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
According to the statistic of the China Mobile of Beijing [1], the number of mobile phone users keeps on increasing at an exponential rate. Therefore, there is an urgent need to study and understand the wireless traffic more deeply.
Traffic models have played a significant role in the design, engineering and performance evaluation of networks. In particular, time-series modeling holds a great promise as a tool for studying network traffic.
Traffic predictions have been used in various fields of networks, such as the long-range forecasting and planning of NSFNET [3], the linear prediction scheme used in the dynamic bandwidth allocation schemes for VBR video [4], and the predictive congestion control for broadband wide area networks [5].

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
The ARIMA (Auto Regressive Integrated Moving Average) model [2] is a good model to capture the behavior of the network traffic. Many variations of the ARIMA models have been broadly applied, e.g. the seasonal ARIMA model in [2].
We summarize the ARIMA process here in order to introduce the notations used in the remainder of the paper. An ARIMA(p,d,q) process is a process where d is the level of differencing, p is the autoregressive order, and q is the moving average order [2]. All three parameters are non-negative integers. In the following, let {Xt: t =..., -1, 0, 1, ...} be an ARIMA(p,d,q) process and B be the backward-shift operator, i.e. BXt = Xt-1.

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 the differencing operator defined as
Then, is the differencing operator and is called the differencing operator defined in the usual binomial expansion, i.e.
where
and  denotes the Gamma function.
An ARIMA(p,d,q) process can be described by the following relationship
where {at: t =..., -1, 0, 1, ...} is a white noise WN(0, 2) with zero mean and variance 2

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
According to the statistic of the China Mobile of Beijing [1], the number of mobile phone users keeps on increasing at an exponential rate. Therefore, there is an urgent need to study and understand the wireless traffic more deeply.
Traffic models have played a significant role in the design, engineering and performance evaluation of networks. In particular, time-series modeling holds a great promise as a tool for studying network traffic.
Traffic predictions have been used in various fields of networks, such as the long-range forecasting and planning of NSFNET [3], the linear prediction scheme used in the dynamic bandwidth allocation schemes for VBR video [4], and the predictive congestion control for broadband wide area networks [5].

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.
It is also important to forecast wireless traffic workload in the planning, design, control and management of wireless networks. Our work in this area includes the fractional ARIMA model in admission control [6], and the seasonal ARIMA model for the prediction of traffic in the dial-up access network of Chinanet-Tianjin [7]. In the work of [7], we have used only one periodicity in this multiplicative seasonal ARIMA model.

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
In general, we say that a series exhibits an s-periodic behavior, when similarities in the series occur after s basic time intervals. For example, s-periodic series in Fig. 1 that we analyze in this paper has an s of 7 days. We summarize the seasonal ARIMA model here. We give a general expression of seasonal ARIMA model with two periodicities.
The periodic effect implies that an observation for a particular time interval is related to the observations for the previous same interval. Suppose that the tth observation Xt is for a particular interval. We might be able to link this observation Xt to observations in the same previous intervals by a model of the form
(3-1)
Where are polynomials in Bs of degrees P and Q respectively similar as (2-4) and (2-5) and satisfying stationarity and invertibility conditions.
Now the error components , … in these models are in general correlated. Therefore, to take care of such relationships, we introduce an ARIMA(p,d,q) model as follows:
(3-2)

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
Substituting (3-1) and (3-2), we finally obtain a general multiplicative model with one period
(3-3)
In (3-3) the subscripts p, q, P, Q are added to distinguish the orders of the various operators. The resulting multiplicative process will be said to be of order
A similar argument can be used to obtain models with three or more periodic components to take care of multiple seasonalities. Therefore we can obtain models with two periodicities s1 and s2 as follows:
(3-4)
In (3-4) the subscripts p, q, P1, Q1, P2, Q2 are added to distinguish the orders of the various operators and s1, s2 express the different periodicities existing in the time series. The resulting multiplicative process will be said to be of order

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
Spectrum analysis is concerned with the exploration of cyclical patterns of data. Previous successful analysis [8] shows that one might uncover just a few recurring cycles of different lengths in the time series of interest. This is the base of building a seasonal model.
Since there are several known ways for fitting ARMA models, we can take advantage of this by first transferring the ARIMA problem to an ARMA problem, and then identifying the necessary parameters.
We propose the following procedure to fit a seasonal ARIMA model to the traffic trace. Explanation is provided where necessary.

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, Q1 and Q2
Step 5: Estimating all the parameters like qi and j
Step 6:Obtaining the fitted multiplicative seasonal ARIMA models from
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 [9].
Step 3: Performing differencing on Xt according to (4-1) to obtain a stationary series.
Step 4: Model identification - Determining all the orders. Since our experience shows that p, P1, P2 and q, Q1, Q2, of fitted traffic ARIMA models are usually small, we propose to begin with candidate parameter sets that have small (p, q), (P1,Q1), (P2,Q2) values such as 0, 1, or 2 but where p, P1,P2 and q ,Q1, Q2 should not be 0 simultaneously in one set. Then, we can select the best (p, q), (P1,Q1), (P2, Q2) combination according to the known model identification such as AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) [10].
Step 5: Estimating all the parameters. Use approximate maxi­mum likelihood parameter estimation methods [11] to obtain all parameters:
Step 6: Obtaining the fitted multiplicative seasonal ARIMA models from (3-4).

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
Since seasonal ARIMA models are linear models, we can use linear prediction to make forecasts based on the minimum mean square error (MMSE).
When the network planning, design, control and manage­ment are based on traffic prediction, new call can be blocked if actual arrivals are continuously greater than predicted traffic value. Therefore, it would be useful to specify the probability limits of a given prediction algorithm [13]. But unlike the upper- and lower-limits used in normal prediction techniques, we only need to calculate the upper probability limit to specify the accuracy of traffic prediction in networks because the lower limit does not contribute to any calls blocking probability. Therefore, we need to adjust prediction technique from before.
To do it, let be the adjusted h-step forecast by 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.
Algorithm B: Procedure to predict traffic of a given upper-bound call blocking probability
Step 1: From the QoS requirement of a particular network (e.g. call blocking probability), determine the value of u.
Step 2: From u, determine the value of u (e.g. from (5-9)).
Step 3: From the operating time scale of a particular network traffic-management mechanism which is based our prediction method, determine the value of the time granularity and the parameter h of the h-step traffic prediction.
Comment: Normally, we choose h = 1 from an engineering point of view.
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 from (5-8).

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
We performed experiments on real-traffic traces to study the feasibility of the proposed algorithms on modeling and prediction. We first fit a seasonal ARIMA model for a trace. Then, we evaluate the performance of the prediction algorithms.
We used real traffic traces measured from the GSM network of China Mobile of Tianjin. We have one original hourly traffic trace from 0:00 June (Friday) to 0:00 April (Saturday), a total of 330 days. Each value of the hourly traffic trace represents the sum of connection time in the past one hour. We accumulated the traffic in each day to obtain the daily traffic trace for the same 330 days. Each value of the daily traffic trace represents the sum of connection time in the past one day. Each sample of traffic is expressed in Erlang. We have used the previous 300 day data trace to do modeling and forecast next 30 day values. We compared the forecasted value with original value to evaluate the performance of the prediction algorithms.

16. 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
Fig. 1 and Fig. 2 show the original traces of the daily and hourly traffic respectively. While the daily traffic shows a periodicity of 7 (one week), the hourly traffic shows two periodicities of 24 (one day) and 168 (one week).

17. Feasibility study ---- Analyzing actual GSM traffic on daily granularity
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
The periodogram based on the daily traffic trace is shown in Fig. 3. The abscissa represents frequency and y-axis represents energy. From Fig. 3 we can see that:
·  There is a peak when the frequency is about 0.14 which is called the main frequency. From this we can infer that there is some periodicity in this network traffic and this period can be gotten from formula: 1/0.14=7, which is to say that the period of this network traffic is one week. This is in accordance with the actual situation.
·  There is a second peak when the frequency is about 0.28 which is called second harmonic. From this we can get another period which is 1/0.28=3.5. This second harmonic is formed because of the asymmetry of network traffic in the seven days period. This is owing to that the traffic of Monday and Sunday is not symmetry and so is the traffic of Friday and Saturday and so on.
·  There is a third peak when the frequency is about which is called third harmonic. This is due to the traffic on Saturday and Sunday that is far below the traffic in workdays. It forms sharp canyon in Fig. 1.

18. Feasibility study ---- Analyzing actual GSM traffic on hourly granulariy
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
The periodogram based on the hourly traffic trace is shown in Fig. 4. From Fig. 4, we can see that the main frequency is about and we can get the periodicity 1/0.042=24. That is to say that there is a periodicity of one day. There are also second and third harmonics. There is another main frequency at 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).

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.
After we have made the certain periodicities of the analyzed GSM traffic, we will then build the seasonal ARIMA model with periodicity of 7 for the daily trace, and the multiple seasonal ARIMA model with two periodicities of 24 and 168 for the hourly trace respectively.
First we preprocess the two traces: We notice over all, the GSM traffic increases linearly over time. However, during long holidays such as Chinese new year and October 1st national day, we see a dramatic drop in traffic. These dips causes our predictions to report lower then the expected values. One solution for this problem is to “patch” these dips with correlating data from before and after the holiday. We do this by taking the average of corresponding date of the week and time of day during the period preceding and following the dip. We then input these values into the dip matching the correct corresponding time interval.
Next we use step 1 ~ 6 of Algorithm A to process the two traces. The fitted seasonal ARIMA model are given in Table I. The ARIMA models of traffic traces show that these traffic traces exhibit periodicity.

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
We have done the "minimum mean square error forecast" experiments on various traces. In the following, we will use the data from the daily and hourly traffic of 300 days to forecast the values of the next 30 days using the model we have built above. The result is shown in Fig. 5 and Fig. 6. We also show the upper probability 98% limit which correspond to a bias u = 2t(1) using adjusted traffic prediction of the daily and hourly traffic trace in Fig. 5 and Fig. 6 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
We compare the forecasted values of the traffic with the actual traffic. These are shown in Fig. 5 and Fig. 6. The comparisons show that they are close. The relative error between forecasting values and actual values are all less than We therefore demonstrate the validity of such prediction method.
We have also used fractional ARIMA models to describe the GSM trace and forecast traffic, but we did not find any improvement which we attributed 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.
In this paper, we have studied a method of fitting multiplicative seasonal ARIMA models to measured traffic traces. We gave a general expression of the multiplicative ARIMA models with two periodicities and we proposed a practical algorithm for building seasonal ARIMA model. We proposed an adjusted traffic prediction method using seasonal ARIMA model. We have repeated the comparison with many prediction experiments on the actual measured GSM traces of China Mobile of Tianjin.