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

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).

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 due to inadequate capacity against operating losses due to inadequate demand.  Strategy of building ahead of demand is often taken to avoid losing customers.

Queuing System Cost Tradeoff

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: Queuing Formulas

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: Queuing Formulas (cont.)

With: Then: Congestion as  1.0

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 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? White & Sons Queuing Analysis

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

1. Assume mechanics work together as a team Mechanics $100 M $500 Ls Total system in Crew (M) Mechanic cost Robot idleness Cost per day 100(1)=$ (1/2)=$250 $ (2)=$ (1/5)=$100 $ (3)=$ (1/8)=$62 $ / /2 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 $ / 12 $ (1/5) 2=$200 $400 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) + 500(.34) = =$370 Determining Number of Mechanics to Serve Robot Line

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 Comparisons of System Performance for Two Mechanics

1. For Exponential Distribution: 2. For Constant Service Time: 3. Conclusion: Congestion measured by L q is accounted for equally by variability in arrivals and service times. Single Server General Service Distribution Model: M/G/1

1.Variability in arrivals and service times contribute equally to congestion as measured by L q. 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, W S. 6.Multiple-servers preferred to single large server (team) if minimizing mean time in queue, W Q. General Queuing Observations

L q for Various Values of C and 