1. Course Description Queuing Analysis This queuing course
is a vital tool used in evaluating system performances
its application covers
A wide spectrum
from Bank automated teller machines
to transportation and communications data networks
This queuing course
focuses on queuing modeling techniques
and its application in data networks

2. Queuing definitions Kleinrock Bose Takagi
“We study the phenomena of standing, waiting, and serving, and we call this study Queueing Theory.”
Bose
“The basic phenomenon of queuing arises whenever a shared facility needs
to be accessed for service by a large number of jobs or customers.”
Takagi
“A queue is formed when service requests arrive at a service facility
and are forced to wait while the server is busy working on other requests”

3. Wrap up The study of queuing is the study of waiting
Customers may wait on a line
Planes may wait in a holding pattern
Jobs may wait for the attention of the CPU in an computer
Packets may wait in the buffer of a node in a computer network
Telephone calls may wait to get through an exchange

4. History lesson History goes back to primitive man
The history of queues goes back to primitive man. If you don’t believe me please take a look at the above presented figure Edward Hicks “Noah’s Ark”

5. Combinatorics Permutations K-permutation of a set of n elements
Combinations
K-combination of a set of n elements
=> k-permutation / k! (where k! is the number of possible ways to permute that combination)

6. Combinatorics (cont’d)
Binomial coefficients
Binomial expansion

7. Poisson distribution Poisson distribution is
Associated with the observation of event occurrences
If N represents the number of events in T
=> N/T = average number of events /minute
interested in answering the following question
How many occurrences of this event take place per minute?
The way it has been done
Either 0 or 1 event occurrence per minute
T=15 min
Event#1
Time 7

8. Poisson distributed random variable
A Poisson random variable X
Characterizes the number of occurrences of an event
Typically an arrival => X = # arrivals per unit time
With parameter λ (average # of arrivals per unit time)
The value of λ (arrival rate) 8

9. Poisson distribution: example 1
If number of accidents occurring on a highway per day
is a Poisson r.v. with parameter λ = 3,
What is the probability that no accidents occur today?
Solution 9

10. Poisson distribution: example 2
Consider an experiment that
counts the number of α-particles emitted in
a one-second interval by one gram of radioactive material.
If we know that ,on average , 3.2 such α-particles are given off
what is a good approximation
to the probability that no more than 2 α-particles appear? 10

11. Expectation: example 3 Calculate E[X]
For Poisson random variable X with parameter λ 11

12. Exponential distribution
is the foundation of most of the stochastic processes
Makes the Markov processes ticks
is used to describe the duration of sthg
CPU service
Telephone call duration
Or anything you want to model as a service time 12

13. Exponential random variable13

14. Link between Poisson and Exponential
Time
Exponentially
distributed with 1/ λ
If the arrival process is Poisson
# arrivals per time unit follows the Poisson distribution
With parameter λ
=> inter-arrival time is exponentially distributed
With mean = 1/ λ = average inter-arrival time 14

15. Proof

16. Memoryless Property 16

17. Example

18. Further properties of the exponential distribution

19. Further properties of the exponential distribution (ct’d)
X1 , X2 , …, Xn independent r.v.
Xi follows an exponential distribution with
Parameter λi => fXi (t) = λi eλit
Define
X = min{X1, X2, …, Xn} is also exponentially distributed
Proof
fX(t) = ?

20. Queuing system A queuing system is a place where customers arrive
According to an “arrival process”
To receive service from a service facility
Can be broken down into three major components
The input process
The system structure
The output process
Customer
Population
Waiting
queue
Service
facility

21. Characteristics of the system structure
λ: arrival rate
μ: service rate λ μ
Queue
Infinite or finite
Service mechanism
1 server or S servers
Queuing discipline
FIFO, LIFO, priority-aware, or random 21

22. Queuing systems: examples
Multi queue/multi servers
Example:
Supermarket
Multi-server/single queue
Bank
immigration .

23. Kendall notation David Kendall
A British statistician, developed a shorthand notation
To describe a queuing system
A/B/X/Y/Z
A: Customer arriving pattern
B: Service pattern
X: Number of parallel servers
Y: System capacity
Z: Queuing discipline
M: Markovian
D: constant
G: general
Cx: coxian

24. Kendall notation: example
M/M/1/infinity
A queuing system having one server where
Customers arrive according to a Poisson process
Exponentially distributed service times
M/M/S/K
M/M/S/K=0
Erlang loss queue K

25. Special queuing systems
Infinite server queue
Machine interference (finite population) λ μ .
S repairmen N
machines

26. M/M/1 queue λn = λ, (n >=0); μn = μ (n>=1) λ μ λ: arrival rate
μ: service rate
λn = λ, (n >=0); μn = μ (n>=1)

27. Traffic intensity rho = λ/μ
It is a measure of the total arrival traffic to the system
Also known as offered load
Example: λ = 3/hour; 1/μ=15 min = 0.25 h
Represents the fraction of time a server is busy
In which case it is called the utilization factor
Example: rho = 0.75 = % busy

28. Queuing systems: stability
N(t)
λ<μ
=> stable system
λ>μ
Steady build up of customers => unstable
busy
idle 1 2 3
Time
N(t) 1 2 3
Time

29. Example#1 A communication channel operating at 9600 bps
Receives two type of packet streams from a gateway
Type A packets have a fixed length format of 48 bits
Type B packets have an exponentially distribution length
With a mean of 480 bits
If on the average there are
20% type A packets and 80% type B packets
Calculate the utilization of this channel
Assuming the combined arrival rate is 15 packets/s

30. Performance measures L Lq W Wq Mean # customers in the whole system
Mean queue length in the queue space W
Mean waiting time in the system Wq
Mean waiting time in the queue 30

31. Mean queue length (M/M/1)

32. Mean queue length (M/M/1) (cont’d)

33. Little’s theorem This result The theorem
Existed as an empirical rule for many years
And was first proved in a formal way by Little in 1961
The theorem
Relates the average number of customers L
In a steady state queuing system
To the product of the average arrival rate (λ)
And average waiting time (W) a customer spend in a system

34. LITTLE’s Formula

35. Graphical Proof

36. Graphical Proof (continued)

37. Graphical Proof (continued)

38. Mean waiting time (M/M/1)
Applying Little’s theorem