academic libraries as a context for teaching mathematical modeling
TRANSCRIPT
This article was downloaded by: [Selcuk Universitesi]On: 20 December 2014, At: 10:55Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK
PRIMUS: Problems, Resources,and Issues in MathematicsUndergraduate StudiesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/upri20
Academic Libraries asa Context for TeachingMathematical ModelingJon WarwickPublished online: 10 Nov 2008.
To cite this article: Jon Warwick (2008) Academic Libraries as a Context for TeachingMathematical Modeling, PRIMUS: Problems, Resources, and Issues in MathematicsUndergraduate Studies, 18:6, 500-515, DOI: 10.1080/10511970701305356
To link to this article: http://dx.doi.org/10.1080/10511970701305356
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all theinformation (the “Content”) contained in the publications on our platform.However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness,or suitability for any purpose of the Content. Any opinions and viewsexpressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verified withprimary sources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses, damages,and other liabilities whatsoever or howsoever caused arising directly orindirectly in connection with, in relation to or arising out of the use of theContent.
This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan,sub-licensing, systematic supply, or distribution in any form to anyone isexpressly forbidden. Terms & Conditions of access and use can be found athttp://www.tandfonline.com/page/terms-and-conditions
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
PRIMUS, XVIII(6): 500–515, 2008
Copyright � Taylor & Francis Group, LLC
ISSN: 1051-1970 print / 1935-4053 online
DOI: 10.1080/10511970701305356
Academic Libraries as a Context for TeachingMathematical Modeling
Jon Warwick
Abstract: The teaching of mathematical modeling to undergraduate students
requires that students are given ample opportunity to develop their own models and
experience first-hand the process of model building. Finding an appropriate context
within which modeling can be undertaken is not a simple task as it needs to be readily
understandable and seen as relevant by students, require an appropriate level of
mathematical training, and reflect the differing learning styles of students. This article
describes the academic library as a context for modeling activity, gives an example of a
typical model based on queuing theory and relates the modeling activity to both
Bloom’s Taxonomy and Kolb’s Learning Cycle. The article also outlines further
model development that students may undertake.
Keywords: Mathematical modeling, academic library, learning styles.
1. INTRODUCTION
Every school, college, and university has a library of some description.
Academic libraries are a vital teaching resource for the provision of physical
learning materials, research literature, multi-media resources, and on-line
services. They are also, however, a useful resource when acting as a context
for setting students’ problems relating to mathematical modeling and statistical
analysis. Modeling the circulation of text books, for example, between the
library and the student body is a problem which can be approached in a variety
of ways of varying complexity and it is often possible to obtain, or to collect
first-hand, data which can be fed into the modeling process.
Many of the problems associated with the efficient running of an
academic library are, of course, interrelated and provide a web of variables,
Address correspondence to Jon Warwick, Faculty of Business, Computing, and
Information Management, London South Bank University, Borough Road, London
SE1 0AA, UK. E-mail: [email protected]
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
parameters, and functional relationships for students to explore. For example,
modeling the circulation of books allows some issues relating to the most
appropriate loan period to be examined. The loan period is related to the
number of duplicate copies of titles provided (providing duplicate copies is
an alternative to short loan periods) and this will relate to the number of
books customers may borrow at one time, the level and pattern of demand for
the books and so on. Furthermore, it is not a context that needs a great deal of
explanation to students as they will probably be quite aware of the relevant
issues and factors through their own interaction with the library system.
In this article we will look at an example model of the borrowing process
that has been used by students as part of their practical modeling work.
Typically, students who would be undertaking such work would be familiar
at least with college algebra and basic probability, and may well be studying
science or engineering at university level. The model described has been used
with students studying operational research and modeling as part of an
undergraduate computing course and also with postgraduate students study-
ing decision science. The article illustrates the value of these types of models
from the learning and teaching perspective in reference to Bloom’s taxonomy
and Kolb’s cycle of learning. An illustration of how students are expected to
engage with the model is given and typical example results produced by the
model are shown. Finally, the article illustrates how the initial model has
been further developed by students and some of the alternative approaches
that are possible that students have explored. This illustrates the flexibility of
academic libraries as settings for modeling exercises.
2. MATHEMATICAL MODELING IN THE ACADEMIC LIBRARY
Much work in management science has been directed towards helping orga-
nizations make the most efficient use of limited resources and a large number
of modeling approaches have been devised and applied over the last 50 years
to help organizations achieve such efficiencies [1]. Examples of well known
approaches that spring to mind range from those that attempt to provide
optimal solutions to specific problems such as mathematical programming to
those that help us understand the dynamic behavior of parts of the organiza-
tion such as simulation or system dynamics. Other modeling approaches have
been developed for specific applications such as the huge spectrum of stock
control models or the plethora of queuing theory models now being devel-
oped for the analysis of distributed computer systems, among other things.
Academic libraries also need to make optimal use of their resources and
in fact they have particularly acute problems in being a vital teaching
resource often under overwhelming pressure from customers, but with
restricted budgets that must replace a book stock becoming rapidly obsolete
in some subject areas. The last of the modeling approaches mentioned above
Academic Libraries as Context for Teaching 501
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
was adopted in what was probably the first comprehensive systematic ana-
lysis of library systems [11]. Morse made extensive use of Markov and
queuing models to examine and predict the circulation rates of books [12,
14] although he also made use of statistical models [13]. Morse’s models
indicated, for example, that the average circulation of a book in any year t is
related to the average circulation in the previous year by the relation
RðtÞ ¼ �þ �Rðt� 1Þ
Where b is a measure of the decrease in popularity of a book from year to
year and a measures the asymptotic circulation rate. Morse suggested a
procedure for deciding when duplicate copies are necessary based on R(t)
and made a number of observations, for example that doubling the number of
copies will not double circulation.
If we restrict ourselves to consideration of the problems of loan and
duplication policy only, then even up to the early 1990s were there analytical,
statistical, stochastic, and simulation models galore [16] and since that time,
with the advent of more powerful and cheaper computing, information, and
decision support systems have made data more accessible and amenable to
manipulation thus encouraging further quantitative analysis. The approaches
adopted have been many and varied, often depending on the particular part of
the system being modeled and a number of classic texts summarise the
application of management science techniques to libraries [9, 15].
One of the major studies undertaken in the UK and probably the most
comprehensive simulation work in this area to date has been that of Buckland
[3]. Buckland’s work related the loan period, duplication policy, book popu-
larity, and satisfaction levels (as measured by the probability of finding a
required book on the shelf). The work resulted in the implementation of a
variable loan and duplication policy in the University of Lancaster library,
the outcome being an increase in the satisfaction level measured six months
later. The increase was not, however, sustainable.
Much of the quantitative modeling work undertaken since the pio-
neering work of Morse has used current or forecast book demand and
circulation rates. Using data collected from the current system assumes no
change in the behavior of users, even though by changing library policy
(e.g., changing the loan period or adding additional copies to the book
stock) managers will have changed the system. Such change might alter
user satisfaction within the system and therefore the way users interact
with the system. In this sense many of the models are essentially static. In
other words, quantitative and simulation models have tended to ignore
human interaction and assume complete objectivity in decision making
and this has caused some questioning of the confidence we should place
in traditional models [2, 5].
502 Warwick
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
A more complete picture would be provided by trying to model and there-
fore predict user behaviour within a system so that the real effect of system
change can be investigated through the changed behavior of humans within the
system. Buckland [4] also recognises this in his mention of the ‘‘feedback
loops’’ that can exist within library systems and we shall now consider the
framework for a model in which such modeling could be incorporated.
3. THE PROBLEM CONTEXT
In this article we look at a relatively simple problem of interest to librarians
responsible for setting library policy, that of deciding on an appropriate loan
period. The situation initially modeled is a very simple version of reality in
which we consider the circulation of a single copy of a text among a group of
people. We shall call these customers and they may be students, researchers,
faculty, or staff. The approach taken here is to use queuing theory, a well
established group of models which have a wide range of applications in the
real world. In the simplest case we have a single queue of customers who are
waiting to be served by a single server. The arrival rate of customers is
assumed to be a random process with an average arrival rate of l per period.
The service time for each customer is assumed to follow an exponential
distribution and the average service rate is m customers per period. Orderly
behavior in the queue is observed by customers at all time. These are the
assumptions of what is termed an M/M/1 queue, the Ms denoting Markovian
(random) arrivals and service times and the 1 denoting a single server. A
good example of a general text covering the essentials of queuing theory is
that of Gross and Harris [6].
In the context of our problem, we have mentioned that the customers are
students, faculty and staff who are queuing to borrow the book copy. The
average time taken to service each customer, 1/m, is the loan period in
operation. Once the book is returned (the customer service is finished) the
next customer in the queue can borrow the book (begin their service).
In conventional queue analysis, the system is described by the number of
customers in the system. Figure 1 shows the state diagram with the individual
states illustrated and since the transitions from state to state are caused by
0 1 2 3
λ λ λ
μ μ μ
Figure 1. State Transition Diagram.
Academic Libraries as Context for Teaching 503
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
customer arrival or departure, the rate of transition from state to state will be
the average arrival and service rates. The first state, 0, represents an empty
system with the book available in the library to be borrowed.
This simple model implicitly assumes that all customers will join the
queue if the book is not available (by making a reservation for the book) and
that the loan period is fixed. Following the earlier comments about making
customer behaviour responsive to system change it is not difficult to make
the model more realistic in two ways. Firstly, customers will respond to the
state of the system so that if reservations already exist for the book (a long
queue exists) then they will not bother to make their own (perhaps going
elsewhere to get the book or finding a substitute) and so the arrival rate
reduces. This is modeled by introducing a factor a (a � 1) which models the
exponential reduction in the arrival rate as the queue grows. This is supported
on a priori grounds since some customers would be put off by the existence
of the queue, but there would be some persistent customers who will make
reservations even if the queue is long. Thus a is a measure of the persistence
of customers in trying to obtain a copy of the book.
Secondly, we can allow for the possibility of a truly variable loan policy
in which the loan period is shortened as the queue grows so that customers
can be serviced more quickly and the circulation rate of the book increased.
The more popular the book, the shorter the loan period becomes. This is
modeled by the factor b (b � 1) which increases the service rate (decreases
the loan period) as the queue grows. Incorporating these factors yields the
state diagram of Figure 2.
We are interested in calculating the steady state behavior of the queue
and in library terms, there are three queue statistics that would be of interest:
(i) What is the average loan period suggested for the book?
(ii) What is the average waiting time to get the book?
(iii) What proportion of the time is the book on loan and in use?
Each of these statistics can be found analytically from the model for
given values of l, m, a and b.
0 1 2 3
λ αλ
μ βμ β2μ
α2λ
Figure 2. Modified State Diagram.
504 Warwick
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
4. SOLVING THE MODEL
Conventional analysis of queues revolves around finding the stationary dis-
tribution of states. This means finding the stationary probability of the system
being in each of the states. We define Pn, as the probability of being in state n
and the stationary distribution is found by equating the flows into and out of
each state. Thus, mP1 = lP0 from which we get that P1 ¼ ��P0 and similarly
P2 ¼ ��
��
� �2
P0 and in general Pn ¼ ��
� �n�1��
� �n
P0. Now by putting � ¼ ��
and � ¼ �� we can sum the stationary probabilities to get:
P0½1þ �þ ��2 þ �2�3 þ . . .� ¼ 1
Summing the series and solving for P0 we obtain:
P0 ¼1� ��
1þ �ð1� �Þ
provided that gr , 1.
We can work out additional queue statistics as follows:
Average Arrival Rate, ��
:
��¼ �P0 þ ��P1 þ �2�P2 þ �3�P3 þ . . .
from which we get that:
��¼ � 1 � ��
1 þ �ð1 � �Þ
� �1 � ��� þ ��
1 � ���
� �
Average Service Rate, ��
:
��¼ �P1 þ ��P2 þ �2�P3 þ . . .
1� P0
from which we get that:
��¼ �ð1� ��Þð1� ��Þ
Expected Queue Length, E[Q]:
E½Q� ¼ 1P2 þ 2P3 þ 3P4 þ . . .
Academic Libraries as Context for Teaching 505
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
from which we get that:
E½Q� ¼ ��2
ð1� ��Þð1þ �ð1� �ÞÞ
Expected Waiting Time, E[W]:
E½W� ¼ E½Q���
from which we get that:
E½W� ¼ �
�
�2ð1� ���Þð1� ��Þ2 ð1� ���þ ��Þ
" #
These formulae are calculated using no more than college algebra and a
knowledge of infinite geometric series and basic probability.
There are two extreme special cases to note here. If we set a = 0 then we
arrive at a model in which no queue forms at all and customers only borrow if
the book is available on the shelf. On the other hand, if we set a = b = 1 then
we have a situation in which every arrival joins the queue and the service
time never changes. In the latter case, the results above reduce to the classic
formulae for the standard M/M/1 queue.
5. SOME EXAMPLES OF MODEL BEHAVIOR
The formulae described above can be input to a spreadsheet in order to
explore the changes to these library operating statistics as the parameters
change. Some examples are given below.
Figures 3, 4, and 5 show the results of applying the queuing model for a
book with a rate of demand (l ) of 0.2 per week and an initial loan period of
four weeks (m = 0.25). The values of a used in each graph vary from 0 to 1 in
steps of 0.2.
We consider Figure 3. In this model, as the queue length increases
the loan period is reduced, so as a increases then we would expect the
average loan period to reduce, which it does. Also, for any given value of
a the average loan period decreases as the value of b increases. Again
this is as expected because as b increases, the response to a lengthening
queue is a faster reduction in loan period. Note also that with a = 0 a
queue never builds up so the average loan period will remain at 4 weeks
for all values of b.
506 Warwick
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
In Figure 4, we can see that the average waiting time increases as aincreases due to the increased demand but decreases as b increases due to the
reduction in loan period and so faster processing of queue members. Here,
the value of the variable loan period can be seen. A fixed 4-week loan period
gives (with a= 1) an unacceptably high average waiting time of 16 weeks.
This is reduced quite dramatically if the loan period is allowed to vary.
Turning finally to Figure 5, we can see that increasing a increases the
book usage (as measured by the proportion of time the book is on loan, 1 – P0)
and increasing b decreases book usage, although the changes are not dramatic.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2Beta
Ave
rage
Loa
n Pe
riod
Increasing α values
Figure 3. Sample Average Loan Periods.
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
1 1.2 1.4 1.6 1.8 2Beta
Ave
rage
Wai
ting
Tim
e Increasing α values
Figure 4. Sample Average Waiting Times.
Academic Libraries as Context for Teaching 507
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
Interestingly, allowing a variable loan period tends to decrease average book
usage, but is beneficial in that it reduces waiting times and increases circula-
tion rates (the circulation rate is defined as book usage divided by the average
loan period and although book usage declines slightly average loan periods
reduce more quickly giving an increase in book circulation).
In general, these types of results illustrate how the model can be used to
assess library performance under different conditions and would allow man-
agers to assess changes to the performance of the library system as a result of
using a fixed loan policy (b= 1) or a variable loan policy (b . 1).
We now consider the benefits of using this approach from the learning
and teaching perspective before considering in more detail how students
engage with this modelling process.
6. CONSIDERATIONS FOR LEARNING AND TEACHING
When planning any kind of teaching activity there are a number of factors
that need to be considered among which are the level of learning which
students are expected to achieve and the learning style of the target students.
When considering the level of learning to be achieved educators often make
reference to Bloom’s Taxonomy [10] which categorizes learning into six
levels, each of which is exemplified with knowledge and skills that students
should be able to demonstrate at each level [7]:
(i) Knowledge: List or recite;
(ii) Comprehension: Explain or paraphrase;
(iii) Application: Calculate, solve, determine, or apply;
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0Beta
Boo
k U
sage
Increasing α values
Figure 5. Average Book Usage.
508 Warwick
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
(iv) Analyse: Compare, contrast, classify, categorize, derive model;
(v) Synthesis: Create, invent, predict, construct, design, improve, propose;
(vi) Evaluation: Judge, select, critique, justify, verify, debate, assess,
recommend.
Learning should be structured in such a way that the desired level of the
taxonomy is reached and that a student’s achievement of the required level is
assessed. In higher education we would be expecting our students to be able
to demonstrate skills at the higher levels of Bloom’s Taxonomy. Using
modeling examples such as the one described allows all levels of the taxon-
omy to be reached and it is the teacher’s decision as to which is the most
appropriate for the level of student being taught. Table 1 below gives an
illustration of how the library modeling activity relates to the levels of
Bloom’s Taxonomy.
Table 1. Mathematical modeling and Bloom’s Taxonomy
Level Descriptor Modeling Activity
1 Knowledge Know and can state formulae, describe queue
parameters, define a Markov process, conditions
for steady state.
2 Comprehension Understand the basic structure of a queue, the
possible arrival and service mechanisms and queue
processing systems. Understand the assumptions of
simple single server queues in the queuing context.
3 Application Be able to formulate a problem in the library domain
as a queue problem, translate the terminology of
queuing theory into the library context and interpret
the assumptions of queuing theory in the library
context. Decide on appropriate data collection.
4 Analysis Specify a mathematical model based on queuing theory
and find a solution in terms of the stationary
distribution of states and queue statistics. Collect
appropriate data, estimate model parameters,
interpret the queue statistics generated within the
context of the library problem.
5 Synthesis Construct experiments based on the model output to
investigate policy, determine the validity of the
model and the assumptions identified, collect data
from the model.
6 Evaluation Critique the model, suggest enhancements. Evaluate
numerical results and their robustness. Make
suggestions for policy, report findings. Assess
alternative modelling approaches.
Academic Libraries as Context for Teaching 509
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
Clearly the activities suggested at each of the levels in Table 1 apply to
any modeling exercise and not just to those set in the library context. The
advantage of the library setting though is that there is scope for expanding the
breadth of the models, for modelling quantitative and qualitative aspects, for
data collection and for investigating alternative modeling approaches. We
shall touch on this a little later.
As well as being able to assess student skills at the different levels of
Bloom’s Taxonomy, these modeling exercises also allow various learning
styles to be experienced by the student. For example, if we consider the Kolb
Cycle of learning [8] our ideal would be ‘‘to structure learning activities that
will proceed completely around the cycle, providing the maximum opportu-
nity for full comprehension’’ [7]. The cycle is described in Figure 6 and there
are four learning styles represented as quadrants on two axes representing
perception (concrete or abstract) and processing (active or reflective). We
can see reflections of all four quadrants in the modeling task set around the
academic library.
Concrete/Reflective: There is a very clear need to establish three things
here. Firstly the real world problem being examined is one which is of direct
relevance to the student and the library context is one which they will all
experience and have a vested interest in improving! Secondly, the modelling
approach adopted is going to be beneficial in attacking this problem and also
has application in many other situations. Thirdly, the higher-level modeling
skills that they will be developing will be of value throughout their working
lives and in a wide variety of problem solving situations.
Abstract/Reflective: This learning style requires that a well organised and
clear theoretical basis be available to the student. There is a clear body of
knowledge associated with queuing theory (and with many other manage-
ment science modelling approaches that might be used) which have been
developed over time and the models are well understood. This provides a
Concrete Experience
Abstract Conceptuali-sation
Reflective Observation
Active Experimentation
Why are we do-ing this? How is it relevant to me?
What is being taught? What is important?
How does this work? What is the underlying theory?
What happens if I do …? How can I use this?
Figure 6. The Kolb Learning Cycle.
510 Warwick
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
firm basis for model development. Furthermore, the context is also relatively
easy to describe and being something with which students are already
familiar should not generate too many initial questions and ambiguities or
be time consuming to grasp.
Abstract/Active: Applying the well understood theoretical approach
within a familiar setting allows modelling to proceed quite rapidly and for
experimentation generating the initial types of output shown earlier to be
undertaken. Analysis of this type of output will assist with an understanding
of the theory (what happens as arrival rates or service rates are changed?
What if we change the level of customer persistence?) and also of the model
in context (is this how customers would really continue to behave if the
average loan period is short? What effect would the type of book have—is
this model valid for all types of books and strength of recommendation?
What other factors might come into play here?). Also there is a powerful
lesson to be learned here regarding the tractability of the model. If the model
is altered (non-random arrivals or service times) then how does the model
solution process change? Modeling students can research the effects of
general service and arrival distributions.
Concrete/Active: Within the context of the library, there is ample scope
for experimentation with the model to decide on a suitable loan policy. Many
libraries already operate a version of the variable loan policy with perhaps
three different loan periods in operation for books with high, medium or low
demand. Very few operate a policy where the loan period could change for
the same book given its demand pattern at any particular time. Is this policy
better from the model results? What are the appropriate performance mea-
sures? How viable is it, with modern computer-based book circulation
systems, to operate such a policy? What are the drawbacks and advantages?
Setting modelling exercises as described within the library context
allows all four learning styles to be addressed so that all students can draw
on their strengths (especially if the work is set in a group context) and also
experience other learning styles for which they are less adapted. As with any
modelling exercise, the teacher’s role is key in providing both technical
expertise and helping with modeling strategy.
7. STUDENT ENGAGEMENT AND PRACTICAL MODELING
The model has been used with both undergraduate and postgraduate students,
with the work students were asked to undertake reflecting different levels of
Bloom’s taxonomy. For both groups of students, the module being taught
introduced a variety of standard model forms including Markov processes
and queuing theory, stock control, population models, and simulation.
Over a two-week period (four hours teaching plus private study time) the
topic of Markov processes was introduced with queuing theory being the first
Academic Libraries as Context for Teaching 511
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
specific example. Different queuing systems were described culminating in
the derivation of the formulae for the operating characteristics of the M/M/1
queue. The library problem context was then introduced to students, and in
the subsequent week all students were guided through the formulation of the
four parameter model given in this article. The students, with the aid of the
teacher, then derived expressions for ��
, ��
, E[Q], and E[W]. Armed with this
general model, the students were then given practical modeling work which
fell into three broad stages. Undergraduate students were expected to com-
plete the first two stages (although just the first stage could be used for
smaller projects) and postgraduate students had to complete all three stages.
7.1. Stage 1
This stage required students to investigate whether they thought the current
loan period in operation in their library was appropriate. This was done by
requiring students to:
� Set b= 1 and 1/m as the current loan period;
� Select an example of a highly used and less used title from the library and
derive an estimate for l in each case;
� Derive an estimate for a;
� Using spreadsheet models to estimate the queue characteristics and inves-
tigate the sensitivity of these characteristics to changes in l and a;
� Try alternative loan periods and assess the effects on the queue character-
istics, including the variable loan period (b . 1);
� Comment on the model’s assumptions and their validity within the library
context.
All students were required to report their findings in written form and would
be given three weeks to complete this stage, working in groups of two or
three. Experience of working with students has shown that in trying to
estimate l, some students took the approach of measuring potential demand
by trying to estimate how many students each of their chosen books had been
recommended to, and then conducting polls of their fellow students to
ascertain the likelihood of students following up the recommendation (this
nearly always ends in overestimations of l!). Others have tried counting the
issues over, say, a year by counting date stamps in the book itself to get a
lower bound on demand, and then using a ‘‘correction factor’’ to adjust for
unsatisfied demand.
In estimating a, students have usually adopted the approach of interview-
ing or using questionnaires with fellow students to find out their likelihood of
making a reservation for the title, given differing lengths of reservation queue.
This usually raises interesting questions about the effect of the type of book
512 Warwick
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
(size or subject area or cost) and the strength of recommendation. At this stage
students would not be expected to produce formal models for l or a, just to
estimate their values for selected books.
7.2. Stage 2
In this stage, lasting two weeks, students were required to suggest how the
unrealistic assumptions of their model might be overcome. Typically, stu-
dents would question the unrealistic service time distribution which, if the
library has an effective fine policy for overdue books, produces too great a
variation in service times and so tends to overestimate waiting times. In
reality, books would not often be kept out long past their due date.
Furthermore, popular books would not often be returned early! Students
consider alternative service time distributions and research for themselves
how the model must change. In the same way, the notion of random arrivals
should also be challenged. Although the M/G/1 and G/G/1 queuing systems
were not explicitly analyzed as part of the taught modules, students generally
researched these results for themselves. As a consequence, a and b drop out
of the model (students were not expected to analyse this) but the more
realistic arrival and service processes are compensation. Some students also
mention multiple servers to deal with multiple book copies and again can
research the results.
7.3. Stage 3
Postgraduate students were asked to consider how they might model the
parameter a in the original model and to produce a critique of their models
produced so far. This would include discussion of the robustness of their
results and how the models might be validated. They were then asked to
devise an alternative modeling approach and to contrast the two approaches
in terms of the validity of assumptions being made. Finally, they contrast the
advantages and disadvantages of each approach as an aid to policy making in
libraries where decisions over the loan period need to be made. Students were
given two weeks to complete this stage.
Examples of other modeling frameworks that could be used include
population models or stock control models. For both of these there is an
extensive literature and the models are tractable and well understood. As an
example, Warwick [17] describes the use of population models in modelling
the circulation of books. In one example, the predator-prey model is used to
represent the populations of books (prey) and customers (predators). The
prey population grows and shrinks as books are borrowed and returned and
the predator population grows and shrinks according to the availability of
Academic Libraries as Context for Teaching 513
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
books dependent on the number of copies and the loan period. These models
are interesting for students to investigate and finding analogies between book
circulation in libraries and other diverse contexts which have already been
modeled is found to be engaging by students.
CONCLUSIONS
Teaching mathematical modeling is a difficult task as the skills required to
generate useful models include technical and mathematical skills, modeling
skills, and very often inter-personal and communication skills. Successful
teaching requires that the material taught and the examples used are at the
appropriate academic level and also engage students in a variety of learning
styles. This article suggests that the context of modeling within the academic
library is a fruitful area of activity as it requires skills at all levels of Bloom’s
Taxonomy and engages students in a variety of learning styles as described
by the Kolb model.
Much analytical work in academic libraries has been described in the
literature and this provides a rich variety of support for modeling activities
and also gives examples of models with differing levels of complexity
(requiring differing levels of general mathematical education) and using a
broad range of modeling approaches.
This article has described a model using queuing theory to illustrate how
a standard modelling framework can provide fairly rapid results for students
to begin exploring, yet allows plenty of scope for further development and
refinement. Experience of using this context with students has been positive
as students have been able to appreciate the value of the modelling exercise
as well as grasp the contextual detail quickly as it is an environment with
which they are familiar.
It is to be hoped that other educators will find the literature available in
this area beneficial in their teaching of mathematical modelling and that
educators will describe other contexts in which they have had success
enthusing budding mathematical modellers.
REFERENCES
1. Anderson, D. R., D. J. Sweeney, and T. A. Williams, 1994. Quantitative
Methods for Business. St Paul, MN: West Publishers.
2. Buchanan, J. T., E. J. Henig, and M. I. Henig, 1998. Objectivity and
subjectivity in the decision making process. Annals of Operations
Research, 80: 333–345.
3. Buckland, M. K., 1975. Book Availability and the Library User. London:
Pergamon Press.
514 Warwick
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14
4. Buckland, M. K., 1988. Library Services in Theory and Context
2nd Edition. Oxford: Pergamon Press.
5. Dahlin, T. C., 1991. Operations Research and organisational decision
making in academic libraries. Collection Management, 14(3/4): 49–60.
6. Gross, D. and C. M. Harris. 1998. Fundamentals of Queuing Theory 3rd
edition. New York: John Wiley and Sons.
7. Jenson, D. and K. Wood, 2000. Incorporating learning styles to enhance
mechanical engineering curricula by restructuring courses, increasing
hands-on activities and improving team dynamics. http://files.asme.org/
asmeorg/Governance/Honors/1094.pdf.
8. Kolb, D. A., 1984. Experimental Learning: Experience as the Source of
Learning and Development. Englewood Cliffs, NJ: Prentice-Hall.
9. Kraft, D. H. and B. R. Boyce. 1991. Operations Research for Libraries
and Information Agencies. San Diego, CA: Academic Press.
10. Krathwohl, D. R., B. S. Bloom, and B. B. Maisa, 1964. Taxonomy of
educational objectives: The classification of educational goals.
Handbook 11, Affective Domain. New York: David McKay Co. Inc.
11. Morse, P. M. 1968. Library Effectiveness—A Systems Approach.
Cambridge MA: MIT Press.
12. Morse, P. M. and C. C. Chen, 1972. Using circulation desk data to obtain
unbiased estimates of book use. Library Quarterly, 45 (2): 179–194.
13. Morse, P. M., 1976. Demand for library materials: An exercise in
probability analysis. Collection Management, 1: 3–4.
14. Morse, P. M., 1979. A queuing theory, Bayesian model for the
circulation of books in a library. Operations Research, 27(4): 693–716.
15. Rowley, J. E. and P. J. Rowley, 1981. Operations Research. A Tool for
Library Management. Chicago: American Library Association.
16. Warwick, J., 1992. A review of some modelling approaches to the loan
and duplication of academic texts. Journal of Librarianship and
Information Science, 24(4): 187–194.
17. Warwick, J., 2006. Model reusability: A question of perspective.
Collection Management, 30(3): 39–48.
BIOGRAPHICAL SKETCH
Jon Warwick completed his first degree in Mathematics and Computing at
South Bank Polytechnic in 1979 and was awarded a PhD in Operations
Research in 1984. He has many years of experience in teaching mathematics,
mathematical modeling, and operations research in the higher education
sector and is currently Professor of Educational Development in
Mathematical Sciences at London South Bank University. He is also the
Faculty Director of Learning and Teaching.
Academic Libraries as Context for Teaching 515
Dow
nloa
ded
by [
Selc
uk U
nive
rsite
si]
at 1
0:55
20
Dec
embe
r 20
14