1. Nonstationarities in teletraffic data which may spoil your statistical tests Piotr Żuraniewski (UvA/TNO/AGH) Felipe Mata (UAM), Michel Mandjes (UvA), Marco Mellia (POLITO)

2. Stationarity Many models assume stationarity: statistical properties do not change over time –strong stationarity: all statistical properties remain the same over time –weak stationarity: statistical properties up to second order (mean, variance, covariance) remain unchanged

3. Nonstationarity – problems Real life: things are changing… Bad news: sample stationarity can not be positively verified Best answer we can get: ‘we found no evidence of given type of nonstationarity’ Some examples: –mean shift –polynomial deterministic trend –variance change

4. Example Change in the number of users in VoIP system Model: load change in M/G/inf queue Sample ACF suggests very high correlation –slow decay? –long range dependency?

5. Example Changepoint detection procedure we developed allows to separate parts with different load There is no significant correlation in either of this parts Sample ACF does not estimate ACF in case of nonstationarity

6. Changepoint detection Window of 50 samples presented to detection procedure Add newest observation, drop oldest and repeat detection procedure In this example: true change in window number 51 Changepoint detection works well – see output of 500 experiments

7. Changepoint detection However, if we add deterministic trend, things go wrong Observe high false alarm ratio after polluting data with trend

8. Work in progress Real VoIP data from Italian service provider and aggregated IP data from Spanish university backbone network Current research: estimate and remove trend from traffic Only than apply changepoint detection procedure(s)

9. Work in progress Trend estimation methods: –moving average? –kernel/wavelets smoothing? –parametric methods? –time series regression? How to judge if estimated trend is really significant? Models different than M/G/inf?

10. Conclusions Different types of nonstationarities may severely influence statistical tests or values of estimators Even if we try to detect one type of nonstationarity, the other type may ruin our original test We always have to pay attention to the assumptions of the theorems used Share your experience!