Transcript Document 7759538

Kleinrock´s Delay Analysis
Telecommunication Network Modeling
Source:
B.R. Haverkort,
Performance of Computer Communication Systems
André Augustyniak, Christian Lohrengel, Ulrike Talbiersky, Holger Wichert
1
Table of Contents:
•
Dr. Leonard Kleinrock
•
System Description
•
Kleinrock´s Independence Assumption
•
A modelling approach, based on
Jackson networks
•
M|M|1 queues  M|G|1 queues (refine)
•
QNA-Method
•
Implementation
•
ARPANET
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Dr. Leonard Kleinrock




Born June 13, 1934 in Manhattan
Professor at the University of Los Angeles,
Computer Science Department since 1963
Member of the American Academy of Arts
and Sciences, of the National Academy of
Engineering
Created the basic principles of packet
switching, the technology underpinning the
internet
3
System description
~1~
^
Mn
Number of nodes
capable of buffering incoming traffic
^
Ml
Number of links
bidirectional with different capacity in each direction
Meshed structures as topology
called: store-and-forward networks
4
System description
~2~





Nodes as concentrators for a large user group,
e.g. university computer centers
Links connect the nodes, e.g. they span a
complete country
Traffic from node i to node j:  ij
Generated as Poisson process
Overall aggregate network traffic (measured in
packets per second):
^
^
M n Mn
    i , j
i 1
j 1
5
System description
~3~



Switching node modelled as a single server
queue
Unidirectional link modelled as a seperate
queueing station
Queueing network with
^
Mn  M n
^
 Ml  2 M l
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Queueing Stations: Characteristics

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

Scheduling order for switching nodes:
FCFS
Service rate i is known
Link l has a certain capacity cl (bits per
second)
The length of a packet generated at node i
and destined for node j is drawn from a
particular packet length distribution
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Kleinrocks Independence
Assumption
~1~


Interarrival times at various queues are
independent
Service times of a given packet at various
queues are independent
(length of the packet is randomly selected each
time it is transmitted over a network link)

Service times do not depend on interarrival
times and vice versa
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Kleinrocks Independence
Assumption
~2~

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Validated with experimental and
simulation results
Good approximation if:
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poisson arrivals at entry points of the network
packet transmission times „nearly“ exponential
densely connected network
moderate to heavy traffic load
9
Given

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All packets have the same mean length
with the value 1/
Transmission of a packet of b bits length
takes b/ci seconds to be transmitted over
link i
Possible throughput over link i: ci
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Workloads
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Traffic matrix  (entries)  ij (packets per second)
R(i,j) set of links visited by packets routed from i to j
N(i,j) set of switching nodes in the route from i to j
Sets are uniquely defined and static

Arrival rate of packets at link l

Arrival rate to switching node n n 
l 




i, j
i , j lR i , j 




i, j
i , j nN i , j 
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Utilisations, Response times
l  l / cl
Utilisations
 n  n / n
E Ri, j  
E R    E R 

 
 
lR i , j



nN i , j
Response time
n
EW   ES   P    EW   ES 

 
 
lR i , j

l
l
l
l
nN i , j
n
n
Expected response time for packets from i to j as
sum over response times at all links and nodes
visited along the way
Splitted in waiting time and service time
Pl propagation delay for link l
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Overall Average Network
Response Time
Expressed as the sum of the expected response times on a route
from node i to j
Weighted by ist relative importance
 i, j
ER   
ERi, j 
i 1 j 1 
Mn Mn
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Evalution using Jackson
queueing networks
~1~
The queueing network model is completely
specified by:
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link and switching nodes parameters (ci und i)
traffic matrix 
mean packet length (1/ )
routing informations (R(i,j) und N(i,j))
additionaly:

packet length and switching time are negative
exponenially distributed random variables
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Evaluation using Jackson
queueing networks
~2~

E S  
per link expected delay:
c
1
 l E Sl 


E Rl 

 E Sl 

cl  l
1  l


 service time
1
l
l
waiting time

E S n  
per node expected delay:
1
 n E Sn 
E Rn  

 E Sn 



n  n
1  n


 service time
1
n
waiting time
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Evalution using networks of
M|G|1 queues
~1~

Advantages of the M|G|1 model:
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use of other than exponential service time and
packet length distribution
different packet length
Disadvantages of the M|G|1 model:
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
computational procedure more complicated
no longer exact
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Evalution using networks of
M|G|1 queues
~2~
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Replace M|M|1 based terms for E[Rl] with
corresponding terms for M|G|1 queue
Use approximate approach, though M|G|1 nonproduct-form (results reasonably accurate,
confirmed by simulation studies)
Simplification for computation:
Fixed packet size
E [ Sl ] 
 i, j 1

( i , j ),lR ( i , j ) l cl i , j
E[ Sl2 ] 
 i, j
1

2
( i , j ),lR ( i , j ) l ( cl i , j )
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QNA Method
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Developed by Kühn, ext. by Whitt
Allows quick analysis of large open queueing
networks
Fixed routing probabilities and FCFS scheduling
Arrival processes need not to be Poisson distributed
Allows multiple customer classes
Approximation
→ Another approach for Kleinrock‘s delay analysis
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Implementation

Arrival rate to a link and a switching node:
l 




i, j
i , j lR i , j 
n 




i, j
i , j nN i , j 
^
Mn
^
Mn


Overall aggregate network traffic     i , j
i 1 j 1
Per link and per node delay:
1
1
E Rl  
E Rn  
cl  l
n  n

Response time for packets from i to j:
E Ri, j    E Rl    E Rn 
lR i , j 

nN i , j 
Overall average network response time:
M
M
 i, j
E R    
E Ri, j 

i 1 j 1
n
n
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ARPANET
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ARPANET:
United States Defense Advanced Research Project Agency
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Precursor of internet
First node 1969 at the University of California in
Los Angeles (UCLA)
Advantage: better communication, availibility,
utilization of resources
Difficulty: to provide effective communication
among a collection of incompatible machines
Innovations: email (1971), FTP file transfer protocol
(1973)
Closed 1990, therefore exists NSFNET
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ARPANET 1969
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ARPANET 1970
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ARPANET 1975
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ARPANET 1977
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ARPANET 1987
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ARPANET 1989
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References
B.R. Haverkort,
Performance of Computer Communication Systems
Dimitri Bertsekas, Robert Gallager,
Data Networks
Prof. Yannis A. Korilis,
Networking Theory & Fundamentals Lecture
Skript SN1
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