More Monte Carlo Demonstrations

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More Monte Carlo Demonstrations

• T. R. Willemain

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Airport Simulation
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Airspace Simulation
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Example Inventory Management

• Scenario
• Demand for a spare part Poisson(?)
• On-hand inventory Binomial(n,p)
• n is our choice for the stock to keep on hand
  each day
• p models our (in)ability to keep n units
  available
• System performance measures
• Lost sales when demandgtsupply
• Excess inventory when supplygtdemand
• These trade off if keep more on hand, get fewer
  stockouts but more excess inventory

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On average, supply equals demand. Average excess
inventory is only 1/2 unit but service level
is only 60.
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Increase p to 0.9 from 0.5, so usually have more
on hand. Lost drops to 0.14 from 0.73, but
excess grows to 3.63 from 0.57.
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Example Maximum Weight

• Scenario
• 100 Loads arrive at random intervals
• Interval Expo(1)
• Loads have random weights
• Weight N(10,1)
• Processing requires a random interval
• Time Expo(10)
• Then the load is removed
• Questions of interest
• What is the distribution of maximum weight over
  time?
• How long does it take to finish processing the
  100 loads?

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One Sample
Variable N Maximum time 200
130.42 weight 200 198.37
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Random 100 c1 Exponential 1. Let c2
pars(c1) Random 100 c3 Normal 10 1.0. Random
100 c4 Exponential 10. Let c5 c2c4 Let c6
-c3 Stack (c2 c3) (c5 c6) (c8 c9). Sort c8 c9 c8
c9 By c8. Let c10 pars(c9) Maximum c8
k1. Maximum c10 k2. Stack k1 c12 c12. Stack k2
c13 c13.
This is the sequence of commands that execute the
simulation and save the maximum time and weight
in C12 and C13. They can be saved as a Minitab
Exec file, e.g., MaxWeight.mtb, and run using the
Minitab command sequence FilegtOther FilesgtRun an
Exec. Then they can be executed
repeatedly. They can also be copy/pasted into
the Command Line Editor using CtrlL and executed
one rep at a time.
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Plots of 9 Samples
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Results of 200 Replications
Descriptive Statistics MaxTime, MaxWeight
Variable N Mean StDev Minimum
Maximum MaxTime 200 130.10 16.59 92.67
178.74 MaxWeight 200 164.88 23.25 117.81
237.05
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In-Class Exercise System Lifetime

• Imagine a set of 5 subsytems in series.
• Each subsystem has 2 components in parallel.
• Assume for simplicity that all 10 components are
  identical
• Component lifetime Exponential(1000).
• Use simulation to estimate the pdf of system
  lifetime.

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Analysis Approach

• Each subsystem has a lifetime equal to the longer
  of its two redundant components lifetimes
• The entire system has series subsystems, so the
  system lifetime is the smallest of the 5
  subsystems lifetimes

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