Decision Maths

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Decision Maths

• Lesson 14 Simulation

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Simulation

• There are many times in real life where we need
  to make mathematical predictions.
• How long should a set of traffic lights stay red.
• How long should appointments be at a doctors
  surgery.
• What is the best queueing system for a bank or
  building society branch to operate to satisfy its
  customers.
• In all the above the situations you could
  observe what would happen in real life and try
  lots of scenarios.
• But in many situations it can be better to create
  a simulation to study.
• This can clearly be less time consuming and cost
  effective.

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Simulation

• Simulation means the imitation of the operation
  of a system.
• You will have clearly heard of flight simulators
  that can be used to train pilots.
• In this lesson we will be looking at Stochastic
  simulations.
• This is where you study situations where chance
  effects the outcome.
• The methods used are commonly called Monte Carlo
  methods.
• Monte Carlo is associated with gambling, dice,
  and roulette wheels.
• Many of the methods use these random devices to
  model situations.

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Simulation

• Think about the following situation
• At a bank
• 20 of customers spend 1 minute at the
• window.
• 30 of customers spend 2 minutes at the window.
• 40 of customers spend 3 minutes at the window.
• 10 of customers spend 4 minutes at the window
• The bank wish to investigate different queuing
  systems by using a computer to simulate the times
  that customers spend at the cashier window.

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Simulation

• They use the following simulation
• The computer generates random numbers from 00
  99 inclusive. (this can also be done on your
  calculator)
• A rule is devised which assigns a service time to
  each randomly generated number that reflects the
  percentages already given.
• 20 of the customers spend 1 minute at the
  window.
• So 20 of the random numbers can be used to
  represent this outcome.
• We could use the numbers 00 19.

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Simulation

• The following table shows how the rest of the
  numbers could be used.
• The way the simulation works is you now generate
  a random number and this will represent 1 person.
• The table will indicate how long that simulated
  person will spend at the cashier.

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Simulation

• A cashier serves 5 people in a row generate a
  possible scenario working out how long it takes
  to serve everyone.
• Example 72, 74, 17, 98, 57
• This queue would take 3 3 1 4 3 14
  minutes to serve.

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Using Random Numbers

• When using random numbers it is important that
  every outcome is uniformly distributed.
• An example to avoid would be rolling two dice and
  adding totals together.
• The P(total 2) 1/36 but the P(total 7) 1/6.
• You should choose your numbers carefully but you
  could be clever and not use all the numbers in
  certain cases.
• Assign the following distributions random
  numbers.

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Exercise
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Exercise
11
The Collectors Problem

• A manufacturer of breakfast cereals is giving
  away cards with pictures of football teams on
  them.
• There are 6 different cards in the set
• Argentina, Brazil, Columbia, Denmark, England,
  France.
• There is only one card in a box at a time and
  they are distributed randomly.
• Naturally a football fan would like to collect
  all 6 cards.

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The Collectors Problem

• i) Simulate the number of packets the fan will
  have to buy in order to collect all six cards.
• ii) Perform the simulation several times,
  recording the number of packets the collector
  buys before having a complete set.
• iii) Display the data and find the mean number of
  packets bought.
• iv) How would you carry out the simulation if
  there where 10 cards in the set.

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Pedestrian Crossing

• A pedestrian crossing in a busy city centre is
  studied.
• At peak times pedestrians arrive at the crossing
  at a rate of 1 every 10 seconds.
• You can model this by assuming that in any
  5-second period there is a 0.5 chance of 0
  pedestrians and a 0.5 chance of 1 pedestrian
  arriving.
• The first pedestrian to arrive at the crossing
  will press the button to request the lights.
• The lights then show dont cross for 25 seconds
  and then cross for 5 seconds.
• During the 5 second period all the people waiting
  can cross.

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Pedestrian Crossing

• i) Use a coin to simulate a period of about 100
  seconds, drawing your results in a table.
• ii) Use the results of your simulation to display
  results on
• a) The number of people crossing each time.
• b) The total lengths of time for which the
  traffic is allowed to flow freely.
• iii) How could this model be made more realistic?
  

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Doctors Surgery

• A Doctor is analysing the amount of time that
  patients spend in her surgery waiting room.
• Her first appointment is at 0900.
• Appointments are made at 10 minute intervals.
• Her last appointment is at 1120
• Each patients visit can vary from 5 to 15
  minutes.
• Patients can arrive up to 5 minutes early, but
  they are never late.

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Doctors Surgery

• i) Making suitable assumptions simulate the
  doctors surgery for 3 mornings.
• (think about using a table to help you).
• ii) Find the patients average waiting time.

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Arrival and service time

• This example can clearly be looked at in more
  detail.
• With any example that involves queuing
  (pedestrian problem) a more complex analysis is
  required.
• One possibility is to study the inter-arrival
  times or intervals.
• This just means the time that elapses between
  successive arrivals.
• We can use random numbers to explore the
  possibilities between arrivals.

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Bank Queue Problem

• A small building society has just one service
  counter.
• During lunchtime hours long queues sometimes
  build up.
• The following tables were drawn up by observing
  customers at the building society during
  lunchtime.
• The first shows the duration of a persons
  service time.
• The second shows the time that elapses between
  customers arriving a the bank.
• Simulate the branch for 30 minutes.

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Bank Queue Problem

• Rule to generate arrival times.
• Rule to generate service times.

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Bank Queue Problem

• Rule to generate arrival times.
• Rule to generate service times.

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Bank Queue Problem

• Use the table to calculate the number of people
  in the queue at any given time.

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Bank Queue Problem
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The queuing discipline

• For any queuing system there is normally a set of
  rules, called the queuing discipline.
• FIFO or First in First out.
• It is possible to have one single queue waiting
  for multiple tills.
• This is commonly seen at petrol stations or the
  self service tills in Sainsburys.
• Finally you can have lots of tills each with
  there own single queue.
• Again this is a common sight in supermarkets.

Till A
Per 4
Per 3
Per 2
Per 1
Till B
Per 8
Per 7
Per 5
Per 1
Till A
Till B
Per 6
Per 4
Per 3
Per 2
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Ex 6B pg 174