ISM 270

1. ISM 270 Service Engineering and Management Lecture 7: Forecasting and Managing Service Capacity

2. Announcements • Project Proposal Due today • Homework 4 due next week • $15 check for ‘Responsive Learning Technologies’ • Final four weeks: • Capacity Planning • Outsourcing • Capacity Management Game • Project Presentations

3. Today • Capacity Management • QueueingModels • Introduction to R

4. Managing Waiting Lines – Queueing Models

5. Essential Features of Queuing Systems Renege Arrival process Queue discipline Departure Calling population Service process Queue configuration No future need for service Balk

6. Arrival Process Arrival process Static Dynamic Random arrivals with constant rate Random arrival rate varying with time Facility- controlled Customer- exercised control Accept/Reject Price Appointments Reneging Balking

7. Distribution of Patient Interarrival Times

8. Temporal Variation in Arrival Rates

9. Poisson and Exponential Equivalence Poisson distribution for number of arrivals per hour (top view) One-hour 1 2 0 1 interval Arrival Arrivals Arrivals Arrival 62 min. 40 min. 123 min. Exponential distribution of time between arrivals in minutes (bottom view)

10. Queue Configurations Multiple Queue Single queue Take a Number Enter 3 4 2 8 6 10 12 7 11 9 5

11. Queue Discipline Queue discipline Static (FCFS rule) Dynamic selection based on status of queue Selection based on individual customer attributes Number of customers waiting Round robin Priority Preemptive Processing time of customers (SPT rule)

12. 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:

13. 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 (Little’s Law for ALL queues):

14. 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

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

16. 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

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.

18. Managing Capacity and Demand

19. Segmenting Demand at a Health Clinic Smoothing Demand by Appointment Scheduling Day Appointments Monday 84 Tuesday 89 Wednesday 124 Thursday 129 Friday 114

20. Hotel Overbooking Loss Table Number of Reservations Overbooked No- Prob- shows ability 0 1 2 3 4 5 6 7 8 9 0 .07 0 100 200 300 400 500 600 700 800 900 1 .19 40 0 100 200 300 400 500 600 700 800 2 .22 80 40 0 100 200 300 400 500 600 700 3 .16 120 80 40 0 100 200 300 400 500 600 4 .12 160 120 80 40 0 100 200 300 400 500 5 .10 200 160 120 80 40 0 100 200 300 400 6 .07 240 200 160 120 80 40 0 100 200 300 7 .04 280 240 200 160 120 80 40 0 100 200 8 .02 320 280 240 200 160 120 80 40 0 100 9 .01 360 320 280 240 200 160 120 80 40 0 Expected loss, $ 121.60 91.40 87.80 115.00 164.60 231.00 311.40 401.60 497.40 560.00

21. Daily Scheduling of Telephone Operator Workshifts Scheduler program assigns tours so that the number of operators present each half hour adds up to the number required Topline profile 12 2 4 6 8 10 12 2 4 6 8 10 12 12 2 4 6 8 10 12 2 4 6 8 10 12

22. LP Model for Weekly Workshift Schedule with Two Days-off Constraint Schedule matrix, x = day off Operator Su M Tu W Th F Sa 1 xx … … … … ... 2 … xx … … … … 3 … ... xx … … … 4 … ... xx … … … 5 … … … … xx … 6 … … … … xx … 7 … … … … xx … 8 x … … … … … x Total 6 6 5 6 5 5 7 Required 3 6 5 6 5 5 5 Excess 3 0 0 0 0 0 2

23. Seasonal Allocation of Rooms by Service Class for Resort Hotel First class Standard Budget 20% 20% 20% 30% 50% 30% 50% 60% Percentage of capacity allocated to different service classes 50% 30% 30% 10% Peak Shoulder Off-peak Shoulder (30%) (20%) (40%) (10%) Summer Fall Winter Spring Percentage of capacity allocated to different seasons

24. Demand Control Chart for a Hotel Expected Reservation Accumulation 2 standard deviation control limits

25. Yield Management Using the Critical Fractile Model Where x = seats reserved for full-fare passengers d = demand for full-fare tickets p = proportion of economizing (discount) passengers Cu = lost revenue associated with reserving one too few seats at full fare (underestimating demand). The lost opportunity is the difference between the fares (F-D) assuming a passenger, willing to pay full-fare (F), purchased a seat at the discount (D) price. Co = cost of reserving one to many seats for sale at full-fare (overestimating demand). Assume the empty full-fare seat would have been sold at the discount price. However, Co takes on two values, depending on the buying behavior of the passenger who would have purchased the seat if not reserved for full-fare. if an economizing passenger if a full fare passenger (marginal gain) Expected value of Co = pD-(1-p)(F-D) = pF - (F-D)

26. Statistical Analysis in R • Homework 4 is designed to introduce you to analysis using R