Kleinrock

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Kleinrocks Delay Analysis
Telecommunication Network Modeling
SourceB.R. Haverkort, Performance of Computer
Communication Systems
André Augustyniak, Christian Lohrengel, Ulrike
Talbiersky, Holger Wichert
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Table of Contents

• Dr. Leonard Kleinrock
• System Description
• Kleinrocks Independence Assumption
• A modelling approach, based on Jackson networks
• MM1 queues ? MG1 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

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System description1

• Number of nodes
• capable of buffering incoming traffic
• Number of links
• bidirectional with different capacity in each
  direction
• Meshed structures as topology
• called store-and-forward networks

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System description2

• 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
• Generated as Poisson process
• Overall aggregate network traffic (measured in
  packets per second)

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System description3

• Switching node modelled as a single server queue
• Unidirectional link modelled as a seperate
  queueing station
• Queueing network with

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Queueing Stations Characteristics

• 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

• Validated with experimental and simulation
  results
• Good approximation if
• poisson arrivals at entry points of the network
• packet transmission times nearly exponential
• densely connected network
• moderate to heavy traffic load

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Given

• 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

• Traffic matrix ? (entries) (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

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Utilisations, Response times

• Utilisations
• 
  
• Response time
• 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

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Evalution using Jackson queueing networks1

• The queueing network model is completely
• specified by
• 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 networks2

• per link expected delay
• per node expected delay

service time
waiting time
service time
waiting time
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Evalution using networks of MG1 queues1

• Advantages of the MG1 model
• use of other than exponential service time and
  packet length distribution
• different packet length
• Disadvantages of the MG1 model
• computational procedure more complicated
• no longer exact

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Evalution using networks of MG1 queues2

• Replace MM1 based terms for ERl with
  corresponding terms for MG1 queue
• Use approximate approach, though MG1 non-
  product-form (results reasonably accurate,
  confirmed by simulation studies)
• Simplification for computationFixed packet size

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QNA Method

• 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 Kleinrocks delay analysis
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Implementation

• Arrival rate to a link and a switching node
• Overall aggregate network traffic
• Per link and per node delay
• Response time for packets from i to j
• Overall average network response time

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ARPANET

• ARPANET
• United States Defense Advanced Research Project
  Agency
• 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