OM2

1. OM2 SUPPLEMENTARY CHAPTER B QUEUING ANALYSIS DAVID A. COLLIER AND JAMES R. EVANS

2. Supplemental Chapter B Learning Outcomes l e a r n i n g o u t c o m e s LO1Describe the key elements and underlying mathematical concepts of analytical queuing models. LO2Explain and compute the operating characteristics associated with the single-server queuing model. LO3Apply the operating characteristic formulas for a multiple-server queuing model. LO4Explain the economic trade-offs associated with designing and managing queuing systems. LO5Explain the psychology of waiting for designing and managing queuing systems.

3. Supplemental Chapter B. Queuing Analysis n electrical utility company uses six customer service representatives (CSRs) at its call center to handle telephone calls and inquiries from its top 350 business customers. The next tier of 700 business customers is also handled by six CSRs. Based on the customer’s code, the call center routes business customers to different queues and CSRs. A manager at the utility explains: “We don’t ignore anyone, but our biggest customers certainly get more attention than the rest.”

4. Supplemental Chapter B. Queuing Analysis What do you think? Do you think that this decision is good or bad? Should all customers be treated the same and be considered as important as any other?

5. Supplemental Chapter B. Queuing Analysis Value-based queuing is a method that allows organizations to prioritize customer calls based on their long-term value to the organization.

6. Supplemental Chapter B. Queuing Analysis Common Types of Queuing Models 1. Single- or multiple-channel with Poisson arrivals and exponential service times. 2. Single-channel with Poisson arrivals and arbitrary service times. 3. Single-channel with Poisson arrivals and deterministic service times. 4. Single- or multiple-channel with Poisson arrivals, arbitrary service times, and no waiting line. 5. Single- or multiple-channel with Poisson arrivals, exponential service times, and a finite calling population.

7. Supplemental Chapter B. Queuing Analysis Arrival Distribution How many customers arrive for service in given periods of time? When arrivals occur in a random pattern and each arrival is independent of each other, the arrival pattern can be described using a Poisson distribution.

8. Supplemental Chapter B. Queuing Analysis Poisson Distribution

9. Exhibit B.1 Poisson Distribution

10. Supplemental Chapter B. Queuing Analysis Service Time Distribution How long does it take to service a customer? Service times are often modeled using the exponential distribution.

11. Supplemental Chapter B. Queuing Analysis Exponential Distribution

12. Exhibit B.2 Service Time Distribution

13. Supplemental Chapter B. Queuing Analysis • Queue Discipline • Shortest processing time (SPT) • Random • Triage • Preemption • Reservations and appointments

14. Supplemental Chapter B. Queuing Analysis Queuing Behavior Reneging is the process of a customer entering the waiting line but later deciding to leave the line and server system. Balking is the process of a customer evaluating the waiting line and server system and deciding not to enter the queue.

15. Supplemental Chapter B. Queuing Analysis • Single Server Queuing Model • 1. The waiting line has a single server. • 2. The pattern of arrivals follows a Poisson probability distribution. • 3. The service times follow an exponential probability distribution.

16. Supplemental Chapter B. Queuing Analysis Single Server Operating Characteristics Probability the service facility is idle: P0 =1 -  [B.4] Probability of n units in the system: Pn = ()nP0 [B.5] Average number in queue:Lq = 2/[ [B.6] Average number in system: L = Lq +  [B.7] Average waiting time in queue: Wq = Lq [B.8] Average time in system: W = Wq+ 1/ [B.9] Probability that an arrival has to wait: Pw =  [B.10]

17. Exhibit B.3 Single-Server Spreadsheet Single Server Queuing Model Spreadsheet

18. Supplemental Chapter B. Queuing Analysis Multiple-Server Queuing Model 1. The waiting line has two or more identical servers. 2. The arrivals follow a Poisson probability distribution with a mean arrival rate of λ. 3. The service times have an exponential distribution. 4. The mean service rate, μ, is the same for each server. 5. The arrivals wait in a single line and then move to the first open server for service. 6. The queue discipline is first-come, first-served (FCFS). 7. No balking or reneging is allowed.

19. Exhibit B.4 Two-Server Queuing System

20. Supplemental Chapter B. Queuing Analysis • Multiple Server Model Formulas • k= number of channels • l= mean arrival rate for the system • m= mean service rate for each channel

21. Supplemental Chapter B. Queuing Analysis Operating Characteristics Probability all k channels are idle: [B.13] Probability of n units in the system: [B.14] Average number in queue: [B.15]

22. Supplemental Chapter B. Queuing Analysis Operating Characteristics Average number in system: Average waiting time in queue: Average time in system:  Probability that an arrival has to wait: [B.16] [B.17] [B.18] [B.19]

23. Exhibit B.6 Multiple-Server Spreadsheet

24. Supplemental Chapter B. Queuing Analysis Economics of Waiting Lines Cw= waiting cost per hour per customer Cs = hourly cost associated with each server Total Cost = CwL+ Csk

25. Bourbon County Court Case Study • Bourbon County Court Case Study • Assuming a Poisson arrival distribution and an exponential service time distribution, apply queuing models to the case situation and evaluate the results. • What are the economics of the situation using queuing • model analysis? • What are your final recommendations using queuing • model analysis?