Queuing Theory – Study of Congestion Operations Research
Queuing Theory – Study of Congestion
Operations Research
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
Where do we find queues?
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?
What do you observe near a queue?
• Conflict• Congestion• Idle counters• Overworked counters• Smart people trying to circumvent the queue
What do we want to know?
• How much time will it take?• How many counters should be there?• How to manage peak hour traffic?
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.
Queuing System
Kendall Notation (a/b/c : d/e/f)
(a/b/c : d/e/f)
Arrival Distribution
Service Time Distribution
Number of concurrent
servers
Service Discipline
Maximum number of
customers in system
Size of source
M/D
M/D
nFIFO/LIFO/
Priority/ Random
n
Infinite/ finite
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
Arrival modeled using Poisson Distribution
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
Types of queuing systems
M/M/1
T1=3T2=7T3=10T4=6T5=6T6=6J=38
Arrival Rate = N/Tt=6/19=.31Mean Time in system = J/N = 38/6=6.3Mean number in system = J/Tt=38/19=2 = (J/N)*(N/Tt)=6.3*.31=2 =Tq