14-1. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 14 Capacity Planning and Queuing Models.

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Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

14

Capacity Planning and Queuing Models

Capacity Planning and Queuing Models

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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.

Learning ObjectivesLearning Objectives

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Capacity Planning ChallengesCapacity 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 DecisionsStrategic 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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C

Queuing System Cost TradeoffQueuing 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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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:

Pnn ( )1

P n k k( )

Ls

Lq

Ws

1

Wq

Queuing FormulasQueuing Formulas

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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:

Lq

2 2 2

2 1( )

Lq

3

24

Ls Lq

Ws Wq

Ws Ls

Wq Lq

1

1

1

Queuing Formulas (cont.)Queuing Formulas (cont.)

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0 1.0

100

10

8

6

4

2 0

With:

Ls 1Then:

Ls

0 00.2 0.250.5 10.8 40.9 90.99 99

Congestion as 1.0Congestion as 1.0

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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 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?

Foto-Mat Queuing AnalysisFoto-Mat Queuing Analysis

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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?

White & Sons Queuing AnalysisWhite & Sons Queuing Analysis

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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.

Capacity Analysis of Robot Maintenance and Repair Service

Capacity Analysis of Robot Maintenance and Repair Service

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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

M2

1 6

1 21 3

1 6

11 6

1 6

3 21 9

/

//

//

/

//

100(1)=$100 500(1/2)=$250 $350

100(2)=$200 500(1/5)=$100 $300

100(3)=$300 500(1/8)=$62 $362

1 1/2

2 1

3 3/2

Ls

1 6

1 2 1 61 2

1 6

1 1 61 5

1 6

3 2 1 61 8

/

/ //

/

//

/

/ //

Determining Number of Mechanics to Serve Robot Line

Determining Number of Mechanics to Serve Robot Line

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2. Assume Robots divided equally among mechanics working alone

Identical $100 n $500 Ls (n) Total SystemQueues (n) Mechanic Robot idleness Cost per day cost

/ n

/

/

/

//

n

1 3

1 12

1 21 6

1 1/ 6 $100 $250 $350

2 1/ 12 $200 500 (1/5) 2=$200 $400

Lsn

n

/

/

/

/

/ //

1 2

1 12

1 2 1 121 5

Determining Number of Mechanics to Serve Robot Line

Determining Number of Mechanics to Serve Robot Line

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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

Ls

3

24

1 6

1 2

1

3

/

/

Determining Number of Mechanics to Serve Robot Line

Determining Number of Mechanics to Serve Robot Line

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System

Single Queue with Team Service 6/ 5 =1.2 days 0.2 days

Single Queuewith Multiple 6 (.34) = 2.06 days 0.06 daysServers

Multiple Queueand Multiple 12 (1/5) =2.4 days 0.4 daysServers

Ls

1 6

1 1 61 5

40 34

1 12

1 2 1 121 5

3

2

/

//

.

/

/ //

WsLs

Wq Ws 1

Comparisons of System Performance for Two Mechanics

Comparisons of System Performance for Two Mechanics

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2 0

Lq

2 2 2

2 1( )

1. For Exponential Distribution:

22

1

Lq

2 2 2 2 2

2 1

2

2 1 1

/

( ) ( ) ( )

2. For Constant Service Time:

Lq

2

2 1( )

3. Conclusion:

Congestion measured by Lq is accounted for equally by variability in arrivals and service times.

Single Server General Service Distribution Model: M/G/1

Single Server General Service Distribution Model: M/G/1

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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.

General Queuing ObservationsGeneral Queuing Observations

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C=1 C=2 C=3 C=40.15 0.026 0.001

0.2 0.05 0.0020.25 0.083 0.004

0.3 0.129 0.0070.35 0.188 0.011

0.4 0.267 0.0170.45 0.368 0.024 0.002

0.5 0.5 0.033 0.0030.55 0.672 0.045 0.004

0.6 0.9 0.059 0.0060.65 1.207 0.077 0.008

0.7 1.633 0.098 0.0110.75 2.25 0.123 0.015

0.8 3.2 0.152 0.0190.85 4.817 0.187 0.024 0.003

0.9 8.1 0.229 0.03 0.0040.95 18.05 0.277 0.037 0.005

1 0.333 0.045 0.007

Lq for Various Values of C and Lq for Various Values of C and

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Topics for DiscussionTopics 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 ExerciseInteractive 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.