1 performance evaluation of computer networks objectives introduction to queuing theory little’s...
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Performance Evaluation of Computer Networks
Objectives Introduction to Queuing Theory Little’s Theorem Standard Notation of Queuing Systems Poisson Process and its Properties M/M/1 , M/M/m , M/M/m/m , and M/G/1
Queuing System Network of queues
Jackson Networks
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Introduction Each one of us has spent a great deal of
time waiting in lines. One example in the Cafeteria Other examples of queues are
Printer queue Packets arriving to a buffer Calls waiting for answer by a technical support
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What makes up a queue?
The System: A collection of objects under study It is important to define the system
boundariesThe Entities: The people, packets, or
objects that enter the system requiring some kind of service
The Servers: The people, resources, or servers that perform the service required
The Queue: An accumulation of entities that have entered the system but have not been served
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Queue Discipline
First Come First Served - FCFS Most customer queues
Last Come First Served - LCFS Packages, Elevator
Served in Random Order - SIRO Entering Buses
Priority Service Multi-processing on a computer Emergency room
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What factors effect system performance The Arrivals Process
The time between any two successive arrivals Does this depend on the number of packets in the system? Finite populations
The Service Process The time taken to perform the service Does this depend on the number of packets in the system?
The number of servers operating in system The Service Discipline System Capacity
Processes waiting + processes being served
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Measuring System Performance The total time an “entity” spends in the
system (Denoted by W) The time an “entity spends in the queue
(Denoted by Wq) The number of “entities” in the system
(Denoted by L) The number of “entities” in the queue
(Denoted by Lq) The percentage of time the servers are busy
(Utilization time)
These quantities are variable over time
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What is Queuing Theory?
Primary methodological framework for analyzing network delay
Often requires simplifying assumptions since realistic assumptions make meaningful analysis extremely difficult
Provide a basis for adequate delay approximation
queue
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Packet Delay
Packet delay is the sum of delays on each subnet link traversed by the packet
Link delay consists of: Processing delay Queuing delay Transmission delay Propagation delay
node
node
node
packet delay
link delay
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Link Delay Components (1)
Processing delay Delay between the time the packet is
correctly received at the head node of the link and the time the packet is assigned to an outgoing link queue for transmission
head node tail node
outgoing link queue
processing delay
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Link Delay Components (2)
Queuing delay Delay between the time the packet is
assigned to a queue for transmission and the time it starts being transmitted
head node tail node
outgoing link queue
queuing delay
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Link Delay Components (3)
Transmission delay Delay between the times that the first and
last bits of the packet are transmitted
head node tail node
outgoing link queue
transmission delay
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Link Delay Components (4)
Propagation delay Delay between the time the last bit is
transmitted at the head node of the link and the time the last bit is received at the tail node
head node tail node
outgoing link queue
propagation delay
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Queuing System (1)
Customers (= packets) arrive at random times to obtain service
Service time (= transmission delay) is L/C L : Packet length in bits C : Link transmission capacity in bits/sec
queue
customer (= packet)
service (= packet transmission)
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Queuing System (2)
Assume that we already know: Customer arrival rate Customer service rate
We want to know: Average number of customers in the system Average delay per customer
customer arrival rate
customer service rate
average delay
average # of customers
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Definition of Symbols (1)
pn = Steady-state probability of having n customers in the system
= Arrival rate (inverse of average interarrival time)
= Service rate (inverse of average service time)
N = Average number of customers in the system
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Definition of Symbols (2)
NQ = Average number of customers waiting in queue
T = Average customer time in the system
WQ = Average customer waiting time in queue (does not include service time)
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Little’s Theorem
N = Average number of customers = Arrival rate T = Average customer time in the
systemN = T
Hold for almost every queuing system that reaches a steady-state
Express the natural idea that crowded systems ( large N ) are associated with long customer delays ( large T ) and reversely
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Application of Little’s Theorem (2) Consider a window flow control system
W : Window size : Packet arrival rate T : Average packet delay
From Little’s TheoremW >= T
If T increases, must eventually decrease If is limited due to congestion, increasing
W merely serves to increase T
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Standard Notation of Queuing Systems (1)
X/Y/Z/K X indicates the nature of the arrival process
M : Memoryless (= Poisson process, exponentially distributed interarrival times)
G : General distribution of interarrival times D : Deterministic interarrival times
Y indicates the probability distribution of the service times M : Exponential distribution of service times G : General distribution of service times D : Deterministic distribution of service times
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Standard Notation of Queuing Systems (2)
X/Y/Z/K Z indicates the number of servers K (optional) indicates the limit on the
number of customers in the system Examples:
M/M/1, M/M/m, M/M/∞, M/M/m/m M/G/1, G/G/1 M/D/1, M/D/1/m
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The Poisson Arrival Model
A Poisson process is a sequence of events “randomly spaced in time”
Examples Customers arriving to a bank Packets arriving to a buffer
The rate λ of a Poisson process is the average number of events per unit time (over a long time)
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Properties of Poisson Process (1)
Interarrival times n are independent and exponentially distributed with parameter
The mean and variance of interarrival times n are 1/ and 1/^2, respectively
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Properties of Poisson Process (2)
If two or more independent Poisson process A1, ..., Ak are merged into a single process A = A1 + A2 + ... + Ak, the process A is Poisson with a rate equal to the sum of the rates of its components
A1
Ai
Ak
A
independent Poisson processes
Poisson process
merge1
i
k
k
ii
1
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Properties of Poisson Process (3)
If a Poisson process A is split into two other processes A1 and A2 by randomly assigning each arrival to A1 or A2, processes A1 and A2 are Poisson
A1
A2
A
Poisson processes
Poisson process
split randomly1
2
with probability p
with probability (1-p)
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M/M/1 Queuing System
A single queue with a single server Customers arrive according to a Poisson
process with rate The probability distribution of the
service time is exponential with mean 1/
Poisson arrival with arrival rate
Exponentially distributed service timewith service rate
single server
infinite buffer
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M/M/1 Queuing System: Results (1) Utilization factor (proportion of time the
server is busy)
Probability of n customers in the system
Average number of customers in the system
(1 )nnp
1N
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M/M/1 Queuing System: Results (2) Average customer time in the system
Average number of customers in queue
Average waiting time in queue
1NT
2
1QN W
1W T
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M/M/m Queuing System
A single queue with m servers Customers arrive according to a Poisson
process with rate The probability distribution of the
service time is exponential with mean 1/
Poisson arrival with arrival rate Exponentially
distributedservice time with rate
m servers
infinite buffer
1
m
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M/M/m Queuing System: Results (1) Ratio of arrival rate to maximal system
service rate
Probability of n customers in the system
m
10
0
0
0
1( ) ( )
! !(1 )
( )
!
!
m
k
n
nm m
k mm mp
k m
mp n m
npm
p n mm
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M/M/m Queuing System: Results (2) Probability that an arriving customer
has to wait in queue (m customers or more in the system)
Average waiting time in queue of a customer
Average number of customers in queue
0( )
!(1 )
m
Qp m
Pm
(1 )
Q QN PW
0 1
QQ m n
n
PN np
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M/M/m Queuing System: Results (3) Average customer time in the system
Average number of customers in the system
1 1 QPT W
m
1
QPN T m
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M/M/m/m Queuing System A single queue with m servers (buffer size
m) Customers arrive according to a Poisson
process with rate The probability distribution of the service
time is exponential with mean 1/
Poisson arrival with arrival rate Exponentially
distributedservice time with rate
m servers
buffer size m
1
m
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M/M/m/m Queuing System: Results Probability of m customers in the
system
Probability that an arriving customer is lost
00
0
11
!
1, 1,2, ,
!
nm
n
n
n
pn
p p n mn
0
( / ) / !
( / ) / !
m
m m n
n
mp
n
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M/G/1 Queuing System
A single queue with a single server Customers arrive according to a Poisson
process with rate The mean and second moment of the
service time are 1/and X2
Poisson arrival with arrival rate
Generally distributed service timewith service rate
single server
infinite buffer
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M/G/1 Queuing System: Results Pollaczek-Khinchin formula
2
1 2(1 )
1
R XW
T W
22
22
2(1 )
2(1 )
QX
N W
XN T
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Network of Queues Network is a model in which jobs departing
from one queue arrive at another queue (or possibly the same queue)
Open Networks: all customers can leave the network
Closed Networks: No customers can leave the network
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Jackson Networks
Jackson Network is named after James R. Jackson
It is the first significant development in the theory of networks of queues
Each node of the queueing network can be analyzed separately
The utilization of all of the queues is less than one
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Open Jackson Networks
There are J queues Customers arrive at queue l according to independent
Poisson processes with rate The service times in queue l are exponential with rates Upon leaving queue l, each customer is sent to queue m with
probability and leaves the network with probability
The routing decision is independent of the past evolution of the network
l
l
lmr
J
mlmr
1
1
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Conclusion
Queuing models provide qualitative insights on the performance of computer networks, and quantitative predictions of average packet delay
To obtain tractable queuing models for computer networks, it is frequently necessary to make simplifying assumptions
A more accurate alternative is simulation, which, however, can be slow, expensive, and lacking in insight
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