Analysis and Design of Asynchronous Transfer Lines as a series of G/G/m queues

1
Analysis and Design of Asynchronous Transfer
Lines as a series of G/G/m queues
2
Topics

• The negative impact of variability in the
  operation of Asynchronous Transfer Lines
• Modeling the Asynchronous Transfer Line as a
  series of G/G/m queues
• Modeling the impact of various operational
  detractors
• Employing the derived models in line diagnosis
• Employing the derived models in line design

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Asynchronous Transfer Lines (ATL)
W2
W3
TH
TH
B2
B3
M2
M3

• Some important issues
• What is the maximum throughput that is
  sustainable through this line?
• What is the expected cycle time through the
  line?
• What is the expected WIP at the different
  stations of the line?
• What is the expected utilization of the
  different machines?
• How does the adopted batch size affect the
  performance of the line?
• How do different detractors, like machine
  breakdowns, setups, and maintenance, affect the
  performance of the line?

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Analyzing a single workstation with deterministic
inter-arrival and processing times
Case I ta tp 1.0
WIP
1
TH 1 part / time unit Expected CT tp
t
1
2
3
4
5
Arrival
Departure
5
Analyzing a single workstation with deterministic
inter-arrival and processing times
Case II tp 1.0 ta 1.5 gt tp
WIP
Starvation!
1
TH 2/3 part / time unit Expected CT tp
t
1
2
4
5
3
Arrival
Departure
6
Analyzing a single workstation with deterministic
inter-arrival and processing times
Case III tp 1.0 ta 0.5
WIP
Congestion!
TH 1 part / time unit Expected CT ? ?
7
A single workstation with variable inter-arrival
times
Case I tp1 ta?N(1,0.12) (ca?a / ta 0.1)
WIP
3
2
TH lt 1 part / time unit Expected CT ? ?
1
t
1
2
3
4
5
Arrival
Departure
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A single workstation with variable inter-arrival
times
Case II tp1 ta?N(1,1.02) (ca?a / ta 1.0)
TH lt 1 part / time unit Expected CT ? ?
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A single workstation with variable processing
times
Case I ta1 tp?N(1,1.02)
TH lt 1 part / time unit Expected CT ? ?
Arrival
Departure
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Remarks

• Synchronization of job arrivals and completions
  maximizes throughput and minimizes experienced
  cycle times.
• Variability in job inter-arrival or processing
  times causes starvation and congestion, which
  respectively reduce the station throughput and
  increase the job cycle times.
• In general, the higher the variability in the
  inter-arrival and/or processing times, the more
  intense its disruptive effects on the performance
  of the station.
• The coefficient of variation (CV) defines a
  natural measure of the variability in a certain
  random variable.

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The propagation of variability
W1
W2
Case I tp1 ta?N(1,1.02)
Case II ta1 tp?N(1,1.02)
WIP
3
2
1
t
1
2
3
4
5
W1 arrivals
W1 departures
W2 arrivals
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Remarks

• The variability experienced at a certain station
  propagates to the downstream part of the line due
  to the fact that the arrivals at a downstream
  station are determined by the departures of its
  neighboring upstream station.
• The intensity of the propagated variability is
  modulated by the utilization of the station under
  consideration.
• In general, a highly utilized station propagates
  the variability experienced in the job processing
  times, but attenuates the variability experienced
  in the job inter-arrival times.
• A station with very low utilization has the
  opposite effects.

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The G/G/1 modelA single-station

• Modeling Assumptions
• Part release rate Target throughput rate TH
• Infinite Buffering Capacity
• one server
• Server mean processing time te
• St. deviation of processing time ?e
• Coefficient of variation (CV) of processing
  time ce ?e / te
• Coefficient of variation of inter-arrival times
  ca

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An Important Stability Condition

• Average workload brought to station per unit
  time
• THte
• It must hold
• Otherwise, an infinite amount of WIP will pile
  up in front of the station.

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Performance measures for a stable G/G/1 station

• Server utilization
• Expected cycle time in the buffer
  (Kingmans
  approx.)
• Expected cycle time in the station
• Average WIP in the buffer
  (by Littles law)
• Average WIP in the station
• Squared CV of the inter-departure times

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Remarks

• For a station with variable job inter-arrival
  and/or processing times, utilization must be
  strictly less than one in order to attain stable
  operation.
• Furthermore, expected cycle times and WIP grow to
  very large values as u?1.0.
• Expected cycle times and WIP can also grow large
  due to high values of ca and/or ce i.e.,
  extensive variability in the job inter-arrival
  and/or processing times has a negative impact on
  the performance of the line.
• In case that the job inter-arrival times are
  exponentially distributed, ca1.0, and the
  resulting expression for CTq is exact (a result
  known as the Pollaczek-Kintchine formula).
• The expression for cd2 characterizes the
  propagation of the station variability to the
  downstream part of the line, and it quantifies
  the dependence of this propagation upon the
  station utilization.

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Performance measures for a stable G/G/m station

• Server utilization
• Expected cycle time in the buffer
• Expected cycle time in the station
• Average WIP in the buffer
• Average WIP in the station
• Squared CV of the inter-departure times

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Analyzing a multi-station ATL
TH

• Key observations
• A target production rate TH is achievable only
  if each station satisfies the stability
  requirement u lt 1.0.
• For a stable system, the average production rate
  of every station will be equal to TH.
• For every pair of stations, the inter-departure
  times of the first constitute the inter-arrival
  times of the second.
• Then, the entire line can be evaluated on a
  station by station basis, working from the first
  station to the last, and using the equations for
  the basic G/G/m model.

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Example ATL Design

• Need to design a new 4-station assembly line for
  circuit board assembly.
• The technology options for the four stations are
  tabulated below (each option defines the
  processing rate in pieces per hour, the CV of the
  effective processing time, and the cost per
  equipment unit in thousands of dollars).

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Example ATL Design (cont.)

• Each station can employ only one technology
  option.
• The maximum production rate to be supported by
  the line is 1000 panels / day.
• The desired average cycle time through the line
  is one day.
• One day is equivalent to an 8-hour shift.
• Workpieces will go through the line in totes of
  50 panels each, which will be released into the
  line at a constant rate determined by the target
  production rate.

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A baseline designMeeting the desired prod. rate
with a low cost
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Reducing the line cycle time by adding capacity
to Station 2
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Adding capacity at Station 1, the new bottleneck
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An alternative optionEmploy less variable
machines at Station 1
This option is dominated by the previous one
since it presents a higher CT and also a higher
deployment cost. However, final selection(s) must
be assessed and validated through simulation.