GB2456894A - Harmonic analysis of computer workload - Google Patents

Abstract

A harmonic analysis of a historical data set for the workload for a given sampling period set is performed, the results of which are analyzed for periodicities (such as daily, weekly, monthly trends) . The harmonic analysis may be a Discrete Fourier Transformation which is used to generate an analytic function approximating the periodic demand pattern. The analytic function can be used for predicting future workload associated with current applications of the computing system. It can be used as part of an autonomous feedback system within computer system's internal workload management system for estimating future work load.

Description

-1-2456894

DESCRIPTION

A method and system for resource optimization

FIELD OF THE INVENTION

The invention relates generally to techniques for analyzing and predicting workload levels for computing systems. Specifically, the invention provides a method for analyzing workload characteristics of a computing system in a proactive way.

BACKGROUND OF THE INVENTION

Computer systems typically comprise workload management systems which manage computing resources (e.g. application servers, database connections, I/O paths etc.) by making use of a variety of workload prediction techniques. The objective of efficient workload management consists in deploying resources in such a way that one or more service objectives can be maintained satisfactorily in the face of variable workload. In a system with variable workload traffic, workload management relies heavily on being able to predict future workloads, so that in case of a workload level increase, appropriate computing resources can be allocated or, conversely, resource under-utilization can be reduced in the event of decreasing workload.

Today's computing systems contain workload management and resource optimization components that typically adjust assignment of resources to workloads based on current and average past usage of resources by the workloads. By their very nature, these algorithms deal with temporal workload changes in a purely reactive way: they respond to sudden spikes in the workload resource usage and suboptimal usage of system resources J-t until they have adjusted to the change in workload. This creates problems in areas which are sensitive to optimized system performance. As an example, a stock trading system becomes overwhelmed when the stock market opens every morning on work days. Due to the reactive nature of today's workload management systems, resources as a rule have to be statically pre-allocated to the sensitive workloads; these pre-allocated resources therefore are not available to other workloads during periods of low demand of the critical workload.

In an effort to obtain more precise predictions of workload demand, a variety of predictive methods such as the Auto-Regressive Integrated Moving Average (ARIMA) method have been developed. These methods, while being capable of furnishing highly accurate predictions, require substantial computing power. Moreover, methods for performing adaptive and robust predictions on a short timescale have been developed, such as the methods described in US 7,039,559 B2. These prediction techniques are adaptive in that they use a minimal amount of historical data to make predictions, the amount being selectable. While being capable of predicting short-term workload changes (of the order of a few seconds to a few minutes) very precisely, these methods are reactive in the sense that they use a short-or medium-term collection of past workload data to calculate a short-term prediction of workload demand.

In view of the foregoing, there is a need for a simple method for analyzing workload levels in a computing system in a systematic way so as to detect underlying temporal characteristics and to use these in predicting future workload demand in a proactive way.

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SUMNARY OF THE INVENTION

It is an objective of the invention to provide a method and a system for analyzing workload characteristics in a computing system, the method being simple to implement and requiring very few computing resources. Moreover, it is an objective of the invention to provide an efficient method for predicting future workload based on the results of this analysis.

These objectives are achieved by the features of the independent claims. The other claims and the specification disclose advantageous embodiments of the invention.

According to a first aspect of the invention, a method for analyzing workload characteristics of a computing system is provided which makes use of a harmonic analysis of past workload data in order to decide whether these data exhibit periodicities. If such periodicities exist, this information can be used in predicting future workload of the computing system.

The method comprises the steps of (1) obtaining a historical data set of workload level data spanning a sampling period, (2) performing a harmonic analysis of the historical data set, and (3) analyzing the results of the harmonic analysis for periodicities. The method enables a workload managing system (either automatic or manual) to dynamically analyze past workload characteristics for regular occurrences/patterns by means of a spectral analysis method, advantageously by performing a Discrete Fourier Transform (DFT). The method allows generating analytical functions that describe the periodical workload patterns. Such analytic functions are easy to use, their computation requiring minimal computing resources.

According to a second aspect of the invention, a method for predicting a future workload of a computing system is provided

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which makes use of an analytic function generated from a harmonic analysis of past workload data. The method comprises the steps of (1) obtaining a historical data set of workload level data spanning a sampling period, (2) performing a harmonic analysis of the historical data set, (3) generating an analytic function based on the result of the harmonic analysis and (4) using the analytic function for predicting future workload of the computing system. The method thus provides a periodic analytic function which can be used by workload management algorithms in Conjunction with current usage characteristics to anticipate periodic workload changes and tune the computing system proactively, thereby eliminating the need to reserve resources for critical workloads due to existing deficiencies in reactive algorithms. By alleviating the need of reserving resources, the method enables making more efficient use of them.

Advantageously, when analyzing the results of the harmonic analysis for periodicities, the leading coefficients of the harmonic analysis are evaluated and compared to the remaining coefficients with respect to some measure in order to determine whether they suffice to characterize the historical workload data. Specifically, the absolute values of the leading coefficients of the harmonic analysis are calculated and compared to the sum of the absolute values of all coefficients; if the sum of the absolute values of the leading coefficients exceeds a pre-determined threshold, the historical workload data is considered to display a periodicity; otherwise, the historical workload data is considered unsuitable for approximation in terms of harmonic functions.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention together with the above-mentioned and other objects and advantages may best be understood from the

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following detailed description of the embodiments, but not restricted to the embodiments, wherein is shown in: Fig. 1 a diagram showing a data set representing variations of workload demand spanning a time period of three days; Fig. 2 a diagram of the absolute values of Fourier coefficients of a Discrete Fourier Transform of the Fig. 3 the diagram of FIG. 1, showing the original data set together with an analytic approximation function; Fig. 4 a schematic method flow diagram of a preferred embodiment of the invention; Fig. 5 a detailed view of measured data representing workload demand as function of time and an analytic approximation function; Fig. 6 a schematic flow diagram of a method for calculating workload predictions using the analytic approximation function; Fig. 7 a diagram of a superposition of several periods of workload demand corresponding to the data set of FIG. 1, the period corresponding to the leading Fourier coefficient; Fig. 8 a computer system implementation of the invention.

In the drawings, like elements are referred to with equal reference numerals. The drawings are merely schematic representations, not intended to portray specific parameters of the invention. Moreover, the drawings are intended to depict only typical embodiments of the invention and therefore should not be considered as limiting the scope of the invention. I -4

DETAILED DESCRIPTION OF EXANPLE EMBODIMENTS

FIG. 1 shows a chart 1 depicting a data set 2 representing a workload demand of a computing system as a function of time over a sampling period of three days. The chart 1 was generated by using sampling data taken by a workload managing system allocating resources of a computing system running typical applications. Workload demand data 2 such as the one shown in FIG. 1 can be obtained e.g. from measurement of queue length or transaction rate, averaged over a time interval ö. The data set 2 was acquired at a sampling rate ho of one data point every ten seconds. Mathematically, the diagram 2 represents a sequence x0, x1, ... x of N discrete data points, with data point x corresponding to the workload demand at time t = n8 and the total number N of data points determined by N = In case the data set 2 displays some kind of periodicity, this information could be used by the workload management system as an indication of when to expect increases/decreases of workload demand. FIG. 4 displays a schematic method diagram 100 of a method geared at detecting periodicity in workload data 2 of a computing system and using it for workload management.

In a first step 102, workload data 2 taken at a sampling rate 1/0 and spanning a certain sampling period is input and may be subjected to a validity check to exclude undesirable data.

In step 104, a Discrete Fourier Transformation (DFT) is performed on the workload data 2. Discrete Fourier Transformation is a well-known method to transform a sequence x0, x1, ... xNl of discrete sampling data 2 from the time domain into the frequency domain: N-i I 2lLikn\ Xk = x exp -J k = 0, . -., N-i n=O where i is the imaginary unit (i2 = -1).

A simple description of this equation is that the complex numbers Xk represent the amplitude and phase of the different sinusoidal components of the input "signal" x. Various very efficient algorithms are known to perform a DFT on a set of sampling data, e.g. the Cooley/Tukey algorithm.

When applying the DFT to the sampling data of FIG. 1 one obtains a frequency spectrum 3, i.e. coefficients Xk as a function of the index k. FIG. 2 shows the absolute values IXkI of the Fourier coefficients Xk corresponding to the workload data x of FIG. 1.

Having calculated the Fourier coefficients Xk, the original sampling data x (n = 0, ... N-i) can be approximated by an analytic function x(t) using the Inverse Discrete Fourier Transformation (IDFT) in the following way: N-i 1 r /2ir /2it x(t) x = __ LRe(Xk) cos k tj + Im(xk) sin k t k= 0 n = 0, . . . , N-i As can be seen from FIG. 2, the frequency spectrum 3 is dominated by a few Fourier coefficients IXkI at low frequencies k while the majority of the remaining Fourier coefficients are quite small. In general, for original sampling data exhibiting a repetitive pattern (such as the data 2 depicted in the graph of FIG. 1), restricting the series expansion to a small number of Fourier coefficients will suffice to approximate the original * I ( diagram 2 fairly well. Thus, if the original data 2 indeed display a periodicity, a superposition of a few sin/cos terms will suffice to describe the functional behaviour of the data fairly well. If, on the other hand, a large number of sin/cos terms is needed to approximate the workload pattern, then the corresponding analytic function obtained from superposing the sin/cos terms will be difficult to handle and time-consuming to evaluate. In step 106, therefore, the Fourier coefficients are analyzed.

In order to determine how many Fourier coefficients Xk are needed to obtain a "good" approximation of the workload demand data 2, the following analysis method has been found to yield good results: Starting off with the set of Xk values characterizing the workload signal 2, the absolute values of the Fourier coefficients IXkI are sorted, the L largest absolute values IXkI are calculated, and their sum is divided by the sum of all absolute values of the coefficients XkI, yielding a ratio aL: L N- i aL = ( max ( I X I}) / ( I Xk I) j=i k=O By increasing L until the ratio a exceeds an empirically determined threshold A (i.e. until aL > A), a suitable number L0 of coefficients Xk needed to approximate the original data is found. If L0 exceeds a certain pre-determined number M to meet the a > A criteria, i.e. if the number of Fourier coefficients Xk needed to characterize the original function is too large, the input data is deemed not to be suitable for the prediction method using Fourier decomposition: If the nu.rnber L of Fourier coefficients Xk needed to characterize the temporal behavior of the data 2 exceeds the limit M, the Fourier decomposition will contain too many terms, thus detracting from the objective of presenting an easy-to-compute approximation and prediction method (step 108). -In practice, a threshold value of A 30% and maximum number of coefficients M 30 has been found to yield good result; this means that unless the sum of the absolute values of the largest 30 Fourier coefficients exceeds 30% of the sum of the absolute values of all Fourier coefficients, the data are considered "non-periodic" in the sense that a Fourier decomposition of the original data 2 will not yield useful information for workload prediction (step 110) Note that the Fourier analysis is characteristic of the historical data set {x} which depends heavily on the applications run on the computing system whose workload is being monitored. Thus, as the applications (or mix of applications) run on the computing system change, the corresponding workload data 2 will change, leading to a change in the results in the Fourier analysis.

Instead of using the absolute values XkI, the applicability criterion can be based on the squares of the Fourier coefficients Xk or on some other measure. Specifically, for data known to be periodic, the m largest coefficients Xk can be selected, with m being an empirically determined number.

If, in step 108, the workload data 2 are found to exhibit a sufficient periodicity, an analytic approximation function x' (t) is calculated, including only the Fourier components with the L largest Fourier coefficients Xk (step 112). For the workload data 2 depicted in FIG. 1, using only the ten largest Fourier coefficients Xk yields a ratio of 31% and can be considered sufficient. An analytic function x' (t) with the five largest Fourier coefficients Xk is shown in FIG. 3 together with the original workload data 2.

Note that according to Nykvist's theorem, in order to detect a periodicity of frequency f in the workload data 2, the sampling -10 -frequency 1/8 of the data 2 must be at least 2*f. Moreover, in order to detect periodicities with a long period (hours, days, weeks, ...), the data should extend over much more than this period. In practice, in order to obtain a good accuracy, workload data will be collected for many periods, e.g. for 1 -2 months, with a sampling rate of 1/6 of one data point per minute.

The amount of data to be used constitutes a compromise of data accuracy and amount of data to be handled.

Once an analytical approximation function x' (t) has been determined as an approximation of the input sampling data 2 as N-l 1 IJi \ x' (t) = LRe(Xk) cos N k tj + Im(Xk) k N k=O (with Xk = 0 if IXkI (max3=i L { IXkI} } , so that the sum effectively contains only L terms corresponding to the largest absolute values IXk of Fourier coefficients), this approximation function x' (t) can be used by a resource optimization algorithm during data processing to predict key performance (or sampling) data for the near future (step 114).

This prediction implies that at some time T (corresponding to the present) the workload demand at a future time T + T is to be calculated. FIG. 5 shows an example of a graph depicting an analytical approximation function x' (t) together with actual workload data 2'. As can be seen, while x' (t) fails to mirror the exact temporal behaviour of the data 2', it correctly predicts the overall rise as well as the decrease of workload demand 2' in the time interval shown.

An exemplary embodiment of a prediction method based on the approximation function x' (t) is depicted schematically in the method diagram of FIG. 6. In a first step 202, the actual value -Ii -x' (T) of the analytic approximation function at time P (corresponding to the present time) is calculated and subjected to a validity check; if, for example, the corresponding value x' (T) is found to be smaller than zero, this is clearly an unphysical result, since workload demand can never fall below zero. Thus, in FIG. 5, the approximation function x' (Ti) at time Ti exhibits a value smaller than zero, indicating that the approximation is not applicable at this time Ti.

If x' (T) meets the validity check, the calculated workload demand approximation function x' (T) at the present time T is compared to the current sampling (i.e. measured) value of workload demand XT at that time T (step 204). If IXT -x' (T) I exceeds a certain pre-defined threshold value Y, the approximation x' (T) is found to be inapplicable at time T (step 208) and is discarded; in the example of FIG. 5, at time T = T2 the value x' (T2) calculated from the approximation function differs considerably from the actual workload demand XT2 and thus the approximation x' (T2) is rejected as invalid. Subsequently, these steps 202 -206 are performed for the next time point T' = T+öT. Only XT is found to differ from x' (T) by less than a pre-defined threshold value in step 204, the approximation x' (T) will indeed be applied (step 206); in the example of FIG. 5, this criterion is fulfilled at time T = T3. Thus, if x'(T) meets this criterion, a forecast value x' (T+öT) for the workload demand at a future time T+T is calculated, with 6T preferably being a short time period, and x' (T+6T) is used by the workload management system as an estimate of the workload demand XT+ST at the future time T+T (step 210). If the difference Ix(T-F6T)-x(T) I exceeds a pre-deterrnjned threshold, the resource optimization algorithm may use a value k*[x(T+&r)_x(T)] as input to the optimization algorithm (where k is an empirically determined constant) and thereby gets proactive to predicted future demand. a I

-12 -Besides using the prediction method outlined in FIG. 6, other evaluation and prediction methods based on the analytical function x' (t) can be used. Specifically, due to the periodicity of the workload demand, the expected point in time and duration of a future rise in workload demand can be approximated quite reliably, thus enabling the workload management system to allocate resources accordingly.

This method is found to work extremely efficiently because the Discrete Fourier Transform can be performed on a regular basis and very efficiently with low resource utilization on sets of sample data ranging back weeks to get regular and good prediction functions for key performance indicators in a system.

In addition, the approximation functions with only a few coefficients are very low profile on resource consumption when used in resource optimization algorithms.

The method is applicable to resource optimization algorithms which use only one set of key input data as illustrated above but also for algorithms that use multiple sets of input data to perform an adaptive prediction.

Besides furnishing an analytic approximation function x' (t) which can be immediately used by the workload management system for prediction of future workloads, the method of performing a Discrete Fourier Transform on a time series of data x (n 0, N-i) reflecting workload demand of a computing system can be used in various other ways to extract workload predictions.

For example, if the workload data x are found to exhibit a leading periodicity corresponding to a frequency value k klead and thus a period t = N/klead, this periodicity klead can be factored out by periodically shifting the data x to obtain a superposition of the original data in the time interval [0, t].

This superposition can be evaluated using a variety of methods in order to detect additional characteristic features which can

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be used for future workload estimates. In the example of FIG. 1, the Discrete Fourier Transform furnishes a leading periodicity corresponding to a period t = N/klead = 1 day. By shifting data with higher periods to the time interval [0, t] and superimposing all data in this time interval [0, tI, one obtains the diagram shown in FIG. 7. By adding up the superimposed data (i.e. x + Xi+t + Xl+2t + . . .; i = 0, . . ., t) , prominent features of the workload data x appear more clearly and can be extracted, e.g. using a x2 fit. For example, by inspecting the superimposed data it is seen that all data exhibit a sharp rise 11 at time 12; while the height of the rise 11 varies, knowledge of the approximate point in time and the approximate duration of this rise 11 can be used by the workload management system to adjust resources in anticipation of this rise 11.

Referring now to FIG. 8, a computer system 300 implementation of the preferred embodiment of the present invention is shown.

Specifically, the present invention can be implemented as a computer system 300 and/or program product 326 for analyzing workload characteristics of a computing system. This allows user 340 to detect periodicities of historical workload data and to use this information in predicting future workload demand for the computing system. Specifically, user 340 may be a workload management system of the computing system which automatically/dynamically controls allocation of resources (CPU, memory, I/O) of the computing system. As depicted, computer system 300 generally comprises memory 312, input/output (I/O) interfaces 314, a central processing unit (CPU) 316, external devices/resources 318, bus 320 and data base 338. Memory 312 may comprise any known type of data storage and/or transmission media, including magnetic media, optical media, random access memory (RAN), read-only memory (ROM), a data cache, a data object etc. Moreover, memory 312 may reside at a single physical location, comprising one or more types of data storage, or can -14 -be distributed across a plurality of physical systems in various forms. CPU 316 may likewise comprise a single processing unit, or be distributed across one or more processing units in one or more locations, e.g. on a client and server.

I/O interfaces 314 may comprise any system for exchanging information from an external source. External devices 318 may comprise any known type of external device, including keyboard, mouse, voice recognition system, printer, monitor, facsimile etc. Bus 320 provides a communication link between each of the components in the computer system 300 and likewise may comprise any known type of transmission link, including electrical, optical, wireless etc. In addition, although not shown, additional components such as cache memory, communication systems, system software etc. may be incorporated into computer system 300.

Database 338 provides storage for information necessary to carry out the present invention. Such information could include, inter alia: (1) historical data of workload characteristics; (2) Fourier coefficients; (3) threshold values etc. Database 338 may include one or more storage devices, such as a magnetic disk drive or an optical disk drive. In another embodiment, database 338 includes data distributed across, for example, a local area network (LAN), wide are network (WAN) or a storage area network (SAN) (not shown in Fig. 9). Database 338 may also be configured in such a way that one of ordinary skill in the art may interpret it to include one or more storage devices. Moreover, it should be understood that database 338 could alternatively exist within computer system 300.

Stored in memory 312 is logic system 326. As depicted, logic system 126 generally includes Transformation System 328, Analysis System 330 and Analytic Function Generating System 332.

-15 -The systems shown herein carry out the functions described above.

Transformation System 328 will perform a harmonic analysis, e.g. a Discrete Fourier Transform, on historical workload data x characteristic of...; Analysis System 330 will evaluate the outcome of the Fourier transformation, notably determine periodicity. Analytic Function Generating System 332 will generate an analytic function x' (t) approximating the historical workload data. The analytic function x' (t) created by Analytic Function Generating System 332 may be used by user 340, e.g. a workload management system, for prediction of future workload.

Specifically, workload management system may utilize the analytic function x' (t) to estimate future workload in an automated process as outlined in FIG. 6.

The invention can take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment containing both hardware and software elements. In a preferred embodiment, the invention is implemented in software, which includes but is not limited to firmware, resident software, microcode, etc. Furthermore, the invention can take the form of a computer program product accessible from a computer-usable or computer readable medium providing program code for use by or in connection with a computer or any instruction execution system.

For the purposes of this description, a computer-usable or computer readable medium can be any apparatus that can contain, store, communicate, propagate, or transport the program for use by on in connection with the instruction execution system, apparatus, or device.

The medium can be an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system (or apparatus -16 - or device) or a propagation medium. Examples of a computer-readable medium include a semiconductor or solid state memory, magnetic tape, a removable computer diskette, a random access memory (RAN), a read-only memory (ROM), a rigid magnetic disk and an optical disk. Current examples of optical disks include compact disk-read-only memory (CD-RON), compact disk-read/write (CD-R/W) and DVD.

Claims (11)

-17 - CLAIMS

1. A method of analyzing workload characteristics of a computing system, the method comprising the steps of -obtaining a historical data set (2) associated with the workload for a given sampling period () (step 102); -performing a harmonic analysis of the historical data set (step 104) -analyzing the results of the harmonic analysis for periodicities (step 106)

2. The method according to claim 1, characterized in that the step of performing a harmonic analysis comprises performing a Discrete Fourier Transform.

3. The method according to claim 1, characterized in that the step of analyzing the results of the harmonic analysis for periodicities (step 106) comprises the steps of -determining the absolute values (Xkj) of the coefficients of the harmonic analysis; -determining a ratio (aL) of the sum of the number (L) absolute values of the leading coefficients and the sum of the absolute values of all coefficients; -determining whether the ratio (aL) is less than a threshold (A).

4. A method of predicting a future workload of a computing system associated with an application, the method comprising the steps of -obtaining a historical data set (2) associated with the application for a given time interval (is) (step 102); -performing a harmonic analysis of the historical data set (step 104) -18 - -generating an analytic function (x'(t)) (step 112); -using the analytic function (x'(t)) for predicting future workload associated with that application (step 114)

5. The method according to claim 4, characterized in that the step (112) of generating the analytic function comprises -determining the leading coefficients of the harmonic analysis; -using the sum of the leading terms of the harmonic analysis as the analytic function (x' (t))

6. The method according to claim 4, characterized in that the step of performing a harmonic analysis comprises performing a Discrete Fourier Transform.

7. A system for analyzing workload characteristics of a computing system, comprising -a Transformation System (328) for performing a harmonic analysis, e.g. a Discrete Fourier Transform, on historical workload data (2) of the computing system -an Analysis System (330) for evaluating the results generated by Transformation System (328).

8. A data processing system for analyzing workload characteristics of a computing system, comprising a machine readable medium containing one or more programs which when executed perform the steps of -performing a harmonic analysis of a historical data set (2) associated with a workload of the computing system (step 104); -analyzing the results of the harmonic analysis for periodicities (step 106)

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9. A data processing program stored on a computer usable medium comprising computer readable program means for execution in a data processing system comprising software analyzing workload characteristics of a computing system, wherein the computer readable program when executed on a computer causes the computer to -perform a harmonic analysis of a historical data set (2) associated with a workload of the computing system (step 104); -analyze the results of the harmonic analysis for periodicities (step 106)

10. A program product comprising a computer useable medium including a computer readable program, wherein the computer readable program when executed on a computer causes the computer to -perform a harmonic analysis of a historical data set (2) associated with a workload of a computing system (step 104); -analyze the results of the harmonic analysis for periodicities (step 106)

11. A program product comprising a computer useable medium including a computer readable program, wherein the computer readable program when executed on a computer causes the computer to -perform a harmonic analysis of a historical data set (2) associated with a workload of a computing system (step 104); -generate an analytic function (x' (t)) based on the results of the harmonic analysis to (step 112); -use the analytic function (x' (t)) for predicting future workload of the computing system (step 114).