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A QUEUING THEORY, BAYESIAN MODEL FOR THE
CIRCULATION OF BOOKS IN A LIBRARY
by
P.M. Morse
OR 061-77
February 1977
- 1
-
A Queuing Theory, Bayesian Model for the
Circulation of Books in a Library.
by Philip M . Morse
Operations Research Center, M. I.
T.
Abstract.
The use of library materials, the borrowing of books
for example, is analogous to a queuing process, where an
arrival is "serviced" when he finds a book on the shelf and
borrows it; when service is busy the arrival is lost to the
system.
Circulation, rate of output of the service channel,
can be measured for each book; the circulation distributions
for a number of classes of related books (homogeneous collections) have been determined.
Circulation rate can be changed
by changing circulation rules, whereas the total rate of
arrival for a book, whether "serviced" or not, is less affected
by changes in rules.
However this total arrival rate (called
demand rate) for a book, cannot be measured directly.
This
paper shows how its expected value may he calculated, using
Bayes' theorem.
From this, the paper develops a self-consistent
model of book circulation, with inter-related probability
distributions, from which one can predict the effect, on
circulation, of a change in loan rules or of the purchase of
duplicate copies of the more popular books, and also can
measure the decline in demand for a book with time.
Results
check available data on six collections, in two libraries.
- 2 -
Demand Rate.
Data taken from a variety of libraries I indicates that
it is possible to develop a self-consistent model of library
material use, as exemplified by book circulation.
assumptions are those of queuing theory.
The basic
A person enters the
library desiring to borrow a book on some subject of interest
to him; he either finds nothing suitable or else he finds a
book and borrows it.
e may have entered the library knowing
exactly which book he wanted, or he may
ave had less definite
desires and proceded to find the book by browsing the shelves
or the catalogue.
In any of these cases, the person's arrival
is counted as an arrival (in the sense of queuing theory) for
the particular book borrowed.
But what of the cases when the book the arriving person
would have borrowed happens to be out of the library, borrowed
by someone else?
Such arrivals also should be counted for the
book, even though^eYdid not result in a circulation.
These
"frustrated" arrivals cannot be measured directly, but they
can be measured probabilistically, if we make the reasonable
assumption that the arrival rate for the particular book is
statistically constant, whether or not the book is in the
library, ready to be borrowed.
That rate would equal the
rate the book would be borrowed if duplicate copies were
available, so that a copy would always be present in the
library, no matter how often it was borrowed.
arrival rate
That limiting
is analogous to the expected arrival rate in
queuing theory; we shall call
rate for the book in question.
the expected yearly demand
As mentioned earlier,
cannot be measured directly.
However its expected value can be determined in terms of the
expected yearly circulation rate r of the book and the mean
the book is out of the library per cir-
length of time 1/
culation (or its reciprocal
, the expected return rate for
the book, the analogue of the service rate in a queuing
problem) both of which can be measured
.
The expected fraction
of the year the book is off the library shelf is equal to the
number r of circulations, on the average ,during the year,
of time (in fractions of a year) the
times the mean length
book is off the shelf per circulation, 1/4.
and the number of arrivals that
between the arrival rate
found the book out,
The difference
(r/4), must be equal to the number of
arrivals that found the book on the shelf and borrowed it
(which is, of course, the expected circulation rate r).
r = -
(r/)
or
r(X) =
(1)
which is a well known queuing result 3.
The demand rate for a book is rarely greater than ten
per year; in general an arrival is as likely to come one week
as the next (if summer rates differ from winter rates, this can
be allowed for).
Thus in general the arrivals are Poisson
distributed, the probability that a book has k arrivals for
it (in the sense of the previous
year is
TTk()
iscussion) in a particular
(kk/k)eX
Therefore, in general, book circulation is
(2)
oisson distributed.
The probability that a book, having demand rate
and return
-4
-
, circulates precisely m times in a particular year is
rate
p(mif ) = TT(r)
(3)
exp(e )
gx +
which is the conditional probability that a book, having the
demand
rate
(and return rate A) has circulation m in a given
year.
Figure 1 displays plots of r(X) for different values of
p, as function of
We see, of course, that mean circulation
.
r(X) of books with mean demand rate
falling further below
is always less than X,
the greater is ttie demand and the
greater is the mean loan period 1/p.
The dashed line
=
is r= k; the vertical distance from this line to the curve
for a given value of
is the expected number of "frustrated"
arrivals per year for a book with lemand rate
.
Curves for the conditional probability p(mif
) are shown
The curves, as functions
in Fig.2.
of X, are sharply peaked for values of m less than
imum coming at r= m, i.e., at
curve for
; the max-
= m/( - m) (for example, the
= 8 and m = 8 has its maximum at
However the
= oo).
more important statement is that, among books with demand rate
X (and return rate
) the more prevalent circulation rate m is
the integer between r and r+
falls more and more below
, where r = Xp/(X+
as X and/or 1/
).
This
is increased. We
also see that there is a finite chance that a book, during some
year, has circulation m greater than its mean demand rate X.
This is characteristic of Poisson distributions; actual
performance can exceed, as well as fall below expectations.
In most queuing situations the mean arrival rate can be
directly measured; this is not the case here.
What can be
--- 55--~ 1t
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Fig.l.
Expected circulation
rate r(X) of book with demand
rate
and return rate
.
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measured for each book is its actual circulation rate m for
a specific year.
rate
Working back from circulation m to demand
must be done probabilistically, for a particular
collection of books.
The typical collection we shall discuss
is that of all books in a library with roughly similar usepatterns, such as all the books on history or on chemistry, or
all the books with Dewey classification between 600 and 700,
for example.
tion.
We call such a collection a homogeneous collec-
Loan period also varies independently; sometimes being
a few days, sometimes, with a renewal, being more than a month.
The average for the collection for the year is the value of 1/
we use for the collection.
The resulting value of
A. is
thus a
constant, characteristic of the collection,whereas the demand
rate
is the stochastic variable in Eqs. (1) and (3).
Though circulation m is the measurable quantity, demand
rate
is the more basic property of the book.
Circulation
rate can be changed by changing the circulation rules (length
of loan period, etc.); the demand rate can be changed only by
Thus
a drastic change in the nature of the borrowing public.
to predict the effect of changes in library circulation, it
is necessary first to estimate the value of the unchanging
quantity, demand, and from it to predict the effect of rule
changes on circulation.
It has been found1 that book circulation is distributed
approximately geometrically in a homogeneous collection.
It
is shown elsewhere4 that this is approximately equivalent to
stating that demand rate is exponentially distributed
homogeneous collection.
In this present paper we shall
n a
-8 -
assume the exponential distribution of demand and explore the
consequences in detail.
In other words we shall assume that
the probability that a book of the homogeneous collection has
demand rate between
+ dX is f(X)dk, where the probab-
and
ility density
-X/D
f(X) = (/D)e
f(X)dX = 1 ;
;
f(k) d
= D
(4)
c
O
The mean value of demand for the whole collection is thus D;
the variance of
for the collection is D
We expect to show
that this assumption results in a set of probability
istrib-
utions for circulation,and relating circulation to demand,
which check satisfactorily with available data and which she4
light on the mechanism of circulation.
Relation between Demand and Circulation.
Having the forms of the unconditional probability f(X)
and the conditional probability p(mif ) of Eqs.(4) and (5),
we can write down the expression for the combined probability
that a book of the collection has circulation rate m and also
has demand rate
;
f(Xkand m) = p(mif )f(X) = f(Xif m)p(m)
1
m
1~(
_kD
mexp(x
-
X)(5)
From this, by integration and by Bayes' theorem, we can obtain
the other two distributions of interest; the unconditional
probability that a book of the collection has circulation rate m,
p(m)
=
f(X and m) d
(6)
and the conditional probability density f(X if m) that a book
of the collection has demand rate
if it circulates m times
-9-
during a year,
)
mexw
f(Xifm) = Bm(X
where
1/Bm
(7)
exp(- -
o
5(
+)
exp(- -
)d
= (Dm')p(m)
0
Figure 3 shows plots of f(X if m) as function of
few values of D and of
.
for a
Note that the curves are more
sharply peaked than those of p(mif
) in Fig.2; in fact they
become more sharply peaked as m is increased, rather than less
so, as is the case with p(mif
).
The value X m of
at the
maximum of f(Xif m) falls more and more below the value of
m as m is increased.
Tiis is understandable; the chance that
a book's circulation m is greater than its demand rate
is
not zero, but that chance gets progressively less the larger
m is; therefore the curves fall off more rapidly on the high-k
side as m increases.
Note also that the combined probability f(Xand m), as a
function of X, is simply proportional to f(Xif m), the conditional probability that a book have demand rate
circulation rate is m.
, if its
The proportionality factor, according
to Eq.(5), is p(m), the probability that a book of the collection circulates m times a year.
over
Since the integral of f(X if m)
must equal unity, the normalizing factor 1/Bm for
f(Xif m), multiplied by the factor 1/Dm', must equal p(m),
i.e., f(A if m) must be BDm' times f(Xand m).
-
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The integral of Eq.(7) must be calculated numerically.
Table I lists values of (1/DB m)
= mp(m), as well as the
cumulative probability
P(>m) =
p(n)
that a book of the
n=:
Note that
collection circulates m or more times per year.
Zp(m) =1
and
2P(
m)
mp(m)
=
=
R
Figure 4 is a plot of P(>im) on a semilog scale, for a few
values of
A,
They are nearly straight lines, indi-
and D.
cating a nearly geometric distribution, as required by the
data (as will be shown later).
As mentioned earlier, demand rate, even the average
demand rate for the whole collection, is not directly measurWhat are measurable are the mean return rate 2
able.
and
the mean circulation rate1
R
I~Dm
D
5
(l
for the collection.
U
+
0
-
-d
)m exp(-
mp(m) = |
=
e X/D
=1
-
D
e/D
eJUe
+
(8)
u/du
The integral in the brackets is the
exponential integral E 1(/D),
which is tabulated5 .
can obtain a value of (R/u) as a function of (D/)
Thus we
or, by
inversion, the mean demand rate for the collection as function
of the collection's mean circulation rate R and its mean return
- 12
L
1.0
____
____
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4
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IN, 3
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.
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i
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= 20
B_
=
_
\.
I
~~~~~~~~~~~~~~
I
!i--
--
KA - -----
! ---------
,-
\.
i
- -.
.......I
0.02.
I
0
I
I ,
!
1
-i
i
N-
I
I
,
I
I
I
L
I
I
.
I.
2
'
, ' .'
.|~~~~~~~~~~
I
I
4
m
Fig.4. Cumulative probability
P(~m) thatabook of the collection circulates m or more times
a year. See Eq.(8).
I
|
z _
!
55
_
6
-15
rate p..
-
The functional relationship between (R/4) and (D/4)
is given in Table II , from which, by interpolation, the
quantities given in Table I
A
and m.
=
can be found as functions of R,
Curves of D against R are shown in Fig.5.
co is the limiting case D= R.
The curve
The vertical distance between
,
this curve and that for the appropriate (finite) value of
times the number of books in the collection, equals the number
of yearly arrivals, for books of the collection, that do not
get their book because of circulation interference (the number
of "frustrated" users of the collection).
As an indication of orders of magnitude, the mean yearly
circulation R for a collection of history books is usually 1
or less, whereas a collection of technical books (physics,
medecine, etc.) can have R between 3 and 4.
For a library with
strict return rules, the mean return rate can be as large as
20 (mean loan period 2 weeks, few renewals; note that a renewal
does not count as a separate circulation).
more relaxed rules,
For a library with
can be less than 10 (mean loan period
greater than a month, including renewals).
Implications of the Analysis.
We thus are able to estimate the effect of the loan period
policy on circulation.
Since demand is generally independent
of the length of the loan period (i.e., on the value of
change in
changes R, not D.
), a
For example, in a collection
that has a mean circulation R= 2 with an allowed loan period
of a month, with renewals allowed (so the return rate is as
low as 8) Fig.5 shows that the mean demand for a book of the
collection is D
3.5.
If now the circulation rules are
12
10
8
6
4
D
2
1
0.8
0.6
1
2
3
R
Fig.5.Mean demand rate D, of a collection of books, as function of mean
circulation rate R, for ifferent values of mean return rate
. See Eq.(3).
4
- 15 -
tightened up, so the mean loan period is reduced from 1/8
to 1/16 of a year (from
= 8 to
= 16), D will remain the
same, but Fig.5 shows that the mean circulation R will increase
from 2 to 2.5 per year.
Instead of (35.5-
2)/3.5 = 3/7 of the
potential borrowers of the books of the collection being
"disappointed" because the book they would have borrowed was
being borrowed by someone else, only (3.5-2.5)/3.5 = 2/7 of
them will be "disappointed".
With a collection of 3500 books,
500 more users per year will find their book and borrow it, if
the tighter circulation rules are put into effect.
The effect
is greater, of course, the more popular is the collection (the
greater D is) and the more lax, originally, are the loan
rules (the smaller the original value of
is).
Thus a measurement of mean yearly circulation R and mean
return rate
, for a homogeneous collection of books, suffices
to determine the circulation characteristics of the collection,
if our assumption of an exponential distribution of demand
is statistically valid for the collection.
This question of validity can be determined only by
reference to actual data; if the formulas correspond to actual
circulation data from a variety of collections, then our
assumption is authenticated.
It should be pointed out that
these required correspondences are more stringent than was
the case with the earlier, less unified theoryl.
In the
earlier version the intimate connection between p(mif
p(m), via f(X), was not worked out.
As a consequence,
) and
in
addition to R and AL,we were free to choose one of the constants
for the assumed geometric circulation distribution p(m).
- 16 -
In the present, more thorozhgoing analysis, once R and
are
determined for a collection of books, all the probability
distributions, p(m) and P(>m) as well as f(X) and f(kif m)
are determined, in addition to D, the mean lemand.
In the
earlier analyses the constant y for the geometric approximation
to the circulation distribution was additionally available,
to be chosen to fit the data.
analysis, once R and
With the present, more complete
are determined the form and magnitude
of p(m) and P(>m) are fixed, as well as the value of D.
In order to facilitate the comparison between theory and
data for circulation distribution, the curves for P(>m) can
be redrawn to make them functions of R, rather than D, for
different values of
and m.
These are shown in Fig.6.
note that, as functions of R, the values of P( m)
depend much on
We
o not
, except for the higher values of m.
This is
not surprising, for the mean circulation is equal to the sum
of all the P(>m)'s for m
R =
0,
mp(m) =
I
P(>m)
(9)
I
Large differences in the small values of P(>m) for m large
can be compensated by small differences in the large values
of P(>m) for m small, to come out with the same value of R.
Thus, from Fig.6 one can quickly find what fraction of
the books of a collection circulate m or more times a year,
once one has determined the mean yearly circulation rate of
the collection and has a rough idea of its return rate
.,
Values of P(.m) were measured for four collections, two from
the
cience Library at M.I.T. and two from the Countway
Library of Medicine at Harvard 1 .
The theoretical curves
re
- 17 1.0
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-i
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T
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.
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--t--·--·
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I~~~~~~
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I
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'i
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/'
I
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-/'I~I
.005
6
8.'
Idiv/
.
I
I
,
· · ·I ··- ·i
;r*
i
It
...1
i
,-----;
..
I
.
?O i .
2
----
[.
0.2
.
~c
I
,
t
i~~
.
-
2
R
Fig.6.Expected fraction P(?m) of the
collection that circulate m or more
times per year, as function of mean
circulation rate R, for values of m
and of p, the mean return rate.
i
--
i
''
I
-
-
-
1
shown in Fig. 7; the related data are the small circles.
The
correspondence is well within the permissible error, particularly
since the theoretical curves are fixed in both slope and value
once R and
have been specified.
In addition to the mean demand rate D for the whole collection, one can obtain the mean demand rate d(m) of those books
of the collection that circulate m times in a year.
It can be
obtained from the fraction f(kif m) of the books of the collection, with circulation rate m, that have demand rate A.
d(m) =
f(
if m) d
= Bm
t
))X(f
exp(-
_
Thus
)dX
However
=
r
=
rl+(r) + (r)2
r 3
]
where r= X/(X+g) has been defined in Eq.(l).
Therefore a
series expansion for d(m) is obtained from Eq.(7),
d(m) )=B.|(
Bm I(rm+l+
++
r+2
m
·
.)exp(--r)dk
e
k
0
Bm(B
m Bm+
Note that
+
1
~
Note
p(m)d(m)
that
= J
o
Bm+2+
fX
(
(10)
(Xandm)dX = D
0o
The quantities d(m) are tabulated in Table III and are
plotted against mean circulation rate R in Fig.8.
We note that,
for collections with mean circulation rate R as low as unity,
the mean demand rate d(l) for books that circulate
only once
a year is a little greater than unity, but d(m) for m
less than m.
Evidently, when R
1 is
1, the books that circulate
more than once per year correspond to exceptions, the occasional
fluctuations above the average of arrivals for a book.
For
19 -
-
I. 0
1.0
---
3--------
- .---------.
~·--
.--- L. --^-.-·I'.------.---
P
·
0.5
1
g
m~
a
..5
~
~
~
4
:t~~~~~~~~~~~s
.~~~~~~~~~~
bO
3
P
·
.2
0.2
o k
0.1
I
1------
I
i
2--Z---.---
4-
I I I
-0.1
I'-'"7
I
I'
I
II
i
1!
L '
' 'l
II~
-1- - -
1.0
0.5
I·--·---·
t
I
I
t
I
Z'
,
----
-
-,-
- "-- -
ql
0.5
k----
A~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
4
.2
1
-
f
t-- - --- -*
0.2
*
4-------
------ 0~t~i -
-
---4
*
~
___
I.
-~~~~~~-,.-.-
0.1
c
L--------
--
---
-----
....
*-T...
--- -----------
-----------
~~--
I
1
r--------------------i
I
i
t ; I
I..
~
-
-·
~---
--
-
9--------...
t\
1
11
-~
Fig.7.Measured values
TT-T
---
-------
-1--
--
.
-4c
0. 1
--
·-
----
t-------
-
-
·---I----- ·-----f
---------- I------,
----
-.------ i
1--c-K--~¾
·-~-1.
:---·- I ---·i
t-------
..
i
------
.
--
of P(;m)(eircles),
fraction of collection that circulates
m or more times a year, compared to the
solid lines from Eq.(7) for the values
of R and i measured for the collection.
-
-0o5
. -
- 20 -
Fig.8. Mean demand rate d(m) for books
of the collection that have circulation rate m, as function of mean circvalues
ulation rate R, for different
of m and return rate . See Eq.(10).
III
'
1
--c
---
C-
II, .. -I
1t
-------I
i,
i
.....
.........
_
t .... +._ ....
CcP12
= 1---
6
,u = 2 0
I
-
/
,
4
-
--
j
-
/
---
-
-t/-..
.
/1 '
/
j
.
- ,.
-
'
'or
_7___-____
;A- _
--
'
__-/-----1------
,'
j
-
/
_
__
d· m)-i
+----
. --
-
-
-
i
,
_- ----
- -4
---
---
-I•- -r -- -
/ ?/
.-.
.........
/
-m
4-'
-
--
-
-
~~1
'''.-:.' -
'
32
- ,,
· ..I
-
O~~~~
2
R
tt
~-"1
3
-
R= 1.6 (D
- 2)
21 -
the mean demand d(m) for m = 1 to 3 exceeds the
circulation rate m; d(4) is somewhat less than 4 for
than 14.
greater
In general d(m) is less than m for m greater than
roughly D 2 ; is greater than m for smaller values of m; with
the d's for A< 10 considerable larger than those for A> 10.
This is understandable; books circulating more often than their
mean demand rate constitute fluctuations out on the "tail" of
the distribution p(mif
).
By similar methods we can obtain a measure of the
ain
in user satisfaction by the addition of duplicate copies of
books in high demand.
As indicated in Eq.(l), the circulation
of a book with demand X is, on the average, r =
/(X+4).
Simple queuing theory6 shows that if the book is duplicated,
the combined circulation of both is
2
1 - P[ - (p/2 )
2p.(X+I)+
rl
+p+p
2
P5
4
where p = (r/p.) = X/(X+4).
p
6
8,+...
(p
1)
The expected gain in per-book
circulation, in other words the expected reduction in the
number of "disappointed" potential borrowers of a book, produced
by the presence of a duplicate copy (which is often, but not
always, on the shelf when the original copy is out) is
For
, the expected
a homogeneous collection2
(11)
For a homogeneous collection, the expected gain in per-book
circulation, if all the books in the collection that circulated
m times in a year were duplicated, would be
- 22 -
g(m)
= m
=
(r
2
- r)f(Xand m) d
(m+2) p(m+2) + (m3)
Dm
(m + 3 )
Bm+
2
-
m+
-.
p(m+5) -.
(12)
The expected gain in mean per-book circulation, when all the
books with circulation m or greater have duplicate copies is
G(>m) = 2Ljg(n)
(15)
rl = yn
Curves for G(Om), as function of mean demand D, for
A= 12, are shown in Fig.9, together with curves for P(>m) and
for the ratio G(>m)/P(> m), for different values of m.
Note
that, even if all the books of the collection were to be duplicated (m= 0), not all potential borrowers would be satisfied,
for G(>O) is smaller than D-R, the mean per-book difference
between demand and circulation.
There are, of course, times
when both books are out of the library and an arrival will be
"disappointed", particularly when mean demand is great.
The curves for G(>m) show that, for D greater than 2,
about half the "unsatisfied" arrivals are "satisfied" by duplicating all the books that circulate 4 or more times per year
(when D is less than 2 the fraction of arrivals not finding
the book is so small that duplication is not worth while).
However the curves for P(>a4) show that duplicating all books
circulating 4 or more times yearly would mean duplicating a
fifth or more of the collection, which may require more funds
than are
eemed
appropriate.
Duplicating all books circulating
5 or more times would involve tuplicating one book in ten,
which may be a more reasonable allocation of book funds; but,
of course this would "satisfy" only about a third of the
expected "unsatisfied" arrivals.
-
cq)
r-H
-
CM
NN
;
4O
LN
E.-
-o
0
r-4
0
(E)d
E
4
I
N
CM
Cu
FrH4
N
0
A\
-
&
CM
-H
L\
o
H4
(m )s
o
0
Fig.9. Mean gain G(f-m) in circulation rate if those books of collection circulating m or more times are duplicated, for
= 12.
Corresponding curves for P(r.m), fraction of collection duplicated,
and for G(~.m)/P(Žm), added circulation per book duplicated.
- 24 -
Perhaps a criterion for the allocation of funds to buying
duplicate copies lies in the expected circulation of the duplicate copies.
If the whole collection were duplicated, the
mean additional circulation would of course be less than the
mean circulation R for a collection of single copies, for there
will always be times when both copies are on the shelf when a
prospective borrower arrives and thus the duplicate copy is
not used.
The ratio G(>m)/P(>m), the expected gain in circ-
ulation divided by the number of duplicates (loosely speaking,
the expected mean per-book circulation of the duplicates) can
provide a criterion, for if this ratio is equal to or greater
than the mean circulation R of the unduplicated collection,
then, on the average, each duplicate copy will be "working as
hard" as the average book in the unduplicated collection.
see that G(t>5)/P(;-5) is roughly equal to R for 2< D
Therefore, for a collection with return rate
We
4.
= 12, a
uplicate
copy of each book circulating 5 or more times would be used
as often as an average copy in the unduplicateA collection,
and thus would be as "useful" as an average new purchase.
When
p. is less than 12, the value of m for breakeven is less than
5; when
> 12, the value is greater than 5.
Change with Time.
A defect in the previous incomplete analysis 6 is that
time was introduced as a discrete variable.
Since circulation
m was considered basic, rather than demand rate
, a change in
circulation with time had to be constructed in terms of a
discrete Markov chain.
This enabled one to include the
- 25 -
variability of circulation with time, but confined the
calculation to yearly intervals, with no simple way to change
the time scale.
With the present analysis, since demand rate
X is itself a probabilistic parameter innerently implying a
variability of circulation, it is possible to consider
(and therefore D for a collection) as changing continuously
with time and ask how the distribution in circulation correspondingly changes.
Thus, for low demand collections, we can
ask how the circulation distribution changes in a 5-year
interval, without having to make the calculation sequentially,
year by year.
Also, for books of high demand, it is possible
to calculate the change in a 6-month period, instead of being
confined to 1-year intervals.
Thus we assume that the mean demand rate D of a given
collection of books (in a given library) is a function of time,
probably decreasing monotonically (though there may be special
exceptions).
If its value at some instant is D, at some later
time the mean demand rate for the same set of books will be
wD, where
is a function of time (perhaps a negative expon-
ential e t/T).
There are not yet enough data to determine
the nature of this time dependence, but a beginning can be
made by studying the data on change in circulation in a year's
time.
A change in D does not imply that the demand rate for
each individual book in the collection varies proportionally
to D.
All it implies is that, as the individual demand rates
change with time, the same fraction of the collection experiences an averafe change in
by D.
proportional to that experienced
In other words if, after an interval of time, the mean
- 26 -
demand rate of the collection (same books, no new books added)
is wD, then the distribution in demand keeps the exponential
form
f'()
d X
-D e-
/wD
dX' -'/D
D
where
d'
D
X'
=- (
Thus, if mean demand rate D becomes D' = D after some
period of time, the mean circulation rate will change from
R(,D) to
= R
according to Eq.(8).
=
1 - (/D')e/DEl(/D
' )
(14)
Figure 10 shows plots of the ratio R'/R
between the mean circulation R' after demand has become D' = (D
and the original rate R, as function of the original rate R,
for different values of w and of
.
From thesecurves one can
from measurements of mean
determine the appropriate value of
circulation rate at the beginning and end of a time interval.
In connection with the earlier, less complete analysis6 ,
the relationship between circulation rate at one time and the
rate a year later, was exhibited by measuring the mean circulation rate N(m) of those books of a collection that had all
circulated m times the year before.
The analogous quantity is
q(m,w): if we consider all the books in the collection that
initially had circulation rate m, q(m,w) is the mean circulation rate of these same books after the mean demand rate of
the collection had changed from D to wD.
The fraction of the collection having circulation rate
m that have demand rate between
according to Eq.(7).
and X + d
is f(X if m)d;A,
At the end of the time period the demand
rate for these books will have changed to o.
According to
Eq.(l) the expected circulation rate for a book of demand
- 27 -
Fig.l0. Proportional reduction R'/R in mean circulation
rate if mean demand rate D has reduced by a factor w
during interval, as function of initial R. See Eq.(14).
0.9 I
i--
·-
r------+-t-···---
8
I!
1
----'-5-
lc- --_-t3- c'lt
_
1-
0
-------
-i
c;
t
0.8
"
c
-t-----~~~-i-~-~~ - -r~~I
!~-~--t~~~--
1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
----
TV
--C-----"
.-- TV
i
i
I
I
'
A,
.... 4
-e
.
_
__.
i~..
___
+-_
._.
"
.
''
.- I
8
.
.
j
,
:
ti
__
_
_
_ _--.-~"~.
II
-it
t
0
0.7 I
*
t::-
___
R'
_
-
_-
I
i
R
--c
.r
FI i_
--.
~
-e
_
_
~~~~~-.
I
_
_
I
,
_
ic~
'
P'
L
_
*
-_.
_
__
I
---
7I
i
I
iI
.
LC
.
- -t'.
i
i
ii
i
i
.
.
I
-
__-
0.6
i,
i
!
i
__
-
_.....
-
I
0.5
f
-
, ____
...
_
_
iLc
;
--- t.A
-
f=
---
I
_
-i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
,
f--i--
i
I
1-
..
..I
'O 00-
--
.
I
. i
/C
,
I
-r--
1·
._
k-
i
I-
--------
'-
......
_
m
_
.E
1
!
i
I~ i
~ _rC~ ~"~~"
I
~cC~
~
-
~
CL·.Ie
i~~~~~~~~~~~~~~~~~~~~~~~~~
i--i
j--
~
;Sp~--'
_
~
~
7 ---- -----
~
~
=;I-
.I
, ... ..
.I
I~~~~~~
Ii
4
I
i
0.4
--
.
_
.-
= 8
I
I
A= 12
_-- 4= 16
=
-7
I
-
II
20
...
...
--
.
I
.+
.
_
i
+
_
_.
,
-
.
.
. II
.
0o
I
I
-C
.
_
-aL
____-
i----1l
0.2
2
R
- 28 -
rate wX will be
wX
r(wX)
r()
- (-w)r
-
-3+
=
r 2 + (l)
ir +
=
where r = r(X) =
2 r3
+ .
/(X+4) as in Eq.(l).
The mean circulation
of these same books (that all hal circulation rate m at the
beginning of the interval) will be, at the end of the period,
q(m,)
= f r(w)f(kif m)dX
=BmS
=
r 2 + ...
r +
B Bm-+
1
Bm+
,
+
rmexp(-
(
where we have utilized Eq.(7) again.
r) dX
(15)
W)
The series converges
fairly rapidly as long as m(l-w) is small compared to
.
Figures 11 and 12 show curves, for different values of
the initial
mean circulation rate of the collection,
of q(m,w)
the final mean circulation rate of those books of the collection
that circulated m times a year, after the collection's mean
demand rate has changed from D to wD, plotted as functions of m.
Note that q(m,l) = (Bm/Bm+ l ) is not equal to m; the mean circulation next year of those books that circulated m times this
year is not necessarily equal to m, as has been pointed out before.
Once R and
one parameter.
are determined for a given collection, only
, is left to be determined by the change in
circulation with time.
This is in contrast to the earlier,
less complete analysis, where two parameters, a and
available to fit the data.
3, were
Consequently it is of interest to
see how well the scanty observational data can be fit by the
- 29 -
-
- 'i
..MUI --
. II II
-
-
f
I
-a,- :
lv
n f
An
-,,n+-,a 0+
h;
--
- 4- I -1
I . ",-
-
.,--
U-
period q(m,w) of books circulating
i times initially, if mean demand ra
-.-.-.----- has decreasedby factor w in interva
-
5
--
4.
~
__ __ _ .
__.
_ ._1
- 20
:
..........
,3
.8
'
1----
-- -----
-- -- ---------
*
=
--
= 12
_
~~....~5"-
---
= 1
I
..
·---- !z
·-- ---i---
I
I
I
----.-- ---i
I
= .6
I
i
3
___··i_
___
i
r --_
.4
,4
___
-.
_ ____.__
1//
;J/
.
.-k
.,-
~~~~~_
W
= .2
I
. x..x.;.I
--
--L-·· ---
1 ./
_
,.s
·--- --·-r--·-
I
_···
-·~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
r
iT~~~~~
0
t
_
!
I
I
l
_
0
i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
--.
.. --- ---
I
I --
.- -
-
1 I -
i
i
i
i
I
t
---
iI
---
-Ti
-i
-- -III
~ ~ R~ ~i 0.9 ~
~i
-1
,
I
II
~ ~~~~~
t
3
_ _
_
-P
-e--
--
-_--
1. I
i
I
i
I
I,
-------- i
f-
-
I---
.e-10
1-c= .. 0-i
Ii
r",
.
..
I
'I
=1
W--
i
I
i
I
.
._
..
i '
'_
-.
.
i
--
i
3
I
I
I
_S
C5-
I
1
d
,
I
I---
-.A
000f
_ ----O
VJ
.Via;
FC-;/
I
TIC·Y
FP-'
---
---
0
0
f
.-
_Z-,
]1
I
I
I
1
__-
.
~
_I
_
1- 1
i
!
2C-
2
I
;·
.I:
.
,-------- 5
i
m
3
~
I~
g
i_
.
.
,-
_ .
-
i-
S
. .
_ _
L.
_ _
------ -------l
f_
I
IT.
!
d
I
/.I . LL
1w-.
I
Ir
, , ...
-
iClc--
-
- - -t
I
i
F" _
_-
~~Z~rrtL1C
0
--- -.---+
---
-.
i I
mr~1C~
-
_____
_
t
=.6
0
0--,
7
=.8
,
5~~~~~~~~~~~~~~~~~
.I,,,
Il
4
i
l
5
6
- 30 --
-1
4
=.8
=.6
-3
W.4
-,.2
1
0
4
1
,.8
3
3
2
1
,.2
0
m
-
single parameter w after R and
collection in question.
1
-
have been determined for the
Figure 13 shows this correspondence
for four different collections in the Science Library at MIT.
The circles indicate measured values of q(m,w), mean circulation
rate a year later, for those books that had circulated m times
in the previous year.
The values of R and
had already been
determined for each collection; the best value of
mined by interpolation of Figs. 11 and 12.
was deter-
It is interesting
to note that the best value for w is 0.9 for all four collections; the differences between the curves come about because
of the differences in R and
.
The data available at present indicates that the analysis
outlined in this paper is valii for many, if not all, homogeneous collections of library books.
More data is needed,
however, particularly data on P(>,m) for collections for which
R and
have been determined, to strengthen tis tentative
conclusion.
As far as dependence on time is concerned, many
more data are needed, particularly on the change with time of
R (and also of
), for the same collection of books, over a
5 to 10 year period.
Further values of q(m,w) would also be
useful, to compare with Figs. 11 and 12 and to see how w
changes with time.
- 32 -
.v0
\
o
0O
·
co,\N
O
co d'
," 0
-o0
-4 0
000zII II 11
) \"
pq
Z
1.3
S
C
.
.
CM
""
.
_
.
0
__
_
-
0
I_
I
cM
0
cM
0
(mn'm)b
(
'wu)b
-
W0
U2O
cO
Co 0
* l-
N
O
WC\CM
0
o0
·)1 i
II
o
II
,, Z 1. 3
4
0
(O
(nru)b
0
CN
C0
0
0
M'
t4
0
() 'tu)b
Fig.13. Measured values (circles) of mean circulation a year
later q(m,o), of those books that circulated m times in the
past year, compared with solid lines obtained from Eq. (15).
- 33 -
References.
1
Morse, P.M., Library Effectiveness, a Systems Approach,
MIT Press, 1958. Also Chen, C.C., Applications of
Operations Research Models to Libraries,
2.
Morse, P.M. Loc.Cit. page 119.
3.
Morse, P.M. Loc.Cit. page 57.
IT Press, 1976.
Se also Morse, P.M.
ueues,
Inventories and Maintenance, Wiley, 1958. page 14.
4.
Morse, P.M. Demand for Library Materials, an Exercise in
Probability Analysis, in process of publication.
5.
Handbook of Mathematical Functions, National Bureau of
Standards, Edited by M.Abramowitz and I.A.Stegun, 1964,
U.S.Government Printing Office, page 227.
6.
Morse, P.M., Loc.Cit. page 71.
7.
Morse, P.M., Loc. Cit. page 87. Also Chen, C.C. Loc.Cit.
page 11.
- 34 -
TABLE I.
Normalising Constant Bm, Fraction p(m) of Collection Circulating
m Times and Fraction P( m) Circulating m or more Times. See Eq.(7).
m
p(m)m'
=1/DB0.P(42
m
p (m)m
= /DBm
p(m)m'
P (t m)
=1/DBm
.
0
1
2
0.5308
.2626
.2415
3
4
5
6
.3135
.5081
.9690
2.095
7
8
5.003
12.97
9
35.98
142
1.00
.469
.207
.086
.034
.012
.004
.002
.001
.000
0.5207
.2590
.2461
.3361
!
f- f5-
* p.po
1.223
2.943
7.943
23.57
75.84
1.( 00
.4 79
20
.2;
.097
/
.u 4 1
P (>m)
i
.6267
1.371
3.487
10.04
1.010
.485
.22 7
.10 I4
.046
.02O
.008
.005
32.03
.oo1
111.7
.00 1
0.5155
.2570
.2475
.5470
l
.0 17
.007
.0 03
.0 01
.0 00
20
i
R = 0. 8942
R = 0.3653
p(m)m
=1/DBm
P(>m)
D = 1
R
C =
16
12
= 8
I
i
I
I
I
i
R = 0.9126
0.5126
.2557
.2436
.3531
.6516
1 .465
1. 0
.487
.252
. 108
.049
.022
.009
.004
11.54
38.553 .002
141.2
.001
3.854
l
D = 2
R = 1. 3969
0
0.3722
1.00
1
.2584
.628
2
.312
.369
3
4
.5902
1.302
.204
5
3.35453
6
9.625
7
8
30.29
102.4
9
367.53 .001
.105
.051
.023
.010
.004
R = 1.5408
0.3606
.2460
1.00
.639
.3225
. 6044
1.441
4.127
.5393
13.31
.057
.o019
.009
.004
48.25
191.0
815.4
.232
.132
.071
R= 1.6284
0.3540
.2401
.5171
.6073
1.498
4.464
15.42
59.98
257.8
1205.
R = 1.6873
1.00
.646
.406
0.3512
.2359
.5127
1.00
.247
.6057
1.523
4.660
.257
.156
.092
16.65
.053
.030
.146
.084
.047
.025
.013
. 007
67.53
304.7
1505.
.649
.413
.017
.009
- 35
TABLE I.
Continued
A= 1 2
L = 8
m
p(m)m'
P( m)
p(m)m'
=
P( m)
0
1
2
0.2913
.2345
.3524
3
4
.7338
1.883
5
5.593
.097
6
18.52
7
66.65
.051
.025
8
9
256.2
1039.
1.00
.709
.474
.298
.176
.012
.005
R = 2.2521
R = 2.0954
0.2802
.2171
1.00I
.720 I
.5302
.505
.?189
1.989
.55338
.218
6.551
24.65
.1355
.08C
103.1
469.5
2295.
.046I
.026 I
.014
I
I
0.2761
.2081
P(~ m)
=1/DBm
D =
R= 1.8500
p(m)m
P( m)
=l/DBm
=1/DB m
=1/DBm
p (m)m'
20
16
I
I
* I
1.0 )0
R = 2.3614
0.2748
1.0(
.72 4
.2024
.72'35
.3167
.7001
.516
. 55
3 18
1.997
6.878
27.45
.24 .1
.15 '8
.1C )O
.06 ,2
.35074
.6847
1.982
6.990
28.80
.52! 5
.·36
.2512
.17;
.11L4.
134.8
701.0
.07d
.04'
7
3991.
.03(0
123.4
611.8
3295.
.038
.02
D = 4
R= 2.2187
0
0.2407
1.00
1
2
3
4
.2119
.3518
.8086
2.284
· 759
.547
5
6
7
7.444
26.95
105.4
.142
8
4539.6
9
1928.
.5372
.237
.079
.042
.021
.010
R = 2.5650
0.2359
.1906
.3170
.7565
2.297
8.300
34.25
156.6
779.2
41535.
1.00
.766
.576
.417
.291
.195
.126
.079
.047
.028
R = 2.7939
0.2359
.1790
.2960
.7131
2.220
8.563
36.50
179.5
972.9
5721.
1.00
R = 2.9578
0.2387
1.010
.585
.1719
.2828
.76 1
. 58'9
.437
.68352
.518
.226
2.151
8.261
8
.44,
.355 4
.156
.105
.070
.046
57.12
189.6
1076.
6692.
.764
.24 5
.17 6
.124
.08 7
.06 O
-
6 -
TABLE
II.
Relationship between Mean Demand Rate D and Mean Circulation Rate
R, relative to Mean Return Rate , for a Homogeneous Collection.
D/A
D/4
R/p
0.00
0.00000
0.26
0.17964
.02
.01925
.28
.04
.06
.0O
.03713
.30
.05385
.06956
.32
.18941
.19881
.20788
.34
.21663
.10
.12
.08437
.09838
.22509
.14
.18
.11167
.12431
.13637
.36
.38
.40
.42
.44
.20
.14789
.26352
.22
.15892
.46
.48
.24
.16949
.50
.16
0.00000
.01
.02
.01020
.02082
.03
.04
.03186
.52
.54
.56
0.10
0.35164
.35648
.356121
.86
.88
.37483
.37919
.38346
.38766
.39176
.36584
.37038
.66
.68
.27053
.70
.72
.32580
.33121
33649
5
.34165
.27734
.74
.34671
.94
.96
.98
1.00
D/ip
R/
D/p
0.30
.21
.22
0.30258
.32478
.34788
0.57074
.60418
.63907
.37193
.39697
.33
.34
.67550
.42303
.35
.45018
.47347
.36
.50795
.53869
·57074
.38
.75330
.79485
. 3830
.838577
.93137
.98120
.23327
.24119
.24886
.25630
0.12239
0.20
.13743
.14
.18621
.05
.06
.04334
.05528
.06768
.23
.24
.15
.20376
.22201
.25
.26
.07
.08
.08058
.09398
.17
.27
.28
.09
.10
.10791
.12239
.19
.20
.24099
.26072
.28123
.18
.28397
.29042
.29670
0.76
.78
.80
.82
.84
.30281
.11
.12
.13
.16
0.27734
.58
.60
.62
.64
D/4i
D/
0.00
0.50
.15307
.16932
.30258
.29
.30
.30877
.31459
.32026
.90
.92
.31
.32
.37
.39
.40
.39580
.39976
.40365
.71355
TABLE III.
Continued
m =
d(m)
0
1.016
xm
1
2
d(m)
0.897
0.838
Am
d(m)
0.800
km
d(m)
1.242
xm
1.020
km
d(m)
0.911
m
d(m)
Am
0.820
6
2.100
4.219
5.153
5.878
6.207
. 732
1.623
2.501
3.599
4.300
5.190
= 8
= 12
1.886
2.883
3. 392
4.878
5.775
6.416
.7726
1.587
2.435
3.308
4. 199
5.099
1.780
D= 3
2.720
3.680
16
4.646
5.577
6.355
.7671
1.566
2.394
3.246
4.117
5.003
1.719
D = 3
2.622
5.407
6.243
.7638
1.554
= 20
3.543
4.479
2. 368
3.203
4.053
4.927
2.631
D=4
4.026
5.362
6.529
7.369
7.653
,8469
1.780
2.730
3.824
4.885
5.944
=8
D = 4
d(m)
5
D= 3
3.180
D= 3
d(m)
xm
4
4 = 12
2.252
3.481
4. 741
5.972
7.063
7. 7853
.8316
1.724
2.670
3.662
4.687
5.736
2.103
= 4
3.242
= 16
4.423
5.620
6.767
7.693
.82358
1.694
2.607
3.560
4.546
5.561
D= 4
3.010
=20
4.098
5.212
2.563
3.496
6.317
7.296
4.450
5.433
1.961
.8191
1.676
See E.
(10).