Queuing Theory

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