1. CDAE 266 - Class 22 Nov. 9 Last class: 4. Queuing analysis and applications Quiz 6 Today: Result of Quiz 6 4. Queuing analysis and applications Project 3 Next class: 5. Inventory analysis and applications Reading: 4. Queuing analysis and application

2. CDAE 266 - Class 22 Nov. 9 Important dates: Problem set 4, due Tuesday, Nov. 14 Problems 20-1, 20-5, 20-6, 20-10, and 20-13. Project 3, Saveway Supermarket (page 4-20) due Thursday, Nov. 16 Final exam: 8:00-11:00am, Thursday, Dec. 14

3. Result of Quiz 6 N = 50Range = 3 – 10Average = 8.25 1 and 2: Queuing STRUCTURES and DISCIPLINES 3. Importance of queuing analysis 4. Cost analysis of a central supply room Average arrival rate =Average service rate = Probability questions: Other questions: Cost analysis:

4. 4. Queuing analysis and applications 4.1. One example 4.2. Basic queuing situations 4.3. Importance of queuing analysis 4.4. Single-server queuing models 4.5. Multiple-server queuing models 4.6. Cost analysis of queuing models 4.7. How to use Excel to solve queuing models?

5. 4.4. Single-server queuing models 4.4.4. Cost analysis (supply room example) Consider the operation of a central supply room in a firm: workers pick up supplies from the supply room & one clerk is there to check out workers. An average of 25 workers arrive in the supply room per hour & it takes the clerk 2 minutes to check out one worker. What is the hourly labor cost for the co. if the hourly labor cost is $15 for a worker and $10 for the clerk? Should the co. rent an electric-controlled "Lazy Susan" storage bin system at a cost of $15 per hour and the system can double the clerk’s check out speed?

6. Class exercise Consider the operation of a central supply room in a firm: workers pick up supplies from the supply room & one clerk is there to check out workers. An average of 20 workers arrive in the supply room per hour & the clerk’s check out speed is 25 workers per hour. (1) What is the total hourly labor cost for the co. if the hourly labor cost is $20 for a worker and $12 for the clerk? (2) Should the co. rent an electric-controlled "Lazy Susan" storage bin system at a cost of $20 per hour and the system can double the clerk’s check out speed?

7. Take-home exercise Consider the operation of a central supply room in a firm: workers pick up supplies from the supply room & one clerk is there to check out workers. An average of 15 workers arrive in the supply room per hour & the clerk can check out one worker every 3 minutes. = 15 workers per hour  = 20 workers per hour (1)What is the total hourly labor cost for the co. if the hourly labor cost is $20 for a worker and $15 for the clerk? Total cost = clerk labor cost + workers’ labor cost = 15 + 0.2*20*15 = $75 (2) What is the probability that there are 3 workers in the supply room? P 0 = 1 – 15/20 = 0.25 P 3 = 0.25 (15/20) 3 = 0.1055 (3) What is the probability that there are more than one worker in the supply room? P [more than 1] = 1 – P [one or less] = 1 – [P 0 + P 1 ] = 1 – [0.25 + 0.19] = 0.56

8. Take-home exercise Consider the operation of a central supply room in a firm: workers pick up supplies from the supply room & one clerk is there to check out workers. An average of 15 workers arrive in the supply room per hour & the clerk can check out one worker every 3 minutes. = 15 workers per hour  = 20 workers per hour (4) Should the co. rent an electric-controlled "Lazy Susan" storage bin system at a cost of $30 per hour and the system can double the clerk’s check out speed?  = 40 workers per hour  W = 1 / (40 – 15) = 0.04 Total cost = clerk cost + workers’ labor cost + rent = 15 + 0.04*20*15 + 30 = $57 (5)If the company rent the Lazy Susan system, what will be the change in the average waiting time per worker? Current Wq = 15 / 20(20-15) = 0.15 hr. New Wq = 15 / 40(40-15) = 0.015 hr. The average waiting time will reduce by 0.135 hour or 8.1 minutes.

9. 4.5. Multiple-server queuing models 4.5.1. Assumptions: Structure: multiple-server & single-stage Discipline: FIFO 4.5.2. Information we need: = mean customer arrival rate  = mean service rate per server s = number of servers 4.5.3. Condition: < s 

10. 4.5. Multiple-server queuing models 4.5.3. Queuing results (pages 4-13 and 4-14) (1) Probability that there are a particular number of customers in the system (2) Average number of customers in the system (L) (3) Average number of customers waiting (Lq) (4) Average time a customer spends in the system (W) (5) Average waiting time of each customer (Wq) (6) Server utilization rate (ρ)

11. The formulas of the multiple- server queuing models are too complex -- that is why we have a computer program!

12. 4.6. How to solve queuing models using Excel? 4.6.1. Excel worksheet 4.6.2. Enter the data: = mean customer arrival rate  = mean service rate per server s = number of servers 4.6.3. Results

13. 4.7. Cost analysis of queuing models: An example A company has a central parts cage for mechanics to pick up supplies. Mechanics arrive at the cage at an average rate of 25 per hour. The only clerk Helen working in the cage can complete a mechanic's part request in an average of 2 minutes. The cost of mechanics’ time is $20 per hour and the cost of the clerk’s time is $12 per hour. The manufacturer wants to evaluate three alternative options to see if the cost can be reduced: (1) To rent an electric-controlled storage bin system at a cost of $15 per hour and the system can double Helen’s speed (2) To add another clerk (Amber) with the same wage rate and same check out speed (3) To replace Helen by Ashley who can check out one person in one minute and her wage rate is $24 per hour.

14. 4.7. Cost analysis of queuing models: An example Hourly cost of the current system: = Cost of Helen + cost of mechanics’ time = $12 + cost per mechanic x number of mechanics = $12 + W x $20 x = $12 + 0.2 x 20 x 25 = $112 Hourly cost of the storage bin system: = Cost of Helen + rent + cost of mechanics’ time = $12 + $15 + cost per mechanics x number of … = $12 + $15 + W x $20 x = $12 + $15 + ? x $20 x 25 = $ Hourly cost of the 2nd option: = Cost of 2 clerks + cost of mechanics’ time = $24 + cost per mechanic x number of mechanics = $24 + W x $20 x = $24 + ? x 20 x 25 = $

15. 4.7. Cost analysis of queuing models: An example Hourly cost of the 3rd option: = Cost of Ashley + cost of mechanics’ time = $24 + cost per mechanic x number of mechanics = $24 + W x $20 x = $24 + ? x 20 x 25 = $ Compare option 2 and option 3 and think about the business applications

16. Take-home assignment Read project 3 (Saveway Supermarket on page 4-20) and think about the following questions: What is the business decision problem? What information do we have? What information does the business need from us? How to analyze the problem? How to present our analysis and results?

17. Project 3 --------------------------------------------------------------------------------------------------------- Labor Construction Goodwill Total  S cost Rent cost cost cost --------------------------------------------------------------------------------------------------------- Current Option 1 Option 2 Option 3a Option 3b Options 1+3a Options 1+3b Options 2+3a Options 2+3b ---------------------------------------------------------------------------------------------------------