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

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Analysis and Design of Asynchronous Transfer
Lines as a series of G/G/m queuesOverview and
Examples
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Topics

• Modeling the Asynchronous Transfer Line as a
  series of G/G/m queues
• Modeling the impact of preemptive,
  non-destructive operational detractors
• Employing the derived models in line diagnosis
• Employing the derived models in line design
• The role of batching in the considered
  manufacturing systems
• An analysis of a workstation involving parallel
  batching

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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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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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Operational detractorsA primal source for the
line variability

• Effective processing time time that the part
  occupies the server
• Effective processing time Actual processing
  time
• any additional non-processing time
• Actual processing time typically presents fairly
  low variability ( SCV lt 1.0).
• Non-processing time is due to detractors like
  machine breakdowns, setups, operator
  unavailability, lack of consumables, etc.
• Detractors are distinguished to preemptive and
  non-preemptive. Each of these categories requires
  a different analytical treatment.

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Preemptive non-destructive operational detractors

• Outages that take place while the part is being
  processed.
• Some typical examples
• machine breakdowns
• lack of consumables
• operator unavailability

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Modeling the impact of preemptive detractors

• X random variable modeling the natural
  processing time (i.e., without the delays due to
  the detractors), following a general
  distribution.
• to EX ?o2VarX co?o / to .
• T random variable modeling the effective
  processing time where
• Ui random variable modeling the duration of the
  i-th outage, following a general distribution,
  and
• N random variable modeling the number of
  outages during a the processing of a single
  part.
• mrEUi ?r2VarUi cr ?r / mr
• Time between outages is exponentially distributed
  with mean mf.
• Availability A mf / (mfmr) percentage of
  time the system is up.
• Then,
• te ET to / A or equivalently re 1/te
  A (1/to) A? ro

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Breakdown Example

• Data Injection molding machine has
• 15 second stroke (to 15 sec)
• 1 second standard deviation (so 1 sec)
• 8 hour mean time to failure (mf 28800 sec)
• 1 hour repair time (mr 3600 sec)
• Natural variabilityco 1/15 0.067 (which is
  very low)

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Example Continued

• Effective variability

Which is very high!
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Example Continued

• Suppose through a preventive maintenance program,
  we can reduce mf to 8 min and mr to 1 min

(the same as before)
Which is low!
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Exampleemploying the developed theory for
diagnostic purposes
Desired throughput is TH 2.4 jobs / hr but
practical experience has shown that it is not
attainable by this line. We need to understand
why this is not possible.
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Diagnostics example continuedCapacity analysis
based on mean values
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Diagnostics example continuedAn analysis based
on the G/G/m model
i.e., the long outages of M1, combined with the
inadequate capacity of the interconnecting
buffer, starve the bottleneck!
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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.
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Lot Sizing

• If affordable, a lot-for-lot (L4L) policy will
  incur the lowest inventory holding costs and it
  will maintain a smoother production flow.
• Possible reasons for departure from a L4L policy
• High set up times and costs gt need for serial
  process batching to control the capacity losses
• Processes that require a large production volume
  in order to maintain a high utilization (e.g.,
  fermentors, furnaces, etc.) gt need for parallel
  process batching
• Selection of a pertinent process batch size
• It must be large enough to maintain feasibility
  of the production requirements
• It must control the incurred
• inventory holding costs, and/or
• part delays (this is a measure of disruption to
  the production flow caused by batching)
• Move or transfer batches The quantities in which
  parts are moved between the successive processing
  stations.
• They should be as small as possible to maintain a
  smooth process flow

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Optimal Parallel Batching A factory physics
approach
Model Parameters k (parallel) batch size B
maximum batch size ra arrival rate
(parts/hr) ca CV of inter-arrival times t
batch processing time (hrs) ce CV for effective
batch processing time
Then CT WTBT CTqt
From the above,
Remark Notice that CT? as u?1 but also as u?0 !
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Determining an optimized batch size
Let um ? rat . Then u um / k ? k um / u .
Substituting this expression for k in the
expression for CT, we get
and we get
Recognizing that
, we set
where
To minimize CT, it suffices to minimize y(u).
This can be achieved as follows
and
which further implies that
Remark If ce2 ? 0, the term ? in the original
expression for u will significant. In that case,
we can set
and obtain u and k as before.