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NBSIR 78-1458

Projecting the Age Distribution

of Full-Time Permanent

Professional Staff at the

National Bureau of Standards

L. S. Joel

Center for Applied Mathematics

National Bureau of Standards

Washington, D C. 20234

March 1978

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NBSIR 78-1458

PROJECTING THE AGE DISTRIBUTION

OF FULL-TIME PERMANENT

PROFESSIONAL STAFF AT THE

NATIONAL BUREAU OF STANDARDS

L. S. Joel

Center for Applied Mathematics

National Bureau of Standards

Washington, D C 20234

March 1978

Final

Prepared for Institute for Materials Research

U S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Dr. Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

ACKNOWLEDGEMENT

This study was undertaken in response to suggestions from within the

Institute for Materials Research (IMR) of the National Bureau of Standards, and was partially funded by that Institute.

The author wishes to acknowledge the enthusiastic and cogent assistance

of members of the Personnel Division, the IMR Office, the Office of the

Associate Director for Programs (OADP), the Office of Standard Referenc

Data (OSRD) and the Office of the Management and Organization Division.

Particular thanks are due to I. Bartky of IMR and to E. Bunten and S.

Newman of OADP for many insights, B. Conrad of Personnel and B. Molino

of OSRD for help in retrieval and interpretation of personnel data,

C. Johnson of the Applied Mathematics Division (AMD) for technical discussions relating to Markov models, to J. Lechner and P. B. Saunders

of AMD who improved the exposition and finally to A. J. Goldman of AMD

who furnished an appendix and was a general technical factotum. The

residual flaws are all mine.

Lambert S. Joel
,,
ABSTRACT

This report presents a simple mathematical model to project the age

distribution of the full-time permanent professional (FTPP) staff of the

National Bureau of Standards. The report includes a brief description of

types of models currently in use for manpower analyses, discussion of the

probable data requirements for reliable models, some staff profile information which supplements material in recent administrative reports, and

the description of our model. The principal projection is that under

"status quo" assumptions (FTPP staff size, age distribution of hires, and

the separation rates from age cohorts remain constant), the FTPP staff

will in about 25 years reach a steady-state age distribution with average

age about 1/2 year higher than its current level (just over h2 yrs . )

.

Comparison of this distribution with the present one shows moderate increases in the fractions of staff in ages 21-30 and 51-60, a fractional

rise in turnover, and a general decrease in the groups 31-50 years old.

A near term effect is the intensified loss of senior scientists.

Keywords: Data analysis; discrete Markov models; manpower planning models

simple Markov models
.
TABLE OF CONTENTS

Page

1. Introduction and Overview 1

1.1 Summary 1

1.2 Related Literature and Analysis Methods 2

1.3 Study of Age Distributions 4

1.4 Concluding Remarks & Possible Extensions 11

2. Data 15

3. The Model 20

3.1 Model Concepts 20

3.2 Discussion of Concepts 23

4. The Working Model and Its Output 30

5. Bibliography 40

Appendix A. The Model in Matrix Form 43

Appendix B. Analysis of the Markov Model... 55

B.l Formulas for Steady State Quantities 55

B.2 Attainability of "Target" Distributions 59
LIST OF TABLES AND FIGURES

Table 1. Hires & Separations, Jan. 1, 1970 - Jan. 31, 1976 18

Table 2. Means and Median Ages of Hires & Separations 18

Table 3. Relevant Data and Model Parameters 31

Table 4. Projected Changes in Age Class Sizes and Distribution 36

Table 5. Rank Orders of Age Classes, 1975 - 1990 38

Table 6. Selected Values from Tables 3 and 4 57

Figure 1. Age-Distribution Shapes for Hires & Separations 19

Figure 2. Median Ages of Hires & Separations, at Years & Half-Years .... 27

Figure 3. FTPP Staff Age Distributions, 1970 - 1974 32

Figure 4: Composition of FTP Staff by Occupational Category,

1964 - 1974 33
1. INTRODUCTION AND OVERVIEW

1 . 1 Summary

This report describes a simple model to project, under "status quo"

assumptions, the age distribution of the full-time permanent professional

(FTPP) staff of the National Bureau of Standards. That particular topic

and the vehicle chosen for its analysis, a "Discrete Time Markov Model",

may be regarded as paradigms for investigating the prospective usefulness

of mathematical models in manpower planning at NBS . The report includes

a cursory description of types of models currently in use for manpower

analyses, discussion of the probable data requirements for reliable models,

some staff profile information (derived from personnel tape records) which

supplements material in recent administrative staff reports, and the description of our model and its output. Facets of the technical analysis

are included in the main text when they are vital to the continuity of the

discussion, but in general they are exiled to an appendix.

The salient output from the Markov model is that the FTPP staff will

arrive at a steady state in about 25 years with an average age about 1/2 year

higher than the current average age, moderate increases in the fractions

of staff aged 21-30 and 51-60, a general decrease in groups 31-50 years old,

and a small increase in turnover, jlf

1. The size of the FTPP staff remains constant.

2. The age distribution of separations and the age distribution of

hires for FTPP hereafter remain constant at their mean levels

during the period JAN 1970 - JAN 1976.

The model's outputs project accelerated loss of senior scientists during

the next 10 years and a slight reduction in departures of professionals of

ages 30-UO.
Further specifics are given in Table 4 on p. 36.

Tabulations of the available data show that while mean and median ages of FTPP hires during 1970-1976 arehhigh relative to conventional expectation (just under 36 and just under 3^ yrs. respectively),

they have held approximately level during this period and are consistent

with mean and median hire ages of FTP technicians, wage board, administrative and clerical staff, all also high. This means that indicators of

concentration of employees at increasing age levels are not peculiar to

professionals but apply to the entire FTP staff, and may be characteristic

of the entire federal establishment or conceivably the entire employed

labor force of the United States.*

Analysis of the model supports the intuitively evident notion that

the age distribution of staff can be controlled to the extent that the hire

and separation age distributions can be controlled. The model (or minor

variants) can be utilized to explore the effects of stipulated changes in

these "input" distributions.

1 . 2 Related Literature and Analysis Methods

In general, models of manpower systems have been developed for the

study of

(1) assignments to jobs and training, [12]

(2) perceived and undesired trends toward "top-heavy" rank

structure in mature organizations** and

*In conversation with the author, a Bureau of Labor Statistics source opined

that the Federal work force has aged subsequent to the employment boom of the

1960's, but that early retirements have had the opposite effect on the national

employed labor force.

**Control of the grade structure of an organization, which depends on promotion paractices as well as on hires and separations, may serve as a method

of indirect management of age distributions because of the typical correlations

between rank and age, especially in systems in which seniority is a factor

in promotion practice.

2
(3) departures one way or the other from target ratios between line

and staff force sizes (or, alternatively, between professional and

administrative force sizes )• [1, 2, 3, 12]

Traditionally, little explicit attention has been paid to the first

type of problem at NBS because at working levels this was regarded as a

casual task of section chiefs which could usually be accomplished informally

(and easily) using educational background as the prime criterion. Determination of job assignments using formal methods usually founders on the difficulty of defining reliable numerical "figures of merit" or scoring standards

for anticipated performance. (Determining post hoc performance standards is

hard enough!) Assignment problems are generally attacked with simple linear

programming models (and a grain of salt), when a formal treatment is attempted. [12]

Normative (i.e., prescriptive) approaches to problems of types (2)

and (3) are subject to the same kind of obstacle: it is difficult to define

objective numerical criteria for target values. The models here tend to

be either complicated mathematical optimization formulations with a heavy

bias in favor of assumptions yielding reduction to (complicated) linear

programs, or else models using the constructs of dynamic programming and

control theory , derived from analogies of the manpower systems with physical

engineering systems, and again characterized by ruthless assumptions of

linearity to make the calculations tractable to known methods of solution

[ 1 , 12 ]

Four kinds of models, sometimes in combination, account for a great

preponderance of the published studies in which analysts' sights are

lowered, from pre scription to description (and estimation, and prediction)

.

3
(A) Regression, or fitting curves to empirical data. Without prior

hypotheses about the functional relationships between causes and effects,

this kind of "model" amounts to fishing for "trends", i.e. linear (or

sometimes log-linear) fits to data points.*

(B) "Atomic" or discrete "Monte Carlo" simulations. These involve

attempts to reproduce numerically the actual behavior and decisions of individuals in an organization and to measure their aggregate consequences.

Such models are usually very complicated and are characterized by extensive

activity to develop inputs. Ironically, model types often classified under

this rubric need not be atomic nor probabilistic, i.e. "Monte Carlo".

For example, the well-known methods of J. Forrester [4] (customarily called

"system dynamics" and widely used today by management analysts) involve

deterministic representation of aggregate behavior.

(C) Renewal theory, primarily Integral Equation models. These have

proven valuable in the study of biological population phenomena and have

become popular for analysis of turnover rates, e.g., when responses to

losses occurring at random points in time are of interest. [l, 12]

(D) Continuous time and Discrete time Markov process models. These

are highly esteemed because they have an analysis-simplifying property:

the future development of a Markov system depends on its current condition

and not on its entire previous history. [1-3, 5-9, 15]

1 . 3 Study of Age Distributions

The previous list of three fundamental problem areas addressed by

manpower analysis omits the one which is the principal topic of this report,

*"Trend fishing", despite the disdain implicit in such a designation, can be

extremely useful in short term analysis.

h
viz. the tendency toward staff aging in organizations — that is, the increasing concentration into higher age groups.*

The composition of an organization’s working force, described in terms

of its distributions into classes according to the members' characteristics:

sex : age, education, rank, seniority, occupational specialty, etc., is a

natural vehicle to carry measures of the organization's suitability for

achieving its perceived missions. A central concern in manpower planning

is the degree to which changes in composition over time can be predicted,

or better, controlled, given the range of political, social and economic

factors affecting them. [1, 11, 13]

At NBS , a current matter of management interest is the age distribution

of the FTPP staff "component". The professional staff is perceptibly "aging";

the popular notion that professional productive output decreases with workers'

ages is supported in research literature, and some investigators have found

evidence that quality of output deteriorates as well. [10, 13]

Various plausible arguments have been advanced** to explain the increasing

concentration of FTPP staff members in high age groups. Among these are:

(a) the termination of "blue sky" research programs that produced a

young-professional recruitment explosion in the late forties and

the fifties.

*Analyzing age distributions is conceptually equivalent to the analysis of

rank distribution in an organizational hierarchy in which all promotions

take place at regular intervals, promotions are always to the next higher

rank, and no demotions can occur. While explicit models of aging are hard

to find in the literature, studies of organizational rank hierarchies are common

.

**Conversat ions between the author, and E. Bunten and S. Newman of the Office

of the Associate Director for Programs.

5
(b) economic contractions that have diminished opportunities outside

NBS for professional (indeed all) staff members, resulting in

longer terms of service,

(c) a tendency for students to remain at universities longer than

heretofore (for a multiplicity of reasons) before seeking first

employment, and a concurrent tendency of project managers to

seek "expert" recruits, both resulting in a high average age of

hires; and finally

(d) the existence at NBS of certain programs involving long term

research goals, leading to progressive refinement of results

along narrow lines, producing a "life's work" attitude in which

workers are reluctant to move to other activities or to relinquish

career activities by retiring.

Accordingly, the primary object of the current study was analysis of

personnel data to check and quantify the current impressions about shifts

in the age distribution of FTPP, with particular emphasis on the rate at

which this distribution would approach a possible steady state, given continuation of current short term trends in the components of change.

A most direct way to project the age distribution would be to trace

over a period of years the number (or proportion) of staff members at each

"age" (or in each of a set of age intervals) and to fit trendlines or other

curves to these "time series" by regression techniques.

The results of the regressions would then be adjusted or "normalized"

to conform to the assumption of constant total staff size in the case of

sizes of the groups, or to the requirement of 100% totals if the regression

is on age-group percentages of the total. In practice, it is desirable to

6
group ages into at least 5-year intervals in order to reduce the number of

series to be treated and to "smooth out" local variations, except in rare

instances in which there are known factors distinguishing the progressions

for finer division. [This might be true, for example, if we were considering

all full time professional and subprofessionals including term appointees,

as the population of interest. Then if the data were recorded on a fiscal

year basis (i.e. staff status on July 1), and there were a recurring summer

Quality High School Student employment program, it would make sense to

distinguish 16-year olds or 16 and 17-year olds from, say, an "under 25"

age interval.] Obviously, a cursory estimate of a trend in age can be

obtained by examining and projecting a single indicator of the distributions,

either the average or median age. This is conceptually equivalent to an

extreme case in which the classes are aggregated into a single component.

In the context of identification of a stable age distribution, the

regression scheme has a minor drawback: it does not ipso facto furnish a

model or hypothesis. When a "trendline" is fitted to data, the regression

procedure finds the slope of that line which furnishes the best fit according

to standard criteria. The "model" here is the (straight) line, and the

regression determines the values of the parameters of the model, in this

case a slope and an intercept. If regression should show very strong (nonhorizontal) straight-line fits to the age distribution data, we would have

to conclude that under present conditions no stable age distribution could

be predicted. (A time series for average age lying along a rising/f ailing

straight line would, if interpreted at face value, indicate an average age

rising/falling indefinitely, precluding arrival at a stable age distribution.

7
Similarly, non level straight trendlines for separate age class sizes

would say that some are rising, others falling, indefinitely, while if the

data consisted of fractions of staff in the various classes, projections

of such lines would be meaningless.)

In general, if observed data values are assumed to lie on a curve of

specified form, regression fixes the best-fitting curve within the confines

of that form. In searching for a steady state we would assume the time

series for each age component to lie along a curve with a more or less horizontal asymptote, i.e. a curve that "levels off", but further pre-analysis

or imaginative examination of plotted data points is needed before the

regression procedure can be really useful.

The model we have chosen to employ instead, does not entail regression

as such and therefore bypasses the need to deduce beforehand an appropriate

shape for the path over time of the values of the age group sizes (or

fractions) in order to arrive at a stable distribution. Our model is a

routine application of the mathematical theory of discrete time Markov processes, which has widespread precedent for the analysis of manpower systems.

[ 1 , 2 , 3 , 7 , 11 , 12 , 15 ]

For projecting basic age trends in a staff of fixed size, the model

requires "only" separation rates by age and an age distribution of hires,

both assumed to be constant . With these assumptions the model yields a

stable age distribution for our FTPP staff and an estimate of how fast it

is attained from the (known) current distribution.

In our case, the "constant" age distribution of hires and age-dependent

rates were derived from averages of distributions for the years 1970-1976 as

8
extracted from an NBS personnel History File data tape. These averages

were informally judged representative, hence usable in the determination

of the constants, on the grounds that the annual distributions (5-year age

classes) for the six years did not "seem" to vary much. Of course, we have no assurance that they will not vary in an unknown manner in the

future, since they depend partially on factors not under the control of

management. The same argument applies to a lesser extent to the total

size of the staff. (With regard to changes in total size, however, there

are relatively simple extensions of the basic method which apply to ex- panding or contracting manpower systems.) [l, 3, 7, 12]

Markov models in the present context are subject to an inconvenient

constraint that does not occur in the curve fitting approach described

earlier. There the age intervals defining classes could be arbitrary, because class sizes were being projected independent of each other (except

for the "consistent total" adjustment). Here, the class boundaries must

be equally spaced because the structural framework is one of flows between

classes, and the core of these flows is the uniform biological process of

aging which could not be so represented if the age intervals varied.* [3]

A third methodology, that of system simulation models, could be used

to study aging and s imult aneo.usly many other problems of importance in

organizational structure and behavior. Usually synthesized using special

computer languages, these models tend to be large and complicated and so

*In addition, if the length of the model’s time periods is not an integral

multiple of the class age interval, the calculations can produce nonsensical

results due to "asynchrony"; an example appears on pp. 25-26, below.

9
none was considered as a practicable candidate for the present analysis. In

contrast to "renewal models" (a class of continuous-time models primarilyapplied to questions of population growth and mentioned again here for

the sake of completeness), simulation models represent philosophically a rationale so different from curve fitting and discrete stage Markov models

that although they are far beyond the scope of the present project, they

are mentioned here as possible tools for an expanded study. The earlier two

methods are essentially schemes for data analysis in which the main questions

are: "What happened?" "What will happen?" or "What is likely to happen?"

but not "Why does it happen?" Simulations, in contrast, generally are intended to incorporate all the factors, behavioral, economic, political etc.

that the analyst can identify and represent numerically. The resulting

model is an attempt to represent realistically the interaction of the

factors or forces defining a system and to generate outputs which can be

studied as substitutes for the output of real (and less observable and controllable) systems. When such models are constructed too naively, and/or

with totally inadequate baseline data, they are useless chimeras

overfull of irrelevant details yet missing vital "components", and formulated

with mathematical absurdities. When they are devised with even moderate

intelligence and used with requisite caution, they can be gold mines of

insight. Frequently, in fact, models developed by research analysts purely

for the purpose of facilitating exposition of technical material to managers,

have uncovered significant information about the modeled system (or organiation) previously known neither to analysts nor to management. [4]

10
1.4. Concluding Remarks & Possible Extensions

While the Markov age model begs the question of arriving at stable

age distributions by assuming constant rates of change, it does tell us

where we are heading given present trends. With moderate additional computational effort, the analysis can be applied, if desired, to staff components

other than FTPP.

It may be of interest to examine, through such a model, the effects

on staff age patterns of policy-induced changes in the hire and separation

age distributions, such as incentives offered for early retirement, or

changes in requirements for entry into service at various grades. These

examples refer to actions mainly outside the scope of Bureau management.

In the area of local policy aimed at such modification, promotion policies

will affect voluntary separations, but of course some study would be required to estimate how "leaver" age distributions are related to promotion

rules.* It would be relatively easy to estimate what modifications in

hire age distributions would result from controlled numbers of appointments

such as post-doctoral fellowships, or others in which there is a strong

correlation between age and educational/experience requirements. This is

one example of how planning for attainment of target age distributions in

a full time staff (not exclusively permanent) is practicable with a Markov

model as an analytical tool.

*A possible point of departure - compare leaver age distributions in or- ganizations which tend to assign positions of responsibility to current

staff members (in effect promoting them) when setting up mission task

forces, to those in organizations which tend to hire task force leaders

from outside.

11
Various other descriptions of manpower staff composition might he

studied usefully with Markov models based on relatively accessible data:

(a) Trends in occupational mix and/or educational level of the professional staff as an indicator of shifts in the scientific

mission of the Bureau.

(b) Trends in the mix of the major occupational categories, that

is, the organizational breakdown into Administrative, Professional,

Technician, Clerical and Wage Board employees, relevant to

mission changes broader than mentioned in (a) above.

(c) Trends in fragmentation if any (here the distribution in question

could be the fractions of staff or professional staff in units

of various size classes)

.

(d) Trends in the educational level and number of skills represented

in the total staff, to estimate growing (or diminishing)

flexibility for organization of in-house task forces in response

to externally imposed research programs.

(e) Trends in service life of equipment - this subject has no direct

bearing on personnel questions , but we include it because of its

suitability for Markov analysis and its putative interest to management.

The emphasis would likely be not on durability but on avoiding

technological obsolescence, with, by the way, a clear analogy tto

the level of continuing training in a personnel staff. [6]

If Markov (or other models) are to be used as guides for management

control, and not just as predictive or projective mechanisms, then three

important questions will demand consideration:

12
Can optimal or desirable distributions of staff according to

characteristics such as age be defined meaningfully? What kinds

of costs are associated with policies intended to modify the

distributions?

(b) To what extent are external (non-management) factors affecting

distribution changes measurable or predictable?

(c) To what extent are distributions for different characteristics

correlated? For instance if two MOUs at the Bureau had differing

current and/or stable age distributions, would the difference

be adventitious or would it be related to measurable structural or

"personality" differences between the MOUs? Could these, in turn,

result from differences in organizational mission?

These questions, unfortunately, fall into areas about which little uselul

information is contained in personnel records, so that study of them,

presumably by simulation, would entail a substantial commitment of time and

effort on data acquisition. For instance, a possible target criterion

for an age distribution is that the mean or median age should not exceed

some fixed value, k. When concern over rising ages is the primary motivation for analysis, such a ("single parameter") measure may be adequate

although even this starkly simple measure begs the question of how the

"critical" value should be determined. More generally one might try to

formulate a staff member "personality profile" whose components are characteristics presumed to be age-dependent, e.g., enthusiasm, energy, currency

of training, imagination, flexibility, prudence, breadth of professional

contact , etc

13
If (l) these characteristics can he quantified, and if, (2) they

can be accumulated over populations so that the sums of traits of several

people is the set of traits of the group, and further if (3) "target"

trait-mixes can be defined relative to organizational missions, then a

linear programming "model" could determine desirable age distributions.

The execution of the formal model is straightforward.* The major problems of judgment and data acquisition occur in determining, quantitatively, how the named characteristics relate to age, and then determining,

again quantitatively, how the characteristics affect expected satisfaction of missions.

*Models which combine Markov processes with optimization procedures such

as linear programming have received some attention recently as possible

approaches to this kind of analysis. [7, 12]

14
2 . DATA

The basic source of "hard" data for the analysis was a copy (edited

for reasons of confidentiality) of a computer-tape compilation of personnel

actions called the History File and maintained by the Personnel Division.

The file contains an entry comprising 30 items (Date of Action, Name, SSN,

Date of Birth, Educational Background, etc.) for each individual personnel

action at NBS (hire, promotion, in-grade promotion, change in occupational

title, etc.) since Jan. 1, 1970. This file is augmented weekly at present.

From the approximately 70,000 records of actions which occurred in

the period Jan. 1, 1970 - Jan. 31, 1976, the 10,598 referring to hires and

separations** were extracted and subjected to statistical procedures aimed

at (a) deriving approximate age distributions for hires and separations of

FTPP staff as input to the Markov model, and (b) determining if these distributions are peculiar to the professional staff. Table 1 (p. l8) gives a

breakdown of the hires and separations according to major occupational

status for the total staff and its FTP subcategory. in light of the known

stability of total-staff and occupational-component sizes there appear to

be discrepancies in the table, particularly for full time permanents who

show over 11/2 times as many separations as hires. These discrepancies

occur in general because of unfilled vacancies at the end of the data period

and, particularly’ as regards FTP, change of status of employees who become

FTP after originally being hired into another tenure category. A continuously

employed staff member whose status so changes will not have been recorded

as a FTP hire, but upon separation will be counted among FTP separations.

^Including the Boulder Laboratories.

**There are also a few entries for calendar 1969- File data include 520

hires and 151 separations in undesignated major occupational categories.

15
Table 2 gives summary data on hire and separation ages of the five

FTP staff occupational categories. Some cautious speculations can be advanced on the basis of the numbers in Table 2, though verification would

require further evidence and/or analysis:

First, the age measures for hires in all groups appear "high", but

' v

- " perhaps only in comparison with those believed typical for a growing

organization with rapidly shifting missions. Second, although the five

occupational categories have distinctly characteristic hire and separation

ages, it does not seem as if any group can be singled out as displaying

age statistics inconsistent with those of the others.

For example, the approximate 3 year age difference between professionals

and technicians is about what would be expected given the difference in

educational requirements; possibly 4 to 5 years for early career appointments and diminishing later, as job experience becomes a significant

factor. Mean and median hire ages for wage board employees fall between

the values for professional workers and technicians, reflecting relatively

demanding preparatory training in, e.g., trades and crafts. (We would

expect the wage board hire ages to be closer to those for clerical employees

if the data in the table were not restricted to FTP, on the grounds that

"Wage Board" would then include a smaller proportion of those trades involving a degree of "craftsmanship" akin to "professionalism".) The

relatively high age figure for Administrative hires is very likely a result of

the fact that until fairly recently there was relatively little emphasis

on development of managerial skills at NBS, so that in this category more

than others appointees have tended to come to NBS in mid career, transferring from jobs elsewhere.

16
The very low age of clerical hires indicates that the "winds of

change" for this group have not blown with significant effect. In the

1970's as in previous decades, hire data are dominated by large numbers of

recent high school graduates, primarily women; turnover rates are high, as

can be deduced from the separation age data. There is reason to believe

that they will remain high even if gender-related events, i.e., marriage

and child rearing, should no longer so frequently precipitate withdrawal

from the employed labor force. This may result from the relative lack of

specific assignment-related skill requirements. That is, since clerktypists generally do not have professional affinities related to the missions

of particular organizations, their effective mobility may be high.

We can make the same superficial analysis of the separation data.

Except for the inversion in rank between the means and medians, for the

wage board and administrative employees, the rankings of these numbers

conform with plausible explanations. For example, technicians should

be expected to remain in given career positions longer than professionals

or "managers" (administrative staff)

because their skill development and

refinement on the job is most closely associated with assignment. Administrative personnel will tend to move more frequently than scientific

professionals because position advancement is for managers a vital component

of the definition of their profession, whereas for scientist professionals it is nominally incidental.

17
Finally, the shapes of the "in" and "out" age distributions for

the groups, sketched from data histograms (fig. 1), seem consistent with

the foregoing conjectures.

TABLE 1

OCCUPATIONAL

CATEGORY HIRES

TOTAL

SEPARATIONS

FULL TIME

HIRES

PERMANENTS

SEPARATIONS

Professional 1371 1389 347 538

Technicians 577 652 59 206

Wage Board 1330 1195 170 272

Administrative 272 404 123 180

Clerical 1368 1369 379 462

Totals 4918* 5009* 1078 1658

Hires and Separations for

(From the History File)

the period Jan . 1, 1970 - Jan. 31, :

OCCUPATIONAL

CATEGORY MEANS

TABLE 2

HIRES

MEDIANS

SEPARATIONS

MEANS MEDIANS

Professionals 35.9 33.8 45.5 45.7

Technicians 32.5 30.1 47.8 52.5

Wage Board 33.4 30.9 43.5 45.6

Administrative 36.9 37.6 44.5 45.0

Clerical 26.6 23.3 33.8 27.0

Mean and Median Ages of Hires and Separations of Full Time

Permanent Employees for the period Jan. 1, 1970 - Jan. 31, 1976.

By occupational category. (From the History File)

*Hires and separations = 9927- Inclusion of the 671 unlabelled hires and

separations mentioned in the footnote on p. 15 gives a total of 10,598.

18
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19
3 . THE MODEL

3 . 1 Model Concepts

We present a simple age distribution analysis to clarify the form

and manner of operation of a Markov model, and its level of realism. Suppose the size of the working staff of an organization is assumed to be

constant, that is, all losses (retirements, etc.) are made up by recruitment, but new hires are limited by "slot" availability. The age distribution of the staff is assumed to be "recorded" every 25 years. Suppose also, that the staff members' ages may range from 21 to 70 (i.e.,

there is a minimum age required for employment and retirement is mandatory at 70 years of age), and we break down the staff into only two classes

21 to 45-year olds and 46 to 70-year olds.* With these ground rules we

would say that the staff is aging if the fraction in the second class

increases over time. We are interested in the conditions which will produce a stable age configuration - one in which the sizes of the two classes

no longer change. Now we make just three more assumptions:

(1) When vacancies occur, the replacements are split between the

two age classes in a constant ratio , so that say, 1/2 of new hires

are 45 or younger and 1/2 are older.

(2) The separation rates for the classes are constants. This means

that in each 25-year period 1/2, say, of the first class will leave

the staff. We do not differentiate among reasons for leaving, so

that for the second class it suffices to say that all will leave,

''Demographers call such classes "age cohorts."

20
daring a 25-year period, because regardless of what fraction leaves

prior to retirement, all those who are members at the start of the

period will by its end reach the age of mandatory retirement.

(3) Separations and hires will be treated as if they all take place

simultaneously at the start of each 25 -year "accounting" period.

Thus in any 25-year period 1/2 the members of the first class leave

and the other 1/2 all age into the second class; all members of the second

class leave the system.

Regular repetition of this accounting sequence, with the new class

sizes replacing the old after each step, defines a discrete time Markov

process or model. The necessary elements are: a fixed time step (in the

example, 25 years), a set of classes or "states" (here, the two age classes)

and constant coefficients relating the state values (the class sizes)

after a time step to their values before it. [8 s 9] *

In equations:

n (t+25) = 1/2 (1/2 n

x (t) + n

2 (t)) = 1/4 n^t) + 1/2 n^t)

n

2 (t+25) = 1/2 n (t) + l/2(l/2 n^t) + n^t)) = 3/4 n^t) + 1/2 n ? (t),

where

t is the current or old year,

t+25 is the forward or new year,

n^(t) is the current size of the first class,

1/2 n (t) + n

2 (t) is t ^ie tota l number of leavers during the 25-year

^Strictly speaking, according to the theory of Markov processes, which

deals with random phenonema of a certain type, the coefficients are probabilities and the rates are thus rates "on the average". In applications, however, that is in analysis of "real" systems using Markov

theorys the distinction between random and deterministic processes is

usually ignored. The computational characteristics of these systems do

not depend on the labels attached to the coefficients, e.g. "constant

rates" or "constant probabilities".

21
period [ cf . (2) above],

n (t) is the current size of the second class,

n^(t+25) is the new size of the first class,

n

? (t+25) is the new size of the second class.

Now if we operate this process on an organizational staff of 1000

members, half of them currently 45 years old or younger and half older —

that is, n^(1975) = 500 and ^(1975) = 500 — then the result is

1^(2000) = 1/4 n

i

(1975) + 1/2 n

2 (1975) = 1/4 • 500 + 1/2 500 = 375,

n

2 (2000) = 3/4 n (1975). + 1/2 n

? (1975) = 3/4 * 500 + 1/2 * 500 = 625;

n

1

(2025)

n

2 (2025)

1/4 n ( 2000) + 1/2 n

2 (2000) = 1/4 • 375 + 1/2 * 625 = 406.25,

3/4 n^ (2000) + 1/2 n

2 (2000) = 3/4 • 375 + 1/2 • 625 = 593.75;

n^ (2050)

n

x (2075)

n^ (2100)

n (2125)

1

398.4375

400.390625

399.9023438 ,

400.0244141 ,

n

2 (2050)

n

2 (2075)

n

2 (2100)

n

2 (2125)

601.5625;

599.609375;

600.0976563;

599.9755859.

The staff (that is, the model) has a stable distribution with 400

persons in the 20-45 year old class and 600 from 46-70. To all intents

and purposes, that distribution is reached in 100 years or 4 cycles of

the process (3 if we aren't fussy).

22
(1) The example has 2 classes and a 25-vear step interval. Our

model for FTP? used 10 age classes with a five-year step interval, and with

moderately small additional effort in preparing data, input could be disaggregated to 50 classes (e.g., ages 21, 22, 70) and an annual time

step. The "mechanics" of the process would be the same in all cases.

(2) With rare exceptions*, aging models of the type studied here

will converge to stable distributions. The stable distribution, bv the

way, is determined entirely by the model parameters (the equation coefficients^ i.e. the "transition probabilities"). That is, all initial

age distributions of a given sized population will converge to the same

stable distribution (but possiblv at differing rates). If, for instance,

in our example we had assumed the initial distribution

n (1975) - 1000 , n (1975) = 0 ,

we would have calculated the sequence

n j ( 2000) =250 n (2000) = 750 ;

n

L

( 202 5) = 4 37.5 n

2 (2025) = 562.5 ;

n (2050) = 390.625 ( 2025) = 609.375 ;

n (2075) = 402.34375 n ( 2050) = 597.65625

n (2100) = 399.4140625 n 0

( 2 1 00 ) = 600.5859375

n (2125) = 400. 1464844 n.,( 2 1 25) = 59b. 85 3515b

L. requiring two additional cycles to reach the (same) stable distribution to

the nearest integer. (Compare the values for year Cl. 1 above to those

for year ;?075 in the previous example.)

^Involving zero hire and/or separation rates for a large subset of the age

classes. In the example, for instance, if there were no separations from t first class and no hires into the second, the two class sizes would interchange endlessly from the start, and thus not converge.
We chose the numerical values of the example's model parameters for

arithmetic simplicity. In general, the hire fractions (in the example, 1/2

for each class) must merely be non negative and add up to one, while the

separation fractions (in the example, 1/2 for the first class and 1 for the

second) must merely be non negative.

The coefficients of the model, which in the example were calculated

from the aging of staff members and the hire and separation fractions,

are similarly free. That is, since the coefficients of any given class,

in total, "account" for the "contribution" of that class to the classes in

the next step, they must merely not be negative or greater than one in

value, and must sum to one for a system with a constant total.* (in

the example we had 1/4 and 3/4 as the coefficients of n^'in the two

equations; 1/2 and 1/2 as the coefficients of n^.)

The non-dependence of the shape of a Markov model's stable distribution on the starting distribution furnishes a crude yardstick for

measuring the validity of "present is like the long-term past" assumptions in a given application.

Specifically, suppose we have estimated the transition probabilities

(coefficients) from recent data and we wish to assume that they have been

unchanged for some extended period of time previously. This would imply

that our application represents a Markov process which has already been

operating for some time , and thus our "starting" current distribution

should not be drastically different from the stable distribution which

represents the expected outcome of the puocess and which we can calculate

from continued application of the model as in the example.

*One of the reasons these coefficients are called "transition probabilities"

in Markov theory is that they have these probability-like properties.

24
On the other hand, we might be led to assume or deduce that the process

defined by our coefficients represents a substantial difference from the

one previously "in effect", that is, that the organization’s staff is

affected by a different set of forces now than previously. In this

case our suspicions would be raised by a stable distribution that resembled

the current one too closely.*

(3) In the sample calculations above, the Markov process operated on

a distribution described by the numbers of staff members in each class.

These numbers should be thought of as "expected values" after each step.

Rounding them to whole numbers to preserve token realism would be improper

because it is equivalent to making small changes in coefficients which

must be constants to satisfy the mathematical definition of the Markov

process. The "psychological" problem of interpreting the stepwise outputs as age-class population sizes can be avoided by stating the distributions in fractional or percentage terms. Thus in the example we

would have used n^(1975) = . 5 or n^(1975) = 50.0% instead of n^(1975) =

500, leading to stable values of .4 or 40 %, respectively.

(4) The reason for requiring that the process step time match the

class age interval can best be explained with an example: If we wished to

pursue the changes in the distribution into two classes more closely by

examining this change every five years instead of every 25, we would need

5-year constant rates in place of 25. Thus we would first have to assume

that the age distribution within the classes be sufficiently uniform so

that each five years exactly 1/5 of those from 21-45 who didn’t leave the

*"Drastically different" and "too closely" are matters of judgment. In

practice simple statistical procedures such as the Chi-square test are pulled

from the shelf and applied.

25
organization could age into the 46-70 year old class, and that similarly

exactly 1/5 of the members of the second class would age into retirement.

The main difficulty begins at the end of the second five year period, be- cause a "Markov process has no memory" so that the people who passed into the

second class five years ago are "diffused" in it and 1/5 of them will

now be "promoted" to age 70 t- that is, the ages of some of our staff

members would increase by over 25 years in a 10 year period!

(5) We specified in our example that the transitions were to be

treated as occurring simultaneously at the starts of time periods. This

was done to avoid a situation in which for instance, during a single

period a 23-year old staff member leaves the staff, is replaced by a 29-

year old, who in turn leaves and is replaced by a 22 -year old, leading

to counting-ambiguities. (The turnover is two, but we see only one new

face at the end of the period.) This artificiality is not a defect peculiar

to this type of model, but is one that is shared by all sorts of accounting

procedures unless they are continuous. A case in point is the administrative tabulation of the total size of the NBS FTPP staff, which has been

quite constant for six years when recorded at calendar-year ends but

shows fluctuations on a July 1 basis. This sort of variation leads to

a basic dilemma in modeling. Discrete, that is, fixed time-step treatment of continuous processes such as aging frequently produces paradoxical

results. A more realistic representation of the flow of people in and

out of NBS would take into account the relatively random nature of the

"departure time" of leavers, and time lags between ' separations and

hires with their possible consequences. Such mathematical models

exist but they are complicated and entail difficult (i.e.

,

costly)

data demands. It is possible that two different sets of constant rates •
FIGURE 2

AGE

1970 1971 19 72 19 73 19 74 19 75 19 76

Median ages of FTPP hires and separations recorded annually (solid lines)

and semiannually (dotted lines) in the period Jan. 1970 - Jan. 1976. (From History File data )

27
could be observed depending on the choice of starting points for there gular intervals; fiscal year end and calendar year end, say, might result in two different Markov processes "existing in tandem." This is one

Of the few cases in which one can usefully introduce a single formal model

which actually permits variation in the rates. Such models, with cyc lic

variation of the parameters, lead to a pair (or more) of alternating "stable"

distributions. [1, 8]^

How the stable distribution will vary, as a result of differing hire

distributions and/or leaver rates, is interesting as an approach to

studying the control of the age distribution, and also as a means of

. ifc ..

. ' , measuring the effect of error in estimating the "true" hire and leaver

parameters. The question of such variations (i.e., sensitivity analysis

of the steady state) will not be treated in any depth in this report,

because specifying suitable "scenarios" of such input changes is a com- plicated problem beyond the scope of the present introductory study. Even

the present two-class example is sufficiently "rich" that with (effectively)

only two quantities subject to modification (the fraction of hires into

and the fraction leaving the first class), it is not immediately obvious

how to program a sequence of meaningful changes. We append an example to

show that in this case, at any rate, moderately small changes in the parameters lead to small changes in results:

If the hire fraction into the first class is changed from .5 to .48

and the leaver fraction from the first class is also changes from .5 to .48,

the coefficients (1/4, 3/4) of n become (.2304, .7696) and the coefficients

(1/2, 1/2) of n

2 become (.48, .52). These lead to a stable fractional

distribution of (.384, .616) instead of (.400, .600).

28
We close this section with some opinions concerning the appropriate

level of detail for a predictive model of the NBS staff age distribution

useful for planning purposes. Obviously, the 25-year chunks of time

of the example can't give us much meaningful information - too much goes

on in the organization during such long intervals, and confining the description of the distribution to a ratio of "young" to "old" (the two

classes) presents a very fuzzy picture. Moreover, since just four cycles

of the process represent 100 years of real time (over 30 years longer than

NBS has existed) , we would be straining credulity too far to expect the

change rates to hold constant over such a period. On the other hand, a

slight amount of fuzziness is desirable, if it militates against the

temptation to regard model outputs as gospel because they "look" very

realistic. This more or less rules out "atomic” monthly or even quarterly

models. An annual model appears to be an attractive alternative, but we

believe that the data at hand are not suitable for the determination of

representative (constant) annual hire and separation rates by one-year

age intervals. The five-year level of detail in the model used as the

basis for this report, was adopted primarily because of the ready availability of five-year summary and profile data in the "Employment Trends"

personnel report [14], but we think that it also affords a reasonable

compromise between an overly crude representation and one for which smoothing

out input-data fluctuations is too difficult. Setting questions of convenience aside, computations with a time period and age class interval

of about three years appear to offer the greatest likelihood of reliable

projections given the now familiar assumptions of constancy in the sets of

(age-dependent) separation and hire rates.

29
4. THE WORKING MODEL AND ITS OUTPUT

The Markov model which was operated to project the age distribution

of the FTPP staff is conceptually the same as the sample model of section

3.1 except that it has 10 classes representing 5-year age intervals, and

a 5-year step size. The assumptions too, are conceptually the same, except that they are applied to observed data (Table 3) , instead of "made

up" numbers. Thus the model's distribution of hires into age classes

was determined by calculating the corresponding fractions of the 347 FTPP

hires recorded in the History File between Jan. 1, 1970 and Jan. 31, 1976.

The separation fractions could not similarly be read directly from

the History File data, because this calculation would give only the distribution into age classes of the total group of leavers in the 6 1/12

year period of recording, whereas what was required was the fraction of

each age class that was separated during a 5-year period. These fractions

were calculated by first estimating a representative number of leavers

in each class during a five year period by multiplying the number of

leavers from the History File by .82 (the approximate ratio between 5 and

6 1/ 12), and then dividing by the size of the class. The class sizes themselves were derived from "informal" averages (i.e. judged rather than calculated) of (a) the percentage breakdowns of FTPP staff into 5-year cohorts

for the years 1970-74 and (b) the breakdown of the FTP staff by sizes of occupational category components given as charts in [l4] and reproduced here

as figs. 3 and 4. It was necessary to "massage" the separation data to avoid

recording a separation rate greater than 100% in the highest age class.

30
STAFF

STAFF

(ADJUSTED)

5-YEAR

5-YEAR

CLASS

CLASS

HIRE

SEPARATION

5-YEAR

SEPARATION

"AGE"

HIRE

CMi—

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31

RELEVANT

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DATA

AND

MODEL

PARAMETERS
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32

Source:

NBS-023,

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&

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ES-01 
the estimated leavers in 5 years having exceeded the size of the class as

read from the chart. (We have previously touched on some of the likely

causes for such a discrepancy: unfilled slots, twice filled slots and

charges of occupational status.)

These details of the crude estimation of hire and leaver rates (assumed representative and constant) have been included to emphasize the

provisional nature of the projections from the Markov model. We remind

the reader again (cf. p. 23 ) that Markov models almost invariably will

produce ultimately stable distributions for the kind of system treated

here and that with some additional expenditure, more precise estimates

of the coefficients can be derived from the recorded data of the past 6

or 7 years. The real catch is that current knowledge gives very little

idea to what extent the forces shaping the hire and separation distributions may vary in the future (and indeed, how they have varied prior to

the recent past). The projections of current trends resulting from the

present exercise are valuable as a reference point, but if they are to be

succeeded by attempts to make accurate predictions, then we believe that

improvements are more likely to derive from analysis of causes of change

than from quantitatively refining the procedures we used for the estimation of parameters. We give three plausible areas for such analysis:

(1) possible relationships over time between the age distributions of various components of the nation's population and

the age distribution of hires in a group of interest, e.g. FTP

or FTPP at HE'S (the notion here is that the composition of an

3h
available "pool" may determine the composition of the recruited

force) ;

(2) relationships between length-of-service distributions and rank

(grade) structures or promotion policies. These might then be

combined with hire-age distributions to estimate the distribution of leaver ages.

(3) A study aimed at amplifying the data base by broadening the

subject population through redefinition. On pp. 2, 16 above we

intimated a suspicion that other components of NBS staff might

be equivalent to FTPP with respect to evolving patterns in age

distribution. If this suspicion were to be substantiated, the

other group or groups could then be amalgamated with FTPP for

the purpose of analysis, furnishing an expanded set of observations (from data files) from which to calculate presumably more

reliable parameters for a Markov model. On the other hand,

careful redefinition of the subject population could result in

a restriction of the data set if, for instance, it turned out

that persons whose status changes after entry to the staff constituted a small but variable fraction with substantially distinct aging characteristics. Even in this case, the reduced

data base might afford improvement if it contained fewer "outliers" to the average trends.

Briefly stated, the Markov model outputs (Table 4) show the staff

age distribution becoming stable (to many decimal places) in 31 cycles

(155 years!) but in practical measures in about 5 cycles or 25 years, with

an increase of less than 1 year in average age.*

*The staff class sizes have been rounded to whole values (integers) for

display only. They were carried in the calculations to "full" computer

significance

35
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The relatively large shifts in the early periods indicate that the

Markov process deduced from the information in the data files is probably

different from one which might have produced the starting (1975) age distribution (cf, p. 24). This conforms with a generally-held perception

that NBS ' current and recent state differs distinctively from that

during a preceding period of growth. It also suggests, among other things,

that in addition to examining the (relatively long term) steady state distributions of the Markov model, a collateral attempt tp extrapolate historical changes and intervals of changes in hire or separation patterns

might be useful in order to estimate parameters for periodic restart

of the model.

The general tenor of the shifts is a moderate increase in the fractions

of staff in ages 21 to 30 and 51 to 60, and a general decrease throughout

the middle aged groups 31-50 years old. The model's turnover (the number of

staff members who can be expected to be separated in a time step) varies

as the process progresses, even though the total staff size remains fixed

and the "loss intensities" (the separation rates relative to the age-class

sizes) are constant. This variation occurs because these rates are applied

to class sizes which vary over time. The stable distribution has a turnover of 482, that is, 32.9% for a 5-year period or 6.8% annually. This

represents a slight increase over the starting value of 441 (31.5% per period,

6.3% annually) which was derived from the History File data and which is

consistent with information graphed on p. 20 of [14].

Variations in the losses from individual classes could be disconcerting

from the point of view of management if the model is realized in flesh and

37
blood terms, particularly since our particular model presents a system

which shifts drastically in the early stages as noted above. The number of

56-60 year olds increases in one period by over 50% (from 91 to 140) to a

value close to its size in the stable distribution (147). Separations from

this age-group, currently numbering about 13 annually, would occur at almost

double that rate* (about 21 annually) during the next ten years. This

points to a fairly heavy exodus of those staff members now of ages 46-55

who constitute the bulk of senior scientific leadership at NBS . To a somewhat lesser extent the "transfer out" of the "young PhD" class (aged 31-40

at separation but now under 30) will decrease during the same period from

approximately 21 annually to 18.

AGE CLASS

YEAR 31-35 36-40 41-45 46-50 51-55

1975 3.5 1 2 3.5 5 1980

1985

53124 5 4 2 1 3.

1995

1990 41235 52413

TABLE 5. RANK ACCORDING TO CLASS SIZE (5 LARGEST CLASSES)

FROM COMPUTER OUTPUT OF MODEL

A notion related to the possible cause for sudden losses of large

numbers of critical senior personnel is the possibility of an "age bulge"

propagating wavelike through the staff over time, as a result of massive hires

of young scientists in the 1950's and early 1960's. We therefore offer table

5 showing progressive class size ranks, but leave the risk of interpreting

it to the reader.

*Note that this is a rate in people/year, not to be confused with the

"separation rate" which is a constant fraction of the class size. In

fact the numbers cited are calculated by multiplying the separation rate

for the class (.7362) by the class size (see tables 3 & 4).

38
As we have implied earlier, the desireabi l i ty ot anv particular age

distributions and strategies for their attainment or avoidance are matters

which cannot be settled by Markov models (although appendix B includes

discussion of the relationships among the hire, separation and stable staff

age distributions). The principal benefit of the Markov model is that it

describes the resolut jon of current trends instead of merely designating

their initial directions. In addition, it can be used to determine the

resolution of specified modifications in these trends.

39
5.

BIBLIOGRAPHY

1. Bartholomew, David J. , Stochastic Models for Social Processes , (New

York: John Wiley & Sons, 1967).

2. Davies, G. S., "Consistent Recruitment in a Graded Manpower System,"

Management Science, Vol. 22, No. 11, pp . 1215-1220, (July 1976).

3. Feichtinger, Gustav and Mehlmann, Alexander, "The Recruitment Trajectory Corresponding to Particular Stock Sequences in Markovian

Person-Flow Models," Mathematics of Operations Research, Vol. 1,

No. 2, pp. 175-184, (May 1976).

4. Forrester, J. W. , Industrial Dynamics , (MIT Press, 1961).

5. Freese, Ralph and Johnson, Charles R., "Comments on the Discrete Matrix

Model of Population Dynamics," Journal of Research of NBS , Vol. 78B,

No. 2, pp. 73-78, (Apr. -June 1974).

6. Gornick, F. and Passaglia, E. , A Dynamic Model for a Population of

Durable Goods , (Draft of a Report from TMR of NBS, 1976).

7. Jaquette , D. L. and Nelson, G. R., "Optimal Wage Rates and Force Composition in Military Manpower Planning," Management Science, Vol. 23,

No. 3, pp . 257-266, (November 1976).

8. Kemeny, John C. and Snell, J. Laurie, Finite Markov Chains , (Van

Nostrand, Princeton, NJ 1960).

9. Pearl, Martin, Matrix Theory and Finite Mathematics , (McGraw Hill,

NY 1973).

10. Simonson, E. and Weiser, P. (eds.). P sychological Aspects & Physiological Correlates of Work and Fatigue , (Charles C. Thomas, Springfield,

IL 1976).

1 1 . Schinnar, Arie P., Eco-demographic Planning and Forecasting

(to appear).

12. Smith, A. P. (ed.). Models of Manpower S ystems - Proceedings of a NATO Conference, (American Elsevier, NY 1971).

13. Stahl, Michael J. , "Organizational Variables Associated With Innovation and Productivity in Research and Development," - Paper //FP2.12,

1976 Joint National Meeting of The Institute for Manavement Science/

Operations Research Society of America, (Phi la. „ PA., April 2, 1976).

14. Unsigned Report, "Employment Trends 1964-1974," National Bureau of

Standards, Office of the Associate Director for Administration,

Personnel Division.

U0
BIBLIOGRAPHY (Contil.)

1 *>

Young, A. and Almond G., "Predicting Distributions ot Stall, " Computer Journal Voi . ), pp. J46-Y0 (1961).
PREFACE TO APPENDICES

Appendix A is a restatement of the material in Sections 3 and 4 of

the report, using matrix notation, augmented by a brief discussion of some

basic properties of discrete time Markov Models. Appendix B is an analysis

of the algebraic relationships among the quantities that define the Markov

transition probabilities (namely, the Hire and Separation age distributions) and the output of .the model (the staff's stable age distribution).

h2
APPENDIX A: THE MODEL IN MATRIX FORM

The Markov Model traces the progressive changes in the age distribution of staff over a sequence of time steps, through linear equations

whose coefficients are rate constants (called transition probabilities

or transition coefficients) , which relate the components of the distribution after each time step to the components of the previous distribution.

According to the customary classifications of Markov models ("Markov

Chains" or "Markov Processes") , the model of this report is a "Finite State

Discrete Time Simple Markov Chain with Stationary Probabilities": the

set of states (age classes) is finite, the process operates (is recorded)

at regular time periods rather than continuously, and the equations that

project the classes one period forward require only the coefficients and

the population's current status. (A "simple" model (in the preceding

sense) is also called a "first order" model. If, instead, information

about both the preceding period and the current period were needed to

project forward, the model would be described as second order; a chain

whose equations include data for k periods including the current one to

project the next is called a k-th order Markov chain.) Finally, the coefficients, as stated previously, are constant.

Most elementary texts simply use the term Markov Chain for the kind

of model described here, and we adopt the abbreviation Markov Model from

this point on. For an introduction to Markov models see references [8, 9]

of the report or Introduction to Finite Mathematics by the authors of [8].

A readable survey of more general Markov models is contained in "Statistical Methods in Markov Chains" by Patrick Billingsley, The Annals of

Mathematical Statistics, Vol. 32, 1961, pp . 12-40.
We have a staff of n members divided into m classes according to

age, with the number of members in the i-th age class denoted by n^(x), where x

is a discrete variable representing time steps or stage intervals over

which the age distribution will be traced.

The values n^(x) define (row) vectors N(x), N(0) being the initial

distribution, with n^(x) _> 0, i = l,2...m, x = 0, 1, 2, ...» and

m

(1) E n . ( x) = n for x = 0, 1, 2, ... i=l 1

The constant transition matrix P = (p..) with

m

p.. > 0, E . p.. = 1 for i = 1, 2, ..., m,

ij ~ 3=1 ij

determines the stage vectors, of expected age-class sizes, N(x), through

(2) N(J) = N(t- 1)P = N(0) p

T

for x = 1, 2, ... where P is the x-th power of P.

Frequently, it is convenient to replace the distribution vector N(x)

with a proportional distribution vector F(x) = (f (x) f (x) . . ., f ( T 12m ))

where f . ( x) is the fraction of the staff in class i at time t. Thus

l f^(t) = n

i

(x)/n, with

m

(la) E f.(x) = 1 for x = 0, L, 2, ... i=l

1

and, analogously to (2),

(2a) F(x) = F(x-l)P = F(0)P T.

The distinction is essentially semantic, but the representation in terms

of F(t) bypasses the necessity of rounding the expected values n^( T

)

to integers for display of intermediate outputs of the model, to

avoid confusion in interpretation of non-integer class sizes.

hk
To determine the entries of P, we first label two artificial

"classes" outside the staff structure to represent a source of hires into

the classes 1, 2, m, and a sink for separations from these classes.

(Note that they do not appear in the age distribution.) We assign these

two artificial classes, respectively, the indices 0 and m+1. Thus p.

l , m+1

(for i=l, 2, . .

.

,

m) is the fraction of the i-th class that will be separated, and p^ (for j=l, 2, ..., m) the fraction of hires into the j-th class,

in a time period. For mnemonic convenience we will now relabel the quantities

p. , as s and p as h. . Then, with a., (for i, j=l, 2, ..., m)the fraction ± 5 Iirr_L x OJ J 1J

of the i-th class that will actually age to the j-th class, in a time period.

(3) p.. = a.. + s.h. i,j = 1, 2, ..., m.

ij iJ i J

In our aging model

a. . will be zero for j ^ i+1 and

a . . , , will be 1-s . l , l+l l

The relationship (3) rests on replacing the total number of separatees

in a period by hires to maintain the constant staff size n. Thus the

products s.h. (j = 1, 2, . . .

,

m) distribute the replacements for the i-th m

class separatees among the age classes—clearly, = 1 by definition.

The remarkable and important fact about Markov models is that if the

matrix P, or merely some power of P, is positive, i.e., has no zero entries

(such a matrix is said to be regular or in algebraic terminology primitive) , then the process defined by the model is guaranteed to converge

as T increases.
In other words, the sequence N(0), N(l) ... approaches a unique vector

N( 00 )=N( 00 )P. Moreover, this "solution" vector does not depend on the starting

distribution W(0). Thus writing N(°°) as N we obtain

oo (4) N = HP = N(0)P

. OO OO OO (5) p p = PP = p

i.e., the powers of P converge.

Equation (1+) is a linear equation. Therefore it is valid for arbitrary scaling of the vector N(0). Thus for any real number a and N(o) =

ctN(O) ,

(1+a) N = aN = N(o)P°°, as well.

Hence the substitution of a fractional distribution for the distribution of

class sizes as in equation 2a could be made before or after the solution

of the model with equivalent results. (a = l/n.) It tarns out that the rows

OO of P are all equal to the stable vector in normalized form, i.e. in terms

of our model, a stable fractional age distribution vector. This means that

while the starting distribution might affect the speed of numerical convergence to a particular number of decimal places, the final distribution

itself is determined entirely by the transition matrix. Transition matrices for our age analysis can be expected to be regular except for very

unlikely hire and separation distributions and thus, for us, the FTPP staff

does approach a stable age distribution determined solely by the separation rates and hire age distribution assuming they are constant. Moreover, whatever the staff age distribution at any point in time, altering

the separation rates and hire age distribution to "new" constant values

will define a "new Markov process at that point, allowing us to determine

1+6
in advance the direction of change in the staff age distribution if the

hire and separation distributions can be controlled by management policy.

Appendix B develops algebraic relationships among the staff distribution,

the hire and separation distributions and the turnover rate.

The specifications of the modeled system were as follows:

(1) The FTPP staff was divided into 10 classes defined by 5-year age

intervals from 20 to .69 • The initial distribution was derived from reference

[1U].

(2) The total FTPP staff size was assumed to remain constant at

1400 persons (from [14]).

(3) Age dependent separation rates and a hire age distribution according to 5-year age intervals were derived from the History Data File

and assumed to remain constant. All hires and separations are assumed to

occur at the beginning of a time step. (In calculating the averaged hire

and separation distributions we did not discard data records outside the

model age limits, but instead included those under 20 in the 20-24 year

old class and those 70 and older in the 65-69 year old class.)

(4) The time step or "stage interval" for the model is the same as

the class age interval, 5 years.

The numerical values for the components of the vectors (h ) and (s^)

for the 5-year class interval FTPP model were determined as follows:
The FTPP hires for the entire 6-year period (Jan. 1970 - Jan. 1976)

covered by the History File were grouped by 5-year classes of age at hire,

and the fractions of the total that fell into each class were assumed to

define a representative hire distribution, i.e. these fractions were considered to define the values for (h.).

J Treated similarly, the FTPP separations from the History File gave

an age distribution of all leavers from the staff during the same period.

Call these fractions g^ (i=l, ...m). The product of g^ by t, the

average 5-year turnover, yields the representative number of "departures"

from the i-th class. Call these quantities d^. Thus

8i

t = d

i

’ 1 = 1 > 2 >

• • • > 10

and

S i =

> i = 1, 2, . . . , 10

where as before the n^(0) are the initial sizes of the age classes, the

average number in each class during (1970-1976)

Two extreme cases for the separations can be analyzed fairly easily.

(1) If s = 1 for all classes, i.e. turnover is total in each period,

then intuitively, the age distribution should be determined solely by

hires. This is indeed true of the model, for the transition matrix becomes

P = (P ±j

) = (h.),

m which since ^h^ = 1, has the property

2

P = P = . . . = P .

It follows that

N(l) = N(2) = ... = N = (h

1

n, h^n, ..., lyi)

(2) If there are no separations except for mandatory retirement, then

hQ
s. = 0 for all classes except s 1

1. P now becomes m

Pr . = 1 i < m

l , l+l

p

ij = 0 i < m; j ^ i+1

p . = h. j = 1, 2, . . . , m.

mj 1

Or in tableau:

P =

0 0 0 1

h

We see that the equation N = WP , which determines the stable vector

N--= N(°°), takes the form

n ) - (h n , n +h n , m 1 m i 2 m

n , + h n ) m-1 m m

or equivalently

n

1

h.. n

1 m

n = n +h n =

2 12m h n +h n

1 m 2 m

<h

i + h

0 )n

2 m

n = n .,+h n = m m-1 m m

= ( h_ + . . .+h )n

1 mm

If n =0, N = (0, 0, ...» 0). Otherwise, we use the notation of

m

equation 4a and a device suggested by that equation, to scale so that

n = m

49

1 and obtain
"l

h

l

n

2 ‘ h1 + h2

%

nm m

Eh.

1

1

Normalizing, i.e. rescaling through division by the sum Zih.5 we

arrive at a fractional distribution f , f„, ...» f (following the nota1 z m

tion of equation (2)). The values n. are determined by n. = nf. where as

i ii

usual, n is the size of the staff.

Thus the fractions of total hires no older than the age thresholds for

the age classes are proportional to the class sizes in the stable age distribution when there are no separations except from the oldest class.

This "top heavy" structure from the model thus confirms intuition as in

the preceding example.

A "pure" hierarchical age model in which there are no losses (separations) except from the highest age group, i.e., mandatory retirements,

and in which al 1 hires are into the lowest age group, is an example of a

Markov model which does not converge to a stationary state because its

transition matrix is not regular. The transition matrix is, in fact, the

cycle matrix

~0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1 J

0 0 0 0

50
Successive powers shift the columns one position to the right with

the last column moving to the first; i.e., the columns cycle so that all

powers of the matrix contain zero elements. The action of this matrix

on a distribution vector permutes its elements cyclically one place in

each time step. If the starting configuration happens to have equal numbers in all ages, the organization will appear to be age stable, but any

other (non uniform) configuration will cycle endlessly.

Thus with a staff of 150 stratified into five age groups with ages

20-29, 30-39, 40-49, 50-59, and 60-69, mandatory retirement at 70, all

hires (replacements) at age 20 and a 10-year time step, if there are 30

individuals in each group initially there will always be 30 in each group,

30 recruits at age 20 replacing the 30 retirees from the high group at

each step. If, instead, the staff were initially distributed into the

groups in numbers

20-29 30-39 40-49 50-59 60-69

10 20 30 40 50

the staff distribution would be successively

50 10 20 30 40

40 50 10 20 30,

completely recycling each 5 periods. This example tacitly illuminates an

important consideration in age models which is sometimes overlooked in

applications, particularly when the transition coefficients are regarded

as probabilities, to wit: the time step size must match the interval

defining the age groups. If not, then serious distortions may occur, as

the following simple example demonstrates.

51
Consider a constant-sized personnel staff stratified by 10-year age

brackets: 20-29, 30-39, 40-49, 50-59, 60-69, with mandatory retirement

at age 70, replacements at age 20 and no other losses. For a simple

Markov transition process with 10-year time-step, the transition matrix

is the matrix

0 10 0 0

0 0 10 0

0 0 0 1 0

0 0 0 0 1

1 0 0 0 0

That is, in each 10 year period the fraction of any 10-year age class

that ages into the next class is 1, or in terms of probability, a person

in any class at time t will be in the next higher class with probability 1

at time t

10 years.

To represent an annual aging process, i.e., to use a one-year time

step with the same structural description (5 classes determined by age

decades) , we might expect the matrix

.9 .1 0 0

0 .9 .1 0

0 0 .9 .1

0 0 0 .9 .10 0 0

0

0

0

. 1 .9

to furnish an annual aging model equivalent to the previous one. Here

the row entries purport to show that on the average, annually, 1/10 of the

individuals in a (10-year) class age into the next higher class while 9/10

"stay put." We see that this requires first, an additional assumption,

namely, that within the classes, the members are divided into equal sized

subclasses corresponding to the 10 years in the decade of ages, and indeed,

because this process is to be iterated, that the condition must persist

52
over time if constant transition rates are to be meaningful. An un- pleasant inconsistency then results: After one year 1/10 of some age

class with 200 members (i.e., 20 individuals), progresses into the next

class. Then, as this class is homogeneously distributed, the tenth

that ages into the next class after another period, includes two of the new members so that in expectation, these two have aged over ten years in two

Moreover, this new model is no longer cyclic but has a regular matrix:

the fifth power of the matrix has no zeros and so the process yields

eventually a stable distribution with any initial vector. Interestingly,

the stable distribution is the uniform distribution, which we recall to

be the stable vector of the cyclic model, the difference, being, of course,

that for stability in the cyclic system, the initial distribution had to

be the stable one.

In formulating the erroneous* model above, we chose an annual time step

for convenience. Let us look now at some time steps larger than the

class interval. A time step of twenty years would induce the transition

matrix

*This age model can be defined as erroneous in terms of Markov theory in

that the stipulated structure with an annual time step is in fact not really

a Markov process because it has "memory" which is being ignored in the representation. That is, the individuals going from, say, the 30-39 year age

class to the 40-49 year age class are in reality 40 years old, but the

model "forgets" this in distributing their ages evenly through the 40-49

interval. The result is a Markov model of something, but not of the intended

age system.

53
0 0 10 0

0 0 0 1 0

0 0 0 0 1 1 0 0 0 0

0 10 0 0

L

which satisfies intuition by cycling the distribution entries two places

each step instead of one as in the base model. More generally: the

matrices representing time steps which are multiples of ten are powers of

the "original" or ten-year matrix. Following the procedure of the "erroneous

model", y-year time step -with y between zero and ten, would result in a matrix

of the form r

1-a

0

0

0a

a1-a

0

0

0

0 0 0 a 0 0

1-a a 0

0 1-a a

0 0 1-a

in which a = y/10. a = 1 corresponds to the base model of the example,

a = 0 gives a model for which every age distribution is stationary (the

identity matrix I) . For 0<a<l all models yield the uniform distribution

asymptotically.* If y is greater than 50 years, the model has substantially

no meaning because it gives a rate matrix identical to that for y

- 50 years.

The "smear" phenomenon exhibited by these models would not necessarily

distort reality in situations revolving about grade or rank (instead of

age), when the coefficients might represent actual promotion policies.

Also, age systems with "proper" time steps, with losses (separations) at

rates 1-a and with hires for direct replacement could be represented by

such models

*It is easy to see that any such matrix is regular, for with 0<a<l each

successive power will add positive entries to a new circulant diagonal.

The (n - l)-st power, i.e. the result of n - 2 matrix multiplications,

will contain no zeros.

5U
APPENDIX B: ANALYSIS OF THE MARKOV MODEL*

B. 1 Formulas for Steady-State Quantities

A principal output of the model is the limiting or "steady-state"

age distribution of FTPP staff. In Section 3, we showed how that distribution could be calculated numerically by repeated application of the

model's transition equations, beginning with an initial age distribution.

In this self-contained technical appendix, we will give an algebraic

treatment, leading to explicit formulas for the limiting distribution

(and some related quantities) in terms of the model's input data. Relations among these inputs, necessary for a consistent model, will be identified. Finally, an analysis will be given of what limiting age distributions are attainable if some of the model inputs are fixed but others are

adjustable.

The steady-state age distribution will be described by a vector

F = (f f ), whose components

1 m

f = limiting fraction of FTPP staff lying in i-th age group

are nonnegative and satisfy

(1) f. = 1.

An auxiliary model output, which will also figure in the analysis, is

t = limiting turnover rate.

Other model outputs, which might for example be of interest should

the issue of age-discrimination arise, are the quantities

A^ = limiting fraction of staff -.leavers lying in i-th age group;

*By A. J. Goldman, Applied Mathematics Division.

55
these quantities form a vector A = (A.,,..., A ) whose components A. are

i m 1 nonnegative and satisfy

(2) ETMA. = 1.

"V Note that t and A are defined only in terms of the steady state; a turnover rate or a leaver-age distribution constant over time is not assumed.

The inputs to the model can be described by two vectors. The first

one, H = (h1 ,...,h ), gives the age distribution of new hires; its components

1 m

h_^ = fraction of new hires lying in i-th age group

are nonnegative and sum to 1. The second vector, s = (s,,...,s ), has as

1 m components the separation rates

s^ = fraction of i-th age group separated during a time period,

which satisfy the conditions 0

' <_ s <_ 1 . Because of the mandatory retirement

age, sm

. 1.

(3)

It follows from these definitions that

t= z“ s.f.

1 li

and

(4) A . t = s.f..

l li

Equation (4) can be used to calculate A, once F and t are determined. The

numerical values of these various quantities, for the particular case of

the NBS FTPP staff, are given in Table 6.

We will first develop formulas for F (and t) in terms of h and g.

For this purpose, note that in the steady state a fraction t of the total

FTPP staff leaves each period; since the total staff size is assumed constant, the same fraction t of the next period's staff consists of new hires,

56
MODEL

PARAMETERS

INITIAL

VS.

STEADY

STATE

("FINAL")

DISTRIBUTIONS

inK

in

W

CO <] W 53 O M

<£ in

Eh25H • o H <H

CO „

25 Cm m

Ph

Cu

in -iH< w M

Pn

25 /'n O -H M CO H ^

COwo

<d

CO

CO<

rJu

00 nO nO CN o o rH LO rH rH o

CO

O CO LO LO LO co o CO o o

rH rH i—iH rH rH 1—

orH o o LO o o LO LO LO LO LO o

CO on LO 00 LO CN NO CM o o rH rH rH rH rH o

rH

<r 00 rH CO o o CO CO i—o oo

lo on '—

t—i—

00 <r o • f—

rH CM CN CN rH rH <r CN »—

<r

CN nO CM oo LO i—

LO o CO

CM rH LO CO rH ON co o • rH CM OJ CM CM rH CN i—

CD00 <

O'* LO LO LO NO CN LO CN o at00 NO CM o 00 00 O rH <N oo o o cfl rH rH 1 1 i—

CN o S-i at>C

ON 00 <y\ nO ON CN NO o

UO rH CN o O rH rH LO i—

o i—

rH rH rH '—

t—

rH o

rH

Ml- C\

CO o o i—

rH i—

00 CN i—rH • • CO nO LO <r LO LO c <r 00 Mr nO 1—

CO

LO LO LO LO LO CO • • • • <—

nO rH nO 00 r>» CN rH <r rH NO

C\J LO LO LO LO NO CO CO

6^5 6^ M 5-i cd at > > o o e e o <d* LO LO CO CN o 5-i 5-i on 00 o o rH NO rH o 3 3

r—

o nO ON CN CN co r^. o H

H

no CM 1—1 rH CM CN ON o 5-J rH

rH a) to

oj 3

>N 3 1 c

lO c

Nf 1—1 co 00 00 LO o LO NO o o

CN LO co CN rv NO CN 00 o o '

1 nO rH 00 co o o o »—

CN CN rH o o o o o o o

rH

lO o LO o lO

O LO o LO o

CN CO CO LO LO NO NO r- 1 rH v£> i—

v£> r—

rH v£> rH CN CN co CO LO LO NO NO

-J

<1 t—

CN CO LO NO oo o H «—

OH

57

TABLE

6.

MODEL

PARAMETERS,

STARTING

VALUES

AND

STEADY

STATE

OUTPUTS
and so a fraction h^t consists of new hires into the i-th age group. Since

the time period's length coincides with the span of each group, such new

hires are the only source of personnel for the first (i.e., youngest) age

group, and so that group's balance equation is

(5)

f

± = hxt.

For j>l, the j-th group consists not only of new hires but also of those

staff members who spent the previous period in the (j-l)-st group and

were not separated; thus the balance equation is

(6)

f = h t + (1-s )f (j>l).

It is convenient to define

s' = 1 - s .

,

i l

so that (6) can be written

(7) f . = h t + s' f (j>l),

and also to define

(8) a.. = } s ' (a.. = 1 for i > j).

ij k=i k ij -

Then the solution to the recurrence (7)

,

with (5) as initial condition, is

(by mathematical induction)

(9) f. = t ^ ,a. . h..

J i=l iJ i

To determine t we sum (9) over j and apply (1) , obtaining

1 = ETM f . = t 5® Z] o..h.

1 j 1=1 i=l ij i m , „m

= t i h. Z. . a...

1=1 l 1 = 1 ii

In the double sum, the coefficient of each contains the positive summand

and so is strictly positive; since at least one fn is strictly positive

58
(they sum to 1), the same is true of the double sum itself and we may write

(10) t = 1/E

1

!

1

n h. Z m

. o. . > 0.

1=1 x 3 = 1 ij

It now follows from (8) and (9) that f . = 0 if and only if (a) h =0 and

J

(b) for each i with i < j and h > 0, there is a k with i £ k < j such

that s =1. For example, f =0 if and only if h =0 (here (b) is satisfied

k 1 i

vacuously)

From (4) and (9) we have

(11

> y

- s.(£./t) - s.X^ o^h..

B.

Attainability of "Target" Distributions

We turn now to a simplest of the questions arising in connection with

using the model for "control" purposes rather than just predictive ones.

That question is: given one of the two vectors (H, S) as fixed, and the

other as freely adjustable, which steady-state age distributions F can be

attained? In other words, we regard F and one of the pair (H, !d) as given,

and ask whether (and which) choice of the other member of the pair will

—

y

* make F the solution of the balance equations. This is of course a highly

idealized situation, since in practice the parameters would be neither precisely fixed nor freely adjustable, and since dynamic as well as limiting

system behavior would be of interest; more realistic "scenarios" may be

formulated and analyzed in subsequent work.

Suppose first that p" and are given. From (3) and (10) it follows that

(12) s f >0 for at least one i

i i

—

y

is a first compatibility condition on the pair (F , S) . By (5) and (6)

,

(13) h

x = f 1

/t , h

j = (f . - s

j

;_ 1

f

j _ 1

)/t (j>D,

59
so that h i s determined uniquely. The requirement that its components

sum to 1 is satisfied automatically, since (13) and (3) give

E 1

h

j - [£

1 + E I=2

(£

J -

- [(f

l + Z

2 V

S

i

£ i

]/t

= [ZTM f ± - ZTM s T f i

] /t (since s^ = 0)

= [£“ (i-spf.]/t = [Z“ = 1.

The requirement that h >_ 0 leads, by (13), to the further compatibility

condition

<14)

on the pair (?, £3 ) . X is uniquely determined by (11)

.

Next, suppose F and H are given. It will prove convenient to introduce

(15)

J = min { j : h^ > 0}.

It follows from (5) and (6) that

(16) f = 0 for j < J,

and thus also that

(17) t = fj/hj,

so that (10) gives

(18) fj > 0;

(16) and (18) are compatibility conditions on the pair (F , H)

s =1 is required, and for j < m, (7) yields m —

U9) - (C - (if f. . > 0), J-l

The choice

so that s. is uniquely determined if f. > 0; if f. =0, then the value of

l i l

s. is immaterial for a steady-state analysis. Since 0 <_ s^ <_ 1, (19) yields

0 < f.-h.t < f. . “ J J - 3“!

6o
as a further compatibility condition on (F, H) ; with the aid of (17),

this reads

(20) h.f < h f. < h f,

1 + h.f (if f . > 0) . J J - J j - J j-l 3 J 3-1

The remaining compatibility condition, also obtained from (5) or (7)

and (17) , is

(21) h f. = h.f (if j=l or f = 0).

0 1 J d J i

A somewhat less natural pair of problems involves stipulating a particular choice of a> rather than F, as the "target" to be attained. In

these problems we regard a and one of the pair ( H, S ) as given, and ask

whether (and which) choice of the other member of the pair will "realize"

A in the steady state.

Suppose first that a and S are given. The compatibility condition

(22) A.=0 if s

. =0

x x

is an immediate consequence of (11). For what follows, it is convenient

to introduce the cumulative sums

(23) A. = E:J X. , H. = Z^h.

J 1 i ill

Using (4), we can rewrite (6) as

(24) f . = (h. - A

j _ 1

) + f._

± (j>l).

This recursion, with (5) as initial condition, has as its solution

(25) f . = (H. - A. ) (A =0)

,

J J J-l o

leading via (4) to

j (26) A. = s.(H. - A. )

J J J J-l

Now let

(27) j (1) < j (2) < ... < j (y) = m

6l
consist of those je{l,2, . .

. ,m} for which s^ > 0; here j (y) = m because

s = 1. It follows from (26) that m

H . , v A . , \ i

""f A , , \ / s . , \ ( y j(v) j(v)-l j(v) j(v)

which by (22) can be written

(28) H . . . = A . / ,v + A. Js

j(v) j(v-l) j(v) j(v) (j(0) = 0)

These values automatically obey the first of the conditions

H = 1 mH„, > H -/ \ j(v+l) - j(v)

which are necessary for the H. ' s to arise from an age' distribution H, while the

J second of these requirements leads via (28) to the further compatibility

condition

(29) A.. ,/s., , N > A . , N s' ,/s.. , j(v+l) j(v+l) — j(v) j (v) j (v)

on the pair (A, s)

If all s

j > 0 (i.e., y=m in (27)) then (28) determines a unique h«

If y<m, then the sets

(hj : j(v) < j _< j(v+l)} (v=0 , 1 ,

. . . , y-1)

are only determined by (28) "up to" their respective sums, so that there

are multiple choices for S, leading via (25) to different (in general)

solutions F of the balance equations. In the context of (25), an interesting

expression for t is obtained by summing over j and applying (1) . This yields

1 = t(Z?H. - ETM

1

A.)

1 J 1 J

= t(Em

Z

j h. - E m

"}z j A ) j=l i=l l j=l i=l i:_

= t(Zm

h.Zm

.1 - Z^A.ZTM"1 !) j=l l j=i 1=1 l j=i

= t[Zm

h . (m+l-i) - Zm

"?-A . (m-i) ] i=l l 1=1 l

,m-l

= t [ (m+1) - Z‘ih. - m( 1-A ) + Z. iA . ] 1 l m 1 l

= t[l + A - A, ] , A 62
where

(3°) A

x = ^«. ,m = l ih.

h lx

are the average steady-state leaving age and the average hiring age, respectively, in age-group span units. Thus

(31) t = 1/(1 + A

x - ^

—y —y Next suppose that A and are given. Then (25) and (31) determine

^ uniquely. We claim that

(32) H >_ A (j=l,2,...,m)

~y ~y is a necessary compatibility condition on the pair (A, h) • To prove this,

first note that if t = 0 then (5) and (24) would imply that all f =0, contradicting (1). Therefore t>0, and so the condition f >0 applied to (25)

yields >_ A ^

. If strict inequality holds in this relation, then (26)

uniquely determines s as

(33) s. = A./(H. - A. J (if H. > A. ) 3 3 3 3-1 3 J-l

and so the requirement s . j< 1 leads to (32). If equality holds in the

relation then s. is not determined but its value is immaterial for the

J steady-state analysis since fj =0 by (25); this implies by (4), so

that (32) holds as an equality. Thus we have the further compatibility

condition

(34) H. = A. implies A. =0.

J J-l J

As the final step in this Appendix, . we . assume that

both ^ and A are given. Interestingly, in this case, it is also possible

to stipulate a "target" value of t, with 0 < t <_ 1 . It follows from (5) that

(35) h

1 = f1

/t ,

63
and (36) from (24) that

hj +1 = A + ( f

j +1 “ f j)/t (j < m) ,

so that H is uniquely determined. The requirement h^. +^

il 0 leads via

(36) to the compatibility condition

(37) A t + (f j+1 - f ) > 0 (j <m)

on ("?, A, t) . It follows from (35) and (36) that

H

m h

+ ir

1 v <£

i

/c) + Vi + »r

1( vi - f

3 )]/t

= (f/t) + (1 - A ) + (f - f )/t

1 m m 1

= 1 - (A - f /t) = 1 , m m

as desired, where the further compatibility condition

(38) f = A t m m

follows from (4) since s^ = 1. Requirement (32), for i = 1, imposes via

(35) the compatibility condition

(39) f 1 At,

for i = 1, while the calculation from (35) and (36) that

H

i+i

* h

i + Ei

h

j+i

" (f,/t) + A. + - £.)J/t

1 i 1 3+1 J

- (f

i

/c) + (A

i+i - h+i> + <£i+i - fi

)/t

- A

i+i + (fi+i

- h+i

c)/t

shows that (39) is also the necessary and sufficient condition for (32)

when i> 1. Given (39), the same calculation shows that (34) will be satisfied. The separation rates s^ are determined to the same extent and in

the same manner as at the end of the previous paragraph; requirement (29)

turns out to be equivalent to (37) .
NBS-114A (REV. M-77)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 78-IU 582. Gov’t Accession

No.

3. Recipient’s Accession No.

4. TITLE AND SUBTITLE

Projecting the Age Distribution of Full-Time Permanent

Professional Staff at the National Bureau of Standards

5. Publication Date March 1978

6. Performing Organization Code

NBSIR 78-1458

7. AUTHOR(S)

L. S. Joel

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2050151 (7120151)

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as Item 9

13. Type of Report & Period Covered Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

This report presents a simple mathematical model to project the age distribution of the full-time permanent professional (FTPP) staff of the National

Bureau of Standards. The report includes a brief description of types of

models currently in use for manpower analyses, discussion of the probable

data requirements for reliable models, some staff profile information which

supplements material in recent administrative reports, and the description

of our model. The principal projection is that under "status quo" assumptions, (FTPP staff size, age distribution of hires, and the separation rates

from age cohorts remain constant) the FTPP staff will in about 25 years

reach a steady-state age distribution with average age about 1/2 year higher

than its current level (just over 42 yrs . )

Comparison of this distribution with the present one shows moderate increases in the fractions of

staff in ages 21-30 and 51-60, a fractional rise in turnover, and a general

decrease in the groups 31-50 years old. A near term effect is the intensified loss of senior scientists.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Data analysis; discrete Markov models; manpower planning models; simple

Markov models

18. AVAILABILITY ITT Unlimited

| For Official Distribution. Do Not Release to NTIS

I Order From Sup. of Doc., U.S. Government Printing Office Washington, D.C. 20402, SD Stock No, SNOO3 -OO3

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