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

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

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 the differencing operator defined as

B 1

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

Outline Introduction

Motivation Objective

Building a seasonal ARIMA model to describe a trace

Traffic Prediction Feasibility study Conclusion

IntroductionStatistics of China Mobile in Tianjin indicates th

at the number of mobile phone users is increasing at an exponential rate need proper modeling important to forecast wireless traffic work-load

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1 41 81 121 161 201 241 281 321Time scale (day)

Tra

ffic

(Erl

ang

)

Previous Work Seasonal ARIMA (Auto Regressive Integrated Movi

ng Average) model linear prediction scheme used in the dynamic bandw

idth 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 traf

fic in the dial-up access network of Chinanet-Tianjin with one periodicity.

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

Seasonal ARIMA ModelExploits the periodic effect, i.e., the relation among va

lues 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

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

Seasonal ARIMA model

sQDPqdp ,,,,

21),,(),,(),,( 222111 ss QDPQDPqdp

1 2 1 2

1 2 1 2

1 2

1 2.

s s D Ddp P P ts s

s sq Q Q t

B B B X

B B B a

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

,tDs

dt XW

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, Q1 and Q2

Step 5: Estimating all the parameters like and Step 6:Obtaining the fitted multiplicative seasonal ARIM

A models from

Algorithm A: Procedure to fit a seasonal ARIMA model to traffic trace

,tDs

dt XW

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

Step 1: Determine the value of u from the QoS requiremente.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.

Algorithm B: Procedure to predict traffic of a given upper-bound call blocking probability

utut hXhX ˆˆ

Feasibility studyExperiments 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

Feasibility study ---- Analyzing actual GSM traffic

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

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Erlan

g

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hour E

rlang

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

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0.0 0.1 0.2 0.3 0.4 0.51/day

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

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0.00 0.05 0.101/hour

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.

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 = 2t(1)

Fig. 5 and Fig. 6 show these result respectively

Feasibility study ----Traffic Prediction for Actual GSM Traffic

Fig.5 Forecast of daily traffic trace

Feasibility study ----Traffic Prediction for Actual GSM Traffic

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Erl

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98% uplimits

forecasts

trace

Fig.6 Forecast of hourly traffic trace

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hour

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ang 98%uplimits

forecasts

trace

Feasibility study ----Traffic Prediction for Actual GSM Traffic

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 val

ues 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

ConclusionStudying 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.