Process Variability and Waiting

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Topic 7 Process Variability and Waiting

• Sources of variability
• Analyzing and estimating variability
• Queuing analysis application
• Practical worst case performance
• Benchmarking example
• Simulation exercise

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A Somewhat Odd Service Process
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A More Realistic Service Process
Patient 1
Patient 3
Patient 5
Patient 7
Patient 9
Patient 11
Patient 2
Patient 4
Patient 6
Patient 8
Patient 10
Patient 12
Time
710
720
730
740
750
800
700
3
2
Number of cases
1
0
2 min.
3 min.
4 min.
5 min.
6 min.
7 min.
Service times
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Variability Leads to Waiting Time
Service time
Wait time
700
710
720
730
740
750
800
5
4
3
2
1
Inventory (Patients at lab)
0
700
710
720
730
740
750
800
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Variability Where does it come from?

• Tasks
• Inherent variation
• Lack of SOPs
• Quality (scrap / rework)

Processing
Buffer

• Input
• Unpredicted Volume swings
• Random arrivals (randomness is the rule,
  not the exception)
• Incoming quality
• Product Mix

• Resources
• Breakdowns / Maintenance
• Operator absence
• Set-up times

• Routes
• Variable routing
• Dedicated machines

• Especially relevant in service operations
• (what is different in service industries?)
• emergency room
• air-line check in
• call center
• check-outs at cashier

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Why are data drawn from a theoretical
distribution viewed as superior to data drawn
from an empirical distribution?
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Example
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Software can also be used to search for the best
fitting Theoretical Probability Distribution
Any statistical or practical reasons for caution
here?
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Calling Population
Service system
Queue
Server
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Poisson Distribution

• Number of events that occur in an interval of
  time
• Example Number of customers that arrive in 15
  min.
• Mean ? (e.g., 5/hr.)
• Probability

? 0.5
? 6.0
Example Serious worker injuries average 2.7 per
year. Assuming injuries follow a Poisson process,
what is the probability that fewer than 2 will
occur next year. Plot calculation assuming
average injuries range from .5 to 5
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P(0) 2.70 e-2.7 / 0! .0672
P(1) 2.71 e-2.7 / 1! .1815
.2487
1.0 - 0.2487 .7513
Fewer than 2 injuries
1.0
.9098
.8
.7358
.6
.5578
P
.4060
.4
.1992
.2
.0916
.0404
1
2
3
4
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Average Injuries
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Negative Exponential Distribution

• Service time and time between arrivals
• Example Service time is 20 min.
• Mean service rate ?
• e.g., customers/hr.
• Mean service time 1/?
• Equation

Exercise Time between telephone calls into a
switchboard is a neg. exponential with l.02
calls per second. What is Probability that
next call is within 20 minutes? Find probability
next call is within 2 standard deviations of
average.
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P(x lt 20 mins)1-e-.02(6020)1-.0000000000377
effectively 100
Mean 1/l 50 seconds Standard Deviation 1/l
50 seconds 50 - (2)50 cant get a call in
negative time (memoryless) 50 (2)50 150
P(x lt 150 seconds)1-e-.02(150)1-.0498 .9502
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Example of an M/M/1 System
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Calling Population
Queue
Dr1
Dr2
Dr1
Dr2
Queue 1
Queue 2
Calling Population
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Managing Waiting Systems Points to Remember

• Variability is the norm, not the exception -
  understand where it comes from and eliminate
  what you can - accommodate the rest
• Variability leads to waiting times although
  utilizationlt100
• Use the Waiting Time Formula to - get a
  qualitative feeling of the system - analyze
  specific recommendations / scenarios
• Adding capacity is expensive, although some
  safety capacity is necessary

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Balancing Efficiency with Responsiveness
Responsiveness
System improvement(e.g. pooling of resources)
High
Increase staff(lower utilization)
Responsive process with high costs
Reduce staff(higher utilization)
Now
Low cost process with low responsiveness
Frontier reflecting current process
Low
Efficiency
High perunit costs(low utilization)
Low perunit costs(high utilization)
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Priority Rules in Waiting Time Systems
Service times A 9 minutesB 10 minutesC 4
minutesD 8 minutes
4 min.
9 min.
19 min.
13 min.
23 min.
21 min.
Total wait time 9192351min
Total wait time 4132138 min

• Flow units are sequenced in the waiting area
  (triage step)
• Shortest Processing Time Rule - Minimizes
  average waiting time - Problem of having true
  processing times
• First-Come-First-Serve - easy to implement -
  perceived fairness
• Sequence based on importance - emergency
  cases - identifying profitable flow units

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Benchmarks for Flow Lines with VariabilityPractic
al Worst Case

• Observation There is a BIG GAP between the Best
  Case and Worst Case performance.
• Question Can we find an intermediate case that
  divides good and bad lines, and is
  computable?
• Experiment consider a line with a given rb and
  T0 and
• single operation workstations
• balanced lines
• variability such that all WIP configurations
  (states) are equally likely

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PWC Example 3 jobs, 4 stations
clumped up states
spread out states
Note average WIP at any station is 15/20 0.75,
so jobs are spread evenly between stations.
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Practical Worst Case

• Let w jobs in system, N no. stations in line,
  and t process time at all stations
• CT(single) (1 (w-1)/N) t
• CT(line) N 1 (w-1)/N t
• Nt (w-1)t
• T0 (w-1)/rb
• TH WIP/CT
• w/(wW0-1)rb

Average time at station from the viewpoint of an
arriving job is (w-1/N)t ave. queue time
t processing time
N stations must be visited
From Littles Law
w/(To(w-1)rb) w/((wo/rb)(w-1)/rb) w/(wWo-1)
rb
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Practical Worst Case Performance

• Practical Worst Case Definition The practical
  worst case (PWC) cycle time for a given WIP
  level, w, is given by,
• The PWC throughput for a given WIP level, w, is
  given by,
• where W0 is the critical WIP.

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TH vs. WIP Practical Worst Case
Best Case
rb
PWC
Good
Worst Case
Bad
1/T0
W0
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CT vs. WIP Practical Worst Case
Worst Case
PWC
Bad
Best Case
Good
T0
W0
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Question, what are the range of performance
benchmarks for the following line?
Assume w 5 jobs N 5 workstations t 10
units of processing time each station
10
10
10
10
10
Rb1/10 To50 Wo(1/10)505
CTTo50
CT5(50)250
THw/To 5/50.1
TH1/50.02
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PWC assuming w 5 jobs N 5 workstations t
10 units of processing time each station
((w-1)/N)t t ((5-1)/5)1010 18 units of
cycle time per station Or ((1(w-1))/N)t
((1(5-1))/5)10 18 CT(line) 18(5)90 or
N((1(w-1))/N)t 90 note processing time
Nt 50 wait time (w-1)t
40 cycle
time 90 TH WIP/CT
5/90 .0555 (w/(wWo-1))rb
(5/(55-1)).1.0555
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PWC assuming w 3 jobs N 4 workstations t
2.5 units of processing time each station
2.5
2.5
2.5
2.5
Rb1/2.5.4 To10 Wo(1/2.5)104
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Internal Benchmarking Example

• Order Processing

The lines maximum throughput is 250 orders per
hour
An orders minimum cycle time is 3.5 minutes
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Example (cont.)

• Critical WIP rbT0 250 ? .0583 14.575
• Actual Values
• CT 31.4 minutes .5233 hours
• WIP 64 orders
• TH 90 orders/hour
• Conclusions
• Throughput is 36 of capacity
• WIP is 4.4 times critical WIP
• CT is 9 times raw process time

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Internal Benchmarking Example (cont.)

• WIP Required for PWC to Achieve TH 0.36rb
• TH Resulting from PWC with WIP 64
• Conclusion actual system is much worse than PWC.

Much lower than actual WIP!
Much higher than actual TH!
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Internal Benchmarking Outcome
Cycle Time
Actual.5233
3.731
0.310
0.256
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Internal Benchmarking Outcome
Throughput
Actual90
250
206.25
17.152
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Current performance
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Current performance
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• Exercise
• How might we simulate the operation at Paquawket
  Water Company?