On the influence of production on utilization functions: Obsolescence or increased use?

Jointly published by Elsevier Science B. V., Amsterdam

and Akad~miai Kiad6, Budapest Scientometrics,

Vol. 34, No. 2 (1995) 285-315

ON THE INFLUENCE OF PRODUCTION ON UTILIZATION FUNCTIONS: OBSOLESCENCE OR INCREASED USE?

L. EGGHE,*.** I. K. RAVICHANDRA RAO,** R. ROUSSEAU**,***

* LUC, Universitaire Campus, B-3590 Diepenbeek, Belgium and UIA, Informatie- en Bibliotheekwetenschap, Universiteitsplein 1, B-2610 Wilrijk (Belgium) /

** DRTC, ISI, 8 tn Mile, Mysore Road 1~ V. College Pos~ Bangalore, 560059 (India) *** KIItW~, Zeedijk 101, B-8400 Oostende, Belgium and UIA, Informatie- en Bibliotheekwetenschap,

Universiteitsplein 1, B-2610 Wilrijk (Belgium)

(Received April 10, 1995)

We study the influence of production on utilization functions. A concrete example of this is

the influence of the growth of literature on the obsolescence (aging) of this literature. Here,

synchronous as well as diachronous obsolescence is studied. Assuming an increasing

exponential function for production and a decreasing one for aging, we show that, in the

synchronous case, the larger the increase in production, the larger the obsolescence. In the

diachronous case the opposite relation holds: the larger the increase in production the smaller

the obsolescence rate. This has also been shown previously by Egghe but the present proof is

shorter and yields more insight in the derived results. If a decreasing exponential function is

used to model production the opposite results are obtained. It is typical for this study that

there are two different time periods: the period of production (growth) and - per year

appearing in the production period - the period of aging (measured synchronously and

diachronously). The interaction of these periods is described via convolutions (discrete as well

as continuous).

1. Introduction

Growth and aging (or 'obsolescence') axe two topics which have attracted a lot of

attent.;on in the informetric literature. 1-7 Reviews have been published by Gapen

and Milner s and, more recently, by Line 9 (see also the chapter on obsolescence in

Ref 10). Some investigators constructed mathematical models describing growth or

aging, 11 while others went one step further and studied the influence of production

(which may or may not grow), or of cumulative production (which always grows) on

Permanent addresses: Egghe: LUC; Rao: DRTC; Rousseau: KIHWV.

0138-9130/95/US $ 9.50

Copyright �9 1995 Akad~miai Kiad6, Budapest

All rights reserved

L. EGGHE et al.: INFLUENCE OF PRODUCTION ON UTILIZATION FUNCTIONS

obsolescence i.e. decreasing u s e . 12-14 Mathematical models to study and explain this

influence were introduced by Hargens and Felmlee, 15 and by Egghe. 16 As the relation

between production and use can be studied from different angles we will clearly

define the notions used in this article and the relations we will investigate.

Consider a group of articles, e.g. all articles published in the year Y in the journal J.

Now, one considers the age distribution, denoted as c(s), of all items cited in the

reference lists of all these publications. Here c(s), the aging or citation function,

denotes the number of references made to items which are s years old. The rate at

which c(s) changes is then called the synchronous utilization rate (The notion of

'rate' will be clarified further on in this introduction).

If, on the other hand, one considers the same group of articles, but now the number

of times these articles are cited t years after publication is studied, then the rate at

which this number of citations changes is called the diaehronous utilization rate (or

obsolescence rate). Also this quantity will be denoted as c(t) and we will make it

clear whether the synchronous or the diachronous case is meant. Further, we have to

define what is meant by 'rate'. In the discrete case, the variables s and t can only

assume integer values (s,t �9 N) and the rate function of a function f, denoted as rf, is

simply:

rf(t) : f(t + 1)/f(t) (1)

Note that, if rf(t) < 1, the smaller this rate, the larger the relative change of the

function f, and vice versa: the larger this rate, the smaller the relative change of the

function f. If, however, rf(t) > 1, this relation is exactly the opposite: the larger this

rate, the larger the relative change of the function f.

Further, for modelling purposes often a continuous parameter such as t is used. Then

the following rate function 17 is appropriate:

f '(t)

f(t) Rf(t) = e (2)

286 $cientometrics 34 (1995)

L. E G G H E et al.: INFLUENCE OF P R O D U C r I O N ON UTILIZATION FUNCTIONS

Of course, instead of considering a fixed year (or a fixed moment in time), one can as

well consider the production over a longer period. During this period, the production

may change. In Ref. 16, 17 Egghe studied the influence of growth (in the sense o f

cumulative production) on the values of the rates r(t) and R(t), defined in (1) and

(2), and this for the synchronous as well as for the diachronous case. There, he used

exponential functions (different ones) both to model cumulative growth, denoted as

G(t), as to model the 'obsolescence' function, c(s):

and

G(t) = G a t , a > 1

c(s) = c b s, 0 < b < 1

(3)

(4)

It is easy to see that in this case rc(t ) = Re(t ) = b, but the point made in Refs 16, 17

is that when considering a bibliography that has grown over a certain period, the

overall synchronous obsolescence (decrease in use) becomes larger when the growth

is faster, i.e. when a in (3) is increasing, but that the opposite effect occurs in the

diachronous case.

In the present paper the same models are studied - and the results are reconfirmed

- but the method is made more transparant by the use of convolutions (for an exact

definition of this notion we refer the reader to Refs 18-19). Convolutions have been

used before in the informetric literature to explain some features of Lotka's

function.20- 21

We will consider the total number of citations to a group of articles published during

a period of length T. It is clear that, the longer we consider the publication period, the

total number of citing or cited articles increases, yet - as time goes on - a fixed article

published in this period will have a decreasing probability to be cited. What then, is

the overall effect of these two trends? Will they balance, or will they reinforce one

another? Denoting by h{s) the total number of citations in the s th period (discrete s)

to articles published during the growth period [0,T], we will show that h(s} is indeed

a convolution. This function h(s} will be referred to as the utilization function. Then

rh(s) is studied as a function of production and aging parameters. A similar study is

made for the continuous case. Moreover, we will show that the same convolution

models apply for the synchronous and for the diachronous case.

Scientometrics 34 (1995) 287


2. The influence of production on utilization: Discrete case

2.1. Synchronous case

We consider a production period of length T, and we study the total number of

citations made to articles published s time units ago {counting from the end of the

production period}.

For simplicity a time unit will be referred to as 'a year', which is the most important

case in practical situations.

: tlme (t)

age (s) 4 I i T j (s) 0

Fig.1 : Time and age axis for the synchronous study

Yeaxly production is denoted as j(s), s = 0,1 ..... T; see Fig. 1. So the variable s will

always denote 'age', not 'time' (which will hereafter be denoted as t). We will further

put j(s) = 0 for s > T. This does not mean that there is no production of articles for

s �9 T; it simply means that these articles are not taken into account in the investigated

period [O,T]. In addition to that, T is fixed but arbitrary.

In the discrete case s is a natural number : s G N (including 0). The age distribution

of references in articles published during the production period is denoted as c(s), s

�9 N. Note that we assume that the function c(s) is the same for every article.

288 Scientometrics 34 (1995)

L. E G G H E ct al.: INFLUENCE OF PRODUCTION ON UTILIZATION FUNCYIONS

Theorem I :

Denoting by h(s) the total number of citations by articles published during the

production period, to articles published s years ago, we have :

I

V s e N : h (s ) = ~E~ j ( i ) c ( s - i ) (5) i .o

The operation used in (5) on the functions j and c is known in the mathematical

literature as the convolution operation, and is denoted as h{s) = {j * c){s}. The function

h is called the synchronous utilization function.

P r o o f :

T h i s f o l l o w s i m m e d i a t e l y f r o m Fig.2.

T+ I T S 2 1 0 S 4 1 1 I I i I I

. . . . . . . 0 0 J(T) . . . . . . . J(s) . . . . . . . j(2) j(1) J(0) p roduc t ion

. . . . . . . c(1) C(O)

. . . . . . . c(T-s+l) c(T-s) . . . . . . . c(o)

. . . . . . . c(T) c(T.1) . . . . . . . c(s-1) . . . . . . . c(1) c(o)

. . . . . . . c(T+I) c(T) . . . . . . . c(s) . . . . . . . c(2) c(1) c(o)

ag ing or c i ta t ion d is t r ibut ions

Fig .2 : I l l u s t r a t i o n o f f o r m u l a {5}



Notes :

1. I t is well-known, and easy to verify, that the convoluUon opera t ion is commutat ive,

i .e. j *c = c*j.

T

2. For s �9 T, h(s) = ~ ] j(i) c(s- i) , as for these values of s, j(s) = 0 and hence, these 8-0

terms can be omit ted from (5).

Next, we consider the case of an exponent ia l product ion function and an exponential

aging (citation) function. If product ion, denoted as p(t), is exponent ia l in time, p(t) is

given as :

P(O = P a t a � 9 (6)

where 0 s t s T. Then we put, with s denoting age :

P a T-s 0 s s s T r

= ~ (71 j(s) t 0 s > T

or

A (a') s 0 s s s T /

its) = { (s) t 0 s > T

with a ' = 1 / a and A = Pa T. Note that for 0 �9 a ' �9 1, i.e. a > 1, the p roduc t ion increases

exponent ia l ly in time; for a ' �9 1, i.e: 0 �9 a �9 1 we have an exponenUal ly decreasing

production; and for a ' = a = 1, the product ion is constant.

The aging distribution c(s) is always assumed to be exponentially decreasing in the

age-variable s �9 N :

c(s)=Cb' , 0 < b < I (91



Theorem 2 :

Assuming exponential functions for the product ion {8) as well as for the aging

function (9), the discrete synchronous utilization function h{s) is given as :

h{s} =,

(__b]'" 1 -

tAC a " ~at) 0 ~ s * T (I0)

,

i f a ' r andifa' =b :

AC b s {s+l} h{s) = {

AC b s (T*I}

O ~ s s T

s > T

{12}

(13)

The utilization rate rh{s) is then given as :

rh(s ) =

�9 '/' -rb '21

1 b'g

if a ' ~ b; and if a ' = b :

0ss<T

s>T

(14}

(15}

s+2 {i- s §

rh{s } =

O ~ s < T

s > T

(16)

[17}


L. EGGHE ct al.: INFLUENCE OF PRODUCTION ON UTILIZATION FUNCFIONS

Note that s = T yie lds an exceptional (and uninteresting) case. This small p roblem will

not occur in a cont inuous setting.

Proof : By (5) :

|

h(s) =f~ A~t| Cb" |

Aa,i CbS-i

s s T

s � 9

i-o s s T

.ACbS s ' T

ACI~' (s+l)

ACb '

A C B ' (T+I)

s l T , a ' , b

s ~; T, a ' = b

s �9 T, a ' * b

s > T , a ' = b

(:8)

(19)

292 Scientomettics 34 (1995)


s s T , a ' = b

ACa" (s+l) s s T, a' = b

AC~" s k~) s > T , a ' = b ,

ACa" (T+I) s > T, a' = b

For the discrete utilization rate, this yields :

_f.l '-'

s<T,a'=b

a/S+2 s + I s<T,a'=b

b s>T,a'~b

rh(s} =

a/=b s>T,a'=b

Rr The utilization rate r h is always independent of the constants A and C. This

means that r h only depends on relative frequencies, i.e. on distr ibutions in the

statistical sense of the word. In particular, a = a' = 1, i.e. a constant production, yields

a uniform distribution on [0,T]. In this case we see that for a constant production, i.e.



a linear comulat ive growth, the utilization rate is independent of the amount of

articles which is produced, and is equal to 1 - b '~ - - ) ~ 1 , 1 - b s' l

Proposition 3 :

Under the assumpt ions of Theorem 2, and taking T -- +*~, we obtain

,am rh(s) = max (a~,b) {20}

Proof :

If a ' = b, the result follows from {16). if b < a', the result follows easily from (14}.

Finally, if a ' < b, we see that (14} is equivalent to (by (18)) :

b 1 -

ra(s) = {21}

from which we derive that

am rh(s) = b ID j -m)

Proposition 3 shows that in the case of a > t (an exponentially increasing product ion

(in time)), the limiting utilization rate is a lways smaller than 1 (and equal to max ( l /a ,

b}}. So, there will - eventually - be obsolescence (= decreased use). However , if a < 1

{an exponentially decreasing production), the limiting utilization rate is a lways larger

than 1, and indeed equal to 1/a. In this situation there will never be obsolescence.

Note that the citation distribution is a lways assumed to be decreasing in s! Note also

that here the function rh(s} is a lways decreasing.


L. E G G H E et al.: INFLUENCE OF P R O D U C T I O N ON UTILIZATION FUNCTIONS

Theorem 4 :

The larger the increase in production, i.e. the larger a, the smaller the synchronous

utilization rate during the production period, i.e.

&h(s) < 0

for fixed 0 s s s T.

Proof :

If a' = b, a = l/b, and hence

a a ~,a 1 ) s § a 2 s + 1

I f a ' , , b

&h(s__.._~) = ~ (1 (I- (ab)'~)l

aa aa (a (I (ab)"1)}

- (s+2)(ab) ''~ ba (I - (ab) ''t) - (I - (ab) ''2) [I - b'*t(s+2) n *'~]

a 2 (I - (ab)"l) 2

Now :

&h(s) m < O

- (s§ *'2 + (s+2)(ab) ~''3

�9 . (s§ ''2 - (s+2)(ab) ,'I

- 1 § (s+2)(ab) s'z * (ab) *'~ - (s+2)(ab) ~''~ < 0

+I>0

,'. ((ab) - I) 2 [(s+l)Cab)' + s(ab) '-I + .... 2(ab) + I] �9 0

which is always the case as ab * 1. []


L. EGGHE et al-: INFLUENCE OF PRODUCTION ON UTILIZATION FUNCTIONS

This theorem shows that in the case of a production parameter a which is larger than

1, the larger the increase in production, the larger the decline in utilization, i.e. the

larger the obsolescence.

On the other hand, the smaller a (0 �9 a �9 I), i.e. the larger the decrease in production

the larger the utilization rate, hence the larger the increase in utilization.

proposition 5 :

Under the assumptions of Theorem 2, we have that the larger the aging, i.e. smaller

b, the smaller the synchronous utilization rate, i.e.

\

& h ( s ) �9 0 ab

for fixed 0 s s s T.

Proof :

I r a ' - - b ,

orh(s) . _.e (b s + 2 ~ . s § 2 Bb Bb I , s + I / s § 1

> 0

I r a ' *b ,

a-~(s) ] a ( ~ _ ( a b ) , . 2 / Bb a ~ (ab) ,'1 )

-_ _~ ( - o § a)(1 - (,b) ''1) (1 - (,b) ''2) (- O+I)(ab)' a) a (1 - (ab)"i ) 2

(aS)' (1 - (ab)"') 2

((at,) ''2 - (s+2)(ab) § (s+l))

,, (ab)' ((ab) - I)) 2 [(ab)' § 2(ab) ':1 + ... § ( s + l ) ] > 0 (1 - (ab)"~) 2 D


L. EGGHE et al.: INFLUENCE OF PRODUCTION ON UTILIZATION FUNCI]ONS

2.2, Diachronous case

As in the synchronous case we consider a production period of length T. We choose

the origin of the time axis at the beginning of this period and we assume that we are

now T o years later. This situation is depicted in Fig.3.

I I I = time (t) 0 T T O

production period

Fig.3 : T i m e axis for the d i a c h r o n o u s s t u d y

The production at time t is denoted as p(t), and similar to the synchronous case, we

assume that p(t} = 0 for t �9 T. The distribution of received citations of an article

published during the production period is denoted as c(t]. Here the variable t denotes

the number of years since publication. Again, we assume that this aging (citation)

distribution is the same for every article.

Theorem 6 :

Denoting by k(t) the total number of citations in the year t to articles published

during the production period, we have :

t

Vt e [0,T o] : k(t) = ~ p(i) c(t-i) (22) t,,o

i.e. k(t) = (p ,c) ( t ) (23}

Similar to the preceding section, the function k is called the diachronous utilization

function. Note that for T < t ~ T O :

T k(t) = ~ p(i) c(t-i) (24}

i..0



, ,P.r0of :

This follows immediately from Fig.4.

0 1 2 . . . . . . . S . . . . . . . T . . . . . . . T o I I I I I ' " i

p(0) p(1) p(2) . . . . . . . p(s) . . . . . . . p(T) . . . . . . . 0

=" t

c(o) . . . . . . . c ( T o - T )

c (o) . . . . . . . c ( T - s ) . . . . . . . c (T o - s )

c(o) c(1) . . . . . . . C ( T o l ) . . . . . . . c (T o - 1)

c(o) c(1) c(2) . . . . . . . c (T) . . . . . . . C(To)

~ r o d u c t i o n

ag ing or citation distr ibut ions

Fig.4 : Illustration of formula [22)

Note that formula (22) has the same structure as formula (5) : the only difference is

that the function j is here replaced by the function p, Moreover the meaning of c is

different. Assuming now that p(t) and c(t) are exponential functions of the form given

by (6) and (9), we have immediately the fo l iowingana logue of Theorem 2.

Theorem 7 :

Assuming exponential functions for the product ion as well as for the citation function,

the discrete diachronous utilization functiofi k(t) is given as :

case a = b :

k(t) =

,

0 < t a T

T < t ~ T O (25)


L. EGGHE et al.: INFLUENCE OF PRODUCT/ON ON UT/LIZAT/ON FUNCT/ONS

and if a = b :

�9 PCb t It*l) / k{t) .W.

/ PCb t (T+ 1 ]

The utilization rate rk(t) is then given as :

rk[t) =,

case a ~ b :

.('-Iv)

0 s t s T

T c t x T o

0 s t c T

b T < t .~To

{26)

(27)

and if a = b

t § rk[t ) =

O - ~ t ~ T

T < t x T O (28)

Theorem 8 :

The larger the increase in production (larger a) the larger the diachronous utilization

rate {during the production period) i.e.

Ork(t) �9 0 (29) Oa

for fixed 0 s t ~ T.

.Proof :

This follows immediately from the proof of Theorem 4, as rk[t) has the same structure

as rk(s), but with a' = I / a replaced by a. This explains the different inequality sign.



Theorem 8 shows that if the production parameter a {> 1) increases, the utilization

rate rk(t) increases too.

As rk{t} �9 I, there is no obsolescence in the sense of a decline in utilization. On the

contrary, there is an increase in utilization. Further, the smaller a {0 < a < I} the

smaller the utilization rate, hence the larger the obsolescence.

Proposition 9 :

Under the assumptions of Theorem 7, we have that the larger the decrease in citation,

i.e. the smaller b (according to time), the smaller the diachronous utilization rate, i.e.

&k(t) > 0 {30) Ob

for fixed 0 �9 t a T.

The proof is exactly the same as that of Proposition 5 : just change-a into 1 /a and s

into t.

Note also that when T (and T o) tend to infinity

lira rk(t ) I, m a x (a,b) | . m

Only if a < 1 (decreasing production) will there be a diachronous obsolescence.



3. Examples

1. S y n c h r o n o u s data : e x p o n e n t i a l f u n c t i o n s a = 2.0, b = 0.75

age

0 1 2 3 4 5

h(s)

1000 750 500

563 375 250

422 281 188 125

4

316 211 141

94 63

5

0.788 r

237 158 105 70 47 31

1000 1250 1188 1016 824 (~ 649 E'l

rh{s) 1.250 0.950 0.855 0.812

The number at the i th column and the jth row shows the number of citations in the

year j (counting backwards) to articles which are published i years ago. This example

shows that rh(s) is decreasing in s and that the limit of rh(s), for increasing s, is equal

to 0.75 (= max (a',b)).

2. Synchronous data : exponential functions a = 1.5, b = 0.75

age

1000

rh{s}

750 667

563 500 A A A

422 375 333 296

316 281 250 222 198

237 211 187 167 148 132

h{s) 1000 1417 1507 1427 1267 1082

0.889 0.854 1.417 1.064 0.947

(*) Note : the small differences between posted values and the addition of the corresponding column is due to the rounding off of the values of the column numbers. The same will occur in the next tables.


L. EGGHE ct al.: INFLUENCE OF PRODUC-qaON ON UTILIZATION FUNCTIONS

3, Synchronous data : exponential functions a t 1.33, b = 0.75

age

0 1 2 3 4 5

his}

rhts}

0

1000

1000

1.502

750 752

1502

1.126

2

563 564 565

1692

1.001

3

422 423 424 425

1694

0.939

4

316 317 318 319 320

1590

0.901

5

237 238 238 239 240 240

1433

The remarks made for example 1 can also be made for examples 2 and 3. Moreover,

comparing examples 1, 2 and 3 illustrates Theorem 4 : the larger the growth, i.e. the

larger a, the smaller the synchronous utilization rate. Examples 4 and 5 show that

nothing really changes when taking a = 1 {constant production} or a < 1 {decreasing

production}.

4. Synchronous data : exponential functions a = 1.0, b = 0.75

age

1000

rh{s}

750 1000

563 750

1000

422 563 750

1000

4

316 422 563 750

1000

5

237 238 422 563 750

1000

h{s} 1000 1750 2313 2734 3051 3288

1.182 1.116 1.078 1,321 1.750

302 Sciemometrics 34 (1995)

L. EGGHE et al.: INFLUENCE OF PRODUCTION ON UTILIZATION FUNCI'IONS

5. S y n c h r o n o u s data : e x p o n e n t i a l f u n c t i o n s a ffi 0.5, b ffi 0.75

age

0 1 2 3 4 5

h[s)

rh(sl

100

100

2.750

75 200

275

2.204

56 150 400

606

2.071

3

42 113 300 800

1255

2.025

4

32 84

225 600

1600

2541

21009

5

24 63

169 450

1200 3200

5106

Example 5 shows that rh(s) tends indeed to max (1/a,b), which in this case is equal

to 1/a : 2.

6. Synchronous data : exponential functions a = 2.0, b -- 0.5

age

1000

rh[S)

500 500

250 250 250

125 125 125 125

4

63 63 63 63 63

5

31 31 31 31 31 31

h(s} 1000 1000 750 500 313 188

1.000 0.667 0.625 0.600 0.750

Examples I and 6 show that the smaller b, the smaller rh(s) [Theorem 5).

We will now give some diachronous examples.



7. Diachronous data : exponent ia l funct ions a = 2.0, b = 0.75

t ime

31

1

23 63

18 47

125

13 35 94

250

4

10 26 70

188 500

7 20 53

141 375

1000

k(t} 31 86 190 392 794 1596

rk(t ) 2.274 2.209 2.063 2.026 2.010

The number at the i th column and the jth row shows the n u m b e r o f citations in the

year j to articles which are published in the year i. This example shows that rk(t) is

decreasing in t and that the limit of rk(t), for increasing t, is equal to 2 (-- max (a,b)}.

8. Diachronous data : exponen t i a l funct ions a -- 1.5, b = 0.75

time

31 23 47

18 35 70

13 26 52

1 0 5

10 20 39 78

157

1.593

7 15 29 59

I18 235

k(t) 31 70 123 196 304 463

rk(t ) 1.757 1.551 1.523 2.258

304 Scientometncs 34 (1995)

L. E G G H E et al.: INFLUENCE OF PRODUCq'ION ON UTILIZATION FUNCTIONS

9. Diachronous data : exponential funct ions a ,, 1.33, b ,, 0.75

time

1 2

3 4 5

31

rk{t)

23 41

2

18 31 55

3 r

13 23 41 73

10 17 31 54 97

7 13 23 41 73

129

k(t} 31 64 104 150 209 286

1.625 1.442 1.393 1.368 2.065

Comparing examples 7, 8 and 9 illustrates Theorem 8 : the larger the growth, i.e. the

larger a, the larger the diachronous utilization rate. Examples 10 and 11 show that

nothing really changes when taking a -- 1 [constant production} or a - 1 [decreasing

production}.

10. Diachronous data : exponential funct ions a = 1.0, b = 0.75

time

18 23 31

0 1

31 23 31

31 54

1.333 1.742

7 10 13 18 23 31

1.181

3 4

13 10 18 13 23 18 31 23

31

85 95

1.074

k(t) 72 102

rk(t} 1.118



11. D iach ronous data : exponent ia l func t ions a = 0.5, b = 0.75

time

k(t}

rk{t}

310

310

1.252

233 155

3 8 8 ~

0.948

174 116

78

368

0.856

3

131 87 58 39

315

0.810

4

98 65 44 29 19

255

0.796

1

74 49 33 22 15 10

203

Example 11 shows that rk{t} tends indeed to max {a,b}, which in this case is equal to

b = 0.75.

12. D iachronous da ta : e x p o n e n t i a l funct ions a = 2.0, b = 0.5

time

0 1 2 3 4 5

k{t}

rk{t}

32

32

2.500

16 64

80

2.100

2 �84

8 32

128

168

2.024

3

4 16 64

256

340

2.006

2 8

32 128 512

682

2.001

1 4

16 64

256 1024

1365

Examples 7 and 12 show that the smaller b, the smaller rk(t} {Theorem 9).



4. The in f luence of p r o d u c t i o n on ut i l i za t ion: C o n t i n u o u s case

4.1. Synchronous case

Let now p and c denote the product ion and aging d e n s i t y functions and let j(s) =

p(T-s), 0 �9 s ~ T. Moreover, we assume, as in the prev ious section, that j(s) = 0 for s

�9 T. We now have, for the overall synchronous ut i l izat ion function :

I

hO) = f j(x) c(s-x) (31) 0

= O,c) (,)

Note that if s �9 T, formula (311 becomes :

T

h(s) = f j (x)c(s-x) dx 0

(32)

As in the previous section, we will e laborate this model in the case p and c are

exponent ial functions. So we assume that

p(t) = Pa' , a �9 0 cf. (6)

where 0 a t ~ T.

c(s) = C b ' , 0 < b < I cf. (9)

cf. {8)

This leads to

I A'(a')S 0 ~ s :i T j(s)

" 0 s > T


L. EGGHE ct al.: INFLUENCE OF PRODUCTION ON UTILIZATION FUNCTIONS

Theorem 10 :

Assuming exponential density functions for the production as well as for the aging

function, the continuous synchronous utilization function h(s) is given as :

AIC (b' h(s) = ~ -a") 0ssaT (33)

AIC b ' ( l - (ab) 4) s �9 T (34)

if a' * b, and

.A'CbSs 0 s s < T {35) J h(s} ~A'Cb~I" s �9 T (36)

i f a ' = I / a = b .

The continuous rate function is then :

1

Rh{S } = b (ab)(*h)'-1 0 a s a T (37)

b s �9 T (38)

if a' * b, and

b e I / s 0 s s s T (39} Rh(S ) -- {

b s �9 T {40)

ifa'=I/a=b.

Proof:

h(s) = f j(x) r dx o

= { i Al(a~ x C bs-x dx 0 ~ s ~ T

Al(a l ) x C b s-x dx s �9 T


L. EGGHE ctal.: INFLUENCE OF PRODUCTION ON UTILIZATION FUNCTIONS

= A I C b ' dx o

A / C b ' f dx o

a'.b

W~

O ~ s s T

s > T

A ' C b ' ~ (1 - (ab)-') 0 ~ s a T {33) In (ab)

A' C b' ~ (I - (ab) "v) s > T [34} In (ab)

if a' = b

A I C b , s

ACbST

To find Rh{S) we also need the derivative of h :

(In b.b' + In a.a-')

In b.b' (I - C.b)'~

h'{s} a' , b

A'C In (at,)

A~C In (sb)

0 ~ s a T [35}

s > T (36)

0 a s a T

s>T

a' =b

A'C (In b.b's + b')

AIC T In b.b s

O ~ s s T

s>T

$cientometrics 34 (1995) 309


= (37}

Consequently, if 0 ~ s ~ T, a' * b :

(h'(s) 1 = e (s) =

H

(.b s l n b + a - S l n a ~ &

exp t b ' - ~ - )

exp l n b § ~ - - - - - ,

I

b . (ab) ( ~ ) ' ' 1 :

If s �9 T, a ' * b,

= exp ( I n b (1 - (ab)"r) / = b Rh(s) L 1 - (ab) "T )

(38)

IfOsssT, a' =b,

Rh(s) =exP( slnb+ l)s l

= b e ' {39}

and if s �9 T, a' = b,

Rh(s) = exp (In b) -- b {40} D

Formulae (37) - (40} show that production influences synchronous utilization in the

production period (s e [0,T]} and has no influence at all beyond this period.

Furthermore, this influence depends on the parameter a.

The next theorem clarifies this.

Theorem I 1 :

The larger the increase in production {larger a} the smaller the synchronous utilization

rate {during the production period}, i.e.

~b(s ) < 0

for fixed 0 ~ s ~ T.

310 $cientometrics 34 (1995)

L, E G G H E ct aL: INFLUENCE OF P R O D U C T I O N ON UTILIZATION FUNCTIONS

P r o o f :

I f a b = 1

1 1 -

= - - - O ' < 0 a 2

(by (39))

If ab # I, i.e. a' * b

log (Rh(s)) ,, log b - I~ (ab) 1 - (at)) '

{by {37))

Se t t i ng , x = ab , i t su f f i c e s to s h o w t h a t :

f1(x) > 0

w h e r e

f(x) ,' - I~ x 1 - X '

(41]

(42)

N o w , since

f~<x) =

_.l _ X._ l + SX._ I Iog X x

(I - x ' ) ~

condition {41} boils down to :

1 n>l - s log x x s

{43}

But, for every y, 1, log y < y - I. Hence taking y = x "s

Scientome~cs 34 (1995) 3 1 1


o r

'log (x") < x- ' - 1

I :, 1 - s log (x) ra X |

This result is in agreement with the discrete case and the results in [16]. We further

note :

Theorem 12 :

If T = *~,

lira Rh(s) - max (al,b)

Proof :

I r a ' = b :

1

lira Rh(s) = lira b �9 = b $ m i w

If a' = I/a < b : 1

Um R~(s) ,, lira b P " I,.~,m | - m

= b

If a ' = 1 / a > b : 1

= b ffi a /

Finally we note (leaving the easy proof to the reader} :

312 Scientomea'ics 34 (1995)

L. E G G H E et al.: INFLUENCE OF P R O D U C T I O N ON UTILIZATION FUNCTIONS

Prooosit ion 13 :

The larger the aging, i.e. the smaller b, the smaller the synchronous utilization rate,

i.e.

~ �9 0

8b

for fixed 0 ~ s ~ T.

4.2. D i a c h r o n o u s c a s e

It is now completely clear that, for t �9 [O, To] , and with the obvious adaptations, the

utilization function kit) is given as :

I

= f p<x) c(t-x) dx = (p * c) (t) (441 o

In the particular case that t �9 [T, To] , (44) becomes :

T

k(O - f p(x) c(t-x) dx o

{45}

As {44} and {45} have the same structure as {31} and {32} {a' is replaced by a} the

conclusions are also similar.

Theorem 14 :

The larger the increase in production {larger a}, the larger the diachronous utilization

rate Rk{t}, i.e.

ORk(t) > 0 8a

We finally note :

S cie~ometrics 34 (1995) 313

L. EGGHE et al.: INFLUENCE OF PRODUCTION ON UTILIZATION FUNCYIONS

ProPos i t ion 15 :

If T {and T o) t e n d to infini ty

tim Rk(t) =- max (a,b)

Prooos i t ion 16 :

The la rger the ag ing , i.e. the smal le r b, the sma l l e r the d i a c h r o n o u s u t i l iza t ion rate,

i.e.

~ , ( 0 �9 0

Part of this research has been carded out at DRTC, ISI, Bangalore (India) where Egghe and Rousseau were visiting professors. They thank DRTC for this appointment and for the hospitality during their stay.

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rates, Information Processing and Management, 30 (1994) 279- 292. 18. W. RUDII~, FunctionalAnatysis, New York, McGraw-Hill, 1973. 19. R. L. GRAHAM, D. E. KI~UTH, O. PATASHNIK, Concrete Mathematics, Reading (MA), Addison-

Wesley, 1989. 20. L. EC~HE, Consequences of Lotka's law in the case of fractional counting of authorships an of first

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Scientome~ics 34 (1995) 315

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On the influence of production on utilization functions: Obsolescence or increased use?