1/31/2007
Industrial Data and Systems Analysis
1
Four Roles of Inventory
Peter L. Jackson
Professor
School of O.R. and I.E.
1/31/2007
Industrial Data and Systems Analysis
2
Pipeline Stock
Cycle Stock
Decoupling Stock
Safety Stock
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Industrial Data and Systems Analysis
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Pipeline Stock
Valueadded process time or transport time, T
Output rate, r
Input rate, r
Little’s Law: Average pipeline stock = rT
To reduce pipeline stock, reduce the duration of
valueadded activities
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Industrial Data and Systems Analysis
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Decoupling Stock
•
Decoupling stock: queues of work
between workstations caused by
variability in processing times
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Industrial Data and Systems Analysis
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A Useful Queueing Model
•
Focus on a single workstation
•
Jobs arrive at rate
λ
•
Jobs can be completed at rate
µ
•
<
so the workstation is sometimes idle
1
Workstation
Jobs
arriving
Completed
jobs
departing
Waiting queue
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Industrial Data and Systems Analysis
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No Decoupling Stock in
Deterministic Case
•
If the time between arrivals is constant
and the processing time is constant then
there will never be a queue
•
Utilization rate: fraction of time the
workstation is busy;
= busy/(busy+idle)
busy
idle
arrival
arrival
arrival
arrival
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Industrial Data and Systems Analysis
7
Variability is the Enemy
•
Queues develop when the arrival rate
temporarily exceeds the processing rate
•
Assume independent and identically
distributed (i.i.d.) interarrival times, and
i.i.d. process times
busy
idle
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Industrial Data and Systems Analysis
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Interpreting the Utilization
Rate
•
Let
ρ
=
λ
/
µ
•
is the utilization rate, the average
fraction of time the workstation is in use
•
1
is the probability you will find the
queue empty and the workstation idle if
you observe it at a random time
•

is the average rate at which queues
decrease, once a queue has formed
•
/(1
)=
/(
) a unitless measure of the
ability of workstation to work off queues
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Industrial Data and Systems Analysis
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How Shall We Measure
Variability?
•
m
a
= mean interarrival time (m
= 1/
)
•
σ
2
= variance of interarrival time
•
c
/ m
, the coefficient of
variation of interarrival time
•
is a unitless measure of variability
•
a
≥
1 corresponds to high variability
•
= 1
is characteristic of the exponential
distribution
•
≤
1 for our examples
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Industrial Data and Systems Analysis
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Variability
•
is called the squared coefficient of
variation (SCV) of the interarrival process
•
Or, “the arrival SCV”
•
Similarly, let
e
= mean effective
processing time (m
•
= variance of effective processing
time
•
Why “effective”? Recall our adjustments
to the throughput rate (speed loss,
breakdowns, etc.)
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Industrial Data and Systems Analysis
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Variability of Effective Process
Time
•
Process times are usually not highly
variable on their own
•
Time to make a part if everything goes well
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 Spring '08
 D.RUPPERT,P.JACKS
 Systems Analysis, Normal Distribution, industrial data, Industrial Data and Systems Analysis

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