1. Queuing Theory –
Study of Congestion
Operations Research

2. What is a queue?
• People waiting for service
– Customers at a supermarket (IVR, railway counter)
– Letters in a post office (Emails, SMS)
– Cars at a traffic signal
• In ordered fashion (who defines order?)
– Bank provides token numbers
– Customers themselves ensure FIFO at Railway
ticket counters

3. Where do we find queues?

4. A thought experiment
• When does a queue form?
• When will it not form?
• When will you not join a queue?
• When will you leave a queue?
• What is the worst case scenario?

5. What do you observe near a queue?
• Conflict
• Congestion
• Idle counters
• Overworked counters
• Smart people trying to circumvent the queue

6. What do we want to know?
• How much time will it take?
• How many counters should be there?
• How to manage peak hour traffic?

7. Origins
• Queuing Theory had its beginning in the research
work of a Danish engineer named A. K. Erlang.
• In 1909 Erlang experimented with fluctuating
demand in telephone traffic.
• Eight years later he published a report addressing
the delays in automatic dialing equipment.
• At the end of World War II, Erlang’s early work
was extended to more general problems and to
business applications of waiting lines.

8. Queuing System

9. Kendall Notation (a/b/c : d/e/f)
(a/b/c : d/e/f)
Arrival Size of
Distribution source
Infinite/
M/D Service Time Maximum
finite
Distribution number of
M/D customers in
Number of Service system
concurrent Discipline n
servers FIFO/LIFO
n / Priority/
Random

10. Identify the queuing system
Railway ticket counter (M/D/3:FIFO/200/∞)
Bank Service Counter
ATM
Airport – Check In
Airport - Security
Traffic Signal
Bus Stop
Train Platform (Boarding)
Paper Correction

12. Arrival modeled using
Poisson Distribution

13. Some parameters
Arrival Rate λ
Service Rate μ
Number of customers in system Ls
Number of customers in queue Lq
Waiting time in system Ws
Waiting time in queue Wq
Utilization ρ

14. Types of queuing systems

19. M/M/1

20. T1=3
T2=7 Arrival Rate = N/Tt=6/19=.31
T3=10 Mean Time in system = J/N = 38/6=6.3
T4=6 Mean number in system = J/Tt=38/19=2
T5=6 = (J/N)*(N/Tt)=6.3*.31=2
T6=6 =λTq
J=38