1. CDAE 266 - Class 20 Nov. 2 Last class: Results of the midterm exam 4. Queuing analysis and midterm exam Today: 4. Queuing analysis and applications Next class: 4. Queuing analysis and applications Quiz 6 (sections 4.1 – 4.4) Reading: 4. Queuing analysis and application

2. CDAE 266 - Class 20 Nov. 2 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. 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?

4. 4.1. One example Consider the operation of a central supply room in a company: 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 min. to check out one worker. (1) What is the average number of workers in the supply room? (2) What is the average time each worker spends in the supply room? (3) Should the company hire another clerk? ……..

5. 4.2. Basic queuing situations 4.2.1. Structures of queuing systems -- Single-server and single stage -- Multiple-server and single-stage -- Single-server and multiple-stage -- Multiple-server and multiple-stage 4.2.2. Queuing discipline -- FIFO -- LIFO -- SIRO -- Some customers have priority 4.2.3. Arrival and service patterns

6. 4.3. Importance of queuing analysis 4.3.1. Profit maximization (cost minimization) (e.g., the operation of a supply room) 4.3.2. Trade-off between service quality and operation cost

7. 4.4. Single-server queuing models 4.4.1. Assumptions Structure: single-server & single-stage Discipline: FIFO 4.4.2. Information we need: = mean customer arrival rate (number of customers per hour) (e.g., 25)  = mean service rate (number of customers per hour) (e.g., 30)

8. 4.4. Single-server queuing models 4.4.4. Queuing results (1) Probability that there are a particular number of customers in the system -- Probability should be between 0 and 1 -- Examples: The average arrival rate ( ) = 25 customers per hour The average service rate (  ) = 30 customers per hour P 0 = Prob. that there is no customer = 1 – 25/30 = 0.167 P 1 = Prob. that there is one customer = 0.167 (25/30) = 0.139 P 2 = Prob. that there are 2 customers = 0.167 (25/30) 2 = 0.116 P 3 = Prob. that there are 3 customers = 0.167 (25/30) 3 = 0.097 Prob. that there are more than 3 customers = Prob. that there are less than 3 customers = Prob. that there are 2 or 3 customers =

9. 4.4. Single-server queuing models 4.4.4. Queuing results (2) Average number of customers in the system (3) Average number of customers waiting

10. 4.4. Single-server queuing models 4.4.3. Queuing results (4) Average time a customer spends in the system (5) Average waiting time (6) Server utilization rate

11. 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?

12. 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?

13. 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 per 3 minutes. (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? (2) What is the probability that there are 3 workers in the supply room? (3) What is the probability that there are more than one worker in the supply room? (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? (5) If the company rent the Lazy Susan system, what will be the change in the average waiting time per worker?