Simulation of Rare Events in Communications Networks

1
Simulation of Rare Events in Communications
Networks

• J. Keith Townsend
• Zsolt Haraszti
• James A. Freebersyser
• Michael Devetsikiotis

2
Background

• Rare event probabilities in communication
  networks.
• Require prohibitively long simulation times
• How to reducing simulation execution time while
  retain the ease and flexibility of simulation?
  --- Importance Sampling based techniques.

3
What is IS?

• Combination of analysis and simulation.
• Modifying (biasing) the underlying probability
  mass so that the rare events occur much more
  frequently.
• Results are weighted to yield a statistically
  unbiased estimator.

4
Objective

• Significant reduction in the number of trials
  while maintain the estimator precision.
• Which parameter(s) of the system to bias?
• How much to bias each of them?
• What is the speedup?

5
Importance Sampling example
6
Techniques

• Modification of Individual Stochastic Elements
• Global Modification via Trajectory Splitting

7
Modification of Individual Stochastic Elements

• Modifying the probability distributions of one or
  more random number generators in the simulation
  model.
• Requires considerable prior knowledge about the
  system.

8
Global Modification via Trajectory Splitting

• Assumption some well identifiable intermediate
  system states are visited much more often than
  the target states and behave as gateway states to
  reach the target states.
• Entering the intermediate states triggers the
  splitting of the trajectory.
• Step-by-step evolution of the system follows the
  original probability measure.

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Trajectory splitting Example - DPR

• DPR - Direct probability redistribution
• Partitions the state-space S into m subsets, S1,
  S2, Sm.
• Oversampling factors, ?1 lt ?2 lt ... lt ?m.
• Every state Si is visited ?i more times.
• Unbiased factors are obtained by weighting a
  subset-dependent factor 1/? ?(Si).

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Tuning/Optimization of Parameters

• Large deviations, effective and decoupling
  bandwidths
• Stochastic optimization
• Conditional biasing
• Iterative balancing for trajectory splitting

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LDT - Large Deviation Theory

• Specify the biased distributions as ? -conjugate
  exponentially twisted versions of original,
  unbiased distribution.
• Effective bandwidths is invoked to determine the
  value of ?.
• A(?) lim n?? (1/n)log E exp? ?ni1 Ai
• d(?) A(?) / ? is the effective Bandwidth.

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LDT Continued

• ? value is equal to the service rate in a
  single queue with deterministic service.
• Additive property of effective bandwidths is used
  to describe multiple streams sharing the same
  queue.
• Decoupling bandwidths is used to provide
  sufficient conditions of a specific tagged stream.

13
Stochastic Optimization

• The mean field annealing (MFA) algorithm is a
  variant of simulated annealing (SA) that avoids
  local minima and arrives at optimal solutions in
  more rapid convergence.
• The stochastic gradient descent (SGD) algorithm
  can potentially zero in on favorable bias
  parameter settings fast by exploiting more
  information at hand.

14
Conditional Biasing

• An important IS technique that is effective in
  uniform probability distributions (UPD).
• Prior knowledge is used to partition the UPD into
  intervals that result in the important events or
  not.
• Requirement occurrence of any sequence of
  random variables resulting in an important events
  not be excluded from the biased random variable
  selection process.

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Iterative Balancing for Trajectory Splitting

• To find appropriate partitioning
• To choose the correct amount of splitting
• Near optimal ? setting is when the subset
  probability masses are equalized.
• A simple iterative procedure can explore subset
  probabilities in a step-by-step fashion.

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Application Examples

• Steady-state simulation of cell loss probability
• Regenerative method or A-cycles
• Application of stochastic optimization
• Tandem ATM network
• Application of Conditional Biasing
• ATM switch is described using operational
  approach
• Application of DPR-based Splitting Simulation
• Systems with internal loop

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Conclusion

• Proves to be effective although it requires
  problem-specific analytical phase
• Simulation will be used to evaluate more
  complicated networks
• More reliable networks will be characterized by
  rarer events
• IS is more important in the future.