A Practice Guide of Predicting Resource Consumption in a Web Server

There is a report that business revenues affected by application performance problems account for overall business revenues by up to 9% when applications suffer from performance issues such as performance degradation [1]. Even worse, about one third (32 percent) of consumers will abandon slow sites between one and five seconds and some of them will go on to tell others about their experience which will prevent more consumers to visit these sites by [2]. One second page load delay means that it will decrease 11% page views, customer satisfaction, and 7% revenue [3].

Several studies [4], [5] have reported that one of the causes of the performance degradation and unplanned software outages is the software aging phenomena. Software aging is a phenomenon observed in software that the long running software system suffers from abnormal state, performance degradation, even hang, and failure. The reasons of software aging are consumption of operating system resources, data corruption and round error accumulation, which could be accompanied by memory leaks [6], unterminated threads, data fragment, unreleased file locks, unreleased database connections and so on. Software aging problems are not only observed in telecommunication systems [7], Web servers [8], enterprise clusters [9], online transaction processing (OLTP) systems [10], and spacecraft systems [11], but also in military system [5] which means loss of lives. In order to counteract these problems, Huang et al. [12] proposed the technique of software rejuvenation that involves occasionally stopping the software application, removing the cumulative error environments and then rebooting the application in a clean environment. Unlike other technologies, rejuvenation technology is a proactive manner, which means the accumulated errors, or defragments can be removed through this process before occurrence of service degradation and failure. Unlike downtime caused by unexpected failure, the downtime by software rejuvenation can be performed at the discretion of the user or administrator, e.g., at midnight or idle time.

The key issue for software rejuvenation is how to find an optimal time, however, the choice of optimal time is dependent on accuracy of time to failure or prediction accuracy of resource consumption, which means the more the accuracy is, the better the rejuvenation is.

In this paper, we compare the effects of linear and nonlinear methods in the aspect of resource consumption predicting in a web server that is suffered from software aging. The resource usage data are collected from a typical long running software system a web server that is not subjected to artificial load, but a true load.

The contribution of this work is: the captured data from running phase are real data, which means workload is not synthetic; through comparisons between the linear and nonlinear time series methods in resource consumption, the most important thing for resource consumption predicting is how to choose a proper data set for training.

The rest of the paper is organized as follows. In Section 2, some related work on software aging is briefly reviewed. In Section 3, we review the ARIMA, ANN, and SVM modeling approaches. In Section 4, the outputs of models are compared with each other through some measurement methods. Section 5 contains the concluding remarks and discusses possible directions for future research.

There are some different methods for resource consumption forecast, which can be generally categorized as traditional statistical models and nonlinear models. Traditional statistical models including moving average, exponential smoothing, and autoregressive integrated moving average are linear in that predictions of future values are constrained to be linear functions of past observations. The Second category of models is nonlinear models. Several nonlinear models have been proposed to overcome the linear limitation of time series models, which include the threshold autoregressive (TAR) model, general autoregressive conditional heteroscedastic (GARCH), the bilinear model, chaotic dynamics, and artificial neural networks, general regression neural networks (GRNNs), support vector machines, and so on. In this section, the basic concepts and modeling process of autoregressive integrated moving average, artificial neural networks, and support vector machines are reviewed.

3.1 ARIMA model

In the 1970s, Box and Jenkins [31] proposed a set of methods for analysis, prediction and controllable method, called ARIMA or Box-Jenkins method. ARIMA model [32] is, in theory, the most general model for forecasting a time series which can be stabilized by transformations such as differencing. In fact, we can think that ARIMA model is an adjusted model of random walk and random trend: the fine adjustment consists of adding lags of the differenced series and/or lags of the forecast errors for the forecasting formula, where removing any last traces of autocorrelation from the forecast errors is needed.

In fact, an ARIMA (p, d, q) model uses autoregressive moving average model (ARMA) to fit time series which is stationary. A stationary time series means that the series has no trend in the long run.

Given a set of resource consumption data series (x₁, …, x_t, …, x_m), where t = 1, …, m, the difference transformation can be expressed as

${{\nabla }^{d}}$     (1)

Where ${{\nabla }^{d}}$ is the difference operator, d is the number of the difference, ${{x}_{t}}$and ${{x}_{t-1}}$ are the actual values of time series at time t and t-1, B is the backward shift operator..

Supposing x_t is the output of a stationary time series, ARMA (p, q) model which is a stationary ARIMA can be described by (2)

${{x}_{t}}={{\varphi }_{1}}{{x}_{t-1}}+...+{{\varphi }_{p}}{{x}_{t-p}}+...+{{\varepsilon }_{t}}-{{\theta }_{1}}{{\varepsilon }_{t-1}}-...-{{\theta }_{q}}{{\varepsilon }_{t-q}}$     (2)

Where ${{\varepsilon }_{t}}$ is a white noise which is an uncorrelated random variable with mean zero and a constant variance $\sigma _{\varepsilon }^{2}$ , p is the order of non-seasonal autoregressive part of the model, q is the order of non-seasonal moving average part of the model, $\varphi$ and $\theta$ are coefficients that satisfy stationary and invertible conditions, respectively.

Using the backward shift operator B with $B{{x}_{t}}={{x}_{t-1}}$ and $B{{\varepsilon }_{t}}={{\varepsilon }_{t-1}}$, we can describe (2) as

$\begin{align}  & \Psi (B){{x}_{t}}=\Theta (B){{\varepsilon }_{t}} \\ & \Psi (B)=(1-{{\varphi }_{1}}B-{{\varphi }_{2}}{{B}^{2}}-...-{{\varphi }_{p}}{{B}^{p}}) \\ & \Theta (B)=(1-{{\theta }_{1}}B-{{\theta }_{2}}{{B}^{2}}-...-{{\theta }_{q}}{{B}^{q}}). \\\end{align}$     (3)

When q is equal to zero, the model turns into AR(p) model.

${{x}_{t}}={{\varphi }_{1}}{{x}_{1}}+...+{{\varphi }_{p}}{{x}_{t-p}}+{{\varepsilon }_{t}}.$     (4)

When p is equal to zero, the model turns into MA(q) model.

${{x}_{t}}={{\varepsilon }_{t}}-{{\theta }_{1}}{{\varepsilon }_{t-1}}-...-{{\theta }_{q}}{{\varepsilon }_{t-q}}.$     (5)

3.2 Artificial neural network

Artificial Neural network (ANN) can be viewed as an attempt by humans to simulate the human brain function and be a stretch computing framework for modeling a broad range of nonlinear problems. The ANNs can process the information from the data in parallel and approximate a large class of functions with a high degree of accuracy over nonlinear data. In the process of model building, there is no prior assumption required. On the contrary, the trained network model is largely determined by the features of the data. Multilayer perceptron (MLP), especially with one hidden layer is one of the most widely used forms of artificial neural networks for modeling and forecasting. The model is characterized by a network of three layers of simple processing units connected by acyclic links. The relationship between the output (x_t) and the inputs (x_t−1, …, x_t−P) can be expressed as

${{x}_{t}}={{w}_{0}}+\sum\nolimits_{j=1}^{q}{{{w}_{j}}g({{w}_{0,j}}+\sum\nolimits_{i=1}^{p}{{{w}_{i,j}}{{x}_{t-i}}})+{{e}_{t}}},$     (6)

Where w_i,j (i=0, 1, 2, …, p, j=1, 2, …, q) and w_j (j=0, 1, 2, …, q) are model parameters called connection weights; p is the number of input nodes; and q is the number of hidden nodes. The sigmoid function is often used as the hidden layer transfer function, that is

$sig(x)=\frac{1}{1+\exp (-x)}.$     (7)

Hence, the ANN model implements a nonlinear functional mapping from the past observations to the future value x_t, i.e.

${{x}_{t}}=f({{x}_{t-1}},...,{{x}_{t-p}},W)+{{e}_{t}}.$     (8)

Where W is a vector of all parameters and f is a function determined by the network structure and connection weights. Therefore, the neural network is equivalent to a nonlinear autoregressive model. Note that (6) implies one output node in the output layer, which is typically used for one-step-ahead forecasting. In practice, simple network structure that has a small number of hidden nodes often works well in out-of-sample forecasting. This may be due to the over-fitting effect typically found in process of neural network modeling. It occurs when the network has too many free parameters, which allow the network to fit the training data well, but typically lead to poor generalization. In addition, it has been experimentally shown that the generalization ability begins to deteriorate when the neural network has been trained more than necessary, that is when it begins to fit the noise of the training data.

The choice of q is dependent on the characteristic of data, which has no systematic rule to decide this parameter. Also, there is no theory that can be used to decide the selection of p. Hence, experiments are often conducted to select an appropriate p as well as q.

3.3 Support vector machine

In a time series, data set can be divided into training set and testing set. A set of training data $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right), \ldots,\left(\mathrm{x}_{\mathrm{m}}, \mathrm{y}_{\mathrm{m}}\right)$ are given, where $\mathrm{x}_{\mathrm{i}} \in \mathrm{R}^{\mathrm{n}}, \mathrm{i}=1, \ldots, \mathrm{m},$ and $\mathrm{y}_{\mathrm{i}} \in \mathrm{R} .$ Each $\mathrm{y}_{\mathrm{i}}$ is the target value for the input vector $\mathrm{x}_{\mathrm{i}}$ and $\mathrm{y}_{\mathrm{i}}$ also comes from the same time series as $\mathrm{x}_{\mathrm{i}}$ comes. Generally, a time series function $\mathrm{f}(\mathrm{x})$ can be written as

$f(x)=\left\langle w,x \right\rangle +b,$      (9)

Where w and b are, respectively, the weight vector and intercept of the model for the optimal regression. Also, $\left\langle \cdot ,\cdot  \right\rangle $ means the inner product of the involved arguments.

Support vector regression (SVR) is one of SVM models, which is an application of SVMs in time series problem. SVR method tries to locate a low-dimensional hyperplane with small risk in high-dimensional feature space based on nonlinear kernel regression method. Among numerous of support vector regression methods, the most commonly used method is $\varepsilon $-SVR which finds a nonlinear hyperplane with an $\varepsilon $-insensitive band [33], [34]. In order to make the method more robust, the mapping from the input data to the target data does not need to lie rigorously inside or on the $\varepsilon $-insensitive band. The mappings outside the $\varepsilon $-insensitive band are punished and slack variables are imported for the mappings. For convenience, in the following, the term SVR is used to present $\varepsilon $-SVR. The objective function with constraints for SVR is as follows

$\begin{align}  & \underset{w,b}{\mathop \min }\,\ \ \frac{1}{2}\left\langle w,w \right\rangle +C\sum\limits_{i=1}^{m}{({{\xi }_{i}}+}{{\xi }_{i}}^{*}) \\ & \text{subject to}\ \ (\left\langle w,\phi ({{x}_{i}}) \right\rangle +b)-{{y}_{i}}\le \varepsilon +{{\xi }_{i}}, \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{y}_{i}}-(\left\langle w,\phi ({{x}_{i}}) \right\rangle +b)\le \varepsilon +{{\xi }_{i}}^{*}, \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\xi }_{i}},{{\xi }^{*}}_{i}\ge 0, \\\end{align}$      (10)

Where i=1, …, m, m is the number of training set, C is a parameter which gives a trade-off between model training error and complexity, ${{\xi }_{i}}$ and ${{\xi }_{i}}^{*}$ are slack variables which are used when the predictable value is above or below the target value more than $\varepsilon $.

One thing to note is that $\phi (x)$ is a nonlinear mapping function from the input space to a feature space. Therefore, we can rewrite the general function (9) into the derived hyperplane which is

$f(x)=\left\langle w,\phi (x) \right\rangle +b.$      (11)

To solve (10), one can introduce the Lagrangian multipliers, take partial derivatives regard to the primal variables and set the results to be zero, and turn the Lagrangian into the following Wolfe dual form

$\begin{align}  & {{\max }_{\alpha ,{{\alpha }^{*}}}}\ \ \sum\limits_{i=1}^{m}{{{y}_{i}}({{\alpha }_{i}}-{{\alpha }_{i}}^{*})}-\varepsilon \sum\limits_{i=1}^{m}{({{\alpha }_{i}}+{{\alpha }_{i}}^{*})} \\ & -\frac{1}{2}\sum\limits_{i=1}^{m}{\sum\limits_{j=1}^{m}{{{y}_{i}}({{\alpha }_{i}}^{*}-{{\alpha }_{i}})}}-\varepsilon \sum\limits_{i=1}^{m}{({{\alpha }_{j}}^{*}-{{\alpha }_{j}})K({{x}_{i}},{{x}_{j}})}, \\ & \text{subject to  }\sum\limits_{i=1}^{m}{({{\alpha }_{i}}^{*}-{{\alpha }_{i}})}=0, \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\le {{\alpha }_{i}},{{\alpha }_{i}}^{*}\le C,\ \ \ \ \ \  \\\end{align}$      (12)

where i is equal to 1, …, m, ${{\alpha }_{i}}$ and ${{\alpha }_{i}}^{*}$ are Lagrange multipliers. $K({{x}_{i}},{{x}_{j}})$ is a kernel function which represents the inner product of $\left\langle \phi ({{x}_{i}}),\phi ({{x}_{j}}) \right\rangle $.

If $0<{{\alpha }_{i}},{{\alpha }_{i}}^{*}<C$, the corresponding training sample is a free support vector.

If ${{\alpha }_{i}},{{\alpha }_{i}}^{*}=C$, the corresponding training sample is a boundary support vector.

If ${{\alpha }_{i}},{{\alpha }_{i}}^{*}=0$, the corresponding training sample is a non-support vector, which doesn’t affect the classification or regression result.

Free support vectors and boundary support vectors are called support vectors.

A widely adopted kernel function also used in this paper is the radial basis function (RBF) which is defined as follows

$k\left(x_{i}, x_{j}\right)=\left\langle\phi\left(x_{i}\right), \phi\left(x_{j}\right)\right\rangle=\exp \left(-\gamma\left\|x_{i}-x_{j}\right\|^{2}\right)$,      (13)

Where $\gamma $ is the width parameter of the RBF kernel. Now, equation (12) can be solved by Sequential Minimal Optimization (SMO) algorithm [35]. Suppose ${{\hat{\alpha }}_{i}}$ and ${{\hat{\alpha }}_{i}}^{*}$ are the optimal values by computation, the hyperplane for the time series problem can be shown as

$f(x)=\sum\limits_{i=1}^{m}{({{{\hat{\alpha }}}_{i}}^{*}-{{{\hat{\alpha }}}_{i}})}K({{x}_{i}},x)+\hat{b}.$      (14)

Software aging analysis and forecasting is an active research area over the last few decades. In this paper, we compare the performances of three models in resource consumption forecasting of a web server. In section 4, we first identify whether the system has degradation trend by using linear regression model. After that, we use the collected data to train three models and use the remainder data to test effect of the models. Through the results of available memory and heap memory prediction, it is shown that nonlinear methods do not better than linear method in some situations. Through two kinds of resource consumption, we believe that choosing a proper data set for training is more important than choosing a linear or nonlinear method.

There are some topics for future research. The relationships between different resource parameters or performance parameters need to be analyzed and new systems, such as platform of Cloud Computing, are also need be considered, when subjected to software aging.