STAGE 2 MATHEMATICS PATHWAYS
FOLIO TASK
Queuing
Topic:
Mathematics and Small Business
Subtopics from the Stage 2 Mathematical Applications Subject Outline:
3.2 - Queuing
A completed investigation should include:
 an introduction that outlines the problem to be explored, including it significance, its features,
and the context
 the method required to find a solution, in terms of the mathematical model or strategy to be used
 the appropriate application of the mathematical model or strategy, including
- the generation or collection of relevant data and/or information, with details of the process of
collection
- mathematical calculations and results, and appropriate representations
- the analysis and interpretation of results
- reference to the limitations of the original problem
 a statement of the results and conclusions in the context of the original problem
 appendices and a bibliography, as appropriate.
Learning Requirements
1.
2.
3.
4.
5.
6.
Assessment Design Criteria
Capabilities
Demonstrate an understanding
of mathematical concepts and
relationships.
Mathematical Knowledge and Skills and Their
Application
Communication
Identify, collect, and organise
mathematical information
relevant to investigating and
finding solutions to
questions/problems.

MKSA1 Knowledge of content and understanding
of mathematical concepts and relationships.

MKSA2 Use of mathematical algorithms and
techniques (implemented electronically where
appropriate) to find solutions to routine and
complex questions.
Recognise and apply the
mathematical techniques needed
when analysing and finding a
solution to a question/problem in
context.
Make informed use of electronic
technology to aid and enhance
understanding.
Interpret results, draw
conclusions, and reflect on the
reasonableness of these in the
context of the question/problem.
Communicate mathematical
ideas and reasoning using
appropriate language and
representations.
The specific features are as follows:

Citizenship
Personal
Development
Work
Learning
MKSA3 Application of knowledge and skills to
answer questions in applied contexts.
Mathematical Modelling and Problem-solving
The specific features are as follows:

MMP1 Application of mathematical models.

MMP2 Development of mathematical results for
problems set in applied contexts.

MMP3 Interpretation of the mathematical results in
the context of the problem.

MMP4 Understanding of the reasonableness and
possible limitations of the interpreted results, and
recognition of assumptions made.
Communication of Mathematical Information
The specific features are as follows:
Page 1 of 3

CMI1 Communication of mathematical ideas and
reasoning to develop logical arguments.

CMI2 Use of appropriate mathematical notation,
representations, and terminology.
Stage 2 Mathematics Pathways Mathematics and Small Business task
Ref: A203776 (revised February 2016)
© SACE Board of South Australia 2010
PERFORMANCE STANDARDS FOR STAGE 2 MATHEMATICS PATHWAYS
Mathematical Knowledge and
Skills and Their Application
A
Comprehensive knowledge of content
and understanding of concepts and
relationships.
Appropriate selection and use of
mathematical algorithms and techniques
(implemented electronically where
appropriate) to find efficient solutions to
complex questions.
Highly effective and accurate application
of knowledge and skills to answer
questions set in applied contexts.
B
Some depth of knowledge of content and
understanding of concepts and
relationships.
Use of mathematical algorithms and
techniques (implemented electronically
where appropriate) to find some correct
solutions to complex questions.
Accurate application of knowledge and
skills to answer questions set in applied
contexts.
C
Generally competent knowledge of
content and understanding of concepts
and relationships.
Use of mathematical algorithms and
techniques (implemented electronically
where appropriate) to find mostly correct
solutions to routine questions.
Generally accurate application of
knowledge and skills to answer questions
set in applied contexts.
D
Basic knowledge of content and some
understanding of concepts and
relationships.
Some use of mathematical algorithms
and techniques (implemented
electronically where appropriate) to find
some correct solutions to routine
questions.
Sometimes accurate application of
knowledge and skills to answer questions
set in applied contexts.
E
Mathematical Modelling and Problemsolving
Development and effective application of
mathematical models.
Complete, concise, and accurate solutions to
mathematical problems set in applied contexts.
Concise interpretation of the mathematical results in
the context of the problem.
In-depth understanding of the reasonableness and
possible limitations of the interpreted results, and
recognition of assumptions made.
Attempted development and appropriate application
of mathematical models.
Mostly accurate and complete solutions to
mathematical problems set in applied contexts.
Complete interpretation of the mathematical results in
the context of the problem.
Some depth of understanding of the reasonableness
and possible limitations of the interpreted results, and
recognition of assumptions made.
Appropriate application of mathematical models.
Some accurate and generally complete solutions to
mathematical problems set in applied contexts.
Generally appropriate interpretation of the
mathematical results in the context of the problem.
Some understanding of the reasonableness and
possible limitations of the interpreted results, and
some recognition of assumptions made.
Application of a mathematical model, with partial
effectiveness.
Partly accurate and generally incomplete solutions to
mathematical problems set in applied contexts.
Attempted interpretation of the mathematical results
in the context of the problem.
Some awareness of the reasonableness and possible
limitations of the interpreted results.
Limited knowledge of content.
Attempted application of a basic mathematical model.
Attempted use of mathematical
algorithms and techniques (implemented
electronically where appropriate) to find
limited correct solutions to routine
questions.
Limited accuracy in solutions to one or more
mathematical problems set in applied contexts.
Attempted application of knowledge and
skills to answer questions set in applied
contexts, with limited effectiveness.
Page 2 of 3
Limited attempt at interpretation of the mathematical
results in the context of the problem.
Limited awareness of the reasonableness and
possible limitations of the results.
Communication of
Mathematical Information
Highly effective communication
of mathematical ideas and
reasoning to develop logical
arguments.
Proficient and accurate use of
appropriate notation,
representations, and
terminology.
Effective communication of
mathematical ideas and
reasoning to develop mostly
logical arguments.
Mostly accurate use of
appropriate notation,
representations, and
terminology.
Appropriate communication of
mathematical ideas and
reasoning to develop some
logical arguments.
Use of generally appropriate
notation, representations, and
terminology, with some
inaccuracies.
Some appropriate
communication of
mathematical ideas and
reasoning.
Some attempt to use
appropriate notation,
representations, and
terminology, with occasional
accuracy.
Attempted communication of
emerging mathematical ideas
and reasoning.
Limited attempt to use
appropriate notation,
representations, or
terminology, and with limited
accuracy.
Stage 2 Mathematics Pathways Mathematics and Small Business task
Ref: A203776 (revised February 2016)
© SACE Board of South Australia 2010
STAGE 2 MATHEMATICS PATHWAYS
FOLIO TASK
Queuing
Introduction
Viktor and Lily are equal partners in a fruit and vegetable business and have noticed that their busiest time is
between 12 noon and 1pm each day. They want to investigate the number of staff needed to serve on an
additional counter that is only open for the one-hour period to cater for the overflow of customers. This counter
can have 1, 2, or 3 different service points each with its own server, and customers would be served at the next
available point.
At 12 noon, there are already 5 customers who need to be served at this counter (5 customers are queuing).
On average, a customer arrives every 2 minutes during this busy hour. The extra counter closes at 1pm, but
anyone still in the queue at that time will be served. Server 1 serves continuously if there are people waiting. A
new server (server 2 or 3) is added to the counter whenever the queue has 4 or more people waiting (and not
yet being served). Servers 2 and 3 close unless there are 4 or more people waiting.
On average, during this one hour, most customers require 3 minutes service time, but every fourth customer
requires 5 minutes of service time.
Read the entire task and write an informative introduction that demonstrates your understanding of the situation
to be investigated.
Mathematical Investigations
1. Draw a table and/or chart to display the results of the number of additional servers needed and when they
are needed.
2. For each server calculate:
 The number of people served by each server
 The time when the final customer is served at one of these service points
 The average customer waiting time
 The percentage of the time that each server spends doing other tasks.
3. Repeat the process above after making changes to your choice of conditions under which the counter is
staffed. At least two changes to the original conditions should be investigated. These could include
variations to: the number of customers initially waiting; the arrival or service interval; the conditions of the
extra servers. Use your sets of results to enable you to complete a comparison of the perceived operation
and efficiency of the queue under different situations.
Analysis/Discussion
Using the results of your calculations as a basis for analysis, decide how many staff need to be available to
serve the overflow of customers during the period from 12 noon to 1pm each day. Discuss the basis for your
decision.
Compare the efficiency of the models.
Discuss the issue of slack time of the servers.
Discuss the reasonableness of the results by considering:


any methods that you could use to improve the accuracy of your answer
any assumptions that you have made, or limitations of the model, in this investigation which could influence
your results.
Conclusion
Explain how many staff Viktor and Lily need to make available during the one hour period. Your answer should
consider which trial offers the most cost effective option for the business.
Notes to Teacher
To assist with the verification of student work each student could be given a variation of the conditions of the
original scenario. For example changes to: the number of customers waiting initially, the arrival interval, the
conditions for an extra server, the service time.
Teachers may consider adding visual aids to the task to assist students who need support in accessing the
requirements of the task.
Page 3 of 3
Stage 2 Mathematics Pathways Mathematics and Small Business task
Ref: A203776 (revised February 2016)
© SACE Board of South Australia 2010