Queuing Simulation

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Example 14.9

• Queuing Simulation

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Background Information

• The Streamlining Company manufactures various
  types of automobile parts.
• Its factory has several production lines, all
  versions of the series system shown on the next
  slide, with varying numbers of stations and
  machines.
• In an effort to improve operations, the company
  wants to gain some insights into how average
  throughput times and other output measures are
  affected by various inputs.

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Background Information -- continued

• Specific questions of interest are
• Is it better to have a single fast machine at
  each station or multiple slower machines?
• How much does the variability of the arrival
  process to station 1 affect outputs? What about
  the variability of processing times at machines?
• The company has experimented with 0 buffers and
  has found that the resulting blocking can be
  disastrous. It now wants to create some buffers.
  In front of which stations should it place the
  buffers?

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SERIESSIM.XLS

• The simulation model in this file allows us to
  experiment as much as we like by changing inputs,
  running the simulation, and examining the
  outputs.
• The inputs section appears on the next slide.
• Note that 1 is the code for constant interarrival
  or processing times, whereas the 2 is the code
  for exponentially distributed times.
• Also, cell B14 is black to indicate that the
  number of buffers in front of station 1 is always
  unlimited.

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Solution

• When we run the simulation, we obtain outputs
  such as those in those shown on the next slide.
• Perhaps the most important part of the outputs is
  in the range B18B21.
• For this particular run, we see that the average
  part took 7.457 minutes to get through the
  system.
• Only 28.09 of this was in processing. The rest
  was spent in queues or being blocked at station 1
  or 2.
• In addition, we see at the top of the output that
  10,090 parts are completed during the run time
  period, and 16 parts were left uncompleted at the
  end of the run time.

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Answering the Questions

• Turning to Streamlinings questions, we first
  examine the trade-off between fast and slow
  machines.
• The outputs on the next slide are typical.
• We keep the arrival rate at 1 part per minute and
  the mean service rate at 1/0.7 parts per minute
  at each station.
• In the first set of runs, there is a single fast
  machine at each station. Each machine has an
  exponential processing time with mean 0.7 minute.

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Answering the Questions -- continued

• In the second set of runs, we triple the number
  of machines at each station and also triple the
  mean processing time for each machine to achieve
  equivalent slow machines.
• The use of three runs per configuration indicates
  that different random numbers can produce
  slightly different results.
• However, if average throughput time is of primary
  interest, it is clear that the fast machines are
  better.

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Answering the Questions -- continued

• Even so, the results are probably not clear-cut
  to a manufacturer.
• So it comes down to a trade-off between a lot of
  time in processing or a lot of time in queues.
• This configuration might be described as low
  utilization.
• Parts arrive at rate 1 per minute, and each mean
  processing time is only 0.7 minute. The next
  slide shows the same type of results when the
  utilization is much higher.

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Answering the Questions -- continued

• Here we increased the mean processing times for
  the fast machines to 0.9. We also increase the
  buffer sizes to 10.
• This system is a disaster take a look at the
  average throughput times and the average times
  spent in queue in front of station 1, for example
  but it does indicate a very interesting result.
• In terms of average throughput time, the slow
  machines are now better by quite a margin.

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Answering the Questions -- continued

• Can you see why intuitively?
• The reason is that when utilization is high, one
  long processing time on a fast machine which
  is always possible with an exponential
  distribution can back up the whole system for
  quite a while.
• If there are multiple machines, however, parts
  can move around a machine experiencing a long
  processing time, and the whole system is not as
  affected.
• We might have guessed this , but with simulation
  it is obvious.

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Answering the Questions -- continued

• Streamlinings next question concerns the
  variability of arrival and processing times.
• Here we examine a 3-station process, with 1
  machine at each station and 5 buffers in front of
  stations 2 and 3.
• Some results are listed on the next slide.
• In columns B and C, interarrival times and
  processing times are exponential. In columns D
  and E, interarrival times are constant and
  processing times are exponential. This might be
  realistic if the company releases one part to
  the line every minute according to a nonrandom
  schedule.

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Answering the Questions -- continued

• In columns F and G, interarrival times are
  exponential and processing times are constant.
• Finally both are constant in column H.
• We made two runs for each of the random cases.
  Of course, only one run is necessary for the
  nonrandom case.
• By this time, these results should not come as a
  surprise. The more the company can do to wipe out
  variability, the better the manufacturing process
  will operate.

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Answering the Questions -- continued

• Finally, we analyze the effects of buffers and
  their placement.
• We now assume a 10-station process with a single
  machine at each station.
• The parts arrive at rate 1 per minute, each
  machine has a mean processing time of 0.5 minute,
  and all times are exponentially distributed.
• You might expect that when parts arrive only half
  as fast as the machine can process them, there
  should be no problem.

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Answering the Questions -- continued

• This is not true, especially if buffers are
  severely limited.
• We made several runs, starting with 0 buffers in
  the system and gradually adding buffers.
• Selected results for average throughput times
  appear on the next slide.
• When no buffers, blocking kills the system. This
  might not be evident from the percentages listed,
  because each part spends only a small amount of
  time being blocked. But there is almost always
  blocking somewhere in the system, and the effect
  is that a long queue eventually builds in front
  of station 1.

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Answering the Questions -- continued

• Suppose Streamlining has enough funds to build
  exactly 1 buffer somewhere. Where should the
  buffer be placed?
• We made nine runs, placing the single buffer in
  front of each station, with results in rows
  19-22.
• It is clear that the single buffer should be
  placed in the middle of the line, in front of
  station 6.
• Placing it in front or the back of the line does
  virtually no good. The reason is probably not
  intuitive, at least until we provide the clue.

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Answering the Questions -- continued

• The basic problem with this serial system is the
  interdependence between stations.
• A long processing time at one station can have
  negative effects throughput the line.
• Upstream stations (to the left) become blocked
  and downstream stations become starved for parts
  to process.
• By placing a buffer in the middle of the line, we
  do the most we can to break the line into two
  less dependent parts.

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Answering the Questions -- continued

• This effect can be seen by continuing to add
  buffers one at a time.
• The bottom section of the last model shown
  indicates the saturation effect of adding more
  buffers.
• The company gets a lot from its money from the
  first few buffers, but after the first few,
  blocking becomes a minor problem and more buffers
  fail to make much of an improvement.
• If buffers entail significant costs, Streamlining
  must trade off these costs against lower average
  throughput times and possibly other
  considerations.