1. CSE808 F'99Xiangping Chen1 Simulation of Rare Events in Communications Networks J. Keith Townsend Zsolt Haraszti James A. Freebersyser Michael Devetsikiotis

2. CSE808 F'99Xiangping Chen2 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. CSE808 F'99Xiangping Chen3 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. CSE808 F'99Xiangping Chen4 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. CSE808 F'99Xiangping Chen5 Importance Sampling example

6. CSE808 F'99Xiangping Chen6 Techniques Modification of Individual Stochastic Elements Global Modification via Trajectory Splitting

7. CSE808 F'99Xiangping Chen7 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. CSE808 F'99Xiangping Chen8 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.

9. CSE808 F'99Xiangping Chen9 Trajectory splitting Example - DPR DPR - Direct probability redistribution Partitions the state-space S into m subsets, S 1, S 2, … S m. Oversampling factors,  1 <  2 <... <  m. Every state S i is visited  i more times. Unbiased factors are obtained by weighting a subset-dependent factor 1/   (S i ).

10. CSE808 F'99Xiangping Chen10 Tuning/Optimization of Parameters Large deviations, effective and decoupling bandwidths Stochastic optimization Conditional biasing Iterative balancing for trajectory splitting

11. CSE808 F'99Xiangping Chen11 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[   n i=1 A i ] –d(  ) = A(  ) /  is the effective Bandwidth.

12. CSE808 F'99Xiangping Chen12 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. CSE808 F'99Xiangping Chen13 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. CSE808 F'99Xiangping Chen14 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.

15. CSE808 F'99Xiangping Chen15 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.

16. CSE808 F'99Xiangping Chen16 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

17. CSE808 F'99Xiangping Chen17 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.