This document discusses queuing theory, which is the mathematical study of waiting lines in systems where demand for service exceeds the available capacity. It covers the key characteristics of queuing systems including arrival patterns, service mechanisms, queue discipline, and number of service channels. Common configurations like single server-single queue and multiple server-multiple queue systems are described. Software used for queuing simulations is discussed along with the Kendall notation for representing queuing models. Limitations of queuing theory are noted.

1. Queuing Theory
AMIT SINGH DAHAL
G5638545

2. Outline:
 Introduction
 Characteristics
 Configuration
 Software
of Queuing System
of Service System
for Simulation
 Limitation
of Queuing Theory

3. Introduction:

a waiting line of "customers" requiring service from one or more servers

Example:

when the short term demand for service exceeds the capacity of
facilities

QT, mathematical study of waiting lines (or queues)

enables mathematical analysis of several related processes, including
arriving at the (back of the) queue, waiting in the queue, and being
served by the Service Channels at the front of the queue
-Waiting for service in bank and at reservation counter
-Waiting for a train or bus
-Waiting at doctor’s clinic

4. Characteristics of Queuing System:
1.
The Arrival Pattern
2.
The Service Mechanism
3.
The Queue Discipline
4.
Number of Service Channels

5. 1. The Arrival Pattern:


Arrivals can be measured as the arrival rate or the inter-arrival time(time
between arrivals)
Inter-arrival time =1/ arrival rate

(i) Balking: The customer may decide not to enter the queue upon
arrival, because it is too long.

(ii) Reneging: The customer may decide to leave the queue after
waiting a certain time in it.

(iii) Jockeying: If there are multiple queues in parallel the customers may
switch between them

6. 2. Service Mechanism:

Service pattern can be deterministic or stochastic

Can also be batched or bulked services

service rate may be state-dependent

Note: difference in Arrival pattern and Service
pattern because services cannot be initiated if the
Queue is empty

7. 3. Queue Discipline

Vital, the way of selecting customers for services

Can be:
i)First in First Out (FIFO)
Ii)Last in First Out (LIFO)
iii)Service in Random Order (SIRO)
iv)Priority Schemes. Priority schemes are :
-Preemptive:
customer with higher priority displaces the customer with low
priority and get services
-Non-preemptive:
customer with the higher priority wait for the current service to
be completed before getting the service

8. 4. Number of Service Channels:

Single Queue:
-have only one queue to get different services
-customer get services based on the availability of the server

9. 4. Number of Service Channels(Contd…)

Parallel Queue
-have multiple queues to get different services
-the customer can change the queue, if the number of customer
deceases in the other queue(Jockeying)

10. Configuration of Service System

Single Server-Single Queue:
-Example: student standing in the library queue

11. 
Parallel Server-Single Queue:
-Example: Bank transaction, many token numbers are served one-byone according to available Server

12. 
Several Server-Several Queue
-Example: different cash counters in electricity office

13. 
Services facilities in a series:
-Example: may be product manufacturing that requires steps like
cutting, drilling, grinding, operations, packaging

14. Software for Simulation:

Model represented using Kendall- Lee Notation
where,
(a/b/c):( d/e/f)
a = arrival rate distribution
b = service rate distribution
c = no. of parallel service channels (identical service)
d = service discipline
e = maximum no. allowed in the system
f = calling source
Common notation:
M =Poissonl/Exponential rates
G = General Distribution of Service Time
Ek-=Erlangian Distribution

15. 
Information Flows:
L
λ
μ
n
Lq
Queuing
Model
W
Wq
ρ
Pn

16. 
Notation:
λc = effective mean arrival rate
λ = λc if queue is infinite
λe = λ - [expected number who balk if the queue is finite]
W = expected waiting time of a customer in the system
Wq = expected waiting time of a customer in the queue
L = expected no. of customers in the system
Lq = expected number of customers in the queue
Po = probability of no customers in the system
Pn = probability of n customers in the system
ρ = traffic intensity= λ/nμ
ρc= effective traffic intensity= λe/μ

17. MULTIPLE CHANNELPOISSON/EXPONENTIAL MODEL
[(M/M/C):(GD/ α /α)]
Characteristics:


1. Input population is infinite.
2. Arrival rate has a Poisson Distribution

3. There is more than one server.

4. Service time is exponentially distributed with mean1/μ. [λ<μ]

5. System capacity is infinite.

6. Balking and reneging are not allowed.

18. Queuing Theory Limitation

Queuing models are quite complex and cannot be easily understood

Form of theoretical distribution applicable to given queuing situation is
not easily known

If the queuing discipline is not FIFO based, then the study of queuing
problem become more difficult

19. References:

DEMO: https://www.stat.auckland.ac.nz/~stats255/qsim/qsim.html

http://homepages.inf.ed.ac.uk/jeh/Simjava/queueing/mm1_q/mm1_q
.html

http://en.wikipedia.org/wiki/Queueing_theory

Zukerman, Moshe: Introduction to Queueing Theory and Stochastic
Teletraffic Models.

http://homes.cs.washington.edu/~lazowska/qsp/

http://people.brunel.ac.uk/~mastjjb/jeb/or/queue.html

http://www.eventhelix.com/realtimemantra/congestioncontrol/queue
ing_theory.htm

20. THE END
Any Questions???