ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Introduction to Queueing Theory
and Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Yunan Liu
Department of Industrial and Systems Engineering
North Carolina State University
ISE Summer Camp, June 24, 2013
Motivation
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Motivation
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Foundation of Operations Research
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
A Little History
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
I
Father of queueing theory: Anger Erlang (1878–1929)
I
Originally used to model telephone exchange
Queues Are Everywhere!
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Queueing Models
Realistic Features
I
Real queues
Decision Making
Useful Tools
Conclusion
I
Virtual queues
I
Systems transformed into queues.
Applications: Real Queues
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
Hospitals
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Applications: Real Queues
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
Transportation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Applications: Real Queues
ISE Summer Camp
Queueing Theory
Yunan Liu
Airport Security Lines
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Applications: Real Queues
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
DMV
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Applications: Real Queues
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
Bank
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Applications: Real Queues
ISE Summer Camp
Queueing Theory
Yunan Liu
Amusement Parks
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Applications: Virtual Queues
Contact Centers
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Applications: Virtual Queues
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
Computer Service Systems
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Applications: Virtual Queues
ISE Summer Camp
Queueing Theory
Yunan Liu
Housing
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Applications: Transformed Queues
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
Manufacturing Systems
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Applications: Transformed Queues
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
Inventory Systems
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Applications: Transformed Queues
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
High Frequency Trading: Order Books
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Queues Are Indeed Everywhere!
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
Mathematical Queueing
Models
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Queueing Models
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
Single-class queues
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Queueing Models
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
Multi-class queues
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Queueing Models
ISE Summer Camp
Queueing Theory
Yunan Liu
Re-entrant queues
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Queueing Models
ISE Summer Camp
Queueing Theory
Yunan Liu
Queueing Networks
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
Realistic Real-World Features
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Realistic Models Features
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Random variables and processes
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
ISE Summer Camp
Queueing Theory
Realistic Models Features
Yunan Liu
Motivation
history
Time-varying arrivals
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
service call center
emergence room
Realistic Models Features
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
Strange customer behavior
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Realistic Models Features
ISE Summer Camp
Queueing Theory
Yunan Liu
Different service disciplines/policies
Motivation
history
I
first-come first-served (FCFS):
used in most service systems
Applications
Queueing Models
Realistic Features
I
last-come first-served (LCFS):
computer stack operations, inventory systems with perishable
products
I
processor sharing (PS):
computer systems
I
Shortest job first (SJF):
computer systems
I
Priority preemptive (PP):
emergency rooms, service systems with multiple classes
Decision Making
Useful Tools
Conclusion
Realistic Models Features
ISE Summer Camp
Queueing Theory
Yunan Liu
Different service disciplines/policies
Motivation
history
I
first-come first-served (FCFS):
used in most service systems
Applications
Queueing Models
Realistic Features
I
last-come first-served (LCFS):
computer stack operations, inventory systems with perishable
products
I
processor sharing (PS):
computer systems
I
Shortest job first (SJF):
computer systems, CPU scheduling
I
Priority preemptive (PP):
emergency rooms, service systems with multiple classes
Decision Making
Useful Tools
Conclusion
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Analyze The Models and
Obtain Quantitative Results
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
I
How? ............ (hereby omit 10,000 pages)
I
The “......” part will be taught in our undergraduate/graduate
courses
ISE Summer Camp
Queueing Theory
Yunan Liu
Help Make Decisions
Motivation
history
Applications
I
Relieve human suffering of waiting
I
Minimize costs/maximize profits
I
Save lives!
I
What else?
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Staffing to Achieve System Stability
Design staffing and shifts
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Meet service level agreements
I
P(waiting < 2 mins)>0.8
I
E (wait) ≈ 3 mins = 0.05 hr
I
P(Abandonment)<0.02
Staffing to Achieve System Stability
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
Bad staffing
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Good staffing
Delay Forecasting and Announcement
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Delay Forecasting and Announcement
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Delay Forecasting and Announcement: An Exercise
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Delay Forecasting and Announcement: An Exercise
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
ISE Summer Camp
Queueing Theory
Yunan Liu
Tools
Motivation
history
Applications
I
Data analysis: analyze data, test hypothesis, abstract
information, etc.
Queueing Models
Realistic Features
Decision Making
I
Computer simulation: discussed earlier today
I
Probability theory: model and predict random events
Useful Tools
Conclusion
ISE Summer Camp
Queueing Theory
Probability Exercise 1: Patterns
Yunan Liu
The pattern HTHH occurs at step 11, 14 and 22 in the sequence:
HHTHTTT HTHHTHHHTTT HTHHTTH . . .
Motivation
history
Applications
Queueing Models
Realistic Features
Now consider 4 patterns:
Decision Making
A ≡ HH,
B ≡ HT ,
C ≡ TT
and
D ≡ TH.
Useful Tools
Conclusion
Q: Which one “on average” appears in the smallest number of
flips?
ISE Summer Camp
Queueing Theory
Probability Exercise 1: Patterns
Yunan Liu
The pattern HTHH occurs at step 11, 14 and 22 in the sequence:
HHTHTTT HTHHTHHHTTT HTHHTTH . . .
Motivation
history
Applications
Queueing Models
Realistic Features
Now consider 4 patterns:
Decision Making
A ≡ HH,
B ≡ HT ,
C ≡ TT
and
D ≡ TH.
Useful Tools
Conclusion
Q: Which one “on average” appears in the smallest number of
flips?
I
NA = NC , NB = ND
ISE Summer Camp
Queueing Theory
Probability Exercise 1: Patterns
Yunan Liu
The pattern HTHH occurs at step 11, 14 and 22 in the sequence:
HHTHTTT HTHHTHHHTTT HTHHTTH . . .
Motivation
history
Applications
Queueing Models
Realistic Features
Now consider 4 patterns:
Decision Making
A ≡ HH,
B ≡ HT ,
C ≡ TT
and
D ≡ TH.
Useful Tools
Conclusion
Q: Which one “on average” appears in the smallest number of
flips?
I
NA = NC , NB = ND
I
NB < NA ? or NB > NA ?
ISE Summer Camp
Queueing Theory
Probability Exercise 1: Patterns
Yunan Liu
The pattern HTHH occurs at step 11, 14 and 22 in the sequence:
HHTHTTT HTHHTHHHTTT HTHHTTH . . .
Motivation
history
Applications
Queueing Models
Realistic Features
Now consider 4 patterns:
Decision Making
A ≡ HH,
B ≡ HT ,
C ≡ TT
and
D ≡ TH.
Useful Tools
Conclusion
Q: Which one “on average” appears in the smallest number of
flips?
I
NA = NC , NB = ND
I
NB < NA ? or NB > NA ?
I
4 = NB < NA = 6.
ISE Summer Camp
Queueing Theory
Probability Exercise 1: Patterns
Yunan Liu
Motivation
The pattern HTHH occurs at step 11, 14 and 22 in the sequence:
history
Applications
HHTHTTT HTHHTHHHTTT HTHHTTH . . .
Realistic Features
Now consider 4 patterns:
A ≡ HH,
B ≡ HT ,
Queueing Models
Decision Making
C ≡ TT
and
D ≡ TH.
Useful Tools
Conclusion
Probability Exercise 2: NYC Subway Problem
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Conclusion
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
I
Observe real-world systems and recognize potential problems
I
Construct mathematical models representing these systems
I
Analyze the models (performance analysis and decision
making)
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
I
Use the analysis to provide strategies, heuristics and insights
I
Solve real-world problems (connect theories and applications)
ISE Summer Camp
Queueing Theory
Yunan Liu
Motivation
history
Applications
Queueing Models
Realistic Features
Decision Making
Useful Tools
Conclusion
Thank You!