1. Operations Management
Queuing (Part 2) - Lecture 7
(Chapter 8)
Dr. Ursula G. Kraus

2. Review Performance Measures for Queuing Systems
Special Queuing Model: M/M/1

3. Learning Objectives
Understand how variability in arrivals and services can lead to a build up of queues.

4. Agenda
Review: General Queuing Models
Special Queuing Models
SofOptics

5. Agenda
Review: General Queuing Models
Special Queuing Models

6. Example: Calling Center
Consider a mail order company that has one customer service representative (CSR) taking calls. When the CSR is busy, the caller is put on hold. The calls are taken in the order received.
Each minute a caller spends on hold costs the company $2 in telephone charges, customer dissatisfaction and loss of future business. In addition, the CSR is paid $20 an hour.
Assume that calls arrive exponentially at the rate of one every 3 minutes. The CSR takes on average 2.5 minutes to complete the order. The time for service is also assumed to be exponentially distributed.
(a) What is the capacity utilization?
(b) How many customers will be on average on hold?
(c) What is the average time spent on hold?
(d) Estimate the average hourly cost of operating the call center.
Source: Managing Business Process Flows (1999)

7. Solution: Calling Center
Source: Managing Business Process Flows (1999)

8. Cost of Service Tradeoff
Total cost
Expected costs
(Service) Capacity Cost
(Customer) Waiting Cost
Level of service

9. Example: Ticket Outlet
Customers arrive at a suburban ticket outlet at a rate of 14 per hour on Monday mornings. Selling the tickets and providing general information takes an average of 3 minutes per customer. There is one ticket agent on duty on Mondays. Assume exponential interarrival and service times.
What percentage of time is the ticket agent busy?
How many customers are in line, on average?
How many minutes, on average, will a customer spend in the system?
What is the average waiting time in queue?
Source: Managing Business Process Flows (1999)

10. Solution: Ticket Outlet
Source: Managing Business Process Flows (1999)

11. Is Idleness a Good Thing?
Base (minimum) Capacity needed: R
Safety Capacity: Rp - R capacity carried in excess of expected demand to cover for system variability
Idle Percentage: 1 - ρ = 1 - R/Rp
Source: Managing Business Process Flows (1999)

12. Performance Measures to Focus on
Sales
Throughput (R)
Abandoning rate (Ra )
Cost
Capacity utilization (r)
Queue length (Iq) ; Total number in process (I)
Customer Service
Waiting time in queue (Tq ); Total time in process (T)
Blocking rate (Rb )
Source: Managing Business Process Flows (1999)

13. Levers for System Improvements
Increase process capacity Rp
adding more servers
working faster (decreasing Tp)
Reduce waiting time Tq and queue length Iq
reduce variability
decrease arrival rate (not desirable in general)
Source: Managing Business Process Flows (1999)

14. Example: Calling Center with 2 CSRs (2 servers)
2 phone numbers
Company hires a second CSR who is assigned a new telephone number. Customers are now free to call either of the two numbers. Once they are put on hold customers tend to stay on line since the other may be worse
1 phone number: pooling
Both CSRs share the same telephone number and the customers on hold are in a single queue
Server
Queue
50%
Servers
Queue
Source: Managing Business Process Flows (1999)

15. Solution: Calling Center with 2 CSRs (servers)
Source: Managing Business Process Flows (1999)

16. Agenda
Review: General Queuing Models
Special Queuing Models
SofOptics