Journal of Software Engineering and Applications, 2011, 4, 710-717
doi:10.4236/jsea.2011.412083 Published Online December 2011 (
Copyright © 2011 SciRes. JSEA
Power-Law Distributions in Hard Drive Behavior
Dominik Strzałka, Piotr Szurlej
Department of Distributed Systems, Rzeszów University of Technology, Rzeszów, Poland.
Received September 13th, 2011; revised October 21st, 2011; accepted November 7th, 2011.
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
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
Copyright © 2011 SciRes. JSEA
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
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
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
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
where x is a some random variable and
is a some constant that is calculated from the normaliza-
tion condition
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
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:
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
Copyright © 2011 SciRes. JSEA
Power-Law Distributions in Hard Drive Behavior713
Figure 1. Number of transferred bytes for Disk Bytes/s
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.
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-
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
Copyright © 2011 SciRes. JSEA
Power-Law Distributions in Hard Drive Behavior 714
Figure 3. Number of read bytes for Disk Bytes Read/s coun-
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 .
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.
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
Parameter Value
Mean 0.07896
Standard Deviation 1.10988
Minimum 0
Maximum 169
Skewness 53.76926
Kurtosis 5084.55531
Copyright © 2011 SciRes. JSEA
Power-Law Distributions in Hard Drive Behavior715
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.
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
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
Copyright © 2011 SciRes. JSEA
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
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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