Simulation of Rare Events in
Communications Networks
J. Keith Townsend
Zsolt Haraszti
James A. Freebersyser
Michael Devetsikiotis
CSE808 F'99
Xiangping Chen
1
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.
CSE808 F'99
Xiangping Chen
2
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.
CSE808 F'99
Xiangping Chen
3
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?
CSE808 F'99
Xiangping Chen
4
Importance Sampling example
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Xiangping Chen
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Techniques
• Modification of Individual Stochastic
Elements
• Global Modification via Trajectory Splitting
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Xiangping Chen
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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.
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Xiangping Chen
7
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.
CSE808 F'99
Xiangping Chen
8
Trajectory splitting Example DPR
• DPR - Direct probability redistribution
• Partitions the state-space S into m subsets,
S1, S2, … Sm.
• Oversampling factors, 1 < 2 < ... < m.
• Every state Si is visited i more times.
• Unbiased factors are obtained by weighting
a subset-dependent factor 1/ (Si).
CSE808 F'99
Xiangping Chen
9
Tuning/Optimization of
Parameters
• Large deviations, effective and decoupling
bandwidths
• Stochastic optimization
• Conditional biasing
• Iterative balancing for trajectory splitting
CSE808 F'99
Xiangping Chen
10
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[ ni=1 Ai]
– d() = A() /  is the effective Bandwidth.
CSE808 F'99
Xiangping Chen
11
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.
CSE808 F'99
Xiangping Chen
12
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.
CSE808 F'99
Xiangping Chen
13
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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Xiangping Chen
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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.
CSE808 F'99
Xiangping Chen
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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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Xiangping Chen
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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.
CSE808 F'99
Xiangping Chen
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