Queuing Analysis

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Queuing Analysis
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An example of a Queue

• Web server handles requests in 1 msec
• If requests arrive at a constant rate of 1000
  req/sec or less, everything works fine.
• In reality, arrival rate is not constant but
  varies.
• Suppose arrival rate is irregular with an average
  of 500 req/sec.

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Queue Behavior

• The behavior of a system with a queue may not be
  according to our intuition.
• When arrival rate is 50, after 50 seconds, an
  average buffer contents of 43 requests, with a
  peak of over 600 requests.
• When arrival rate is 95 (2 times 50), the
  average buffer contents rises to 1859 (40 times
  43).
• When arrival rate is 99 (tiny increase), the
  average buffer contents rises to 2583 (40
  increase).

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Why Queuing Systems?

• Projection of performance
• A theoretical method of getting some idea of what
  the actual situation will be in the real world.
• Assumptions made result in discrepancies between
  theoretical and actual outcomes.

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Queuing Models

• Single Server Queue
• Multiserver Queue

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Single Server Queue
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Example
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Multiserver Queue
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Difference with Multi Single Server Queue
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Parameters

• Theoretical maximum input rate that can be
  handled by the system is
• In practice 70-90.

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Basic Queuing Relationships
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Queuing Formulas
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Example

• Messages arrive at a switching center for a
  particular outgoing communication line in a
  Poisson manner with a mean arrival rate of 180
  messages per hour. Message length is distributed
  exponentially with a mean length of 14,400
  characters. Line speed is 9600 bps.

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Example (contd)

• What is the mean waiting time in the switching
  center?
• mean message length 14400 X 8 115200 bits
• average service time Ts 115200 / 9600 12
  sec
• arrival rate ? 180 / 3600 0.05 message/sec
• utilization ? 0.05 X 12 0.6
• mean waiting time T? 0.6 X 12 / (1-0.6) 18
  sec

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Example (contd)

• How many messages will be waiting in the
  switching center for transmission on the average?
• messages waiting ? 0.6X0.6/(1-0.6)
• 0.9 messages

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Self Similar Traffic
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Self Similarity

• The idea is that something looks the same when
  viewed from different degrees of magnification
  or different scales on a dimension, such as the
  time dimension.
• Its a unifying concept underlying fractals,
  chaos, power laws, and a common attribute in many
  laws of nature and in phenomena in the world
  around us.

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Cantor
Each left portion in a step is a full replica of
the preceding step
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Network Traffic in the Real World

• For years, traffic was assumed to
• be based on Poisson.
• It is now known that this traffic has a self
  similar pattern.
• Characterized by burstiness.

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Self Similarity of Ethernet Traffic

• Seminal paper by W. Leland et al published in
  1993, examined Ethernet traffic between 1989 and
  1992, gathering 4 sets of data, each lasting 20
  to 40 hours, with a resolution of 20
  microseconds.
• Paper shattered the illusion of Poison
  distribution being adequate for traffic analysis.
• Proved Ethernet traffic is self similar with a
  Hurst factor of H 0.9
• 0 lt H lt1 the higher H, the more self similar
  the pattern

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Nature of self-similar traffic

• Burstiness small variations over small time
  periods, big variations over big time periods (as
  seen in the figure of slide 23)
• As a result of this If the traffic averaged over
  longer periods is plotted, one sees the same
  percentage of variation as when averaged over
  short time periods (see figure on slide 25,
  columns left and right)
• Note In the case of Poisson traffic, the
  percentage variations decrease as the time period
  over which the traffic values are averaged
  increases (see middle column)

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Self Similar Traffic in Simulation

• A superposition of many Pareto-distributed ON-OFF
  sources can be used to generate self similar
  traffic.
• Pareto distribution is a heavy-tailed
  distribution the tail decays much more slowly
  than the exponential distribution.
• Typical sample includes many small values and a
  few very large values (bursty).

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How Inaccurate Are Older Models?
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Why is the Internet traffic self-similar ?

• It took long time to understand why the Internet
  traffic is rather self-similar. It appears that
  the TCP protocol, which is currently used by most
  applications over the Internet, introduces this
  traffic property. The speed of data transmission
  of TCP is influenced by congestion control when a
  packet gets lost. Through this mechanism, the
  many independent TCP connections that run over
  the Internet become dependent on one another. The
  interaction is quite complex and involves the
  retransmission process after time-out. The net
  result is that the traffic becomes self-similar.
• Note that voice and video streaming does not use
  TCP. As these types of applications become more
  important over the Internet, it can be expected
  that the traffic will become less self-similar.
• It is to be noted that the arrival pattern of new
  sessions (e.g. TELNET sessions or Web server
  sessions) have been observed to follow a Poisson
  distribution.