Power-Law Distributions in Hard Drive Behavior

doi:10.4236/jsea.2011.412083

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Journal of Software Engineering and Applications, 2011, 4, 710-717

doi:10.4236/jsea.2011.412083 Published Online December 2011 (http://www.SciRP.org/journal/jsea)

Power-Law Distributions in Hard Drive Behavior

Dominik Strzałka, Piotr Szurlej

Department of Distributed Systems, Rzeszów University of Technology, Rzeszów, Poland.

Email: [email protected], [email protected]

Received September 13th, 2011; revised October 21st, 2011; accepted November 7th, 2011.

ABSTRACT

Taking into account the fact that the computer systems, as the implementations of Turing machine, are physical devices,

the paper shows considerations in which hard drive behavior will be presented in terms of statistical mechanics. Be-

cause computer is a machine, its analysis cannot be based only on mathematical models apart of physical conditions. In

the paper it will be presented a very narrow part this problem—an analysis of hard drive behavior in the context of the

power-law distributions. We will focus only on four selected hard drive parameters, i.e. the rate of transfer bytes to or

from the disk during the read or write, the number of pending requests to the disk and the rate of read operations. Our

research was performed under the Windows operating system and this allows to make a statistical analysis for the pos-

sible occurrence of power-laws representing the lack of characteristic scale for considered processes. This property

will be confirmed in all analyzed cases. A presented study can help describing the behavior of the whole computer sys-

tem in terms of physics of computer processing.

Keywords: Power Laws, Hard Drive Behavior, Performance Monitor, Windows Operating System, Physics of

Computer Processing

1. Introduction

It can be observed that nowadays a high computing power

and the great possible efficiency are required from mod-

ern computer systems. This applies not only to special-

ized computers, but also to personal ones. As a rule, this

problem is realized by the increase of the processor clock

frequency or by the number of its cores, eventually by

the memory capacity expansion. Meanwhile, the com-

puter system exhibits the characteristics of the complex

systems and hence its parameters are not merely the sum

of the features of individual components, but the results

of both: the characteristics of these elements properties

and the phenomena (processes) that occur inside them as

well. There is no doubt that the computer system, as the

implementation of Turing machine, is a physical device,

which is governed by the laws of physics. Because it’s a

machine in which the processing can be considered as a

transformation of energy for a useful form of work (that

is for example to perform calculations), its analysis can-

not be based only on mathematical models apart of phy-

sical conditions.

The considerations and results of research presented in

this paper will be related to the wide context of systems

analysis. Taking into account the historical background it

can be said that the problem of each system analysis can

be based on two possible approaches. The first one (and

so far the most popular) is the approach based on the

rules given by Rene Descartes, who in his Discourse de

la methode [1] proposed a rule that can be related to the

ancient Romans: divide and conquer [2]. In the case of

systems analysis this require to divide each system into

parts (subsystems), analyze their behavior and basing on

this making a simple sum of part components properties

to have the picture of the whole system. But nowadays it

is known [3-5] that this approach can be used only in the

case of simple systems. On the other hand we have sys-

tems that can be considered as the complex ones. In the

case of these ones we can said, according to Aristotle [6],

that they follow a rule in which the whole is more that a

sum of its parts. For such systems it’s not only important

what kind of properties exist for each of their parts but

also how they act together [2]. In the case of computer

systems such an approach can be also used—this is sug-

gested by their complexity: for example this fact was

noted by Dijkstra [7] and Murray Gell-Mann [4]. Thus in

this paper this approach will be taken it in the case of

hard drive behavior analysis. Obviously, only a part of

this more general problem will be shown, but presented

considerations will be related to a different approach than

the so far is usually taken where the hard disks behavior

(and further performance) is related only to their physical

Power-Law Distributions in Hard Drive Behavior711

parameters, i.e., access time, interleave, seek time, rota-

tional speed and latency, buffer size, data transfer rate,

number of clusters, power consumption, etc.

We will follow a way, in which complex systems ap-

proach will be the main motive. It is based on the idea of

holism and the research is connected with such terms as:

statistical self-similarity, long-range dependencies, per-

colation, non-extensive thermodynamics, thermodynamic

non-equilibrium, power-laws, phase transitions, small wor-

lds, scale-free networks, motifs, hierarchy, etc. Taking

into account the (presented above) definition of system

the existence of power-laws in the case of hard drive

behavior will be related not only to hardware properties

but also to the processes that appear inside it during proc-

essing. To be more precise: a hard drive behavior will be

described in terms of physical phenomena [8] and their

properties but basing on the approach that from one hand

will take the hard drive physical properties (a more gen-

erally: hardware features) and processed on hard drive

tasks (a more generally: software features). This will be

done, because if in the complex systems approach we are

forced to focus on how each component behaves and acts

together with other components thus in the case of com-

puter systems, for example, we cannot separate the hard-

ware behavior form the software behavior, the network

topology from the packets flow, algorithm from the input

data, etc. Generally, we cannot separate the processed

tasks from the processing environment. They cooperate

giving us the picture of the whole system behavior.

The paper is divided into four Sections. After the In-

troduction, in Section 2 we have the description of ex-

periment, and further in Section 3 the results of research.

They present a basic properties of probability distribu-

tions with scale free property and possible consequences

of some parameters values interpretation. The paper is

crowned in Section 4. The approach presented in this pa-

per is a continuation of work [8] and can be considered

as a further evidence of complex behavior of computer

systems thus the paradigm change for their analysis is

needed.

2. Experiment

The research presented in this paper was done basing on

one personal computer, which worked under Windows 7

system. The configuration of computer was as follows:

 Dualcore Intel® Pentium® processor T2390 with

f = 1.86 GHz;

 Cache L2 level: 1 MB;

 RAM 2 GB, DDR Technology;

 Hard drive: Hitachi® Travelstar 5K250 with capac-

ity 250 GB, 5400 rpm, SATA Interface; average latency:

5.5 ms, average time of seeking: 11 ms; max data transmi-

ssion rate (buffer-host) 150 MB/s with buffer size 8 MB.

As it can be seen these parameters locate this disk

among the average ones, but we are interested in its dy-

namical behavior in terms of physics not in terms of its

technological properties. Obviously, these paremeters are

very important ones because they establish its limitations,

but in our approach a hard drive performance will be

presented in relation to the processing that is performed

in computer system.

In order to collect the necessary data for analysis there

was used a Performance Monitor, i.e., an inner monitor

that is available in Administration Panel in Windows

operating system starting from Windows ME edition. This

program (called perfmon) allows tracking many different

parts of the system basing on the idea of different count-

ers that can be configured for the computer system as a

whole and also for its particular parts and even for par-

ticular processed computer programs. It is a very inter-

esting tool in which the system administrator (but also

operating system itself) basing on Windows properties

not only can trace its actual behavior but also record dif-

ferent data sets for further statistical analysis. The time

interval can be set starting from 1 s thus during one hour

of system tracing 3600 samples can be obtained. Some of

the counters represent the average values, but most of

them show real data. One of the most important property

of this monitor is a fact that its usage almost doesn’t in-

fluence the overall systems performance and behavior,

because perfmon shows information that is normally

collected for Windows work. In other words: no matter if

perfmon works or not such data are always traced be-

cause this ensures normal, stable work of Windows oper-

ating system.

During the tracing of computer system a workload was

generated, but a short remark about this is needed. There

are two approaches for workload generation—both of

them have their own advantages and disadvantages. In

the first one it can be assumed that the workload will be

given basing on special tests (for example benchmarks)

or other techniques—such an approach allows for differ-

ent combinations of this workload generation and also

guarantees that the experiment can be repeated for dif-

ferent configurations of the system hardware level. But it

should be also noted that such a high and extreme work-

load can be considered as an artificial one, because nor-

mally during work the user doesn’t use any special ben-

chmarks or computer programs that constantly generate

such a workload. To be more precise: if we observe a

typical user that is working with the computer we can say

that she/he is using a set of applications, for example:

office applications, web browser, internet communicator,

mail program, video player, peer-to-peer system, etc.

which generate a “normal” (average) workload. Obviously,

during the work a way of each program usage is dependent

Power-Law Distributions in Hard Drive Behavior 712

on the user behavior, which isn’t repeatable (the weak-

ness of this approach) but on the other hand it seems that

it better reflects the typical workload for a computer. In

presented analysis a second approach was taken.

The research was conducted for about 3 days, using

the Performance Monitor (perfmon ver. 6.1.7600) and

also written program that collects information about the

active processes during the experiment. The data were

recorded to a file in the floating-point format and there

were collected 256,064 samples from selected counters

related to the physical hard disk drive. There were the

following counters (a list with short description):

 Disk Bytes/sec—it’s an indicator of the rate of

transfer bytes to or from the disk during the read or write

operations;

 Disk Bytes Read/sec—it’s an indicator of the rate of

transfer of bytes from the disk during read operations;

 Current Disk Queue Length—it indicates the num-

ber of pending requests to the disk at the time of data

collection. It also includes requests handled at the time of

data collection. It shows the instantaneous value, not the

average one. This counter may reflect the temporarily

high or low queue length, but if the drive is under con-

stant workload, this value is consistently high;

 Disk Reads/sec—it’s an indicator of the rate of read

operations on disk.

The above counters were taken to show a wide spec-

trum of physical phenomena that appear during process-

ing in computer system. Obviously, they show only the

behavior of hard drive but it is possible to configure

other counters that can reflect the behavior of cache, pro-

cessor, RAM, network, etc. but this is not the aim of this

paper.

3. Results of Experiment

As it was mentioned in the Introduction the problem of

hard drive behavior will be given basing on complex

systems approach. We are interested in disk behavior

description but our analysis will start from the simple

question. Is it always possible to measure everything what

we want? This seems to be a naïve question but it may

turn out that it’s one of the most important questions in

the case of results presented in this paper and, more gen-

erally, in the case of computer system performance and

behavior.

3.1. Problems with the Dimension

In the case of many aspects of engineering it seems that

everything what we see can be measured. Unfortunately,

this belief is not entirely justified—there are some objects

in our surrounding that can’t be measured. One of the

most known examples are the fractals and the problems

with their measurement is well-known in the literature

[9,10]. This allows to conclude that there are some ob-

jects that are scale-free. This term is also connected with

the idea of power-law distributions sometimes called hea-

vy tailed. There are many evidences of existence power-

laws in different parts of science, to mention a few: eco-

nomy [11], Internet and www [12,13], network traffic

[14], queue lengths [15], etc.

This distribution is given by [16]:





Px Cx





 (1)

where x is a some random variable and





min

1Cx







is a some constant that is calculated from the normaliza-

tion condition



min

min min

Px xCxx



 











 (2)

and xmin stands for the smallest x value. Because for such

distributions the average value x is given by



min min

xxPxxCx











dx (3)

it diverges for α ≤ 2, thus if we have power-law distri-

butions with such exponent they don’t have the mean

value. For α > 2 the mean value exists but if α < 3 one

cannot calculate the standard deviation. Generally, if one

wants to calculate for power-law distributions a central

moment of order m it must be: m ≥ α – 1.

There are several problems with the interpretation whe-

ther the distribution is a power-law or not but in the case

of any considerations about power-law distributions the

most important problem are the calculations of α para-

meter [17]. In this paper the method that usually is con-

sidered as one of the frequently used, i.e. maximum like-

lihood estimator will be taken [16,17]. This estimator is

given by:

1min

ˆ1ln





















 (4)

where {xi} are the n data points with xi ≥ xmin.

3.2. Disk Analysis

In this subsection for all counters mentioned above we

will show the analysis in which one will see: a graph

showing counter behavior, basic statistics, a frequency

count in log-log scale presenting possible existence of

heavy tails and calculated power-law exponent basing on

maximum likelihood estimator. This estimator is taken

because the calculations of α parameter basing on fre-

quency count give only a rough estimate.

3.2.1. Disk Bytes/s Counter

Figure 1 shows the number of bytes read from and write

Power-Law Distributions in Hard Drive Behavior713

Figure 1. Number of transferred bytes for Disk Bytes/s

counter.

to the disk in each second during the observation. As one

can see during the experiment there were moments with

the extremely high number of transferred bytes. The ba-

sic statistics for this counter are presented in the Table 1

where we can see that the average value of transferred

bytes is ≈230 kB but the standard deviation is above 1

MB indicating that the process has large fluctuations.

The skewness indicates that the distribution has long

right tail and the mass of the distribution is concentrated

on the left; the kurtosis indicates the possible existence of

heavy tailed distribution.

The presumptions presented above can be confirmed

basing on process frequency count (see Figure 2) plotted

on log-log scale. Because the possible existence of power-

law can be in the simplest (and usually rough) approach

indicated basing on the linear fit obtained by the least

mean square method, Figure 2 shows also this fit, how-

ever the answer weather the distribution has or not the

power-law behavior and what is the real value of α pa-

rameter was calculated basing on the Equation (4).

Taking into account that the xmin value for Equation (4)

will be equal to the min value form the Table 1 the esti-

mated value of α parameter is indicating that

for this process even the calculations of mean value can

be uncertain because α < 2. Because the problem of an

appropriate xmin choice is quite difficult to solve [16] it

was empirically checked that α > 2 when 44,000 < xmin <

110,000—this is the result of specific frequency count

behavior (see Figure 2) in this interval, but when xmin >

110,000 again we can see that α < 2. As we can see bas-

ing on this short analysis it can be assumed that this dis-

tribution is governed by the power-law showing a possi-

ble scale-free behavior of disk in this experiment.

ˆ1.387





3.2.2. Disk Bytes Read/s Counter

The next analyzed counter indicates the number of read

Table 1. Basic statistics for counter Disk Bytes/s.

Parameter Value

Mean 222,945.25

Standard Deviation 1.12856E6

Minimum 5055.18484

Maximum 4.32981E7

Skewness 14.73172

Kurtosis 282.45005

Figure 2. Frequency count of transferred bytes for Disk

Bytes/s counter in log-log scale.

bytes from the disk. It’s directly related to the previous

counter because Disk Bytes/s = Disk Bytes Read/s +

Disk Bytes Write/s, thus having this we can conclude

whether the read or write operations are those which

generate the workload for the disk. Figure 3 shows the

number of bytes read from the disk in each second during

the observation. As one can see during the experiment

there were moments with the extremely high number of

bytes read from the disk but also moments where bytes

weren’t read (Table 2, min value = 0). The basic statis-

tics for this counter are presented in the Table 2 where

we can see that the average value of read bytes is ≈92 kB

that is less in the case of previous counter and the stan-

dard deviation ≈720 kB—that is below in comparison to

the previous counter but again the skewness parameter

(almost the same like in Disk Bytes/s) and the high value

of kurtosis parameter (greater than for Disk Bytes/s) indi-

cate that the distribution can be governed by a po-

wer-law.

Taking into account the process frequency count (Fi-

gure 4) plotted on log-log scale and the information gi-

ven in the Table 2 again it can be supposed the free-scale

nature of this process. Basing on the Equation (4) with

the assumption that for calculation the xmin value will be

Power-Law Distributions in Hard Drive Behavior 714

Figure 3. Number of read bytes for Disk Bytes Read/s coun-

ter.

Table 2. Basic statistics for counter Disk Bytes Read/s.

Parameter Value

Mean 92184.8208

Standard Deviation 720120.1298

Minimum 0

Maximum 4.31896E7

Skewness 14.46726

Kurtosis 315.64668

Figure 4. Frequency count of transferred bytes for Disk

Bytes Read/s counter in log-log scale.

set to 1, the estimated value of α parameter is .

ˆ1.086





However, because the min value for this process is 0

there is a small problem with some calculations because

for the Equation (4) there is xmin ≠ 0. If the value of xmin

will be set close to 0, e.g. 10–3 or even 10–6, the estimated

. In the case of this counter we can see (Figure

4) that the behavior of frequency count plot is different that

on Figure 2 and this influence the values of α parameter

because they are smaller than 2 in a wide range of xmi n

values, even when xmin > 200,000 thus the existence of

power-law seems to be a reliable assumption.

ˆ1.086





3.2.3. Current Disk Queue Length

The next counter seems to have the main influence on

disk performance because it shows the number of re-

quests to be handled and its high value shows that the

drive is under constant workload. Usually it is assumed

that if a disk in computer has one spindle this queue

length shouldn’t be greater than 3 (generally no more than

the number of spindles + 2), but as one can see form Fi-

gure 5 sometimes it’s even greater than 100. Taking into

account Table 3 we can see that the average value for

this counter is less than 0.1 (standard deviation 1.1) and

it may seem that in this disk we don’t have any situations

where there are the extreme cases but the skewness and

kurtosis (especially) again indicate that if one wants to

base its opinion about this disk behavior should be careful.

In the case of this counter it should be noted that the

random variable x for Equation (1) is a discrete one. This

implies that the estimation of α parameter can be a little

bit difficult and there is no exact closed-form for α cal-

culations but an approximate expression to Equation (4)

can be used [16]:

Figure 5. Current Disk Queue Length counter.

Table 3. Basic statistics for counter Current Disk Queue

Length.

Parameter Value

Mean 0.07896

Standard Deviation 1.10988

Minimum 0

Maximum 169

Skewness 53.76926

Kurtosis 5084.55531

Power-Law Distributions in Hard Drive Behavior715

min

ˆ1ln























(5)

If in our analysis it will be assumed that xmin = 1 the

value of α parameter basing on the Equation (5) is ≈1.817

and for xmin = 2 it is ≈1.943. But the estimator given by

the Equation (5) is very sensitive on the value of xmin,

because this number also influence n. Nevertheless, this

doesn’t imply that the power-law do not exists (see Fig-

ures 6(a) and (b))—the main problem is the real value of

α parameter. In the Table 4 on can see the estimated

value of α parameter depending on the xmin value—in this

process a standard deviation can be calculated only if for

power-law exponent estimation xmin > 8.

(a)

(b)

Figure 6. (a) Frequency count of Current Disk Queue Leng-

th counter in log-log scale; (b) Frequency count of Disk Reads

counter in log-log scale.

Table 4. Dependence of α parameter on xmin value.

xmin n ˆ



1 6850 1.817

2 2476 1.943

3 1668 2.083

4 1218 2.163

5 998 2.301

6 867 2.497

7 738 2.659

8 640 2.850

9 551 3.03

10 469 3.189

3.2.4. Disk Reads/s Counter

It is not only important how many bytes are read from

the disk but also how often such reads should be done.

Because in nowadays computer systems there is a possi-

bility of many task simultaneous processing it is obvious

that appropriate amount of memory should be guaran-

teed for each process. But usually the amount of memory

is less than is needed thus the mechanism of virtual me-

mory is used. It expands the RAM memory on hard drive,

but every time when there is a need to read something

from the virtual memory a request to disk is send by op-

erating system. This obviously implies that the hard drive

will be used and we have the read operation that wasn’t

caused directly by the user (he/she doesn’t directly read

the data from the disk) but was caused by operating sys-

tem. This is especially well visible when in computer

system there is a running program, which for a long time

is in an idle state and suddenly a user wants to use it. It

can be expected that such a situation will cause a high

number of reads from disk. Obviously, sometimes the

user can create such a situation when the number of reads

will be high but if he/she doesn’t perform batch process-

ing or doesn’t copy many files from the disk such a

situation is rather rare. In our experiment it can be seen

(see Figure 7) that generally the number of read is quite

often high. There are several reasons for such a state but

here we will consider only statistic properties of this

counter (see Table 5).

Taking into account Table 6 for values of xmin < 50 it

can be seen that the process is scale-free with no mean

value and for xmin < 180 with no standard deviation.

However, again we can see the scale-free property of the

process.

4. Conclusions

In this paper there has been shown that in the case of

computer hard disk drive analysis when the workload is

generated basing on the “average” (normal) user behav-

ior it can be seen that the problem of disk performance

not always can be considered in the context of classic

Power-Law Distributions in Hard Drive Behavior 716

Figure 7. Current disk queue length counter.

Table 5. Basic statistics for counter disk reads.

Parameter Value

Mean 4.79824

Standard Deviation 33.7096

Minimum 0

Maximum 1429.88049

Skewness 12.61225

Kurtosis 238.02054

Table 6. Dependence of α parameter on xmin value.

xmin n ˆ



1 27,758 1.375

2 19,686 1.422

5 12,686 1.44

10 9981 1.488

20 8427 1.609

35 7514 1.808

50 6888 2.02

80 5240 2.396

100 4202 2.561

180 1859 3.004

statistics, because in the case of all counters shown in the

paper a possible existence of scale-free property was in-

dicated. Obviously, this was only a simple experiment

performed on one computer with unchanged configura-

tion and the further research is needed, but it might be

supposed that such phenomena can appear in many other

cases. This can be confirmed basing on further experi-

ments where the different configurations of the system

hardware will be used or different types of workload will

be generated. For example the following procedure can

be used: configure as much as possible different con-

figurations of hardware basing on some number of dif-

ferent processors (in the simplest approach only with

changing frequency, but the amount of cache memory or

the number of cores can be taken into account), for each

it will be available some amount of memory (e.g., 1 GB,

2 GB, 4 GB) and some number of hard drives with dif-

ferent parameters. For each such a configuration perform

similar experiments (they can last for a longer time in

order to obtain sets of data that will have at least 106 ob-

servations) or even experiments were the workload will

be generated by benchmarks or special programs. In each

case trace analyzed counters and calculate similar statis-

tics like in this paper. Depending on the obtained results

it will be possible to have:

 Further conclusions about hard disk drives behavior;

 Evidences of possible existence of power-laws in

the case of probability distributions;

 Measure and comparison of influence of different

systems parameters on hard disk drive behavior and vice

versa;

 Comparison of obtained statistics.

Basing on this approach it is also possible to analyze

cache, RAM or virtual memory, processor, operating sys-

tem (e.g. semaphores, threads, sections, events, etc.) be-

havior in a similar manner.

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