Fractal Analysis of Resource Exhaustion in Operating Systems:

Initial Steps

Mark Shereshevsky, Bojan CukicMark Shereshevsky, Bojan CukicJonathan Crowell

Computer Sc. and Electrical Engineering DepartmentWest Virginia University

(smark, cukic, crowell)@csee.wvu.edu

Software aging phenomenon “Software aging" phenomenon implies that the state

of the software system degrades with time. The degradation manifests itself in performance

decline (excessive paging and swapping activity etc.), possibly leading to crash/hang failures or both.

Degradation is caused, in particular, by the exhaustion of the operating system resources, such as the number of unused memory pages, the number of disk blocks available for page swapping, etc.

Earlier studies of resource exhaustion

Vaidyanathan and Trivedi describe the behavior of operating system recourses as a function of time Slope (trend) depends on the workload state of the system, Workload dynamics is modeled as semi-Markov process.

However, in many workload states the dynamics of the resources demonstrates very high variance resulting in very broad confidence intervals. The highly irregular and oscillatory behavior of the data makes any linear trend model insufficient.

Our research objectives

Investigate correlation between fractal properties of the resource data and the system’s workload

Develop fractal-based model of the resource exhaustion process Apply it to real-time operating systems.

Investigate possibility of using such model for predicting system outages and for preventive maintenance planning

Data collection

Two sets of data collected from WVU machines, plus a set of data from Duke University.

WVU1: resources realMemoryfree and freeSwapSpace on a SUN server, Data collected once per second for 488,051 seconds

(approximately 5 days and 18 hours between June 13 and June 19).

WVU2: resources realMemoryfree, freePageBytes and freeVirtualMemory from an NT server, 106 data points were collected (using a fairly simple C++

application) from June 26, 2:50:50 PM to June 27, 6:47:45 PM.

Fractal signals: theory

Most popularly known fractals are two-dimensional “beautiful pictures” (Mandelbrot, Julia sets Koch snowflake, Serpinski triangle, “ferns” etc.) exhibiting complex structure at every scale and self-similarity: zoom any fragment of it with any resolution and get a picture which resembles the whole set.

Functions may also be fractal, i.e. have fractal graphs. Such functions are non-smooth, do not have a derivative at any point.

Example of fractal function Weierstrass function: , a = 0.5, b = 3.

0

)2cos()(k

kk xbaxw

Fractal signals: applications

The study of fractal functions (i.e. those whose plots exhibit fractal properties) has proved important in several scientific domains. The examples of processes described by such functions include • the heartbeat oscillations;

• the speech signals;

• the network traffic;

• Solar wind;

• the stock index charts etc.

Holder exponent: theory and computation

Fractal functions have local singularities exhibiting wild oscillations at every scale.

Mathematical tool for studying the fractal functions: Hölder exponent measures the intensity of these oscillations

The Hölder exponent of a function f(t) at the point t is defined by the formula:

The exponent ranges from 0 to 1. It equals 1 wherever the function is smooth, and it decreases to zero as the the oscillatory singularities intensify. For Weirstrass: Hw(t) = log2/log3.

||log

|)()(|loginflim)(0

tftftH f

Using Hölder exponent to Analyze time series

The study of experimental time series with fractal structure is often done by computing and analyzing its Hölder exponent.

As we deal with a discrete sequence of measurements we cannot apply the definition of Hölder exponent directly. We have developed an algorithm for computing the Hölder exponent of the discrete time series.

Our algorithm showed very robust performance on several test examples.

Hölder exponent computations for memory free data

Two experiments performed on the realMemoryFree time series from the SUN server.  SET1: A three-hour set of data collected in early morning hours: from 2:30

AM to 5:30 AM;  SET1: Collected in the midday: from 12 noon to 3 PM.  The measurements taken once per second (10,800 points in each time

series)

Both time series show multifractal behavior. Since the Hölder exponent plots are extremely chaotic, we also computed

the histograms of the Hölder exponents, in order to visualize their distribution over the range [0, 1].

Plots of data with Hölder exponent (1)

realMemoryFree from 2:30 AM to 5:30 AM (“low workload”);

The Hölder exponents for the data.

Hölder exponent hystogram (1) The histogram (distribution) of the values of Hölder exponent (SET1: low workload).

Plots of data with Hölder exponent (2)

realMemoryFree data from SUN server (high workload);

Hölder exponents for the data sets.

Hölder exponent hystogram (2) The histogram of Hölder exponent for realMemoryFree (high workload).

Fractality of memory time series reflects workload

The histograms show clear differences in the shape of Hölder exponents distribution for these the two sets of data. The midday hours yield realMemoryFree data with very few Hölder

exponents in the upper part of the range (very few points where data are relatively smooth).

realMemoryFree time series obtained in the night hours is quite “fat” in the upper part of the range (i.e. significant portions of the signal are relatively “smooth”).

The higher workload data gives a less dispersed spectrum of Hölder exponents which are mainly concentrated in the [0.4, 0.6] range. This is sign of higher self-similarity (a perfectly self-similar signal has a

strictly constant Hölder exponent).

Future work• Recognition of the systems behavioral patterns from

fractal properties of resource data;• Correlation between fractal characteristics of the system

resource data and malfunction/failure events.• Experimental application to a real-time application

• Substantial benefit from preventive maintenance.