1. Author: Chengchen, Bin Liu Publisher: International Conference on Computational Science and Engineering Presenter: Yun-Yan Chang Date: 2012/04/18 1

2. Flow statistics is a basic task of passive measurement and has been widely used to characterize the state of the network. Adaptive Non-Linear Sampling (ANLS) is one of the most accurate and memory-efficient flow statistics method proposed recently. This paper studies the parameter setting problem for ANLS. Proposed a parameter self-tuning algorithm which enlarges the parameter to a equilibrium tuning point and renormalizes the counter when it overflows. 2

3. A method that adjusts the sampling rate according to the counter value. No prediction or estimation of flow size distribution is required beforehand. The counting process can be presented as c c + 1 with probability p(c) c : the counter value, 0 < a < 1. Estimate flow length can be formulated as f(c)=[(1+a) c -1]/a. Relative error of flow size can be presented as. n: real flow size 3

4. Two major performance metrics for flow statistics: Relative error Measures the accuracy of a flow statistics method and can be quantified by coefficient of variation as shown in (3) for ANLS Counting range (related to memory consumption) The largest flow size that a flow statistics method could record. For ANLS, the counting range is B=[(1+a) c -1]/a. 4

5. Tradeoff between small relative error and large counting range 5 Fig. 1. Relative error vs. parameter a when flow size n = 5000. Fig. 2. Counting range vs. parameter a when the largest counter value is 256 (the counter is 8-bit width).

6. When a counter overflows, a is adjusted from a 1 to a larger value a 2, and renormalized to a smaller value according to the reconfigured a 2. 6

7. Determine the equilibrium point for tuning the parameter. Accuracy utility (U e ) e 1 and e 2 are the relative error before and after the parameter tuning. Counting Range Utility (U b ) B 1 and B 2 are the counting range before and after the parameter tuning. 7

8. 8 Fig. 3. Finding the equilibrium tuning point.

9. To keep the inverse estimations before and after the tuning is the same. Suppose the counter value and parameter before tuning is c 1 and a 1, after tuning is c 2 and a 2, the estimations are: (13) and (14) provide the same estimation 9

10. To keep the inverse estimations before and after the tuning is the same. 10 The counter is renormalized to with probability x -, and is reset to with probability 1 (x ).

11. Prove that the renormalization process does not introduce error and ensure the estimation is the same before and after renormalization. Theorem: The expect error in renormalization process is zero. Proof: Let From the Algorithm 3, the expected value of c 2 can be formulated as E(c 2 ) = (X ) + (1 X + ) = X Namely, f(E(c 2 )) = f(c 1 ). 11

12. Growth of ANLS with and without counter renormalization. 12 Fig.4 Grow of the counter value

13. Relative error of ANLS with and without parameter tuning under different traffic scenarios. 13