This document provides an introduction to queueing theory. It defines key terms like queues, congestion, and queueing systems. Queueing theory is applied in many fields like telecommunications, healthcare, and manufacturing to model waiting lines and predict performance. Little's Law relates the average number of tasks in a system to the arrival rate and average time spent in the system. The document gives examples of applying queueing theory to model retail checkout lines and computer networks. It outlines the key characteristics used to describe queueing systems, such as arrival and service processes, and the Kendall notation used to specify queueing models.
2. 2
Queueing theory definitions
(Bose) “the basic phenomenon of queueing arises
whenever a shared facility needs to be accessed
for service by a large number of jobs or
customers.”
(Wolff) “The primary tool for studying these
problems [of congestions] is known as queueing
theory.”
(Kleinrock) “We study the phenomena of
standing, waiting, and serving, and we call this
study Queueing Theory." "Any system in which
arrivals place demands upon a finite capacity
resource may be termed a queueing system.”
(Mathworld) “The study of the waiting times,
lengths, and other properties of queues.”
3. 3
Applications of Queuing Theory
Telecommunications
Traffic control
Determining the sequence of computer
operations
Predicting computer performance
Health services (ex. control of hospital
bed assignments)
Airport traffic, airline ticket sales
Layout of manufacturing systems.
4. 4
Example application of
Queueing theory
In many retail stores and banks
multiple line/multiple checkout system a queuing
system where customers wait for the next available
cashier
We can prove using queuing theory that : throughput
improves increases when queues are used instead
of separate lines
6. 6
Queueing Theory for studying
networks
View network as collections of queues
FIFO data-structures
Queuing theory provides probabilistic
analysis of these queues
Examples:
Average length
Average waiting time
Probability queue is at a certain length
Probability a packet will be lost
7. 7
Little’s Law
Little’s Law:
Mean number tasks in system = mean arrival rate x
mean response time
Observed before, Little was first to prove
Applies to any system in equilibrium, as long as
nothing in black box is creating or destroying tasks
Arrivals Departures
System
8. 8
Proving Little’s Law
J = Shaded area = 9
Same in all cases!
1 2 3 4 5 6 7 8
Packet
#
Time
1
2
3
1 2 3 4 5 6 7 8
# in
System
1
2
3
Time
1 2 3
Time in
System
Packet #
1
2
3
Arrivals
Departures
9. 9
Definitions
J: “Area” from previous slide
N: Number of jobs (packets)
T: Total time
l: Average arrival rate
N/T
W: Average time job is in the system
= J/N
L: Average number of jobs in the system
= J/T
10. 10
1 2 3 4 5 6 7 8
# in
System
(L) 1
2
3
Proof: Method 1: Definition
Time (T)
1 2 3
Time in
System
(W)
Packet # (N)
1
2
3
=
W
L T
N
)
(
NW
TL
J
W
L )
(l
11. 11
Proof: Method 2: Substitution
W
L T
N
)
(
W
L )
(l
)
)(
( N
J
T
N
T
J
T
J
T
J
Tautology
12. 12
Model Queueing System
Server System
Queuing System
Queue Server
Queuing System
Use Queuing models to
Describe the behavior of queuing systems
Evaluate system performance
13. 13
Characteristics of Queueing
Systems
Arrival Process
The distribution that determines how the tasks
arrives in the system.
Service Process
The distribution that determines the task
processing time
Number of Servers
Total number of servers available to process the
tasks
14. 14
Kendall Notation 1/2/3(/4/5/6)
Six parameters in shorthand
First three typically used, unless specified
1. Arrival Distribution
2. Service Distribution
3. Number of servers
4. Total Capacity (infinite if not specified)
5. Population Size (infinite)
6. Service Discipline (FCFS/FIFO)