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									An Extended ME Methodology
  for General QNMs and its
 Application to ATM Switch
         Demetres Kouvatsos
  Presented by : Ravindra Vaishampayan
• Queuing Network Model. A complex
  network of Queues and Servers
• Useful for analysis of computer,
  communication and manufacturing systems
• We are interested in the QNM which
  models an ATM switch
    QNM modelling ATM Switch
•   Finite Capacity Queues
•   Multiple classes
•   Time and Space priority
•   Non-Exponential interarrival and service
    time distributions.
      Time and Space Priority
• Time priority refers to the fact that some
  services can tolerate lower delay that others
  e.g. voice Vs data
• Space priority controls allocation of buffer
  space to an arriving cell
• A sequence of buffer class thresholds (N1,
  N2 … NR) defines the buffer class
  thresholds of priority classes 1-R
      Time and Space Priority
• Ni < N where i = 1 .. R and N is the total
  buffer capacity of the Queue
• Packets for a class i are dropped if the
  number of packets of class i exceeds Ni.
  This scheme is also known as PBS i.e.
  Partial Buffer Sharing.
     Contribution of the Paper
• The problem is that an exact closed form
  solution for such a QNM (which models an
  ATM switch) is not known except for
  special cases.
• Paper provides a numerically cost-effective
  approximation for the solution of such a
      Approach of the solution
• sGGeo (shifted Generalised Geometric)
• GGeo ( Generalised Geometric)
• The author first provides a solution for a single
  sGGeo/GGeo/1/N1,N2….NR queue.
• He then describes how such a solution can be
  used as a building block for an open QNM with an
  arbitrary configuration i.e. any number of
  independent queues with any transition probability
  between different queues.
• Batch Renewal Process : Defined by two
  distributions : E(s) and K(s) where s is the
  batch number. Both E(s) and K(s) are
  independent and identically distributed. E(s)
  gives the time between sth and (s-1)th
  batch, and K(s) gives the count of the sth
• P[E(s)=t] = a(t) and P[K(s) = n] = b(n)
• Given only correlation of counts and
  correlation of intervals the least biased
  choice of process is a batch renewal
• sGGeo Batch renewal process :
• Given first two moments of message size and first
  two moments of interarrival time between
  messages and that the measured covariances are
  geometric, sGGeo batch renewal process is the
  least biased choice of the traffic model
• However an sGGeo process may be modelled as a
  GGeo process so that both processes satisfy the
  same first two moments of counts distribution.
ME Analysis of a censored
 sGGeo/Ggeo/1/N Queue
ME Analysis of a censored
 sGGeo/Ggeo/1/N Queue
ME Analysis of arbitrary queuing
• 1. Initialize blocking probabilities per class
• 2. Solve P(n) separately for all queues
• 3. For each queue, calculate the flow
  transition probabilities
• 4. Calculate flow balance eqns per class
• 5. Calculate first two moments of service
  time, and first two moments of interarrival
  time per class
ME Analysis of arbitrary queuing
• 6. Obtain new blocking probabilities by applying
  Newton-Raphson method
• 7. Calculate the first two moments of the
  interdeparture times per class.
• 8. Calculate new value for interarrival time SCV
  and return to step 2 till convergence
• 9. Apply the Universal ME solution once more
  under effective service time parameters.

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