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# Traffic modeling and Prediction ----Linear Models

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1. Traffic modeling and Prediction ----Linear Models Traffic models are important in • the design, engineering and performance evaluation of networks. • studying network traffic • generating linear processes • traffic modeling using linear models • predicting traffic in various fields of networks • Minimum mean square error forecast

2. ARIMA(p,d,q) Models (Auto Regressive Integrated Moving Average) • Let {at: t =..., -1, 0, 1, ...} be a white noise WN(0, 2) with zero mean and variance 2 • Then Xt is an ARIMA(p,d,q) process if • B is the backward-shift operator, i.e. BXt = Xt-1 •  (B) and (B) are polynomials in complex variables with no common zeroes, and in addition  (B) has no zeroes in the unit disk

3. ARIMA (p,d,q) Models • p -- autoregressive order, non-negative integer • p = 0 : MA (q) models • q -- moving average order, non-negative integer • q = 0 : AR (p) models • d is the level of differencing • d = 0: stationary • d is non-negative integer: nonstationary • is thedifferencing operator defined as

4. Wireless Traffic Modeling and Prediction Using Seasonal ARIMA Model Yantai Shu1 Minfang Yu1 Jiakun Liu1 Tianjin University1 Presenter: Oliver W.W. Yang2 University of Ottawa2 May 2003

5. Outline • Introduction • Motivation • Objective • Building a seasonal ARIMA model to describe a trace • Traffic Prediction • Feasibility study • Conclusion

6. Introduction Statistics of China Mobile in Tianjin indicates that the number of mobile phone users is increasing at an exponential rate  need proper modeling  important to forecast wireless traffic work-load

7. Previous Work • Seasonal ARIMA (Auto Regressive Integrated Moving Average) model • linear prediction scheme used in the dynamic bandwidth allocation schemes for VBR video • Predictive congestion control for broadband WAN Our work on • the fractional ARIMA model in admission control • the seasonal ARIMA model for the prediction of traffic in the dial-up access network of Chinanet-Tianjin with one periodicity.

8. Objective • Studying the characteristic of wireless traffic • provide a general expression for the wireless traffic in China • Fitting seasonal ARIMA model to capture the properties of real wireless traffic • Seasonal model with two periodicities • Using the model to forecast wireless traffic • Provide guidance in designing, engineering and performance evaluating of networks

9. Seasonal ARIMA Model Exploits the periodic effect, i.e., the relation among values of different observation time intervals. Let • Xt be the tth observation in an interval • s be the period • be the error (noise) components (general correlated) Then using relationship we obtain

10. Seasonal ARIMA model General multiplicative model • with one period of order • with two period of order • can similarly obtain models with three or more periodic components with similar argument

11. Building a seasonal ARIMA model to describe a trace • Use spectrum analysis to uncover different periodicities in the time series • basis of building a seasonal model • Transfer the ARIMA problem to an ARMA problem • Make use of the several known ways for fitting ARMA models to traffic traces • Identify the necessary parameters (d and D) • Obtain from the ARMA model on process

12. Algorithm A: Procedure to fit a seasonal ARIMA model to traffic trace Step 1: Obtaining the periods such as s1 and s2 through spectrum analysis. Step 2: Obtaining an estimate of d, D1 and D2 according to incremental analysis of the trace, determining d, D1 and D2 using ADF test. Step 3: Performing differencing on Xt according to to obtain a stationary series. Step 4: Model identification - Determining all the orders p, P1, P2, q, Q1and Q2 Step 5: Estimating all the parameters like qiandj Step 6:Obtaining the fitted multiplicative seasonal ARIMA models from

13. Prediction: Using seasonal ARIMA model to forecast time series • Using linear prediction to make forecasts • since seasonal ARIMA model is linear model • based on the minimum mean square error (MMSE) • Useful to specify the probability limits of a given prediction algorithm • new call can be blocked if actual arrivals are continuously greater than predicted traffic value  obtaining the traffic prediction based on upper probability limit after • adding a bias u to the minimum mean square error forecast

14. Algorithm B: Procedure to predict traffic of a given upper-bound call blocking probability Step 1: Determine the value of u from the QoS requirement e.g. call blocking probability Step 2: From u, determine the value of u Step 3: Determine the time granularity and the step-parameter h Step 4: Use Algorithms A to construct a seasonal ARIMA models to fit the traffic trace. Step 5: Predict the next value of the time series using h-step minimum mean square error forecast. Step 6: Obtain the predicted traffic by adding a bias u i.e.

15. Feasibility study Experiments of proposed algorithms on modeling and prediction using real traffic trace • measured from the GSM net of China Mobile Tianjin • we have original hourly traffic trace from 0:00 June 1, 2001 (Friday) to 0:00 April 27, 2002 (Saturday), a total of 330 days • accumulating the traffic in each day to obtain the daily traffic trace for the same 330 days ** using the previous 300 day data trace to do modeling and forecast next 30 day values • comparing the forecasted value with original value to evaluate the performance of the prediction algorithms

16. Fig. 1 Original traces of daily traffic Abscissa represents the accumulated time length, and unit is day y-axis represents the sample of traffic and unit is Erlang Fig. 2 Original traces of hourly traffic Abscissa represents the accumulated time length, and unit is hour y-axis represents the sample of traffic and unit is Erlang Feasibility study ----Analyzing actual GSM traffic

17. From Fig. 3, we can see that: A peak occurs at about 0.14 getting the period 1/0.14=7 in accordance with the actual situation A second peak occurs at about 0.28, because of the asymmetry of network traffic in the seven days period A third peak occurs at about 0.42 due to the traffic on Saturday and Sunday is far below the traffic in workdays Fig. 3 Periodogram based on daily trace abscissa represents frequency, unit is 1/day y-axis represents energy Feasibility study ----Analyzing actual GSM traffic on daily granularity

18. Form Fig. 4, we can see that: main frequency is about 0.042 getting the period 1/0.042=24 there are also second and third harmonics. another main frequency at 0.006 with second and third harmonics. this corresponds to the periodicity of 168 i.e. one week. Thus, the hourly traffic shows two periodicities of 24 (one day) and 168 (one week) Fig. 4 Periodogram based on hourly trace abscissa represents frequency, unit is 1/hour y-axis represents energy Feasibility study ----Analyzing actual GSM traffic on hourly granulariy

19. Feasibility study ----Building Seasonal ARIMA Model for Actual GSM Traffic • From Fig.1,We notice: • the GSM traffic increases linearly over time • during long holidays • e.g.Chinese new year and October 1st national day • we see a dramatic drop in traffic. • These dips has effect on our predictions • Before building model for actual traffic trace, we preprocess the two traces • use the average of corresponding date of the week and time of day during the period preceding and following to replace the dip in the corresponding time interval values • use Algorithm A to process the two traces.

20. Feasibility study ----Traffic Prediction for Actual GSM Traffic • Using the model built above to forecast • using the daily and hourly traffic of 300 days to forecast the values of the next 30 days • also showing the upper probability 98% limit • using adjusted traffic prediction • correspond to a bias u= 2t(1) • Fig. 5 and Fig. 6 show these result respectively

21. Feasibility study ----Traffic Prediction for Actual GSM Traffic • Fig.5 Forecast of daily traffic trace

22. Feasibility study ----Traffic Prediction for Actual GSM Traffic • Fig.6 Forecast of hourly traffic trace

23. Feasibility study ---- Comparing the Forecasts with the Actual Traffic Traces The comparison was repeated with many prediction experiments on the actual measured GSM traces of China Mobile of Tianjin. • the relative error between forecasting values and actual values • all less than 0.02 • lend a strong support to our prediction method • our experiments showed that the seasonal ARIMA model is a good traffic model capable of capturing the properties of real traffic. Have used fractional ARIMA models to describe the GSM trace and forecast traffic • did not find any improvement • attribute to the weakness of the long-range dependency in the traffic characteristics

24. Conclusion Studying a method of fitting multiplicative seasonal ARIMA models to measured wireless traffic traces. • gave a general expression of the multiplicative ARIMA models with two periodicities • proposed a practical algorithm for building seasonal ARIMA model. • proposed an adjusted traffic prediction method using seasonal ARIMA model. Future work • extend the seasonal ARIMA model based traffic prediction to network design, management, planning and optimization.