MATH 564 Project Report. Analysis of Desktop Virtualization Capacity with. Linear Regression Model

Transcription

1 MATH 564 Project Report Analsis of Desktop Virtualization Capacit with Linear Regression Model Hongwei Jin CWID:A Dec. 1 st, 2012

2 1. Problem Describe a) Background Information At the beginning, let me declare the terminolog of virtualization. In computing, virtualization is the creation of a virtual (rather than actual) version of something, such as a hardware platform, operating sstem (OS), storage device, or network resources. What s more, the so-called desktop virtualization is separating a personal computer desktop environment from a phsical machine using the client server model of computing. It means ou can use our computer just as normal tpe wherever and whenever ou are with kinds of devices, such as laptop, TV, tablets, and even mobiles. All our data will be saved on server safel. With the development of cloud computing, more and more companies using virtualization technolog to improve the efficienc of work and cut off devices budgets. In the business field, almost all the companies want to build their own virtualization sstem efficientl and economicall. So a direct question from the manager is that if deploing a server, how man users can it provide at the same time? b) Data Description 1 The most important data I care about is the average dela time (ms) of all users which denoted as, the number of users at the same time, which denoted and the cores of server provided for those users, which denoted as. Here I get some kind of data from our human testing under control of the server, which means I ask some people to operate their own virtual desktop at the same time with a sequence of operations. However, it takes a long time to finish a round test. So I develop software to simulate the dail operation of humans. It will speed up to get the goal. c) Objective In m prior work, I have analsis the performance of server. However, there still exist some omits. How can I prove that the software is trustable? If not, all the analsis will be fault. The report of this report will provide evidences statisticall b using the linear regression model that the software is trustable when measuring the capacit of server. 2. Analsis 1 The project is a subproject of a real project conducted b ICT (China Academ of Sciences) and Huawei. All the data are come from real measurement from m work there. [2] -1-

3 a) Method [1] The problem of this project can be viewed as reliabilit engineering, which is one of the fields of engineering statistics. Compared with linear regression model and time series models, the report will choose linear regression model to convince the goal. b) Numerical Output and Figures We can first look at the original data of testing from Table 1. Here are some comments: first, some data are not available due to either the dela time is too large, or the dela time is so small. For example, when 1 core of server supports the service with more than 7 users, it will dela more than ms, and when 3 cores of server support less than 3 users is a waste of resource. It doesn t make an sense; then a general idea from the data is that for each fixed number of cores, the dela time will increase exponentiall or quadraticall; at last, with the increase of cores, the average dela time of users will decrease exponentiall or quadraticall; what s more, range of data is quite large. The data of x2 1,2,3, 4 is come from the human test result. It reflects the pattern of benchmark of human normal operation on the server. While the data of 5 is the simulated one from software I developed. Table 1 Original data from testing result x2=1 x2=2 x2=3 x2=4 x2=5 x1= N/A N/A N/A x1= N/A N/A x1= N/A x1= x1= x1=7 N/A x1=8 N/A x1=9 N/A N/A x1=10 N/A N/A N/A x1=11 N/A N/A N/A x1=12 N/A N/A N/A x1=13 N/A N/A N/A N/A x1=14 N/A N/A N/A N/A x1=15 N/A N/A N/A N/A According to the comment above, we can do Box-Cox transformation of the original data, what more we should take the relation between core number and dela time. So we have our first model log and x x So the data will become much more reasonable for fit model. Then the data will -2-

4 become like Table 2. Table 2 Data after Box-Cox transformation Then start fit the model with simplest one with least square estimation: Here are some results from JMP. β0 β1x1 β2x2 ε Figure 1 Actual b Predicted Plot From the Figure 1, we can obtain that the predicted probabilit is smaller than Table 3 Summar of Fit RSquare RSquare Adj Root Mean Square Error Mean of Response Observations (or Sum Wgts) 28 From the Table 3, we can obtain that the RSquare is larger than 81.9%, which means almost 81.9% can be explained b this model. -3-

5 Table 4 Analsis of Variance Source DF Sum of Squares Mean Square F Ratio Prob > F Model <.0001 Error C. Total From the ANOVA table above, we can obtain that the model has large F-ration and small Probabilit. Table 5 Parameter Estimates Term Estimate Std Error t Ratio Prob> t Intercept <.0001 x <.0001 X <.0001 From the parameter estimate table, we can obtain that all the parameters are all significant. Figure 2 Residual b Predicted Plot From the residual plot, we can obtain that there are some outlier plots, which are row 6 and row 13. Obviousl, we should remove the outlier rows to fit the model again. Figure 3 Residual b Predicted Plot after removing outliers After removing the outlier plot, the model improved a litter. RSquare= , all -4-

6 the estimated parameter are significant with probabilit < The residual plot indicates that the residual are randoml distributed. Can we improve the model with interactions? So we consider a model with parameter interactions. β0 β1x1 β2x2 β3x1 x2 ε However, if we fit the model with interactive parameters, we will find that the interactive parameters have small F-ratio, large probabilit, which means there is no need to build an interactive model. Table 6 Parameter Estimates with interactives Term Estimate Std Error t Ratio Prob> t Intercept <.0001 x <.0001 x <.0001 (x )(x ) There is still no need to add self-interactive parameters. The probabilit of such parameter is large, so it is not an significant parameter in the model. Then we analsis how trustable the software simulation is? First, we get the 95% point prediction intervals of x 1 1,...,11 and 5, which can compare with the result of software estimation. Table 7 Model PI v.s. Software simulation result 95% P.I. Software simulation result From the Table 7, we can obtain that ever data of come from the software are in the range of 95% prediction interval of the model. Which means the software is almost trustable, and it can be used in the later works. c) Summar of the Result First, we declare the model again. -5-

7 e ε e β0 β1( x1 x2 1) β2x2 ( β0 β1 ) β1x1 ( β2 β1 ) x2 ε Where, means the dela time (ms), means the number of users at the same time and means cores of server provided for those users. Then through some analsis the residual and compare interactive models, we have the final model above. Even though it is a simple model, but it ma be the best model to fit the data. At last, I analsis the results of software simulation and the 95% P.I. to draw the conclusion that the software, which developed to simulations the work of human being, is trustable. I used the software to test a single server (16 cores) can contain almost 27 users at the same time, if we assume that one user can bear the dela time of 8000ms. 3. Conclusion a) Pros and Cons i. Pros This is a simple model using linear regression model. We should build up models as simple as possible. It statisticall proves that such software can be used in the sstem. For engineering statistics, it is an innovative one to analsis some kind of software is trustable in a particular work. ii. Cons There is not such mass data from the test both in human being or software. It should compare the data both in C.I. and P.I. It ma use another engineering statistic method, though I don t know. b) Improvement and Notation i. Method Choose There are other kind of method related to engineering work [3], such as DOE, qualit control, time and methods engineering, reliabilit engineering, probabilistic design, and sstem identification. It is obvious that the design of testing is dnamic, so a much more reliable method can be choose to measure it. ii. Testing data Even though there is a set of data, it is not large enough such that I can measure more patterns and characteristics from the data. Since it is conducted in other institution, I cannot do it again to improve it. -6-

8 Reference [1]. Bowerman, O Connell, Koehler. Forecasting, Time Series, and Regression (fourth edition). Thomson Brooks/Cole, [2]. Tao Jiang, Rui Hou, Lixin Zhang etc, Micro-architectural Characterization of Desktop Cloud Workloads. IISWC: San Diego, 2012 [3]. Engineering statistics, Wikipedia, -7-