * Y=**F3.0** AND PV::»*F4.2

1 (WHICH MEANS A BUFFER OF **F3.0** ) THE RESULTS FOLLOW.*/

2 4(/20X #*FOR* *F5.2t * HOUR(S) THE MEAN ARRIVAL RATE WAS**F8.3*

3 • WITH A standard DEVIATION 0F**F6.3))

ITEST^O

450 CONTINUE

500 CONTINUE

STOP

END

- 134 -
APPENDIX G: EXTENSION TO A DUAL-USE RUNWAY

The capacity concept developed and applied in the body of this report

dealt with a facility serving a single stream of customers, e.g. a runway

serving a stream of IFR landings. An Important next step, in bringing

this approach nearer to practical usefulness, is clearly its extension to

the case of a runway serving both arrivals (landings) and departures (takeoffs) in a broader context, to facilities which serve a pair of customerstreams. The present appendix records our extremely tentative and

exploratory thoughts on this topic.

As in the single-use case, the initial problem is not that of deriving

a mathematical formula for a well-defined property of the runway. Rather,

it is that of deciding what intuitive notion we are seeking to express in

a quantitative way.^^^

It seems reasonable that the desired "capacity" should be descriptive

of something like "maximum throughput rate" for departures, arrivals, or

both, under some appropriate assumptions about the "service discipline"

which sets priorities for the processing of the two streams. And once

the "something like" and "some appropriate" are pinned down, it seems

likely that translation first into a rigorous description and then into

a mathematical formula can Indeed be carried out using suitable generalizations (which may present technical difficulties) of the techniques

employed in Sections 2 and 3. This technical development, involving the

various "separation rules" Involved, will not be taken up here; our aim

is to get a start on the "pinning down" of the basic ideas involved.

( 1 ) "It comes as a bit of a shock when one first realizes that the real

problem of science is not so much *what is the answer? ' as what is

the question?’" [21], p.89.

- 135 -
For a simple symbolic notation, let

A = throughput rate for arrivals

,

D = throughput rate for departures.

Going back to fundamentals, we can safely say that a major role of a

"capacity” concept is to distinguish those numerical pairs (A, D) which

tlie facility can turn out from those it isn’t capable of producing. This

can for example be accomplished (at least in principle) by listing the

"feasible" combinations (A, D) , or by giving an algebraic or geometric

description of the corresponding "feasible region" in the first quadrant

of a plane with (A, D) as Cartesian coordinates.

However, "capacity" suggests not only the notion of "feasible" (i.e.,

attainable), but also the notion of "feasible and unimprovable," as in

the qualifying adjective of "maximum throughput rate." To capture this

additional idea, we fix attention not on the whole feasible region, but

rather its "upper frontier," the set of feasible points (A, D) which would

become infeasible were either coordinate increased further. Economists, using

a similar diagram to discuss an economy or a plant which turns out two

( 2 ) products, term such points efficient ^ q we will therefore refer to the

upper frontier as the efficient curve for the facility ("production

possibility curve" would be the technical phrase) . Its points represent

combinations of throughput rates for landings and takeoffs respectively,

such that neither can be increased without detriment to the other; such

limitations arise because the same "scarce resource" (use of the runway

and its environs) is required in the production of both "goods."

^^^See [22], p. 307.

- 136 -
At this point our capacity concept involves not a single number

("scalar") but rather a vector with an A-component and a D-component; and

moreover not just one such two-dimensional vector, but rather the infinite

set of them which comprise the efficient curve. On the basis of our very

limited consideration to date of the dual-use case, it does not seem

possible to obtain a more definitive kind of "answer" without introducing

additional considerations or assumptions (see the next paragraph) . Thus

it presently appears that one of the analytical tasks for future work,

perhaps a principal one, is to determine the efficient curve. Its equation

might be written

E(A, D) = 0 (G.l)

or perhaps in the solved-for-D form

D = e(A) (G.2)

giving the maximum throughput rate for departures when A is the specified

throughput level for arrivals. The functions E in (G.l) and e in (G.2)

would contain as parameters the various quantities characterizing the

situation treated, e.g.: relative proportions of different aircraft

types among arrivals and departures; relevant speeds, lengths and times;

required separation distances and/or times; violation probability thresholds

and associated error distributions like those in the body of this report.

In search of possible clues to what sort of "additional considerations"

might be introduced to single out a definite point on the efficient curve,

we might consider how the economist treats the analogous problem when

considering a plant \diich can turn out two products in various quantity

- 137 -
combinations. His (quite natural) approach is to postulate that plant

management will elect to operate at that point on the efficient curve

which maximizes the function

P(A, D) = net profit from production and sale

of the output mix (A, D) . (G.3)

That is, P(A, D) is to be maximized subject to the constraint given by eq.

(G.l). Under the shnplifying assumptions that the two products are sold

in independent markets, and that production costs can be assigned

unambiguously to one or the other of the products (i.e., "joint costs" are

negligible) , the simpler functional form

P(A, D) = P^(A) + P2(D) (G.4)

results; with the further assumption of constant per unit profits in each

market (P^ and Pj^ , we have the further simplification

P(A, D) = P^ A + Pq D. (G.5)

Although "profit" is presumably not the appropriate criterion for our

runway situation, the last paragraph does suggest that the selection of a

single point from the efficient curve must involve the optimization of

some "scoring function" analogous to P(A, D) ; its optimized value might

then serve as a single "capacity number." There might be some inclination

to choose as scoring function the total operation rate A+D (summing components

is certainly one way to convert a vector into a scalar!), but no real

justification is presently apparent for this choice, especially since its

compatibility with the preference actually accorded arrivals is dubious.

Tsl ^ ^ ^We omit discussing the "more production is more profitable" condition

needed to ensure that the chosen operating point will be on the

efficient curve rather than interior to the feasible region.

- 138 -
A similar idea, which singles out a point on the efficient curve though

not yielding a scalar "capacity," is the minimization of some penalty

function over that curve. The sort of function which suggests itself most

naturally is the sum of two terms, one for each of the two streams; each

term is the average cost of delay for that stream, perhaps obtained as the

product of average delay by a constant (stream-specific) cost per unit

( 4 ) delay. ^ ^ This approach seems to incorporate many realistic aspects of the

situation, yet raises many difficulties: the complications (noted in

^pendix E) associated with the use of "delay" concepts; the need to specify

demand patterns before aggregate delay effects become meaningful

;

the

fact that "capacity," if taken as the minimizing point on the efficient

curve, can depend on cost factors not necessarily related to the runway

and its surroundings (e.g., fuel costs and crew wage rates).

With matters in this inconclusive state, we shift to a discussion of

what "scenario" should be employed in determining the efficient curve

(i.e., what are the "appropriate assumptions" alluded to in the third

paragraph of this appendix) . Since our interest is capacity-focused,

some version of the assumption of continuous demand (see subsection 2,2)

should surely be imposed; as a minimum, whenever the facility is free to

handle a customer, a customer from at least one of the two streams should

be at hand.

Should the "continuous demand" condition be applied to either stream

separately? Here we must take account of the current requirement that

arrivals take priority over departures . If this rule is absolute, and

[ 23 ].

^^^This is analogous to the need to know supply-demand curves in the

A-market and D-market before a profit function P(A, D) can be determined.

(6) Except of course for emergencies, e.g., a Coast Guard plane taking off

on a rescue mission.
if "continuous demand" for arrivals obtains, the "production rate" for

r 71 departures will clearly be zero. ^ This outcome being unacceptable, we

must assume either that the priority-for-arrivals rule is not absolute,

or that the traffic load of arrivals is less than continuous -demand, or

both.

Consider first whether our scenario should treat the priority-for-arrivals

rule as less than absolute. This raises the question of whether our goal is

fidelity to current official rules, or fidelity to actual practice (in

which "rules" may become "guidelines" to be tempered by the professional

judgement of controllers) , or the ability to encompass possibly quite

different future situations to which it might be desired to apply the

capacity concept. Tnere are clearly subtleties involved; for example a

departure may have been fitted into a "gap" in the stream of arrivals,

but did the gap in fact arise out of "permissiveness", a deliberate failure

to close up the arrival stream as tightly as possible? We will have to

seek further information and guidance on these points.

Representing an absolute rigidly-followed rule in a mathematical

model may present technical difficulties, but is at least unambiguous. If

the priority-for-arrivals rule is not of this nature, how should degrees

and kinds of deviations from it be modeled? Presumably the decision

criteria (whether articulated or not) for exceptions to the rule are not of

some mathematically convenient random nature, but rather refer to how long

and long-suffering is the queue of aircraft waiting to depart. While

TTl ^ ^We assume that departures cannot be interspersed between arrivals subject to

both priority and "continuous demand" for the latter, but have not

checked out this plausible impression.

- 140 -
further study is required, contemplation of these potential difficulties

yields a fairly strong predisposition to treat the priority rule as "hard"

should this prove a factually acceptable option.

Given the prospective treatment of the priority rule as rigid, we are

obliged to turn to the second alternative posed above, that of dropping

the "continuous demand" assumption on the stream of aircraft arriving for

landings. This stream will therefore exhibit gaps, which (if long enough)

can accommodate takeoffs. It seems reasonable to impose the continuous

demand assumption on the departure stream \dien determining the function in

(G.2), so that the remaining major problems involve (a) arriving at a

mathematical description of the stream of arrivals, and (b) deriving from

this description the necessary information on the frequency of gaps of

various sizes in this stream.

Regarding (b) , there is a modest literature on "gapology" developed

mainly in connection with road traffic, e.g. a pedestrian seeking to cross

a traffic stream, or a vehicle from a side road seeking to enter or cross

such a stream. We cite here only [24] - [26], as well as the runwayoriented [27]

;

a careful study of the potential usefulness of these

papers and the further references they contain, as well as later work on

the topic, had to be postponed for a possible follow-on project.

Concerning (a)

,

one might hypothesize Poisson arrivals as in the work

underlying the Handbook [1] (see for example [2], p. 8-1), but this would

be subject to the criticisms noted earlier on p. 66. The use of less

tractable probability distributions will surely conplicate the mathematics,

- 141 -
but a quick appraisal of the references cited above offers hope that these

difficulties can be coped with.

This winds up the account of our present ideas on extending the

capacity concept to a runway used both for landings and for takeoffs

Quite obviously, our progress on this topic has not reached the point

(jubilantly announced on p. 24 when attained in connection with the single

stream runway) at which conceptual problems give way to mathematical

problems. However, considerable conceptual structuring and clarification

have been accomplished, and initially preferred lines for further work

have been identified: these include seeking formulas or numerical solution

methods for determining the efficient curve under the assumptions of

strict priority for landings and of continuous demand for departures.

We conclude by recalling, for contrast and comparison, the approach

adopted in the Handbook [1] . On the technical side, both streams are

regarded as Poisson -generated ([2], p. 8-1; [15], p. 2) and the priority-ofarrivals rule is assumed to hold strictly ([15]; pp. 1-2). The conceptual

aspects can be inferred from sample exercises (Examples 1-3, [1], pp. 17-1,

2) : using the mix of aircraft types (specifically, the Class A proportion)

,

a value of A+D is read from Figure 17.1. The user must supply the desired A/D

ratio by which to convert this (A+D) -value into a pair (A, D) with the

indicated sum. However, this point (A, D) may not be attainable. Figure

17.6 is entered with the mix of A/C types to obtain a value A*, and

A =min(A, A*) is taken as the "arrival capacity." Define D by A /D =A/D;

c c ^ ^

- 142 -
then apparently A^+D^ is taken as the total capacity. Average delays for

both streams are obtained by entering Figure 17.7 using A+D, A^+D^, A^ and

A.

The look-up in Figure 17.1 (together with the user-supplied ratio

A/D) can be interpreted as yielding a point (A, D) which may fail to be

in the feasible region; in that case it is replaced by the point (A^, D^)

in which the efficient curve meets the line joining (A, D) to the origin

(0, 0). The rationale for this graphical construction, which involves

maintenance of the original A/D ratio, is not evident.

143 -
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