1. Capacity Planning and Queuing Models
McGraw-Hill/Irwin
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.

2. 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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3. 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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4. 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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5. 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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6. Single Server Model with Poisson Arrival and Service Rates: M/M/1
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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7. 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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8. Congestion as With: Then: 0 0 0.2 0.25 0.5 1 0.8 4 0.9 9 0.99 99 1006 4 2
With:
Then:
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9. 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
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 customer’s 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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10. White & Sons Queuing Analysis
White & Sons wholesale fruit distributions employ a single crew whose
job is to unload fruit from farmer’s 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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11. 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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12. Determining Number of Mechanics to Serve Robot Line
1. Assume mechanics work together as a team
Mechanics $100 M $500 Ls Total system
in Crew (M) Mechanic cost Robot idleness Cost per day /2 /2
100(1)=$ (1/2)=$ $350
100(2)=$ (1/5)=$ $300
100(3)=$ (1/8)=$ $362
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13. 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
/ $ $ $350
/ $ (1/5) 2=$ $400
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14. Determining Number of Mechanics to Serve Robot Line
3. Assume two mechanics work alone from a single queue.
Note: =
= 0.34
Total Cost/day = 100(2) (.34) = =$370
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15. Comparisons of System Performance for Two Mechanics
Single Queue
with Team Service / 5 =1.2 days days
with Multiple (.34) = 2.06 days days
Servers
Multiple Queue
and Multiple (1/5) =2.4 days days
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16. 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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17. 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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18. Lq for Various Values of C and
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19. Houston Port Authority
Prepare a queuing analysis to evaluate the alternative of adding another unloading crew or purchase of a pneumatic handling system. The recommendation should include a total cost analysis of each alternative.
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20. Freedom Express
During periods of bad weather, as compared with periods of clear weather, how many additional gallons of fuel on average should FreeEx expect its planes to consume because of airport congestion?
Given FreeEx’s policy of ensuring that its planes do not run out of fuel more than 1 in 20 times while waiting to land, how many reserve gallons (i.e, gallons over and above expected usage) should be provided for clear-weather flights? For bad-weather?
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21. Freedom Express (cont.)
3. During bad weather, FreeEx has the option of instructing the Washington air controller to place its planes in a holding pattern from which planes are directed to land either at Reagan National or at Dulles International, whichever becomes available first. Assume that Dulles landing rate in bad weather also is 30 per hour, Poisson distributed, and that the combined arrival rate for both airports is 40 per hour. If FreeEx must pay $200 per flight to charter a bus to transport its passengers from Dulles to Reagan National, should it exercise the option of permitting its aircraft to land at Dulles during bad weather? Assume that if the option is used, FreeEx’s aircraft will be diverted to Dulles about one-half the time.
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22. Renaissance Clinic Patient Flow
Nurse Clinician
µN = 30
ρ1ρ2λ
ρ1λ
Receptionist
µR =40
Physician
µD = 15
λ = 30
(1-ρ1)λ
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23. Renaissance Clinic (A)
Description
Value λ
Arrival rate of patients
30/hr ρ1
Fraction of patients to nurse clinician
2/3 ρ2
Fraction of nurse clinician patients to MD
0.15 µR
Service rate of receptionist
40/hr µN
Service rate of nurse clinician µD
Service rate of doctor
15/hr
See Figure 16.6 for Renaissance Clinic Schematic
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24. Renaissance Clinic (A) Questions
Assume the waiting line at the receptionist, the nurse clinician, and the physician are managed independently with FCFS priority. Using queuing formulas and the assumption that patients exiting from an activity follow a Poisson distribution, estimate the following statistics:
a. Average waiting time in each of the queues (i.e., receptionist, nurse clinician, physician).
b. Average time in the entire system for each of the three patient paths.
c. Overall average time in the system (i.e., expected time for an arriving patient).
d. Average idle time in minutes per hour of work for the receptionist, nurse clinician, and physician.
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25. Renaissance Clinic (A) Questions
What are the key assumptions involved in your analysis above? Discuss the appropriateness of each in this situation.
What would be the impact of the above calculations of adding a second doctor to the clinic and sharing the doctor queue between them on a “first MD available” basis?
The clinic is considering adopting a queue priority system that is determined by the time of entry into the system at the receptionist. How might patients waiting for the doctor react to this policy?
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26. Topics for Discussion
Discuss how one could determine the economic cost of keeping customers waiting.
Example 16.1 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?
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27. Interactive Exercise
Go to ServiceModel on the OLC 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.
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