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Traffic concept, measurements, statistics
Lecturer: Dmitri A. Moltchanov
E-mail: moltchan@cs.tut.fi
http://www.cs.tut.fi/˜moltchan/modsim/
http://www.cs.tut.fi/kurssit/TLT-2706/
Traffic modeling D.Moltchanov, TUT, 2005
OUTLINE:
• Traffic concept;
• Traffic measurements;
• Step-by-step traffic modeling procedure;
• Points of interest in traffic modeling;
• Observations from Internet traffic measurements;
• What statistics to capture;
• Estimation of the statistics;
• Choosing a candidate model;
• Fitting parameters of the model;
• Testing for accuracy of approximation;
• Example of the traffic modeling procedure.
Lecture: Traffic concept, measurements, models 2
Traffic modeling D.Moltchanov, TUT, 2005
1. Importance of the trafficThe cost of any telecommunication system depends on the amount of traffic.
The main aims when planning a telecommunication system is to:
• adjust the amount of equipment so that the required quality if satisfied;
• use the equipment as efficient as possible;
• keep costs as small as possible.
Teletraffic theory deals with developing methods suitable for:
• optimization of the structure of the network to satisfy a given traffic;
• adjustment of the amount of equipment to satisfy a given traffic.
Since these both tasks depends upon the amount of traffic we have to define:
• what is the traffic?
• what is the unit of traffic?
We distinguish between circuit-switched and packet switched networks.
Lecture: Traffic concept, measurements, models 3
Traffic modeling D.Moltchanov, TUT, 2005
2. Traffic in circuit-switched networksDefinition: traffic intensity in a pool of resources at t is the number of busy resources at t.
Mean traffic intensity is given by:
Y (T ) =1
T
∫ T
0
n(t)dt, (1)
• where n(t) denotes the number of occupied resources (servers) at the time t.
Note the following:
• a pool of resources may be any group of certain servers (i.e. number of trunks);
• statistical moments of the traffic intensity may be calculated for a given period of time T ;
• traffic intensity is usually referred to as average traffic intensity.
Definition: carried traffic AC is the traffic carried by the group of servers during interval T .
Lecture: Traffic concept, measurements, models 4
Traffic modeling D.Moltchanov, TUT, 2005
n(t), number of busy trunks
t, time
instantaneous traffic intensity
average traffic intensity
Figure 1: Illustration of the average traffic intensity.
The following is important:
• the carried traffic cannot exceed the number of trunks;
• a single trunk can at most can carry one Erlang of the traffic!
The total traffic carried in a period of time T is called a traffic volume (Erlang-hours).
Lecture: Traffic concept, measurements, models 5
Traffic modeling D.Moltchanov, TUT, 2005
Note the following:
• carried traffic AC is often proportional to income of a network operator;
• losses are usually due to inability to carry all traffic!
Definition: offered traffic A:
• traffic which would be carried if no arrivals were rejected due to a lack of capacity;
• this concept is usually used in theoretical studies:
How to compute offered traffic:
A = λ × s. (2)
• λ: mean number of arrivals per a time unit;
• s: mean service time of arrival.
Lecture: Traffic concept, measurements, models 6
Traffic modeling D.Moltchanov, TUT, 2005
So far we defined two different types of traffic. These are:
• offered traffic A;
• carried traffic AC .
• volume of these traffics (A, AC) may not be equal to each other.
Lost (rejected, blocked) traffic: difference between offered traffic and carried traffic:
AL = A − AC . (3)
• the value of lost traffic is reduced by increasing the capacity of the system;
• when the capacity of the system tends to infinity AC → A.
Example: arrival intensity is 10 arrs/m.; mean service time is 2 minutes:
A = 10 · 2 = 20(Erlangs). (4)
Lecture: Traffic concept, measurements, models 7
Traffic modeling D.Moltchanov, TUT, 2005
2.1. Traffic variations
Traffic in circuit-switched networks varies according to activity of subscribers:
• traffic is generated by single sources – subscribers;
• subscribers are assumed to be independent.
Measurements have shown that traffic is characterized by two major components:
• stochastic component:
– random generation of calls by subscribers.
• deterministic component:
– nearly deterministic variability of number of calls over days, weeks, months and even years
– cause: subscribers’ needs to make more calls in a certain period of time.
Variations in traffic can be divided into:
• variations in service times;
• variations in number of calls.
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Figure 2: Average number of voice calls: 10 workdays averages, taken from V.B. Iversen.
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Figure 3: Average service times for voice calls: taken from V.B. Iversen.
Lecture: Traffic concept, measurements, models 10
Traffic modeling D.Moltchanov, TUT, 2005
Peaks in average number of calls and service time usually depends on:
• whether the exchange is located in residential, rural, or business area;
• on what traffic we look at.
Deterministic nature: traffic patterns looks very similar for a different days:
• traffic patterns are similar during week-days;
• traffic patterns are similar during week-end days;
• traffic patterns are different during week-days and week-end days.
Natural question:
• when the peak number of calls occurs?
• is this peak the same for each day?
Lecture: Traffic concept, measurements, models 11
Traffic modeling D.Moltchanov, TUT, 2005
Generally, deterministic variations in the traffic can be divided to:
• 24 hours variations:
– as those we considered previously.
• weekly variations:
– highest traffic: Monday, then on Friday, Tuesday, Wednesday, Thursday, Saturday, Sunday.
• year variations:
– for example: there is a very low traffic in vacation times (July in Finland).
• large scale variation:
– traffic increases depending on technology development and economic state of the society.
The following is important:
• we considered a traditional PSTN traffic;
• other traffic types or other circuit-switched networks have their own patterns and variations.
– dial-up Internet via modems;
– voice calls in GSM/IS-95/UMTS mobile networks.
Lecture: Traffic concept, measurements, models 12
Traffic modeling D.Moltchanov, TUT, 2005
Figure 4: Average number of modem calls: single day, taken from V.B. Iversen, year 1999.
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Figure 5: Average service times for modem calls: taken from V.B. Iversen.
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2.2. The concept of busy hour
Time consistent busy hour (TCBH):
• time period of 60 minutes during which, measured on a long time, the highest traffic occurs.
Note the following:
• the highest traffic may not occur during the same time every day!
time0 4 8 12 16 20 24
calls/minute
time0 4 8 12 16 20 24
calls/minute
Lecture: Traffic concept, measurements, models 15
Traffic modeling D.Moltchanov, TUT, 2005
2.3. Blocking concept
Circuit-switched telecommunications systems:
• are dimensioned so that subscribers are sharing the expensive equipment:
– never dimensioned so that all subscribers can connect at the same time;
– equipment which is separate for each subscriber should be made as cheap as possible.
• there is a concentration from the subscribers towards exchange....
custo
mers
Exchange
...
to domestic concentrator
...to international concentrator
Figure 6: Sketch of the telephone exchange.
Lecture: Traffic concept, measurements, models 16
Traffic modeling D.Moltchanov, TUT, 2005
What are usual dimensioning rules applied:
• about 5 − 8% of subscribers should be able to make domestic calls at the same time;
– note that each phone is usually used 10 − 16% of the time.
• about 1% of subscribers should be able to make international calls at the same time.
How it is made and what it gives:
• statistical multiplexing at the call level;
• subscriber feels that he has an unrestricted access to all resources.
There should be some problems:
• resources are shared with many others;
• it is possible that a subscriber cannot establish a call.
Lecture: Traffic concept, measurements, models 17
Traffic modeling D.Moltchanov, TUT, 2005
When it is not possible to to establish a call it:
• has to wait;
• has to be blocked.
Depending on how system operates we distinguish between:
• loss systems: arrival is lost when there are insufficient resources in the system;
• waiting systems: arrival waits when there are insufficient resources in the system;
• mixed loss-waiting systems: depending on arrival it can wait of get lost.
Networks performance measures in loss systems can be expressed using:
• call congestion B: fraction of call attempts that observes all servers busy;
• time congestion E: fraction of time when all servers are busy;
• traffic congestion C: the fraction of the offered traffic that is not carried.
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3. Traffic in packet-switched networksIn data networks we talk about transmission needs:
• any packet can be of s units in length (bits or bytes);
• any link is characterized by a capacity φ (units per second).
Then the service time for a customer (so-called transmission time) is:
s
φ. (5)
Utilization ρ of the link is:
ρ =λs
φ, 0 < ρ < 1. (6)
• λ is arrival rate of packets per time unit.
Lecture: Traffic concept, measurements, models 19
Traffic modeling D.Moltchanov, TUT, 2005
4. Traffic measurementsTo provide quantitative analysis of telecommunication system we have to:
• provide adequate traffic model:
– determine important statistical parameters of input traffic:
∗ measure traffic patterns;
∗ compute statistical parameters of the patterns.
– match these parameters using appropriate traffic model.
• provide model of the service process;
• carry out analysis of the system under different conditions.
Any traffic measurement is characterized by the following three parameters:
• period of measurement;
• type of measurement;
• parameters under consideration.
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Traffic modeling D.Moltchanov, TUT, 2005
4.1. Why do we need traffic measurements?
There are four main reasons why network traffic measurements are very useful:
• network troubleshooting:
– malfunctioning equipment may disrupt the operation of the network;
– examples: broadcast storm, illegal packet sizes, incorrect addresses, security attacks;
– measurements: may help to locate this equipment.
• protocol/application debugging:
– developers want to test a new, improved version of protocol/application;
– measurements: may reveal ’hidden problems’ of the protocol/applications.
• traffic characterization:
– what is the current workload, what are the future trends?;
– measurements: are required to answer these questions.
• performance evaluation:
– what is the performance of the router, application?
– measurements: are required to characterize the workload.
Lecture: Traffic concept, measurements, models 21
Traffic modeling D.Moltchanov, TUT, 2005
4.2. Methods of measurements
Traffic measurements can be implemented using the following operations:
• observing number of events:
– collecting a number of events over a constant time intervals;
– it corresponds to number representation of arrival process;
• observing time intervals:
– collecting data about the lengths of time intervals between events;
– this corresponds to an interval representation of arrival process.
Using these operations we may obtain any characteristic of a traffic process to:
• apply these characteristics to develop a traffic model;
• directly to dimension a system under consideration.
Traffic measuring methods can be divided into two major categories:
• continuous measuring methods: measuring equipment is activated at the instant of the event;
• discrete measuring methods.
Lecture: Traffic concept, measurements, models 22
Traffic modeling D.Moltchanov, TUT, 2005
4.3. Discrete measurements
Discrete measurement (so-called scanning method):
• time points are chosen;
• measuring equipment tests whether there have been changes at the measuring time points;
• time points are usually equally separated;
• events between two time points are considered as happened together.
time
time
points of measutements
1 1 2 1 2 1 1
Figure 7: Discrete measurements.
Lecture: Traffic concept, measurements, models 23
Traffic modeling D.Moltchanov, TUT, 2005
4.4. Active and passive measurements
Whether we monitor real network traffic or some kind of ’artificial’ traffic:
• active traffic measurements;
• passive traffic measurements.
Active measurements:
• packets are generated by a tool to probe the network and measure characteristics:
– ping: tool to estimate network latency to a particular destination in the Internet;
– tracert: tool to determine routing paths;
– pathchar: tool to estimate link capacities and latencies along the Internet path.
Passive measurements:
• network monitor is used to observe and record traffic on an operational network;
• most free measurement tools fall into this category:
– tcpdump;
– Etherial.
Lecture: Traffic concept, measurements, models 24
Traffic modeling D.Moltchanov, TUT, 2005
4.5. Software and hardware-based measurements tools
Depending on implementation we distinguish between:
• hardware-based traffic measurement tools;
• software-based traffic measurement tools.
Hardware-based measurement tools:
• equipment (device) with specific functionality;
• often referred to us as traffic analyzers;
• expensive: depends on the number of network interfaces, storage capacity, analysis capabilities;
• usually provide on-line statistical traffic analysis.
Software-based measurement tools:
• specific programs developed for collection and analysis of data;
• are not so expensive sometimes providing the same functionality;
• some examples: tcpdump, Etherial, ping;
• non-specific examples: web servers, proxies, firewalls, providing log files.
Lecture: Traffic concept, measurements, models 25
Traffic modeling D.Moltchanov, TUT, 2005
Figure 8: An example of the main window of Etherial with captured trace.
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5. Step-by-step traffic modeling procedureStep-by-step procedure:
• determine the level of interest;
• measure traffic at the point of interest;
• decide what statistics should be captured;
• estimate statistics of traffic observations:
• choose a candidate model;
• fit parameters of the model;
• test accuracy of the model.
We will be dealing with:
• traffic in packet-switched networks;
• different levels of aggregation.
Lecture: Traffic concept, measurements, models 27
Traffic modeling D.Moltchanov, TUT, 2005
6. Level of interest for traffic modelingTraffic can be represented:
• at the session level:
– request for downloading files from ftp server;
– request for downloading pages from www server.
• at the packet level.
Which level to choose:
• depends on particular task;
General notes:
• session level:
– usually claimed for follow Poisson process;
– reality might be quite different!
• packet level: any behavior should be expected.
Lecture: Traffic concept, measurements, models 28
Traffic modeling D.Moltchanov, TUT, 2005
7. Points of interest in traffic modelingYou have to take into account:
• where you are asked to model the traffic (evaluate performance)?
customer side network side
2
3
1
Figure 9: Points at which traffic is usually measured and modeled.
Lecture: Traffic concept, measurements, models 29
Traffic modeling D.Moltchanov, TUT, 2005
7.1. Point 1: particular application:
We distinguish between:
• voice application;
• video application;
• data transfers:
– ftp information;
– http information;
– ssh information.
What is important: properties of transport layer protocol and application:
• UDP: no specific pattern:
– does not affect much properties of application;
– you may model traffic of application only.
• TCP: very specific pattern:
– affect data transmission;
– should be taken into account.
Lecture: Traffic concept, measurements, models 30
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Co
ng
estio
nw
ind
ow
,M
SS
s
time
1
4
8
16
18
TCP Tahoe
TCP Reno
Figure 10: TCP traffic pattern: TCP Reno and TCP Tahoe.
• how much traffic does the application have?
• ftp: large files; ssh, e-mail, http: large and small transfers.
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7.2. Point 2: aggregated traffic from a number of applications
We distinguish between:
• heterogenous applications;
• homogenous applications.
customer side
voip
voip
voip
voip
customer side
voip
video
ftp
voip
Figure 11: Homogenous and heterogenous traffic aggregates.
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Traffic modeling D.Moltchanov, TUT, 2005
7.3. Point 3: aggregated network traffic
What is that:
• aggregation of a large number of flows.
access router
access router
access router
backbone router
Figure 12: Aggregated backbone traffic.
• may have quite sophisticated properties;
• practically, cannot be obtained as superposition of individual flows.
Lecture: Traffic concept, measurements, models 33
Traffic modeling D.Moltchanov, TUT, 2005
8. Observations from the Internet traffic measurementsThe most important fact: Internet traffic changes in time.
Recent observations, trends and facts on the Internet traffic:
• TCP accounts for most of the packet traffic in the Internet;
• traffic flows are bidirectional, but often asymmetric;
• most TCP sessions are short-lived;
• the packet arrival process in the Internet is not Poisson;
• the session arrival process may be approximated by Poisson distribution;
• packet sizes are bimodally distributed;
• packet traffic is non-uniformly distributed;
• aggregate network traffic may have multi-fractal nature;
• Internet traffic continues to changes.
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Traffic modeling D.Moltchanov, TUT, 2005
8.1. Domination of TCP
TCP accounts for most of the packet traffic in the Internet:
• beginning of 90th:
– it was firstly observed that TCP dominates.
• middle of 90th:
– introduction of multimedia services;
– development of RTP, RTCP, RTSP...;
– UDP share was expected to grow.
• beginning on 2000:
– TCP still dominant protocol;
– multimedia content also extensively use TCP.
Reasons of TCP dominance:
• multimedia is usually place of web pages;
• availability of TCP.
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Traffic modeling D.Moltchanov, TUT, 2005
8.2. Bidirectional asymmetric traffic flows
Traffic flows are bidirectional, but often asymmetric:
• bidirectional exchange of data:
– ftp, http, ssh, e-mail, etc.
• asymmetric traffic pattern:
– ftp, http, ssh, e-mail are all request-response based protocols.
• what are the current trends:
– p2p applications may generate bidirectional asymmetric traffic;
– p2p applications: napster, kazaa, etc.
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8.3. Short-lived TCP sessions
Most TCP sessions are short-lived:
• almost 90% of TCP connections exchange fewer than 10Kbytes of data in few seconds:
– WWW service: request - response;
– http v1.0: separate connection for an object on a page;
– http v1.1: single connection for a page;
– most pages and objects are less than 10Kbytes in length.
• what the effect:
– heavy-tailed distribution of session sizes.
– heavy-tail: a lot of frequencies corresponding to large histogram bins;
– reasons for heavy-tail: most of the sessions are small, some a big (ftp).
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Traffic modeling D.Moltchanov, TUT, 2005
8.4. Packet arrivals are not homogenous Poisson
The packet arrival process in the Internet is not homogenous Poisson:
• common belief was:
– aggregated traffic is Poisson (or at least Markovian) in nature;
– a lot of studies have been made with Poisson assumption.
• reality:
– arrival process is not homogenous Poisson;
– interarrival times are at least correlated;
– packet arrival process may not be event covariance stationary;
– there can be so-called packet ’clumps’ or ’batches’
• result: packet arrival process is far from common assumptions.
Lecture: Traffic concept, measurements, models 38
Traffic modeling D.Moltchanov, TUT, 2005
8.5. Session arrivals are Poisson
The session arrival process may be approximated by Poisson distribution:
• what are the reasons:
– there are a lot of users getting access to a certain site;
– users can be assume the be independent;
– situation is similar to telephone network where Poisson assumption holds.
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Traffic modeling D.Moltchanov, TUT, 2005
8.6. Special distribution of packet sizes
Packet sizes are bimodally distributed:
• around 50% of packets are as large as possible:
– these are TCP data packets;
– recall, it is determined by MTU of Ethernet: 1500 bytes;
– around 50% of packets are 1500 bytes in length.
• around 40% of packets are as small as possible:
– these are TCP ACKs;
– recall, it is determined by headers of TCP (20 bytes), IP (20 bytes): 40 bytes;
– around 40% of packets are 40 bytes in length.
• around 10% of packet lengths are uniformly distributed between 40 and 1500;
• additional peaks: fragmentation of IP packets.
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8.7. Non-uniform distribution of the traffic
Note: it is related to flows in the network!
Packet traffic is non-uniformly distributed:
• around 90% of traffic is between 10% of nodes:
– explanation: client-server configuration of most services.
• this property may change:
– p2p applications.
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8.8. Unknown patterns of the packet traffic
What was suggested over decades:
• 80th: Poisson nature of the aggregated packet traffic:
– common agreement: this assumption is no longer valid!
• 90th: self-similar nature of the aggregated traffic:
– the most respected hypotheses today;
– seems a little bit strange.
• 2000: is it simply non-stationary?
– probably the correct answer:
– small timescales: stationary;
– long timescales: non-stationary.
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8.9. Changing nature of Internet traffic
Internet traffic continuous to changes:
• what applications dominated over decades:
– 80 − 95: e-mail, remote access;
– 95−: WWW, large file transfers;
– predictions: 2010: p2p, WWW.
• how to deal with:
– you cannot rely upon ’old measurements’;
– new measurements are required.
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8.10. Find more about Internet traffic
Where you may find more about current Internet traffic:
• Internet traffic archive: http://ita.ee.lbl.gov;
• Internet traffic report: http://www.internettrafficreport.com;
• National laboratory for applied network research (NLANR): http://www.nlanr.net;
• NLANR measurement and operations analysis team (MOAT): http://moat.nlanr.net;
• National Internet measurement infrastructure (NIMI): http://www.ncne.nlanr.net/nimi;
• tcpdump measurements software: http://www.tcpdump.org;
• etherial software: http://www.etherial.com;
• research papers:
– free search engine: http://researchindex.org/;
– free search engine: http://scholar.google.com/;
– ieee: http://ieeexplore.ieee.org/Xplore/guesthome.jsp.
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9. What statistics to captureGeneral answer is not straightforward:
• what are the aims of traffic modeling:
– just propose a new, better traffic model?
– carry out performance evaluation?
– what kind of performance evaluation simulation/analytic?
• how close you are going to describe the traffic:
– trade-off between accuracy and complexity!
– is it sufficient just to get basic ideas?
– is there interest in precise parameters?
• what statistics are important:
– mean, variance, distribution, ACF?
– you can never say before you get results;
– you can never say before you capture a certain parameter.
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Traffic modeling D.Moltchanov, TUT, 2005
9.1. Description of stochastic processes
Note the following for process {S(n), n = 0, 1, . . . }:• full information is given by N -dimensional distribution:
S(t1, s1, t2, s2, . . . ) = Pr{S(t1) ≤ s1, S(t2) ≤ s2, . . . }. (7)
– impossible to deal with;
– never considered in teletraffic theory.
• in most general case we operate with:
– empirical distribution (in terms of the histogram);
– autocorrelation function.
• note the following:
– distribution and ACF: does not fully describe arbitrary process;
– distribution and ACF: gives full description for processes with Normal distribution.
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Traffic modeling D.Moltchanov, TUT, 2005
fi,E(�)
0 5 10 15 20 25 30 350
0.028
0.055
0.083
0.11
i�
(a) Empirical distribution
KY(i)
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
i, lag
(b) NACF
Figure 13: Empirical distribution and NACF with approximations.
It is advisable to construct both:
• you get a picture how distribution behaves;
• you get a picture how ACF behaves.
Lecture: Traffic concept, measurements, models 47
Traffic modeling D.Moltchanov, TUT, 2005
fX(x)
x
normal
exponential, hyperexponential
gamma, beta, Erlang, Weibull
Pareto
Figure 14: Forms of distribution.
Note: form may significantly affect results of performance analysis!
Lecture: Traffic concept, measurements, models 48
Traffic modeling D.Moltchanov, TUT, 2005
If distribution is hard to capture, capture moments:
• 1st moment: mean;
• 2nd moment: variance;
• 3rd moment: skewness;
• 4th moment: kurtosis;
• higher moments.
If ACF is hard to capture, capture:
• lag-1 ACF only;
• short-range behavior of ACF;
• long-range behaviour of ACF.
Lecture: Traffic concept, measurements, models 49
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9.2. Importance of moments
fX(x)
xmean
variance
Figure 15: Mean and variance of distribution.
• mean: measure of central tendency;
• variance: measure of dispersion.
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fX(x)
x
sk = 0sk < 0 sk > 0
Figure 16: Skewness of the distribution.
• skewness: normalized third central moment (for symmetric sk = 0);
• skewness: measure of the lopsidedness of the distribution.
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fX(x)
x
kurt 1
kurt 2
kurt 2 > kurt 2
Figure 17: Kurtosis of the distribution.
• kurtosis: normalized fourth central moment - 3;
• kurtosis: whether the distribution is tall and skinny or short and squat compared to normal.
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Notes on fitting moments:
• if 4 first moments are fitted you may expect fair approximation of histogram;
• sometimes it is easier to fit histogram than more than 2 moments.
fX(x)
x
Figure 18: Distribution to be approximated.
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fX(x)
x
Figure 19: Fitting mean and variance results in a number of forms of a distribution.
• different skewness;
• different length on fX(x) axis.
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fX(x)
x
Figure 20: Fitting mean, variance and skewness limits a number of forms of a distribution.
We still have differences:
• different length on fX(x) axis.
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fX(x)
x
Figure 21: Fitting mean, variance, skewness and kurtosis may result in a desired distribution.
• if we fit kurtosis we are sure that the length on fX(x) axis is the same;
• 4 moments are fairly enough to get good fitting.
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9.3. Importance of the ACF
Note the following:
• autocorrelation affects results of performance analysis:
The effect may not be straightforward
• autocorrelation in packet arrivals usually leads to more losses;
• autocorrelation in bit errors of the wireless channel may lead to less losses.
ACF manifests itself in two effects:
• short-range dependence:
– more losses and larger delays compared to no autocorrelation.
• long-range dependence:
– more losses and larger delays compared to short-range dependent models.
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K(i)
i
short-range dependence
long-range dependence
i < 10-20 i > 50
Figure 22: Long and short-range dependence.
• short-range dependence: K(i) = 0 already for some i < 20 ∼ 30;
• long-range dependence: K(i) �= 0, i > 50.
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i
single exponential/geometric component
power decay (long-range dependence)
two exponential/geometric components
K(i)
Figure 23: Common forms of the normalized ACF in traffic models.
• exponential/geometric decay: short-range dependence;
• power decay: long-range dependence.
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i
mixture of exponential terms
1
( )k
i
j j
j
K i � ��
� �
K(i)
Figure 24: Power decay can be approximated by sum of exponentials (to the some extent).
• some models have such a kind of ACF;
• example: Markov modulated processes.
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Note on long-range dependence:
• may exists as a consequence of non-stationarity!
• you have to be pretty sure in stationarity:
– anomaly behavior of ACF may be the sign of non-stationarity;
– example: ACF is not strictly decreasing.
– example: ACF is slowly decreasing.
The following is important:
• observations of stationary traffic usually have strictly decreasing ACF;
• however, note the following:
– some anomalies can be due to outbursts that may not be important;
– some anomalies are important.
• you have to have intuition!
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Y(i)
0 25 50 75 1005
0
5
10
15
i, timeKY(i)
0 25 50 75 1001
0
1
i, lag
Y(i)
0 25 50 75 1000
5
10
15
i, timeKY(i)
0 25 50 75 1002
1
0
1
i, lag
Figure 25: Traces and NACFs of non-stationary observations.
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Y(i)
0 25 50 75 1002
0
2
4
i, timeKY(i)
0 25 50 75 1000.5
0
0.5
1
i, lag
Y(i)
0 25 50 75 10010
0
10
20
i, timeKY(i)
0 25 50 75 1000.5
0
0.5
1
i, lag
Figure 26: Traces and NACFs of stationary observations.
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9.4. What statistics are important?
Note the following:
• mean value:
– must be captured;
• variance:
– must be captured;
– one may use standard deviation or coefficient of variation instead.
• lag-1 ACF:
– was found to be important;
– may have unexpected effect.
• structure of the ACF:
– sometimes may affect significantly (e.g. long-range dependence).
• histogram of relative frequencies:
– captures all moments of one-dimensional distribution;
– required when you have to be pretty sure.
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9.5. Example of importance of parameters
We consider: SBP+SPP/D/1/K queuing system:
• SBP: switched Bernoulli process;
• SPP: switched Poisson process;
• arrivals of SBP have priority over SPP;
• constant service time of one slot;
• K waiting positions;
• parameters of interest: pdf of waiting time fL(l), pdf of losses fQ(q).
Application: frame transmission process over wireless channels:
• SBP: frame error process;
• SPP: frame arrival process;
• limited buffer of the mobile terminal;
• single wireless channel.
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Figure 27: Effect of the lag-1 autocorrelation coefficient of SPP.
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Figure 28: Effect of the variance of SPP.
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Figure 29: Effect of the form of the distribution of SPP.
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Figure 30: Effect of the lag-1 autocorrelation coefficient of SBP.
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Figure 31: Effect of the variance of SBP.
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9.6. Common matching schemes
Common matching:
• mean and variance:
– usually easy to do;
– used to get mean performance parameters.
• mean, variance and lag-1 ACF:
– there are a number of models and algorithms;
– relatively easy to do.
• mean and ACF:
– there are a number of models and algorithms;
– sometimes not easy to do.
• histogram:
– one may look for analytical distribution;
– usually easy to do using discrete distribution.
• histogram and ACF.
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9.7. Classes of models and characteristics
Classes of models:
• renewal class of models:
– distribution can be arbitrary;
– ACF is zero for all lags: no autocorrelation.
• autoregressive class:
– distribution is normal;
– ACF is a sum of exponential/geometric terms.
• Markov-modulated models:
– distribution can be arbitrary;
– ACF is a sum of exponential/geometric terms.
• models with self similar properties:
– distribution can be either normal (FBM) or arbitrary (F-ARIMA);
– ACF non-zero for large lags: long-range dependence.
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9.8. Receipts
You may use the following when you are to capture:
• first two moments:
– Erlang, hyperexponential, exponential distributions;
– approximation by discrete distribution: p1, p2, . . . , pk such that∑
i pi = 1.
• first m moments:
– special case of phase-type distribution;
– approximation by discrete distribution.
• first two moments and lag-1 of ACF:
– Markov modulated processes;
– autoregressive processes.
• first two moments and ACF:
– Markov modulated processes;
– autoregressive processes.
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10. Estimating statistics of traffic observationsWhat is special in teletraffic:
• usually we have enough statistics to estimate;
• that is, we can capture for days...
• see, Internet Traffic Archive: http://ita.ee.lbl.gov/
What does it mean:
• recall, unbiased and consistent estimate for variance:
σ2[X] =1
N − 1
N∑i=1
(Xi − m)2. (8)
– we may not care about 1/(N − 1) and just use 1/N when N is sufficiently large.
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General questions you have to answer at this step:
• is the traffic process ergodic:
– practically, there are no means to test for ergodicity;
– look for reasons for ergodicity of the traffic process;
– ergodic: a single sufficiently long observation can be further used;
– not ergodic: a set of observations must be obtained.
• are there reasons for stationarity of the traffic process?
– practically, there are no means to test for stationarity;
• if stationary (first- or second-order):
– estimate important statistics;
• if not stationary:
– try to change representation of observations;
– another representation may be stationary.
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11. Choosing a candidate modelInput information
• parameters of the traffic that have to be captured;
• a set of traffic models.
What you have to know:
• traffic models and their properties;
• analytical tractability of models:
– simulation: any model is suitable;
– analytical: only tractable models are suitable.
Examples:
• analytically tractable: renewal models, Markovian models;
• analytically intractable: most non-Markovian models.
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11.1. Example of the problem
Assume we have:
• observations of RV X that is defined on [0,∞);
• we have to capture first two moments of observations:
E[X], C2
< 1. (9)
What one may guess:
• Erlang distribution (E2):
– defined on [0,∞);
– has C2 < 1.
• shifted exponential distribution:
– defined on [d,∞).
– has C2 < 1.
• what to choose?
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fX(x)
xd
Shifted exp: fX(x) = be-b (x-d)
E2: fX(x) = bxe-bx
Figure 32: pdfs of shifted exponential and E2 distributions.
Conclusion:
• shifted exponential does not satisfy implicit requirement X ∈ [0,∞)
• we choose Erlang distribution;
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12. Self-similar trafficNote the following:
• it is a common belief nowadays:
– may not be true.
• a way to deal with aggregated network traffic:
– may not be the only approach.
Simple example of deterministic self-similar behavior: Cantor set:
• take a certain subspace of the space (assume rectangle [0, 1][0, 1] in R2);
• scale its size by 1/3 and place in corners on initial subspace;
• do the same for each obtained rectangle;
• continue...
Note: self-similar structures are sometimes called fractals.
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take 2 of length 1/3 and place in corners
arrivals
Figure 33: Illustration of the 2D Cantor set and 1D Cantor set as ON/OFF traffic.
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arrivals
Figure 34: Weighted Cantor set with weights 2/3 and 1/3 (we preserve the whole weight).
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Figure 35: Coast of England is an example of fractals: length scales with a timescale.
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Figure 36: Stochastic self-similarity in the network traffic.
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12.1. Definition of self-similarity
Assume we are working with:
• {Y (t), t = 0, 1, . . . }: cumulative arrival process:
– may not be stationary!
• {X(t), t = 0, 1, . . . }, X(t) = Y (t + 1) − Y (t): process of increments:
– first order difference process: X(t) = ∇Y (t) (recall ARIMA);
– this process is should be covariance stationary with zero mean.
Y(t) X(t)
0 2 4 6 8 1040
20
0
20
40
60
0 2 4 6 8 1050
0
50
t, time t, time
Figure 37: Example of processes {Y (t), t = 0, 1, . . . } and {X(t), t = 0, 1, . . . }.
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Define aggregated averaged process of {X(t), t = 0, 1, . . . } as:
X(m)(n) =1
m(Xnm−m+1 + · · · + Xnm) =
1
m
mn∑t=m(n−1)+1
X(t). (10)
t
t
t
{X(n),n = 0,1,..}
{X(5)
(n),n = 5,10,..}
{X(10)
(n),n = 10,20,..}
Figure 38: Averaging the process {X(t), t = 0, 1, . . . }.
Note: X(m)(n) is the sample mean of the sequence (Xnm−m+1 + · · · + Xnm).
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A process {X(t), t = 0, 1, . . . } is exactly second-order self-similar if:
γ(k) =σ2
2[(k + 1)2H − 2k2H + (k − 1)2H ], k = 1, 2, . . . , (11)
• γ(k) is ACF of {X(t), t = 0, 1, . . . };• such structure implies that γ(k) = γ(m)(k) for all m ≥ 1;
• H is called Hurst parameter;
• for self-similar processes 0.5 < H < 1.
A process {X(t), t = 0, 1, . . . } is asymptotically second-order self-similar if:
γ(m)(k) = limm→∞
σ2
2[(k + 1)2H − 2k2H + (k − 1)2H ], k = 1, 2, . . . . (12)
• γ(m)(k) is ACF of {X(m)(n), n = 0, 1, . . . }.Note the following:
• exactly self-similar: correlation structure is strictly preserved over timescales;
• asymptotically self-similar: correlation structure is preserved under time-aggregation.
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12.2. Self-similarity in terms of distribution
Consider the following:
• continuous time process {Y (t), t ∈ };
Process {Y (t), t ∈ } is self-similar with certain 0 < H < 1 if:
Y (t) = a−HY (at), a > 0, t ≥ 0. (13)
• it means: {Y (t), t ∈ } and {Y (at), t ∈ } normalized by a−H have the same distribution;
• we usually interpret {Y (t), t ∈ } as cumulative arrival function.
What is important:
• {Y (t), t ∈ } cannot be stationary du to normalization factor a−H !
• its increment process can be (does not mean it must) covariance stationary.
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12.3. Long range dependence
Determine variance of {X(m)(n), n = 0, 1, . . . } via variance of {X(t), t = 0, 1, . . . }:
σ2[X(m)] = E2
[1
m(Xnm−m+1 + · · · + Xnm)
]−(
E1
m[Xnm−m+1 + · · · + Xnm]
)2
=
=σ2[X]
m+
2
m2
m∑k=1
(m − k)r(k) =
= σ2[X]
[1 + 2
m∑k=1
(1 − k
m
)r(k)
]m−1. (14)
• r(k), k = 1, 2, . . . is the NACF of {X(t), t = 0, 1, . . . }.Note: if process {X(n), n = 0, 1, . . . } is uncorrelated, {X(n)(m), n = 0, 1, . . . } is uncorrelated.
r(k) = 0, k = 1, 2, . . . , σ2[X(m)] = D[X]m−1. (15)
Note: if process {X(n), n = 0, 1, . . . } is correlated, then for large m we have:
σ2[X(m)] = σ2[X]
(2
m∑k=1
r(k)
)m−1. (16)
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Consider the case when r(k) �= 0 and∑∞
k=−∞ r(k) < ∞:
σ2[X(m)] = σ2[X]Cm−1, (17)
• C is some constant;
• sample variances decay to zero as fact as m−1.
Models that MAY have such behavior:
• Markovian models;
• ARMA(p, q) and its special cases (MA(q), AR(p)).
Where sample variance may decay at slower rate than m−1:
• {X(t), t = 0, 1, . . . } is self-similar process;
• {X(t), t = 0, 1, . . . } is non-stationary process.
Where practically: aggregated traffic.
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How to model slower than m−1 decay?
• one should have decay of σ2[X(m)] proportional to m−a, a ∈ (0, 1), why?
– if a = 1 we have limited serial correlation or no correlation at all;
– if a = 0 the process degenerates to constant case.
• it requires the sum in expression for σ2[X(m)] must be proportional to m1−a:
m∑k=1
r(k) = Cm1−a, a ∈ (0, 1). (18)
When α < 1, previous implies that the ACF decays so slowly that it is not summable:
∞∑k=−∞
r(k) → ∞. (19)
An example of such ACF is a power decaying ACF:
r(k) ∼ Ck−α, α ∈ (0, 1). (20)
• in this case we say that ACF decays as power function.
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KX(i)
0 20 40 60 80 1000.2
0
0.2
0.4
0.6
0.8
empirical ACF
error: +2/sqrt(n)
error: -2/sqrt(n)
i, time
KX(i)
0 20 40 60 80 1000.2
0
0.2
0.4
0.6
0.8
empirical ACF
error: +2/sqrt(n)
error: -2/sqrt(n)
i, time
Figure 39: NACFs of short-range dependent and long-range dependent processes.
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12.4. Values of H
We have the following possibilities:
• H = 0.5: process is completely uncorrelated,∑∞
k=−∞ r(k) is finite;
• 0 < H < 0.5:∑∞
k=−∞ r(k) = 0 that is rarely observed in applications;
• H = 1 leads to r(k) = 1, k = 1, 2, . . . there is linear dependence in the series;
• H > 1: prohibited due to stationarity exposed on {X(t), t = 0, 1, . . . }.
Self-similarity and long-range dependence:
• there are long-range dependent processes that are not self-similar;
• there are self-similar processes that are not long-range dependent;
• asymptotic self-similar:
– self-similarity leads to long-range dependence;
– long-range dependence leads to self-similarity.
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12.5. Heavy-tailed distributions
Observe the following:
• heavy-tailed distribution is when CPDF is:
Pr{Z > z} ≈ x−a. (21)
– 0 < a < 2 is some constant.
– tails of distribution decreases slowly.
• short-tailed distribution is when CPDF is:
Pr{Z > z} ≈ e−z. (22)
– tails of distribution decreases quickly.
Mote the following:
• 0 < a < 1: infinite mean, infinite variance;
• 1 < a < 2: infinite variance.
We are interested: 1 < a < 2 when only variance is infinite.
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Common heavy-tailed distribution is Pareto:
Pr{Z ≤ z} = 1 −(a
x
)a
, b ≤ x, (23)
• 0 < a < 2 is the scale parameter;
• b is the location parameter;
• mean is given by:
E[Z] =ab
a − 1. (24)
• variance is infinite.
Note the following:
• gamma distribution has subexponential tail but has a finite variance;
• weibull distribution has subexponential tail but has a finite variance.
Note: main characteristic is high variability:
• may take very large values with non-negligible probabilities;
• sample: a lot of small values and some values are extremely big.
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fY(y)
50 8.33 33.33 75 116.67 158.33 2000
0.01
0.02
0.03
0.04
Weibull distributiony
fY(y)
50 8.33 33.33 75 116.67 158.33 2000
0.01
0.02
0.03
0.04
Normal distribution
Exponential distributiony
Figure 40: Examples of short-tailed and heavy-tailed distributions.
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12.6. Heavy-tails and predictability
Assume the following:
• we have duration of a lifetime of a certain thing with RV Z;
• time is discrete t = 0, 1, . . . ;
• {A(t), t = 0, 1, . . . } is an indicator process:
– A(t) = 1 something is in;
– A(t) = 0 something already expired.
• we are interested in: something still persists in the future given that it persists now:
U(τ) = Pr{A(τ + 1) = 1|A(τ) = 1}, 1 ≤ t ≤ τ. (25)
We can express U(τ) as:
U(τ) = 1 − Pr{Z = τ}Pr{Z ≥ τ} . (26)
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Assume short-tailed distribution in the form Pr{Z > x} ≈ c1e−c2x:
U(τ) = 1 − Pr{Z = τ}Pr{Z ≥ τ} ≈
≈ 1 − c1e−c2τ − c1e
−c2(τ+1)
c1e−c2τ= 1 − (1 − e−c2) = e−c2 . (27)
• prediction is the same for all τ !
• recall exponential distribution.
Assume long-tailed distribution:
U(τ) = 1 − Pr{Z = τ}Pr{Z ≥ τ} ≈
≈ 1 − cτ−a − c(τ + 1)−a
cτ−a= 1 −
(1 −
(τ
τ + 1
)a)=
(τ
τ + 1
)a
. (28)
• prediction is different for all τ !
• when τ → ∞, U(τ) → 1!
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12.7. Heavy-tails as a cause of LRD
Let us introduce the following:
• FBM: fractional Brownian motion:
– non-stationary Gaussian process with 0 < H < 1.
• FGN: fractional Brownian noise.
– stationary increment process of FBM with 0 < H < 1;
– distribution is also Gaussian.
Note that when H = 0.5:
• FBM reduces to Brownian motion:
– also non-stationary process.
• FGN reduces to Gaussian noise:
– stationary process;
– completely uncorrelated.
Note: for Gaussian processes distributional and second-order self-similarity are equivalent!
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Consider the following:
• N independent on/off sources Xi(t), i = 1, 2, . . . , N ;
• aggregated process SN(t) =∑N
i=1 X(i)(t).
Figure 41: Examples of on/off sources and their aggregation.
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Consider cumulative process YN(Tt):
YN(Tt) =
∫ Tt
0
(N∑
i=1
X(i)(s)
)ds. (29)
• T > 0 is a scale factor.
If the following holds:
Pr{τon > x} ≈ cx−a, x → ∞, 1 < a < 2, c > 0, (30)
For large T and N process YN(Tt) behaves as:
E[τon]
E[τoff ] + E[τon]NTt + CN1/2THBH(t) (31)
• H = (3 − a)/2 is Hurst parameter;
• BH(t) is FBM with Hurst parameter H;
• C > 0 is a constant depending on distributions of τon and τoff ;
• distribution of the off times can be arbitrary (short-tailed or heavy-tailed).
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What we can say about YN(Tt):
• long range dependent (0.5 < H < 1) if 1 < a < 2: on intervals are heavy-tailed:
– distribution of off intervals does not matter.
• long range dependent (0.5 < H < 1) if 1 < a < 2: off intervals are heavy-tailed:
– distribution of on intervals does not matter.
• if off and on intervals are short-tailed it is short-range dependent:
– heavy-tails are required to have self-similarity.
Practice:
• file sizes distribution has heavy-tail;
• generator: web site with downloading capabilities.
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12.8. Estimating H: variance-time plot
What property we are going to use:
• self-similar process has a slowly decaying variances with increasing of m:
σ2[X](m) = σ2[X]m−β. (32)
• where H = 1 − β/2.
You have to do the following:
• determine several m (it is better to use m = 1, 10, 100, 1000, . . . );
• compute log10 σ2[X(m)/m] and log10 m for each m;
• ignore small values of m;
• fit a least squares fit line through a rest of resulting points in the plane;
• estimate the Hurst parameters as:
H = 1 − β
2(33)
– where β is the value of estimated asymptotic slope.
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0 1 2 3 4 55
4
3
2
1
0
log10(�2[X(m)/m])
log10(m)
KX(i)
0 20 40 60 80 1000.2
0
0.2
0.4
0.6
0.8
empirical ACF
error: +2/sqrt(n)
error: -2/sqrt(n)
i, time
Figure 42: Example of variance-time plot and ACF of self-similar process (H ≈ 0.82).
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KX(i)
0 20 40 60 80 1000.2
0
0.2
0.4
0.6
0.8
empirical ACF
error: +2/sqrt(n)
error: -2/sqrt(n)
i, time0 1 2 3 4 5
4
2.4
0.8
0.8
2.4
4
log10(�2[X(m)/m])
log10(m)
Figure 43: Example of variance-time plot and ACF of a process without self-similarity (H ≈ 0.48).
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12.9. Estimating H: R/S statistics
What we use here:
• R/S (rescaled/adjusted range) statistics:
– practically, measure of decline of variance.
• R/S statistics for self-similar process:
limn→∞
E[R(n)/S(n)] ≈ cnH . (34)
• estimate of H: slope of log-log plot of R/S statistics.
How to compute R/S statistics: for each value d = 10, 20, 30, . . . do:
• compute K points of R/S statistics R(ti, d)/S(ti, d);
• starting points must satisfy (ti − 1) + d ≤ N ;
• estimate H as the slope of log-log graph of R/S statistics.
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Figure 44: Algorithm to compute R/S statistics.
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log(R/S)
logd
Figure 45: Estimating Hurst parameter using R/S statistics (H ≈ 0.9).
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12.10. Caution!
Note the following:
• self-similar processes:
– cumulative process {Y (t), t = 0, 1, . . . } may not be stationary;
– increment process {X(t), t = 0, 1, . . . }, Xt = Yt+1 − Y (y) must be stationary;
– note: increment process is just fist difference process.
• some traffic seems to be non-stationary at all:
– deterministic variations: hourly, daily (recall PSTN traffic)!
Stationarity of the process:
• in real traffic depends of the timescale at which traffic is measured;
• recent hypothesis:
– short timescales: stationary behavior;
– long timescales: non-stationary behavior.
Another problem: distinguishing between self-similarity and non-stationarity.
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Illustrative example:
• we generate process segments of which have different mean and variance;
• is this process self-similar?
– NO: both {Y (t), t = 0, 1, . . . } and X(t) = Y (t + 1) − Y (t) are non-stationary!
– can you say that observing only left figure?
Figure 46: First 5E5 and 1E4 observations of non-stationary trace.
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What people usually do:
• estimate Hurst parameter or NACF;
• incorrect conclusion about self-similarity and non-stationarity!
KX(i)
0 20 40 60 80 1000.2
0
0.2
0.4
0.6
0.8
empirical ACF
error: +2/sqrt(n)
error: -2/sqrt(n)
i, time0 1 2 3 4 5
3
1.8
0.6
0.6
1.8
3
log10(�2[X(m)/m])
log10(m)
Figure 47: NACF and variance-time plot for non-stationary trace (H ≈ 0.76!!!).
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13. Fit parameters of modelsNote the following:
• no general algorithms;
• algorithms are specific for a class of models;
• there could be more than a single algorithm for a chosen model;
• there could be no algorithm for a chosen model.
General procedure:
• determine parameters of the model:
– these parameters must completely characterize a model;
– not only parameters you are going to capture.
• derive equation relating measuring statistics and parameters:
– note that some parameters can be free.
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14. Tests accuracy of the modelNote the following:
• sometimes is not needed:
– when you exactly match parameters.
• sometimes is needed:
– when approximation is used at a certain step.
Tests:
• compare distribution and empirical data:
– χ2 test;
– Smirnov’s test.
• compare autocorrelations:
– just visually comparing;
– Q-Q graph.
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15. ExampleWhat we have to do:
• propose a model of the aggregated traffic;
• capture histogram and ACF as close as possible;
• model should be further used in simulation study.
network side
1..
.
Figure 48: The point at which traffic is to be modeled.
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15.1. Measuring the traffic at the point of interest
We carried out two sufficiently long measurements:
• reality: 2000 observations may not sufficiently long!
• disclaimer: these observations do not represent real traffic of any kind!
Y(i)
0 500 1000 1500 20000
6
12
18
24
30
i
(a) Experiment 1
Y(i)
0 500 1000 1500 20000
6
12
18
24
30
i
(b) Experiment 2
Figure 49: Traffic observations obtained in two experiments.
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15.2. Estimating statistics
What you may guess?
• are they stationary ergodic?
• what kind of distribution these traces come from?
• is the same approximating distribution the same for both traces?
• which model to use to capture statistics?
What to do to get basic knowledge:
• compute statistics;
• analyze statistics to identify properties.
What statistics we usually start in MODELING:
• histogram of relative frequencies;
• normalized autocorrelations function.
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Histograms looks like as follows:
• are they really normal?
• testing using χ2: yes with level of significance α = 0.1!
fi,E(�)
0 5 10 15 20 25 30 350
0.028
0.055
0.083
0.11
i�
(a) Experiment 1
fi,E(�)
0 5 10 15 20 25 30 350
0.028
0.055
0.083
0.11
i�
(b) Experiment 2
Figure 50: Histograms of presented traces with normal approximations.
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NACFs look like as follows:
• we have no anomalies;
• such NACFs are inherent for stationary processes.
KY(i)
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
i, lag
(a) Experiment 1
KY(i)
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
i, lag
(b) Experiment 2
Figure 51: Normalized ACFs of presented traces with geometric approximations.
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15.3. Choosing a candidate model
What are our observations:
• observations are stationary ergodic: assumption;
• empirical distribution is normal;
• ACF is distributed according to a single exponential/geometric term.
Which model to guess:
• autoregressive model or order 1: AR(1):
Y (n) = φ0 + φ1Y (n − 1) + ε(n), n = 1, 2, . . . , (35)
– φ0 and φ1 are some parameters, ε ∼ N(0, σ2);
– marginal distribution is normal, NACF K(i) = φi1, i = 0, 1, . . . .
• Markov modulated model:
– may approximate Normal distribution;
– NACF is a sum of exponential/geometric terms.
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15.4. Fitting AR(1) model
What we have to do: estimate the following:
φ0, φ1, σ2[ε]. (36)
Properties of AR(1) model:
• if AR(1) process is covariance stationary we have:
E[Y ] = µY , σ2[Y ] = γY (0), Cov(Y0, Yi) = γY (i). (37)
• µY , σ2[Y ] and γY (i) of AR(1) are related to φ0, φ1 and σ2[ε] as
µY =φ0
1 − φ1
, σ2[Y ] =σ2[ε]
1 − φ21
, γY (i) = φi1γY (0). (38)
• φ0, φ1 and σ2[ε] are related to statistics of observations as:
φ1 = KX(1), φ0 = µX(1 − φ1), σ2[ε] = σ2[X](1 − φ21), (39)
– KX(1), µX and σ2[X] are the lag-1 value of ACF, mean and variance of observations.
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15.5. Testing for accuracy of fitting
Why we need it:
• we were asked to capture histogram and NACF;
• we fit only first two moments and lag-1 value of ACF!
Is there a case when we need not to do testing:
• assume we were to capture only µX , σ2[X] and KX(1);
• since we explicitly fit them, AR(1) model exactly represents them.
What allows us to assume we get fair approximation:
• AR(1) model is characterized by only three parameters that all were matched;
• distribution of AR(1) model is normal;
• NACF of AF(1) models is geometrically distributed.
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The first step:
• generate trace from the model:
– for simplicity you may generate exactly the same amount of observation.
Test histograms using χ2 or Smirnov’s test for two samples:
• first sample: empirical observations;
• second sample: generated from model;
• hypotheses to be tested:
– H0: distributions of two samples are the same;
– H1: distributions of both samples are different.
Test NACFs:
• you may carry out visual test by plotting NACFs of both samples;
• you may test for significant correlation using Box-Ljiung statistics.
Lecture: Traffic concept, measurements, models 121
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