CSE 550 Computer Network Design

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CSE 550Computer Network Design

• Dr. Mohammed H. Sqalli
• COE, KFUPM
• Spring 2007 (Term 062)

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Outline

• Queuing Models
• Application to Networks
• Traffic Flow Analysis

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Queuing Models - Single-Server Queue -

• ? average number of packets arriving per second
  pps
• Utilization, fraction of time the facility is
  busy ? ?Ts
• Theoretical maximum input rate that can be
  handled by the system is ?max 1/Ts
• Queues become very large near system saturation,
  growing without bound when ? 1
• Practical considerations limit the input rate for
  a single server to 70-90 of the theoretical
  maximum
• Little's formula (general relationship) r ?Tr
  and w ?Tw

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Queuing Models - Multiserver Queue -

• Utilization ? ?Ts/N
• Theoretical maximum input rate that can be
  handled by the system is ?max N/Ts
• Traffic intensity u N?

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Queuing Models - Multiple Single-server queues -

• Example of a Network of Queues
• Traffic Partitioning
• Traffic Merging
• Queues in Tandem

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Queuing Models - Notation

• The notation X/Y/N is used for queuing models
• X distribution of the inter-arrival times
• Y distribution of service times
• N number of servers
• The most common distributions are
• G general independent arrivals or service times
• M negative exponential distribution
• D deterministic arrivals or fixed length
  service
• Example M/M/1

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Queuing Models - Single-server queues -

• M/G/1 model
• The arrival rate is Poisson and the service time
  is general
• M/M/1 model
• The standard deviation is equal to the mean, the
  service time distribution is exponential, i.e.,
  service times are essentially random
• M/D/1 model
• The standard deviation of service time is equal
  to zero, i.e., a constant service time
• The poorest performance is exhibited by the
  exponential service time (M/M/1), and the best by
  a constant service time (M/D/1)
• Usually, the exponential service time can be
  considered to be the worst case
• An analysis based on this assumption will give
  conservative results

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Queuing Models - Single-server queues -

• Coefficient of variation sTs/Ts
• Zero Constant service time (M/D/1)
• Example all transmitted messages have the same
  length
• Ratio less than 1 Using M/M/1 model would give
  answers on the safe side it will give queue
  sizes and times that are slightly larger than
  they should be
• Example a data entry application for a
  particular form
• Ratio close to 1 This is a common occurrence and
  corresponds to exponential service time (M/M/1)
• Example message sizes varying over the full
  range, shared LAN, and packet-switching networks
• Ratio greater then 1 Need to use the M/G/1 model
  and not rely on the M/M/1 model
• Example a system that experiences many short
  messages, many long messages, and few in between

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Queuing Models- Network of Queues -

• Jackson's theorem states that
• In such a network of queues, each node is an
  independent queuing system, with a Poisson input
  determined by the principles of partitioning,
  merging, and tandem queuing
• Each node may be analyzed separately from the
  others using the M/M/1 or M/M/N model
• Results may be combined by ordinary statistical
  methods, e.g., mean delays at each node may be
  added to derive system delays

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Application to a Packet-Switching Network

• Consider a packet-switching network
• Consists of nodes interconnected by transmission
  links
• Each node acts as the interface for zero or more
  attached systems, each of which functions as a
  source and destination of traffic
• Each link is seen as a service station servicing
  packets

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Component Models

• Simplifications
• Packets (requests) arrive according to a Poisson
  process (exponential interarrival times)
• Infinite buffer size
• Independent queues (just add delays induced in
  the different queues encountered on the path)

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Inside a Router
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Traffic Flow Analysis - Objective

• Estimate
• Delay
• Utilization of resources (links)
• Traffic flow across a network depends on
• Topology
• Routing
• Traffic workload (from all traffic sources)
• Desirable topology and routing are associated
  with
• Low delays
• Reasonable link utilization (no bottlenecks)

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Traffic Flow Analysis - Assumptions

• Topology is fixed and stable
• Links and routers are 100 reliable
• Processing time at the routers is negligible
• Capacity of all links is given (in bps)
• Traffic workload is given ? ?jk (in pps)
• Routing is given
• Average packet size is given

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Analyzing Throughput

• The capacity of the network can also limit the
  number of connections/users it can handle for a
  particular type of service
• This is determined by finding out the narrowest
  available bandwidth in the path
• This is the network bottleneck
• The narrowest bandwidth can be a router, switch,
  or link

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External Workload

• The external workload offered to the network is
• Where
• ? total workload in packets per second
• ?jk workload between source j and destination k
• N total number of sources and destinations

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Internal Workload

• The internal workload on link i is
• ?i Si ? ?jk ?jk
• Where
• ?jk workload between source j and destination
  k
• ?jk path followed by packets to go from source
  j and destination k
• The total internal workload is
• Where
• ? total load on all of the links in the
  network
• ?i load on link i
• L total number of links

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Link Utilization

• Utilization of link i is ?i ?i Tsi
• Service time for link i is Tsi M / Bi
• Where
• M Average packet length (in bits)
• Bi Data rate on the link (in bps)
• Average service rate 1/Tsi Bi / M
• ?i ?i M / Bi
• ?b max (?i) Link b is the primary bottleneck
• Stability condition of a network is ?b lt 1

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Path Length and Packets Waiting

• Average length for all paths
• Average number of packets waiting and being
  served for link i is
• Number of packets waiting and being served in the
  network can be expressed as (using Little's
  formula)
• ?T

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Link Delay

• Because we are assuming that each queue can be
  treated as an independent M/M/1 model, we have
• The service time for link i is Tsi M / Bi
  ,Then

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Network Delay

• Average delay experienced by a packet through the
  network
• Putting all of the elements together, we get

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Applying M/M/1 Results to a Single Network Link

• Poisson packet arrivals with rate ? 2000 pps
• Fixed link capacity C 1.544 Mbps (T1 Carrier
  rate)
• We approximate the packet length distribution by
  an exponential with
• mean L 515 bits/packet
• Thus, the service time is exponential with mean
• Ts L/C  0.33 ms/packet              
• i.e., packets are served at a rate of µ 1/Ts
  M / C  3000 pps
• Using our formulas for an M/M/1 queue
•                       ? ?/µ   ?Ts 0.67
• So,
• r ?/(1- ?) 2 packets                
• and
• Tr r/ ? 1 ms

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Exercise 1

• The problem consists of 3 Routers A, B, C, and 6
  Switches, a, b, c, d, e, and f
• Assume that the three Routers are connected
  according to a unidirectional ring topology
  (A-B-C-A) and that all links have the same
  capacity of 2 Mbps
• Assume that the Switches are connected as
  follows (a, C), (b, C), (c, A), (d, A), (e, B),
  (f, B)
• The average packet size has been estimated equal
  to 2000 bits
• It has also been observed that the traffic
  generated by the various switches is Poissonian
  with rates as indicated in the following table
  showing the Inter-switches traffic in pps
• Question Find T, the average delay per packet

a b c d e f
a - 20 50 10 30 20
b 20 - 10 20 40 60
c 50 10 - 80 20 10
d 10 20 80 - 50 50
e 30 40 20 50 - 100
f 20 60 10 50 100 -
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(No Transcript)
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Animation of a Transmission Link

• Play with animation of a transmission link at
  http//poisson.ecse.rpi.edu/vastola/pslinks/perf/
  hing/mm1animate.html

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References

• William Stalling, Queuing Analysis, 2000
• Dr. Khalid Salah (ICS, KFUPM), CSE 550 Lecture
  Slides, Term 032