Queuing Models and Capacity Planning

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.

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 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.

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 TotalCost/hour = Cs C + Cw Lq Note: Only consider systems where

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:

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:

Congestion as With: Then: 0 0 0.2 0.25 0.5 1 0.8 4 0.9 9 0.99 99 100 6 4 2 With: Then: 0 0 0.2 0.25 0.5 1 0.8 4 0.9 9 0.99 99 0 1.0

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?

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?

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.

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

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) 2=$200 $400

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

Comparisons of System Performance for Two Mechanics Single Queue with Team Service 6/ 5 =1.2 days 0.2 days with Multiple 6 (.34) = 2.06 days 0.06 days Servers Multiple Queue and Multiple 12 (1/5) =2.4 days 0.4 days

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.

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.

Lq for Various Values of C and

Topics for Discussion Example 18.1 presented a naïve capacity planning exercise criticized for using averages. Suggest other reservations about this 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.