Queueing theory models systems with queues that form due to demand for services exceeding the system's capacity. It was originally developed to model telephone traffic congestion. Key components of queueing systems include the arrival process, queue discipline, service mechanism, and outlet. Common models include the M/M/1 queue with a single server and Poisson arrivals, and the M/M/m queue with multiple servers. Queueing theory aims to minimize waiting times and costs by understanding the tradeoff between service levels and waiting.

1. QUEUEING THEORY
Muthuganesan N
Prakash S.P

2. Murphy's Law
If you change queue, the one that you
left will start to move faster than
the one you are in now.
Your queue always goes the slowest.
Whatever queue you join, no matter
how short it looks, it will always take
the longest to you to get served.

3. History of queueing theory
 Developed to provide models for forecasting
behaviors of systems subject to random demand
 The first problems addressed concerned congestion
of telephone traffic
 Erlang observed that a telephone system can be
modeled by Poisson customer arrivals and
exponentially distributed service times
 Molina, Pollaczek, Kolmogorov, Khintchine, Palm,
Crommelin followed the track

4. Importance of Queueing system

5. Queues/ waiting line
▪Common phenomenon of everyday life
▪Line maybe People / Items
▪Examples
– Grocery shop, Bank, Petrol refilling units, Automobile Service
station, Airplane, Train etc.

6. Queue
The line that forms in front of
service facilities is called a queue or
a waiting line
It involves arriving customers (or
items) who wait to utilize the
services at the facility which provide
the service they require
Departure of served
customers
Departure of impatient customers
Customer
arrivals

7. Queueing theory
A branch of the operations research providing probabilistic modelling tools
for Queueing Systems
 Defined as the construction of mathematical models of various types of
queuing system so that predictions may be made about how the
systems will cope with demands made upon it

8. Objective of queueing theory
▪Minimize total waiting time and service costs

9. WAITING COST AND SERVICE LEVEL TRADE-OFF

10. Queueing system
 A very simple mechanism
 Customers arrive at service counter and are attended by
one or more of the service counters
 As soon as a customer is served, he departs from the
system
 Arrivals
 Queue
 Service
 Outlet

11. Components/Characteristics of queueing system
1.The input process or arrival process
2.The Queue discipline or Waiting line
3.The Service mechanism
4.The Outlet of the queue

12. The input process or arrival process
Arrival from the input population may be classified based on
a.Size of the calling population
b.Pattern of arrivals at the system
c.Behavior of the arrivals

13. a. Size of the calling population
▪Can be finite / infinite
▪Infinite
–Cars arriving at toll booth
–Shoppers arriving at a supermarket
▪Finite
–Fleet of 4 cars visiting a repair shop
–A milling unit with three mills which require service from time to time

14. b. Pattern of arrivals at the system
▪Organized pattern/ random order
▪No. of arrivals per unit time estimated by Poisson
distribution
P(x) = e-λ λx
x!
◦ λ = the average arrival rate per time unit
◦ P(x) = the probability of exactly x arrivals occurring during one
time period

15. Arrival rate
▪The average number of customers arriving per time
period
▪λ

16. c. Behavior of the arrivals
▪Customer attitude about getting into line differs
▪Balking customer refuse to join the line because it is too long to suit
their needs or interest
▪Reneging customer enters the queue but they become impatient and
leave without completing their transaction
▪Jockeying customers move from one queue to another hoping to
receive service more quickly

17. 2. The Queue discipline or Waiting line
▪Length of line unlimited/ limited
▪A queue is limited when it cannot by law or physical
restrictions increase to an infinite length. Ex. Small
restaurant
▪A queue is unlimited when its size is unrestricted, as in case of
the toll booth serving arriving automobiles

18. Order of service
▪FCFS
▪LCFS
▪SIRO
▪Priority service
▪Pre-emptive priority
▪Non pre-emptive priority

19. 3. The Service mechanism
◦ Two aspects
a.Structure of the service system
b.Speed of service

20. Structure of the service system
How the service facilities are arranged
i.A single facility
ii.1 queue – several service facility
iii.Multichannel facility
iv.Multi-stage channel facility

21. b. Speed of service
Speed of service can be expressed in 2 ways:
i.Service rate
–No. of customers during a particular period of time
ii.Service time
–The amount of time required to service a customer
–Generally assumed to be exponentially distributed about
average service time

22. Service rate
▪The average number of customers that can be served
per time period
▪μ

23. 4. The Outlet of the queue
▪No problem in single channel facility whereas,
▪Problem in multistage channel facility
▪Service station breakdown can have effect on the queues
▪Line before the breakdown will lengthen and the line
following the breakdown will diminsh

24. Queuing Models
▪Grouped as
▪Deterministic (Fixed arrival and fixed service time)
▪Probabilistic (Random arrivals)
– Use assumptions

25. Kendall’s Notation
A/B/m/K/N/D
A Arrival Process M: Markovian D: Deterministic Er:
Erlang G: General
B Service Process M: Markovian D: Deterministic Er:
Erlang G: General
m Number of servers m=1,2,…
K Storage Capacity K= 1,2,… (if œ †hen i† is omified)
N Number of customers N= 1,2,…
(for closed networks otherwise it is omitted)
D Queueing Discipline FCFS,LCFS,SIROetc. Default is FCFS

26. Commonly used Queueing models
Model Queue characteristic Example
M/M/1/∞ Single server, discouraged
Arrivals
Single help Desk or counter in
store
M/M/m/∞ m servers, infinite number of
waiting positions
Airline ticket counter
M/M/m/m m server loss system, no waiting Automated car wash facility
M/M/1/K Single server queue with K
waiting positions
Mechanic shop
M/M/1/-/N Single server, infinite number of
waiting positions, finite customer
population N
Receiving degree in
Convocation

27. THANK YOU