Capacity Planning and Queuing Models

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Capacity Planning and Queuing Models

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Learning Objectives

• Discuss the strategic role of capacity planning.
• Describe a queuing model using A/B/C notation.
• Use queuing models to calculate system
  performance measures.
• Describe the relationships between queuing system
  characteristics.
• Use queuing models and various decision criteria
  for capacity planning.

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Capacity Planning Challenges

• Inability to create a steady flow of demand to
  fully utilize capacity
• Idle capacity always a reality for services.
• Customer arrivals fluctuate and service demands
  also vary.
• Customers are participants in the service and the
  level of congestion impacts on perceived quality.
• Inability to control demand results in capacity
  measured in terms of inputs (e.g. number of hotel
  rooms rather than guest nights).

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Strategic Role of Capacity Decisions

• Using long range capacity as a preemptive strike
  where market is too small for two competitors
  (e.g. building a luxury hotel in a mid-sized
  city)
• Lack of short-term capacity planning can generate
  customers for competition (e.g. restaurant
  staffing)
• Capacity decisions balance costs of lost sales if
  capacity is inadequate against operating losses
  if demand does not reach expectations.
• Strategy of building ahead of demand is often
  taken to avoid losing customers.

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Queuing System Cost Tradeoff

• Let Cw Cost of one customer waiting in
  queue for an hour
• Cs Hourly cost per server
• C Number of servers
• Total Cost/hour Hourly Service Cost Hourly
  Customer Waiting Cost
• Total Cost/hour Cs C Cw Lq
• Note Only consider systems where

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Queuing Formulas
Single Server Model with Poisson Arrival and
Service Rates M/M/1 1. Mean arrival rate 2.
Mean service rate 3. Mean number in service 4.
Probability of exactly n customers in the
system 5. Probability of k or more customers
in the system 6. Mean number of customers in the
system 7. Mean number of customers in
queue 8. Mean time in system 9. Mean time in
queue
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Queuing Formulas (cont.)
Single Server General Service Distribution
Model M/G/1 Mean number of customers in queue
for two servers M/M/2 Relationships among
system characteristics
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Congestion as
100 10 8 6 4 2 0
With
Then
0 0 0.2 0.25 0.5 1 0.8
4 0.9 9 0.99 99
0
1.0
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Foto-Mat Queuing Analysis
On average 2 customers arrive per hour at a
Foto-Mat to process film. There is one clerk in
attendance that on average spends 15 minutes per
customer. 1. What is the average queue length
and average number of customers in the
system? 2. What is the average waiting time in
queue and average time spent in the
system? 3. What is the probability of having 2
or more customers waiting in queue? 4. If the
clerk is paid 4 per hour and a customers
waiting cost in queue is considered 6 per
hour. What is the total system cost per hour? 5.
What would be the total system cost per hour, if
a second clerk were added and a single queue
were used?
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White Sons Queuing Analysis
White Sons wholesale fruit distributions employ
a single crew whose job is to unload fruit from
farmers trucks. Trucks arrive at the
unloading dock at an average rate of 5 per hour
Poisson distributed. The crew is able to unload
a truck in approximately 10 minutes with
exponential distribution. 1. On the average,
how many trucks are waiting in the queue to be
unloaded ? 2. The management has received
complaints that waiting trucks have blocked the
alley to the business next door. If there is room
for 2 trucks at the loading dock before the
alley is blocked, how often will this problem
arise? 3. What is the probability that an
arriving truck will find space available at the
unloading dock and not block the alley?
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Capacity Analysis of Robot Maintenance and Repair
Service

• A production line is dependent upon the
  use of assembly robots that periodically break
  down and require service. The average time
  between breakdowns is three days with a Poisson
  arrival rate. The average time to repair a robot
  is two days with exponential distribution. One
  mechanic repairs the robots in the order in which
  they fail.
• 1. What is the average number of robots out of
  service?
• 2. If management wants 95 assurance that
  robots out of service will not cause the
  production line to shut down due to lack of
  working robots, how many spare robots need to be
  purchased?
• 3. Management is considering a preventive
  maintenance (PM) program at a daily cost of 100
  which will extend the average breakdowns to six
  days. Would you recommend this program if the
  cost of having a robot out of service is 500 per
  day? Assume PM is accomplished while the robots
  are in service.
• 4. If mechanics are paid 100 per day and the PM
  program is in effect, should another mechanic be
  hired? Consider daily cost of mechanics and idle
  robots.

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Determining Number of Mechanics to Serve Robot
Line
1. Assume mechanics work together as a team
Mechanics 100 M
500 Ls
Total system in Crew (M)
Mechanic cost
Robot idleness Cost per day

1 1/2 2
1 3 3/2
100(1)100
500(1/2)250 350 100(2)200
500(1/5)100
300 100(3)300
500(1/8)62 362
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Determining Number of Mechanics to Serve Robot
Line
2. Assume Robots divided equally among mechanics
working alone Identical
100 n
500 Ls (n) Total System Queues (n)
Mechanic
Robot idleness Cost per
day
cost
1 1/ 6
100 250
350 2
1/ 12 200
500 (1/5) 2200 400
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Determining Number of Mechanics to Serve Robot
Line
3. Assume two mechanics work alone from a single
queue. Note
0.01 0.33

0.34 Total Cost/day 100(2) 500(.34)
200 170 370
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Comparisons of System Performance for Two
Mechanics
System Single Queue
with Team
Service
6/ 5 1.2 days 0.2 days Single
Queue with Multiple
6 (.34) 2.06 days
0.06 days Servers Multiple Queue and Multiple

12 (1/5) 2.4 days 0.4 days Servers
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Single Server General Service Distribution Model
M/G/1
1. For Exponential Distribution
2. For Constant Service Time
3. Conclusion Congestion measured by Lq is
accounted for equally by variability in
arrivals and service times.
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General Queuing Observations
1. Variability in arrivals and service times
contribute equally to congestion as measured
by Lq. 2. Service capacity must exceed
demand. 3. Servers must be idle some of the
time. 4. Single queue preferred to multiple
queue unless jockeying is permitted. 5.
Large single server (team) preferred to
multiple-servers if minimizing mean time in
system, WS. 6. Multiple-servers preferred to
single large server (team) if minimizing mean
time in queue, WQ.
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Lq for Various Values of C and
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Topics for Discussion

• Example 14.1 presented a naïve capacity planning
  exercise criticized for using averages. Suggest
  other reservations about this planning exercise.
• For a queuing system with a finite queue, the
  arrival rate can exceed the capacity. Explain
  with an example how this is possible.
• What are some disadvantages associated with the
  concept of pooling service resources?
• Discuss how one could determine the economic cost
  of keeping customers waiting.

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Interactive Exercise

• Go to ServiceModel on the CD-ROM and select
  the Customer Service Call Center demo model. Run
  the animated simulation and display the results.
  Have the class explain in terms of queuing theory
  why the revised layout has achieved the
  remarkable reductions in average and maximum hold
  times.