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A. Polynomial Fitting

1) Prediction based on Polynomial Fitting.

From our observations we found that 2nd or 3rd order polynomial fitting achieves much better (and the best) fitting effect than the 1st order (linear) fitting

2阶3阶多项式拟合效果更好

1阶线性拟合

3 (V_(T-2), V_(T-1), V_T) or 4 (V_(T-3), V_(T-2), V_(T-1), V_T)

Polynomial fitting function and then use the function value at polynomial the next time point as P_(T+1)

2) Prediction based on Similar Patterns

when the load traces vary smoothly and monotonously,多项式拟合取得好的效果，

We use the value of last turning point（拐点） as the prediction value for the current point, P_(T+1).

We think that next point will be a turning point.

If we can't find a successively increasing series we will predict using a "conservative" strategy: we use V_T as P_(T+1)

For the other cases we also choose a conservative prediction strategy(比较保守的预测) that set the increment (decrement增减) between V_T and P_(T+1) as 0.

Begin

if

if  V_{T-5}是拐点，在它之前的五个点也相继递减，

则 P_{T+1}=V_{T-5}#直接使用拐点值作为预测结果，结合单调性

else 预测P_{T+1}使用多项式拟合

if V_T是拐点

搜索V_T之前最后一个拐点，使用递增度量VT预测PT+1

if VT-1是拐点

if  最后模式改变方向在五个点之后

使用点递增两步要求最终拐点为VT递增

else PT+1=VT

if VT>VT-1>VT-2>VT-3

IF VT-3>VT-4 PT+1=VT

ELSE P_T+1=VT-(VT-VT-1)

end

IV.预测任务运行时间

A.连续时间的任务运行时间估计

t_nom：nominal time

t_exp: we want to predict the expected execution time of the task on the loaded host

at(t_exp)=t_nom

The available CPU time until time t, at(t) (t > 0), can be expressed as

V(t)为当前加载，可用时间递减同平均负载递增

B. Discrete-time Task Running Time Estimation

Suppose the sample interval is delta, the available time until the ith sample (i delta-th second) in the future can be expressed as:

the continuous-time signal which represents the available time until t then can be estimated

We substitute the predicted load signal PT+j for VT+j so that we obtain the
predicted value for ali using (7), and then we calculate the predicted discrete-time available time using (6) and its corresponding continuous-time approximation using (8).

- Choose a random value between 100 ms and 10seconds from a uniform distribution as the nominal time of a task, t_nom.

- Select a CPU load prediction system from our method and AR(16) model.

- Calculate the expected task running time texp using load predictions derived from the above selected load prediction system.

- Calculate the actual task running time t_act on the host with the load measurements in the trace.
- Calculate the prediction error for the selected load prediction system.

The results of experiments that we conducted demonstrate that this new prediction strategy outperforms the task running time prediction method using