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RFC2330 - Framework for IP Performance Metrics

王朝other·作者佚名  2008-05-31
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Network Working Group V. Paxson

Request for Comments: 2330 Lawrence Berkeley National Lab

Category: Informational G. Almes

Advanced Network & Services

J. Mahdavi

M. Mathis

Pittsburgh Supercomputer Center

May 1998

Framework for IP Performance Metrics

1. Status of this Memo

This memo provides information for the Internet community. It does

not specify an Internet standard of any kind. Distribution of this

memo is unlimited.

2. Copyright Notice

Copyright (C) The Internet Society (1998). All Rights Reserved.

Table of Contents

1. STATUS OF THIS MEMO.............................................1

2. COPYRIGHT NOTICE................................................1

3. INTRODUCTION....................................................2

4. CRITERIA FOR IP PERFORMANCE METRICS.............................3

5. TERMINOLOGY FOR PATHS AND CLOUDS................................4

6. FUNDAMENTAL CONCEPTS............................................5

6.1 Metrics......................................................5

6.2 Measurement Methodology......................................6

6.3 Measurements, Uncertainties, and Errors......................7

7. METRICS AND THE ANALYTICAL FRAMEWORK............................8

8. EMPIRICALLY SPECIFIED METRICS..................................11

9. TWO FORMS OF COMPOSITION.......................................12

9.1 Spatial Composition of Metrics..............................12

9.2 Temporal Composition of Formal Models and Empirical Metrics.13

10. ISSUES RELATED TO TIME........................................14

10.1 Clock Issues...............................................14

10.2 The Notion of "Wire Time"..................................17

11. SINGLETONS, SAMPLES, AND STATISTICS............................19

11.1 Methods of Collecting Samples..............................20

11.1.1 Poisson Sampling........................................21

11.1.2 Geometric Sampling......................................22

11.1.3 Generating Poisson Sampling Intervals...................22

11.2 Self-Consistency...........................................24

11.3 Defining Statistical Distributions.........................25

11.4 Testing For Goodness-of-Fit................................27

12. AVOIDING STOCHASTIC METRICS....................................28

13. PACKETS OF TYPE P..............................................29

14. INTERNET ADDRESSES VS. HOSTS...................................30

15. STANDARD-FORMED PACKETS........................................30

16. ACKNOWLEDGEMENTS...............................................31

17. SECURITY CONSIDERATIONS........................................31

18. APPENDIX.......................................................32

19. REFERENCES.....................................................38

20. AUTHORS' ADDRESSES.............................................39

21. FULL COPYRIGHT STATEMENT.......................................40

3. Introduction

The purpose of this memo is to define a general framework for

particular metrics to be developed by the IETF's IP Performance

Metrics effort, begun by the Benchmarking Methodology Working Group

(BMWG) of the Operational Requirements Area, and being continued by

the IP Performance Metrics Working Group (IPPM) of the Transport

Area.

We begin by laying out several criteria for the metrics that we

adopt. These criteria are designed to promote an IPPM effort that

will maximize an accurate common understanding by Internet users and

Internet providers of the performance and reliability both of end-

to-end paths through the Internet and of specific 'IP clouds' that

comprise portions of those paths.

We next define some Internet vocabulary that will allow us to speak

clearly about Internet components such as routers, paths, and clouds.

We then define the fundamental concepts of 'metric' and 'measurement

methodology', which allow us to speak clearly about measurement

issues. Given these concepts, we proceed to discuss the important

issue of measurement uncertainties and errors, and develop a key,

somewhat suBTle notion of how they relate to the analytical framework

shared by many ASPects of the Internet engineering discipline. We

then introduce the notion of empirically defined metrics, and finish

this part of the document with a general discussion of how metrics

can be 'composed'.

The remainder of the document deals with a variety of issues related

to defining sound metrics and methodologies: how to deal with

imperfect clocks; the notion of 'wire time' as distinct from 'host

time'; how to aggregate sets of singleton metrics into samples and

derive sound statistics from those samples; why it is recommended to

avoid thinking about Internet properties in probabilistic terms (such

as the probability that a packet is dropped), since these terms often

include implicit assumptions about how the network behaves; the

utility of defining metrics in terms of packets of a generic type;

the benefits of preferring IP addresses to DNS host names; and the

notion of 'standard-formed' packets. An appendix discusses the

Anderson-Darling test for gauging whether a set of values matches a

given statistical distribution, and gives C code for an

implementation of the test.

In some sections of the memo, we will surround some commentary text

with the brackets {Comment: ... }. We stress that this commentary is

only commentary, and is not itself part of the framework document or

a proposal of particular metrics. In some cases this commentary will

discuss some of the properties of metrics that might be envisioned,

but the reader should assume that any such discussion is intended

only to shed light on points made in the framework document, and not

to suggest any specific metrics.

4. Criteria for IP Performance Metrics

The overarching goal of the IP Performance Metrics effort is to

achieve a situation in which users and providers of Internet

transport service have an accurate common understanding of the

performance and reliability of the Internet component 'clouds' that

they use/provide.

To achieve this, performance and reliability metrics for paths

through the Internet must be developed. In several IETF meetings

criteria for these metrics have been specified:

+ The metrics must be concrete and well-defined,

+ A methodology for a metric should have the property that it is

repeatable: if the methodology is used multiple times under

identical conditions, the same measurements should result in the

same measurements.

+ The metrics must exhibit no bias for IP clouds implemented with

identical technology,

+ The metrics must exhibit understood and fair bias for IP clouds

implemented with non-identical technology,

+ The metrics must be useful to users and providers in understanding

the performance they eXPerience or provide,

+ The metrics must avoid inducing artificial performance goals.

5. Terminology for Paths and Clouds

The following list defines terms that need to be precise in the

development of path metrics. We begin with low-level notions of

'host', 'router', and 'link', then proceed to define the notions of

'path', 'IP cloud', and 'exchange' that allow us to segment a path

into relevant pieces.

host A computer capable of communicating using the Internet

protocols; includes "routers".

link A single link-level connection between two (or more) hosts;

includes leased lines, ethernets, frame relay clouds, etc.

routerA host which facilitates network-level communication between

hosts by forwarding IP packets.

path A sequence of the form < h0, l1, h1, ..., ln, hn >, where n >=

0, each hi is a host, each li is a link between hi-1 and hi,

each h1...hn-1 is a router. A pair <li, hi> is termed a 'hop'.

In an appropriate operational configuration, the links and

routers in the path facilitate network-layer communication of

packets from h0 to hn. Note that path is a unidirectional

concept.

subpath

Given a path, a subpath is any subsequence of the given path

which is itself a path. (Thus, the first and last element of a

subpath is a host.)

cloudAn undirected (possibly cyclic) graph whose vertices are routers

and whose edges are links that connect pairs of routers.

Formally, ethernets, frame relay clouds, and other links that

connect more than two routers are modelled as fully-connected

meshes of graph edges. Note that to connect to a cloud means to

connect to a router of the cloud over a link; this link is not

itself part of the cloud.

exchange

A special case of a link, an exchange directly connects either a

host to a cloud and/or one cloud to another cloud.

cloud subpath

A subpath of a given path, all of whose hosts are routers of a

given cloud.

path digest

A sequence of the form < h0, e1, C1, ..., en, hn >, where n >=

0, h0 and hn are hosts, each e1 ... en is an exchange, and each

C1 ... Cn-1 is a cloud subpath.

6. Fundamental Concepts

6.1. Metrics

In the operational Internet, there are several quantities related to

the performance and reliability of the Internet that we'd like to

know the value of. When such a quantity is carefully specified, we

term the quantity a metric. We anticipate that there will be

separate RFCs for each metric (or for each closely related group of

metrics).

In some cases, there might be no obvious means to effectively measure

the metric; this is allowed, and even understood to be very useful in

some cases. It is required, however, that the specification of the

metric be as clear as possible about what quantity is being

specified. Thus, difficulty in practical measurement is sometimes

allowed, but ambiguity in meaning is not.

Each metric will be defined in terms of standard units of

measurement. The international metric system will be used, with the

following points specifically noted:

+ When a unit is expressed in simple meters (for distance/length) or

seconds (for duration), appropriate related units based on

thousands or thousandths of acceptable units are acceptable.

Thus, distances expressed in kilometers (km), durations expressed

in milliseconds (ms), or microseconds (us) are allowed, but not

centimeters (because the prefix is not in terms of thousands or

thousandths).

+ When a unit is expressed in a combination of units, appropriate

related units based on thousands or thousandths of acceptable

units are acceptable, but all such thousands/thousandths must be

grouped at the beginning. Thus, kilo-meters per second (km/s) is

allowed, but meters per millisecond is not.

+ The unit of information is the bit.

+ When metric prefixes are used with bits or with combinations

including bits, those prefixes will have their metric meaning

(related to decimal 1000), and not the meaning conventional with

computer storage (related to decimal 1024). In any RFCthat

defines a metric whose units include bits, this convention will be

followed and will be repeated to ensure clarity for the reader.

+ When a time is given, it will be expressed in UTC.

Note that these points apply to the specifications for metrics and

not, for example, to packet formats where octets will likely be used

in preference/addition to bits.

Finally, we note that some metrics may be defined purely in terms of

other metrics; such metrics are call 'derived metrics'.

6.2. Measurement Methodology

For a given set of well-defined metrics, a number of distinct

measurement methodologies may exist. A partial list includes:

+ Direct measurement of a performance metric using injected test

traffic. Example: measurement of the round-trip delay of an IP

packet of a given size over a given route at a given time.

+ Projection of a metric from lower-level measurements. Example:

given accurate measurements of propagation delay and bandwidth for

each step along a path, projection of the complete delay for the

path for an IP packet of a given size.

+ Estimation of a constituent metric from a set of more aggregated

measurements. Example: given accurate measurements of delay for a

given one-hop path for IP packets of different sizes, estimation

of propagation delay for the link of that one-hop path.

+ Estimation of a given metric at one time from a set of related

metrics at other times. Example: given an accurate measurement of

flow capacity at a past time, together with a set of accurate

delay measurements for that past time and the current time, and

given a model of flow dynamics, estimate the flow capacity that

would be observed at the current time.

This list is by no means exhaustive. The purpose is to point out the

variety of measurement techniques.

When a given metric is specified, a given measurement approach might

be noted and discussed. That approach, however, is not formally part

of the specification.

A methodology for a metric should have the property that it is

repeatable: if the methodology is used multiple times under identical

conditions, it should result in consistent measurements.

Backing off a little from the Word 'identical' in the previous

paragraph, we could more accurately use the word 'continuity' to

describe a property of a given methodology: a methodology for a given

metric exhibits continuity if, for small variations in conditions, it

results in small variations in the resulting measurements. Slightly

more precisely, for every positive epsilon, there exists a positive

delta, such that if two sets of conditions are within delta of each

other, then the resulting measurements will be within epsilon of each

other. At this point, this should be taken as a heuristic driving

our intuition about one kind of robustness property rather than as a

precise notion.

A metric that has at least one methodology that exhibits continuity

is said itself to exhibit continuity.

Note that some metrics, such as hop-count along a path, are integer-

valued and therefore cannot exhibit continuity in quite the sense

given above.

Note further that, in practice, it may not be practical to know (or

be able to quantify) the conditions relevant to a measurement at a

given time. For example, since the instantaneous load (in packets to

be served) at a given router in a high-speed wide-area network can

vary widely over relatively brief periods and will be very hard for

an external observer to quantify, various statistics of a given

metric may be more repeatable, or may better exhibit continuity. In

that case those particular statistics should be specified when the

metric is specified.

Finally, some measurement methodologies may be 'conservative' in the

sense that the act of measurement does not modify, or only slightly

modifies, the value of the performance metric the methodology

attempts to measure. {Comment: for example, in a wide-are high-speed

network under modest load, a test using several small 'ping' packets

to measure delay would likely not interfere (much) with the delay

properties of that network as observed by others. The corresponding

statement about tests using a large flow to measure flow capacity

would likely fail.}

6.3. Measurements, Uncertainties, and Errors

Even the very best measurement methodologies for the very most well

behaved metrics will exhibit errors. Those who develop such

measurement methodologies, however, should strive to:

+ minimize their uncertainties/errors,

+ understand and document the sources of uncertainty/error, and

+ quantify the amounts of uncertainty/error.

For example, when developing a method for measuring delay, understand

how any errors in your clocks introduce errors into your delay

measurement, and quantify this effect as well as you can. In some

cases, this will result in a requirement that a clock be at least up

to a certain quality if it is to be used to make a certain

measurement.

As a second example, consider the timing error due to measurement

overheads within the computer making the measurement, as opposed to

delays due to the Internet component being measured. The former is a

measurement error, while the latter reflects the metric of interest.

Note that one technique that can help avoid this overhead is the use

of a packet filter/sniffer, running on a separate computer that

records network packets and timestamps them accurately (see the

discussion of 'wire time' below). The resulting trace can then be

analyzed to assess the test traffic, minimizing the effect of

measurement host delays, or at least allowing those delays to be

accounted for. We note that this technique may prove beneficial even

if the packet filter/sniffer runs on the same machine, because such

measurements generally provide 'kernel-level' timestamping as opposed

to less-accurate 'application-level' timestamping.

Finally, we note that derived metrics (defined above) or metrics that

exhibit spatial or temporal composition (defined below) offer

particular occasion for the analysis of measurement uncertainties,

namely how the uncertainties propagate (conceptually) due to the

derivation or composition.

7. Metrics and the Analytical Framework

As the Internet has evolved from the early packet-switching studies

of the 1960s, the Internet engineering community has evolved a common

analytical framework of concepts. This analytical framework, or A-

frame, used by designers and implementers of protocols, by those

involved in measurement, and by those who study computer network

performance using the tools of simulation and analysis, has great

advantage to our work. A major objective here is to generate network

characterizations that are consistent in both analytical and

practical settings, since this will maximize the chances that non-

empirical network study can be better correlated with, and used to

further our understanding of, real network behavior.

Whenever possible, therefore, we would like to develop and leverage

off of the A-frame. Thus, whenever a metric to be specified is

understood to be closely related to concepts within the A-frame, we

will attempt to specify the metric in the A-frame's terms. In such a

specification we will develop the A-frame by precisely defining the

concepts needed for the metric, then leverage off of the A-frame by

defining the metric in terms of those concepts.

Such a metric will be called an 'analytically specified metric' or,

more simply, an analytical metric.

{Comment: Examples of such analytical metrics might include:

propagation time of a link

The time, in seconds, required by a single bit to travel from the

output port on one Internet host across a single link to another

Internet host.

bandwidth of a link for packets of size k

The capacity, in bits/second, where only those bits of the IP

packet are counted, for packets of size k bytes.

routeThe path, as defined in Section 5, from A to B at a given time.

hop count of a route

The value 'n' of the route path.

}

Note that we make no a priori list of just what A-frame concepts

will emerge in these specifications, but we do encourage their use

and urge that they be carefully specified so that, as our set of

metrics develops, so will a specified set of A-frame concepts

technically consistent with each other and consonant with the

common understanding of those concepts within the general Internet

community.

These A-frame concepts will be intended to abstract from actual

Internet components in such a way that:

+ the essential function of the component is retained,

+ properties of the component relevant to the metrics we aim to

create are retained,

+ a subset of these component properties are potentially defined as

analytical metrics, and

+ those properties of actual Internet components not relevant to

defining the metrics we aim to create are dropped.

For example, when considering a router in the context of packet

forwarding, we might model the router as a component that receives

packets on an input link, queues them on a FIFO packet queue of

finite size, employs tail-drop when the packet queue is full, and

forwards them on an output link. The transmission speed (in

bits/second) of the input and output links, the latency in the router

(in seconds), and the maximum size of the packet queue (in bits) are

relevant analytical metrics.

In some cases, such analytical metrics used in relation to a router

will be very closely related to specific metrics of the performance

of Internet paths. For example, an obvious formula (L + P/B)

involving the latency in the router (L), the packet size (in bits)

(P), and the transmission speed of the output link (B) might closely

approximate the increase in packet delay due to the insertion of a

given router along a path.

We stress, however, that well-chosen and well-specified A-frame

concepts and their analytical metrics will support more general

metric creation efforts in less obvious ways.

{Comment: for example, when considering the flow capacity of a path,

it may be of real value to be able to model each of the routers along

the path as packet forwarders as above. Techniques for estimating

the flow capacity of a path might use the maximum packet queue size

as a parameter in decidedly non-obvious ways. For example, as the

maximum queue size increases, so will the ability of the router to

continuously move traffic along an output link despite fluctuations

in traffic from an input link. Estimating this increase, however,

remains a research topic.}

Note that, when we specify A-frame concepts and analytical metrics,

we will inevitably make simplifying assumptions. The key role of

these concepts is to abstract the properties of the Internet

components relevant to given metrics. Judgement is required to avoid

making assumptions that bias the modeling and metric effort toward

one kind of design.

{Comment: for example, routers might not use tail-drop, even though

tail-drop might be easier to model analytically.}

Finally, note that different elements of the A-frame might well make

different simplifying assumptions. For example, the abstraction of a

router used to further the definition of path delay might treat the

router's packet queue as a single FIFO queue, but the abstraction of

a router used to further the definition of the handling of an RSVP-

enabled packet might treat the router's packet queue as supporting

bounded delay -- a contradictory assumption. This is not to say that

we make contradictory assumptions at the same time, but that two

different parts of our work might refine the simpler base concept in

two divergent ways for different purposes.

{Comment: in more mathematical terms, we would say that the A-frame

taken as a whole need not be consistent; but the set of particular

A-frame elements used to define a particular metric must be.}

8. Empirically Specified Metrics

There are useful performance and reliability metrics that do not fit

so neatly into the A-frame, usually because the A-frame lacks the

detail or power for dealing with them. For example, "the best flow

capacity achievable along a path using an RFC-2001-compliant TCP"

would be good to be able to measure, but we have no analytical

framework of sufficient richness to allow us to cast that flow

capacity as an analytical metric.

These notions can still be well specified by instead describing a

reference methodology for measuring them.

Such a metric will be called an 'empirically specified metric', or

more simply, an empirical metric.

Such empirical metrics should have three properties:

+ we should have a clear definition for each in terms of Internet

components,

+ we should have at least one effective means to measure them, and

+ to the extent possible, we should have an (necessarily incomplete)

understanding of the metric in terms of the A-frame so that we can

use our measurements to reason about the performance and

reliability of A-frame components and of aggregations of A-frame

components.

9. Two Forms of Composition

9.1. Spatial Composition of Metrics

In some cases, it may be realistic and useful to define metrics in

such a fashion that they exhibit spatial composition.

By spatial composition, we mean a characteristic of some path

metrics, in which the metric as applied to a (complete) path can also

be defined for various subpaths, and in which the appropriate A-frame

concepts for the metric suggest useful relationships between the

metric applied to these various subpaths (including the complete

path, the various cloud subpaths of a given path digest, and even

single routers along the path). The effectiveness of spatial

composition depends:

+ on the usefulness in analysis of these relationships as applied to

the relevant A-frame components, and

+ on the practical use of the corresponding relationships as applied

to metrics and to measurement methodologies.

{Comment: for example, consider some metric for delay of a 100-byte

packet across a path P, and consider further a path digest <h0, e1,

C1, ..., en, hn> of P. The definition of such a metric might include

a conjecture that the delay across P is very nearly the sum of the

corresponding metric across the exchanges (ei) and clouds (Ci) of the

given path digest. The definition would further include a note on

how a corresponding relation applies to relevant A-frame components,

both for the path P and for the exchanges and clouds of the path

digest.}

When the definition of a metric includes a conjecture that the metric

across the path is related to the metric across the subpaths of the

path, that conjecture constitutes a claim that the metric exhibits

spatial composition. The definition should then include:

+ the specific conjecture applied to the metric,

+ a justification of the practical utility of the composition in

terms of making accurate measurements of the metric on the path,

+ a justification of the usefulness of the composition in terms of

making analysis of the path using A-frame concepts more effective,

and

+ an analysis of how the conjecture could be incorrect.

9.2. Temporal Composition of Formal Models and Empirical Metrics

In some cases, it may be realistic and useful to define metrics in

such a fashion that they exhibit temporal composition.

By temporal composition, we mean a characteristic of some path

metric, in which the metric as applied to a path at a given time T is

also defined for various times t0 < t1 < ... < tn < T, and in which

the appropriate A-frame concepts for the metric suggests useful

relationships between the metric applied at times t0, ..., tn and the

metric applied at time T. The effectiveness of temporal composition

depends:

+ on the usefulness in analysis of these relationships as applied to

the relevant A-frame components, and

+ on the practical use of the corresponding relationships as applied

to metrics and to measurement methodologies.

{Comment: for example, consider a metric for the expected flow

capacity across a path P during the five-minute period surrounding

the time T, and suppose further that we have the corresponding values

for each of the four previous five-minute periods t0, t1, t2, and t3.

The definition of such a metric might include a conjecture that the

flow capacity at time T can be estimated from a certain kind of

extrapolation from the values of t0, ..., t3. The definition would

further include a note on how a corresponding relation applies to

relevant A-frame components.

Note: any (spatial or temporal) compositions involving flow capacity

are likely to be subtle, and temporal compositions are generally more

subtle than spatial compositions, so the reader should understand

that the foregoing example is intentionally naive.}

When the definition of a metric includes a conjecture that the metric

across the path at a given time T is related to the metric across the

path for a set of other times, that conjecture constitutes a claim

that the metric exhibits temporal composition. The definition should

then include:

+ the specific conjecture applied to the metric,

+ a justification of the practical utility of the composition in

terms of making accurate measurements of the metric on the path,

and

+ a justification of the usefulness of the composition in terms of

making analysis of the path using A-frame concepts more effective.

10. Issues related to Time

10.1. Clock Issues

Measurements of time lie at the heart of many Internet metrics.

Because of this, it will often be crucial when designing a

methodology for measuring a metric to understand the different types

of errors and uncertainties introduced by imperfect clocks. In this

section we define terminology for discussing the characteristics of

clocks and touch upon related measurement issues which need to be

addressed by any sound methodology.

The Network Time Protocol (NTP; RFC1305) defines a nomenclature for

discussing clock characteristics, which we will also use when

appropriate [Mi92]. The main goal of NTP is to provide accurate

timekeeping over fairly long time scales, such as minutes to days,

while for measurement purposes often what is more important is

short-term accuracy, between the beginning of the measurement and the

end, or over the course of gathering a body of measurements (a

sample). This difference in goals sometimes leads to different

definitions of terminology as well, as discussed below.

To begin, we define a clock's "offset" at a particular moment as the

difference between the time reported by the clock and the "true" time

as defined by UTC. If the clock reports a time Tc and the true time

is Tt, then the clock's offset is Tc - Tt.

We will refer to a clock as "accurate" at a particular moment if the

clock's offset is zero, and more generally a clock's "accuracy" is

how close the absolute value of the offset is to zero. For NTP,

accuracy also includes a notion of the frequency of the clock; for

our purposes, we instead incorporate this notion into that of "skew",

because we define accuracy in terms of a single moment in time rather

than over an interval of time.

A clock's "skew" at a particular moment is the frequency difference

(first derivative of its offset with respect to true time) between

the clock and true time.

As noted in RFC1305, real clocks exhibit some variation in skew.

That is, the second derivative of the clock's offset with respect to

true time is generally non-zero. In keeping with RFC1305, we define

this quantity as the clock's "drift".

A clock's "resolution" is the smallest unit by which the clock's time

is updated. It gives a lower bound on the clock's uncertainty.

(Note that clocks can have very fine resolutions and yet be wildly

inaccurate.) Resolution is defined in terms of seconds. However,

resolution is relative to the clock's reported time and not to true

time, so for example a resolution of 10 ms only means that the clock

updates its notion of time in 0.01 second increments, not that this

is the true amount of time between updates.

{Comment: Systems differ on how an application interface to the clock

reports the time on subsequent calls during which the clock has not

advanced. Some systems simply return the same unchanged time as

given for previous calls. Others may add a small increment to the

reported time to maintain monotone-increasing timestamps. For

systems that do the latter, we do *not* consider these small

increments when defining the clock's resolution. They are instead an

impediment to assessing the clock's resolution, since a natural

method for doing so is to repeatedly query the clock to determine the

smallest non-zero difference in reported times.}

It is expected that a clock's resolution changes only rarely (for

example, due to a hardware upgrade).

There are a number of interesting metrics for which some natural

measurement methodologies involve comparing times reported by two

different clocks. An example is one-way packet delay [AK97]. Here,

the time required for a packet to travel through the network is

measured by comparing the time reported by a clock at one end of the

packet's path, corresponding to when the packet first entered the

network, with the time reported by a clock at the other end of the

path, corresponding to when the packet finished traversing the

network.

We are thus also interested in terminology for describing how two

clocks C1 and C2 compare. To do so, we introduce terms related to

those above in which the notion of "true time" is replaced by the

time as reported by clock C1. For example, clock C2's offset

relative to C1 at a particular moment is Tc2 - Tc1, the instantaneous

difference in time reported by C2 and C1. To disambiguate between

the use of the terms to compare two clocks versus the use of the

terms to compare to true time, we will in the former case use the

phrase "relative". So the offset defined earlier in this paragraph

is the "relative offset" between C2 and C1.

When comparing clocks, the analog of "resolution" is not "relative

resolution", but instead "joint resolution", which is the sum of the

resolutions of C1 and C2. The joint resolution then indicates a

conservative lower bound on the accuracy of any time intervals

computed by subtracting timestamps generated by one clock from those

generated by the other.

If two clocks are "accurate" with respect to one another (their

relative offset is zero), we will refer to the pair of clocks as

"synchronized". Note that clocks can be highly synchronized yet

arbitrarily inaccurate in terms of how well they tell true time.

This point is important because for many Internet measurements,

synchronization between two clocks is more important than the

accuracy of the clocks. The is somewhat true of skew, too: as long

as the absolute skew is not too great, then minimal relative skew is

more important, as it can induce systematic trends in packet transit

times measured by comparing timestamps produced by the two clocks.

These distinctions arise because for Internet measurement what is

often most important are differences in time as computed by comparing

the output of two clocks. The process of computing the difference

removes any error due to clock inaccuracies with respect to true

time; but it is crucial that the differences themselves accurately

reflect differences in true time.

Measurement methodologies will often begin with the step of assuring

that two clocks are synchronized and have minimal skew and drift.

{Comment: An effective way to assure these conditions (and also clock

accuracy) is by using clocks that derive their notion of time from an

external source, rather than only the host computer's clock. (These

latter are often subject to large errors.) It is further preferable

that the clocks directly derive their time, for example by having

immediate Access to a GPS (Global Positioning System) unit.}

Two important concerns arise if the clocks indirectly derive their

time using a network time synchronization protocol such as NTP:

+ First, NTP's accuracy depends in part on the properties

(particularly delay) of the Internet paths used by the NTP peers,

and these might be exactly the properties that we wish to measure,

so it would be unsound to use NTP to calibrate such measurements.

+ Second, NTP focuses on clock accuracy, which can come at the

expense of short-term clock skew and drift. For example, when a

host's clock is indirectly synchronized to a time source, if the

synchronization intervals occur infrequently, then the host will

sometimes be faced with the problem of how to adjust its current,

incorrect time, Ti, with a considerably different, more accurate

time it has just learned, Ta. Two general ways in which this is

done are to either immediately set the current time to Ta, or to

adjust the local clock's update frequency (hence, its skew) so

that at some point in the future the local time Ti' will agree

with the more accurate time Ta'. The first mechanism introduces

discontinuities and can also violate common assumptions that

timestamps are monotone increasing. If the host's clock is set

backward in time, sometimes this can be easily detected. If the

clock is set forward in time, this can be harder to detect. The

skew induced by the second mechanism can lead to considerable

inaccuracies when computing differences in time, as discussed

above.

To illustrate why skew is a crucial concern, consider samples of

one-way delays between two Internet hosts made at one minute

intervals. The true transmission delay between the hosts might

plausibly be on the order of 50 ms for a transcontinental path. If

the skew between the two clocks is 0.01%, that is, 1 part in 10,000,

then after 10 minutes of observation the error introduced into the

measurement is 60 ms. Unless corrected, this error is enough to

completely wipe out any accuracy in the transmission delay

measurement. Finally, we note that assessing skew errors between

unsynchronized network clocks is an open research area. (See [Pa97]

for a discussion of detecting and compensating for these sorts of

errors.) This shortcoming makes use of a solid, independent clock

source such as GPS especially desirable.

10.2. The Notion of "Wire Time"

Internet measurement is often complicated by the use of Internet

hosts themselves to perform the measurement. These hosts can

introduce delays, bottlenecks, and the like that are due to hardware

or operating system effects and have nothing to do with the network

behavior we would like to measure. This problem is particularly

acute when timestamping of network events occurs at the application

level.

In order to provide a general way of talking about these effects, we

introduce two notions of "wire time". These notions are only defined

in terms of an Internet host H observing an Internet link L at a

particular location:

+ For a given packet P, the 'wire arrival time' of P at H on L is

the first time T at which any bit of P has appeared at H's

observational position on L.

+ For a given packet P, the 'wire exit time' of P at H on L is the

first time T at which all the bits of P have appeared at H's

observational position on L.

Note that intrinsic to the definition is the notion of where on the

link we are observing. This distinction is important because for

large-latency links, we may obtain very different times depending on

exactly where we are observing the link. We could allow the

observational position to be an arbitrary location along the link;

however, we define it to be in terms of an Internet host because we

anticipate in practice that, for IPPM metrics, all such timing will

be constrained to be performed by Internet hosts, rather than

specialized hardware devices that might be able to monitor a link at

locations where a host cannot. This definition also takes care of

the problem of links that are comprised of multiple physical

channels. Because these multiple channels are not visible at the IP

layer, they cannot be individually observed in terms of the above

definitions.

It is possible, though one hopes uncommon, that a packet P might make

multiple trips over a particular link L, due to a forwarding loop.

These trips might even overlap, depending on the link technology.

Whenever this occurs, we define a separate wire time associated with

each instance of P seen at H's position on the link. This definition

is worth making because it serves as a reminder that notions like

*the* unique time a packet passes a point in the Internet are

inherently slippery.

The term wire time has historically been used to loosely denote the

time at which a packet appeared on a link, without exactly specifying

whether this refers to the first bit, the last bit, or some other

consideration. This informal definition is generally already very

useful, as it is usually used to make a distinction between when the

packet's propagation delays begin and cease to be due to the network

rather than the endpoint hosts.

When appropriate, metrics should be defined in terms of wire times

rather than host endpoint times, so that the metric's definition

highlights the issue of separating delays due to the host from those

due to the network.

We note that one potential difficulty when dealing with wire times

concerns IP fragments. It may be the case that, due to

fragmentation, only a portion of a particular packet passes by H's

location. Such fragments are themselves legitimate packets and have

well-defined wire times associated with them; but the larger IP

packet corresponding to their aggregate may not.

We also note that these notions have not, to our knowledge, been

previously defined in exact terms for Internet traffic.

Consequently, we may find with experience that these definitions

require some adjustment in the future.

{Comment: It can sometimes be difficult to measure wire times. One

technique is to use a packet filter to monitor traffic on a link.

The architecture of these filters often attempts to associate with

each packet a timestamp as close to the wire time as possible. We

note however that one common source of error is to run the packet

filter on one of the endpoint hosts. In this case, it has been

observed that some packet filters receive for some packets timestamps

corresponding to when the packet was *scheduled* to be injected into

the network, rather than when it actually was *sent* out onto the

network (wire time). There can be a substantial difference between

these two times. A technique for dealing with this problem is to run

the packet filter on a separate host that passively monitors the

given link. This can be problematic however for some link

technologies. See [Pa97] for a discussion of the sorts of errors

packet filters can exhibit. Finally, we note that packet filters

will often only capture the first fragment of a fragmented IP packet,

due to the use of filtering on fields in the IP and transport

protocol headers. As we generally desire our measurement

methodologies to avoid the complexity of creating fragmented traffic,

one strategy for dealing with their presence as detected by a packet

filter is to flag that the measured traffic has an unusual form and

abandon further analysis of the packet timing.}

11. Singletons, Samples, and Statistics

With experience we have found it useful to introduce a separation

between three distinct -- yet related -- notions:

+ By a 'singleton' metric, we refer to metrics that are, in a sense,

atomic. For example, a single instance of "bulk throughput

capacity" from one host to another might be defined as a singleton

metric, even though the instance involves measuring the timing of

a number of Internet packets.

+ By a 'sample' metric, we refer to metrics derived from a given

singleton metric by taking a number of distinct instances

together. For example, we might define a sample metric of one-way

delays from one host to another as an hour's worth of

measurements, each made at Poisson intervals with a mean spacing

of one second.

+ By a 'statistical' metric, we refer to metrics derived from a

given sample metric by computing some statistic of the values

defined by the singleton metric on the sample. For example, the

mean of all the one-way delay values on the sample given above

might be defined as a statistical metric.

By applying these notions of singleton, sample, and statistic in a

consistent way, we will be able to reuse lessons learned about how to

define samples and statistics on various metrics. The orthogonality

among these three notions will thus make all our work more effective

and more intelligible by the community.

In the remainder of this section, we will cover some topics in

sampling and statistics that we believe will be important to a

variety of metric definitions and measurement efforts.

11.1. Methods of Collecting Samples

The main reason for collecting samples is to see what sort of

variations and consistencies are present in the metric being

measured. These variations might be with respect to different points

in the Internet, or different measurement times. When assessing

variations based on a sample, one generally makes an assumption that

the sample is "unbiased", meaning that the process of collecting the

measurements in the sample did not skew the sample so that it no

longer accurately reflects the metric's variations and consistencies.

One common way of collecting samples is to make measurements

separated by fixed amounts of time: periodic sampling. Periodic

sampling is particularly attractive because of its simplicity, but it

suffers from two potential problems:

+ If the metric being measured itself exhibits periodic behavior,

then there is a possibility that the sampling will observe only

part of the periodic behavior if the periods happen to agree

(either directly, or if one is a multiple of the other). Related

to this problem is the notion that periodic sampling can be easily

anticipated. Predictable sampling is susceptible to manipulation

if there are mechanisms by which a network component's behavior

can be temporarily changed such that the sampling only sees the

modified behavior.

+ The act of measurement can perturb what is being measured (for

example, injecting measurement traffic into a network alters the

congestion level of the network), and repeated periodic

perturbations can drive a network into a state of synchronization

(cf. [FJ94]), greatly magnifying what might individually be minor

effects.

A more sound approach is based on "random additive sampling": samples

are separated by independent, randomly generated intervals that have

a common statistical distribution G(t) [BM92]. The quality of this

sampling depends on the distribution G(t). For example, if G(t)

generates a constant value g with probability one, then the sampling

reduces to periodic sampling with a period of g.

Random additive sampling gains significant advantages. In general,

it avoids synchronization effects and yields an unbiased estimate of

the property being sampled. The only significant drawbacks with it

are:

+ it complicates frequency-domain analysis, because the samples do

not occur at fixed intervals such as assumed by Fourier-transform

techniques; and

+ unless G(t) is the exponential distribution (see below), sampling

still remains somewhat predictable, as discussed for periodic

sampling above.

11.1.1. Poisson Sampling

It can be proved that if G(t) is an exponential distribution with

rate lambda, that is

G(t) = 1 - exp(-lambda * t)

then the arrival of new samples *cannot* be predicted (and, again,

the sampling is unbiased). Furthermore, the sampling is

asymptotically unbiased even if the act of sampling affects the

network's state. Such sampling is referred to as "Poisson sampling".

It is not prone to inducing synchronization, it can be used to

accurately collect measurements of periodic behavior, and it is not

prone to manipulation by anticipating when new samples will occur.

Because of these valuable properties, we in general prefer that

samples of Internet measurements are gathered using Poisson sampling.

{Comment: We note, however, that there may be circumstances that

favor use of a different G(t). For example, the exponential

distribution is unbounded, so its use will on occasion generate

lengthy spaces between sampling times. We might instead desire to

bound the longest such interval to a maximum value dT, to speed the

convergence of the estimation derived from the sampling. This could

be done by using

G(t) = Unif(0, dT)

that is, the uniform distribution between 0 and dT. This sampling,

of course, becomes highly predictable if an interval of nearly length

dT has elapsed without a sample occurring.}

In its purest form, Poisson sampling is done by generating

independent, exponentially distributed intervals and gathering a

single measurement after each interval has elapsed. It can be shown

that if starting at time T one performs Poisson sampling over an

interval dT, during which a total of N measurements happen to be

made, then those measurements will be uniformly distributed over the

interval [T, T+dT]. So another way of conducting Poisson sampling is

to pick dT and N and generate N random sampling times uniformly over

the interval [T, T+dT]. The two approaches are equivalent, except if

N and dT are externally known. In that case, the property of not

being able to predict measurement times is weakened (the other

properties still hold). The N/dT approach has an advantage that

dealing with fixed values of N and dT can be simpler than dealing

with a fixed lambda but variable numbers of measurements over

variably-sized intervals.

11.1.2. Geometric Sampling

Closely related to Poisson sampling is "geometric sampling", in which

external events are measured with a fixed probability p. For

example, one might capture all the packets over a link but only

record the packet to a trace file if a randomly generated number

uniformly distributed between 0 and 1 is less than a given p.

Geometric sampling has the same properties of being unbiased and not

predictable in advance as Poisson sampling, so if it fits a

particular Internet measurement task, it too is sound. See [CPB93]

for more discussion.

11.1.3. Generating Poisson Sampling Intervals

To generate Poisson sampling intervals, one first determines the rate

lambda at which the singleton measurements will on average be made

(e.g., for an average sampling interval of 30 seconds, we have lambda

= 1/30, if the units of time are seconds). One then generates a

series of exponentially-distributed (pseudo) random numbers E1, E2,

..., En. The first measurement is made at time E1, the next at time

E1+E2, and so on.

One technique for generating exponentially-distributed (pseudo)

random numbers is based on the ability to generate U1, U2, ..., Un,

(pseudo) random numbers that are uniformly distributed between 0 and

1. Many computers provide libraries that can do this. Given such

Ui, to generate Ei one uses:

Ei = -log(Ui) / lambda

where log(Ui) is the natural logarithm of Ui. {Comment: This

technique is an instance of the more general "inverse transform"

method for generating random numbers with a given distribution.}

Implementation details:

There are at least three different methods for approximating Poisson

sampling, which we describe here as Methods 1 through 3. Method 1 is

the easiest to implement and has the most error, and method 3 is the

most difficult to implement and has the least error (potentially

none).

Method 1 is to proceed as follows:

1. Generate E1 and wait that long.

2. Perform a measurement.

3. Generate E2 and wait that long.

4. Perform a measurement.

5. Generate E3 and wait that long.

6. Perform a measurement ...

The problem with this approach is that the "Perform a measurement"

steps themselves take time, so the sampling is not done at times E1,

E1+E2, etc., but rather at E1, E1+M1+E2, etc., where Mi is the amount

of time required for the i'th measurement. If Mi is very small

compared to 1/lambda then the potential error introduced by this

technique is likewise small. As Mi becomes a non-negligible fraction

of 1/lambda, the potential error increases.

Method 2 attempts to correct this error by taking into account the

amount of time required by the measurements (i.e., the Mi's) and

adjusting the waiting intervals accordingly:

1. Generate E1 and wait that long.

2. Perform a measurement and measure M1, the time it took to do so.

3. Generate E2 and wait for a time E2-M1.

4. Perform a measurement and measure M2 ..

This approach works fine as long as E{i+1} >= Mi. But if E{i+1} < Mi

then it is impossible to wait the proper amount of time. (Note that

this case corresponds to needing to perform two measurements

simultaneously.)

Method 3 is generating a schedule of measurement times E1, E1+E2,

etc., and then sticking to it:

1. Generate E1, E2, ..., En.

2. Compute measurement times T1, T2, ..., Tn, as Ti = E1 + ... + Ei.

3. Arrange that at times T1, T2, ..., Tn, a measurement is made.

By allowing simultaneous measurements, Method 3 avoids the

shortcomings of Methods 1 and 2. If, however, simultaneous

measurements interfere with one another, then Method 3 does not gain

any benefit and may actually prove worse than Methods 1 or 2.

For Internet phenomena, it is not known to what degree the

inaccuracies of these methods are significant. If the Mi's are much

less than 1/lambda, then any of the three should suffice. If the

Mi's are less than 1/lambda but perhaps not greatly less, then Method

2 is preferred to Method 1. If simultaneous measurements do not

interfere with one another, then Method 3 is preferred, though it can

be considerably harder to implement.

11.2. Self-Consistency

A fundamental requirement for a sound measurement methodology is that

measurement be made using as few unconfirmed assumptions as possible.

Experience has painfully shown how easy it is to make an (often

implicit) assumption that turns out to be incorrect. An example is

incorporating into a measurement the reading of a clock synchronized

to a highly accurate source. It is easy to assume that the clock is

therefore accurate; but due to software bugs, a loss of power in the

source, or a loss of communication between the source and the clock,

the clock could actually be quite inaccurate.

This is not to argue that one must not make *any* assumptions when

measuring, but rather that, to the extent which is practical,

assumptions should be tested. One powerful way for doing so involves

checking for self-consistency. Such checking applies both to the

observed value(s) of the measurement *and the values used by the

measurement process itself*. A simple example of the former is that

when computing a round trip time, one should check to see if it is

negative. Since negative time intervals are non-physical, if it ever

is negative that finding immediately flags an error. *These sorts of

errors should then be investigated!* It is crucial to determine where

the error lies, because only by doing so diligently can we build up

faith in a methodology's fundamental soundness. For example, it

could be that the round trip time is negative because during the

measurement the clock was set backward in the process of

synchronizing it with another source. But it could also be that the

measurement program accesses uninitialized memory in one of its

computations and, only very rarely, that leads to a bogus

computation. This second error is more serious, if the same program

is used by others to perform the same measurement, since then they

too will suffer from incorrect results. Furthermore, once uncovered

it can be completely fixed.

A more subtle example of testing for self-consistency comes from

gathering samples of one-way Internet delays. If one has a large

sample of such delays, it may well be highly telling to, for example,

fit a line to the pairs of (time of measurement, measured delay), to

see if the resulting line has a clearly non-zero slope. If so, a

possible interpretation is that one of the clocks used in the

measurements is skewed relative to the other. Another interpretation

is that the slope is actually due to genuine network effects.

Determining which is indeed the case will often be highly

illuminating. (See [Pa97] for a discussion of distinguishing between

relative clock skew and genuine network effects.) Furthermore, if

making this check is part of the methodology, then a finding that the

long-term slope is very near zero is positive evidence that the

measurements are probably not biased by a difference in skew.

A final example illustrates checking the measurement process itself

for self-consistency. Above we outline Poisson sampling techniques,

based on generating exponentially-distributed intervals. A sound

measurement methodology would include testing the generated intervals

to see whether they are indeed exponentially distributed (and also to

see if they suffer from correlation). In the appendix we discuss and

give C code for one such technique, a general-purpose, well-regarded

goodness-of-fit test called the Anderson-Darling test.

Finally, we note that what is truly relevant for Poisson sampling of

Internet metrics is often not when the measurements began but the

wire times corresponding to the measurement process. These could

well be different, due to complications on the hosts used to perform

the measurement. Thus, even those with complete faith in their

pseudo-random number generators and subsequent algorithms are

encouraged to consider how they might test the assumptions of each

measurement procedure as much as possible.

11.3. Defining Statistical Distributions

One way of describing a collection of measurements (a sample) is as a

statistical distribution -- informally, as percentiles. There are

several slightly different ways of doing so. In this section we

define a standard definition to give uniformity to these

descriptions.

The "empirical distribution function" (EDF) of a set of scalar

measurements is a function F(x) which for any x gives the fractional

proportion of the total measurements that were <= x. If x is less

than the minimum value observed, then F(x) is 0. If it is greater or

equal to the maximum value observed, then F(x) is 1.

For example, given the 6 measurements:

-2, 7, 7, 4, 18, -5

Then F(-8) = 0, F(-5) = 1/6, F(-5.0001) = 0, F(-4.999) = 1/6, F(7) =

5/6, F(18) = 1, F(239) = 1.

Note that we can recover the different measured values and how many

times each occurred from F(x) -- no information regarding the range

in values is lost. Summarizing measurements using histograms, on the

other hand, in general loses information about the different values

observed, so the EDF is preferred.

Using either the EDF or a histogram, however, we do lose information

regarding the order in which the values were observed. Whether this

loss is potentially significant will depend on the metric being

measured.

We will use the term "percentile" to refer to the smallest value of x

for which F(x) >= a given percentage. So the 50th percentile of the

example above is 4, since F(4) = 3/6 = 50%; the 25th percentile is

-2, since F(-5) = 1/6 < 25%, and F(-2) = 2/6 >= 25%; the 100th

percentile is 18; and the 0th percentile is -infinity, as is the 15th

percentile.

Care must be taken when using percentiles to summarize a sample,

because they can lend an unwarranted appearance of more precision

than is really available. Any such summary must include the sample

size N, because any percentile difference finer than 1/N is below the

resolution of the sample.

See [DS86] for more details regarding EDF's.

We close with a note on the common (and important!) notion of median.

In statistics, the median of a distribution is defined to be the

point X for which the probability of observing a value <= X is equal

to the probability of observing a value > X. When estimating the

median of a set of observations, the estimate depends on whether the

number of observations, N, is odd or even:

+ If N is odd, then the 50th percentile as defined above is used as

the estimated median.

+ If N is even, then the estimated median is the average of the

central two observations; that is, if the observations are sorted

in ascending order and numbered from 1 to N, where N = 2*K, then

the estimated median is the average of the (K)'th and (K+1)'th

observations.

Usually the term "estimated" is dropped from the phrase "estimated

median" and this value is simply referred to as the "median".

11.4. Testing For Goodness-of-Fit

For some forms of measurement calibration we need to test whether a

set of numbers is consistent with those numbers having been drawn

from a particular distribution. An example is that to apply a self-

consistency check to measurements made using a Poisson process, one

test is to see whether the spacing between the sampling times does

indeed reflect an exponential distribution; or if the dT/N approach

discussed above was used, whether the times are uniformly distributed

across [T, dT].

{Comment: There are at least three possible sets of values we could

test: the scheduled packet transmission times, as determined by use

of a pseudo-random number generator; user-level timestamps made just

before or after the system call for transmitting the packet; and wire

times for the packets as recorded using a packet filter. All three

of these are potentially informative: failures for the scheduled

times to match an exponential distribution indicate inaccuracies in

the random number generation; failures for the user-level times

indicate inaccuracies in the timers used to schedule transmission;

and failures for the wire times indicate inaccuracies in actually

transmitting the packets, perhaps due to contention for a shared

resource.}

There are a large number of statistical goodness-of-fit techniques

for performing such tests. See [DS86] for a thorough discussion.

That reference recommends the Anderson-Darling EDF test as being a

good all-purpose test, as well as one that is especially good at

detecting deviations from a given distribution in the lower and upper

tails of the EDF.

It is important to understand that the nature of goodness-of-fit

tests is that one first selects a "significance level", which is the

probability that the test will erroneously declare that the EDF of a

given set of measurements fails to match a particular distribution

when in fact the measurements do indeed reflect that distribution.

Unless otherwise stated, IPPM goodness-of-fit tests are done using 5%

significance. This means that if the test is applied to 100 samples

and 5 of those samples are deemed to have failed the test, then the

samples are all consistent with the distribution being tested. If

significantly more of the samples fail the test, then the assumption

that the samples are consistent with the distribution being tested

must be rejected. If significantly fewer of the samples fail the

test, then the samples have potentially been doctored too well to fit

the distribution. Similarly, some goodness-of-fit tests (including

Anderson-Darling) can detect whether it is likely that a given sample

was doctored. We also use a significance of 5% for this case; that

is, the test will report that a given honest sample is "too good to

be true" 5% of the time, so if the test reports this finding

significantly more often than one time out of twenty, it is an

indication that something unusual is occurring.

The appendix gives sample C code for implementing the Anderson-

Darling test, as well as further discussing its use.

See [Pa94] for a discussion of goodness-of-fit and closeness-of-fit

tests in the context of network measurement.

12. Avoiding Stochastic Metrics

When defining metrics applying to a path, subpath, cloud, or other

network element, we in general do not define them in stochastic terms

(probabilities). We instead prefer a deterministic definition. So,

for example, rather than defining a metric about a "packet loss

probability between A and B", we would define a metric about a

"packet loss rate between A and B". (A measurement given by the

first definition might be "0.73", and by the second "73 packets out

of 100".)

We emphasize that the above distinction concerns the *definitions* of

*metrics*. It is not intended to apply to what sort of techniques we

might use to analyze the results of measurements.

The reason for this distinction is as follows. When definitions are

made in terms of probabilities, there are often hidden assumptions in

the definition about a stochastic model of the behavior being

measured. The fundamental goal with avoiding probabilities in our

metric definitions is to avoid biasing our definitions by these

hidden assumptions.

For example, an easy hidden assumption to make is that packet loss in

a network component due to queueing overflows can be described as

something that happens to any given packet with a particular

probability. In today's Internet, however, queueing drops are

actually usually *deterministic*, and assuming that they should be

described probabilistically can obscure crucial correlations between

queueing drops among a set of packets. So it's better to explicitly

note stochastic assumptions, rather than have them sneak into our

definitions implicitly.

This does *not* mean that we abandon stochastic models for

*understanding* network performance! It only means that when defining

IP metrics we avoid terms such as "probability" for terms like

"proportion" or "rate". We will still use, for example, random

sampling in order to estimate probabilities used by stochastic models

related to the IP metrics. We also do not rule out the possibility

of stochastic metrics when they are truly appropriate (for example,

perhaps to model transmission errors caused by certain types of line

noise).

13. Packets of Type P

A fundamental property of many Internet metrics is that the value of

the metric depends on the type of IP packet(s) used to make the

measurement. Consider an IP-connectivity metric: one obtains

different results depending on whether one is interested in

connectivity for packets destined for well-known TCP ports or

unreserved UDP ports, or those with invalid IP checksums, or those

with TTL's of 16, for example. In some circumstances these

distinctions will be highly interesting (for example, in the presence

of firewalls, or RSVP reservations).

Because of this distinction, we introduce the generic notion of a

"packet of type P", where in some contexts P will be explicitly

defined (i.e., exactly what type of packet we mean), partially

defined (e.g., "with a payload of B octets"), or left generic. Thus

we may talk about generic IP-type-P-connectivity or more specific

IP-port-HTTP-connectivity. Some metrics and methodologies may be

fruitfully defined using generic type P definitions which are then

made specific when performing actual measurements.

Whenever a metric's value depends on the type of the packets involved

in the metric, the metric's name will include either a specific type

or a phrase such as "type-P". Thus we will not define an "IP-

connectivity" metric but instead an "IP-type-P-connectivity" metric

and/or perhaps an "IP-port-HTTP-connectivity" metric. This naming

convention serves as an important reminder that one must be conscious

of the exact type of traffic being measured.

A closely related note: it would be very useful to know if a given

Internet component treats equally a class C of different types of

packets. If so, then any one of those types of packets can be used

for subsequent measurement of the component. This suggests we devise

a metric or suite of metrics that attempt to determine C.

14. Internet Addresses vs. Hosts

When considering a metric for some path through the Internet, it is

often natural to think about it as being for the path from Internet

host H1 to host H2. A definition in these terms, though, can be

ambiguous, because Internet hosts can be attached to more than one

network. In this case, the result of the metric will depend on which

of these networks is actually used.

Because of this ambiguity, usually such definitions should instead be

defined in terms of Internet IP addresses. For the common case of a

unidirectional path through the Internet, we will use the term "Src"

to denote the IP address of the beginning of the path, and "Dst" to

denote the IP address of the end.

15. Standard-Formed Packets

Unless otherwise stated, all metric definitions that concern IP

packets include an implicit assumption that the packet is *standard

formed*. A packet is standard formed if it meets all of the

following criteria:

+ Its length as given in the IP header corresponds to the size of

the IP header plus the size of the payload.

+ It includes a valid IP header: the version field is 4 (later, we

will expand this to include 6); the header length is >= 5; the

checksum is correct.

+ It is not an IP fragment.

+ The source and destination addresses correspond to the hosts in

question.

+ Either the packet possesses sufficient TTL to travel from the

source to the destination if the TTL is decremented by one at each

hop, or it possesses the maximum TTL of 255.

+ It does not contain IP options unless explicitly noted.

+ If a transport header is present, it too contains a valid checksum

and other valid fields.

We further require that if a packet is described as having a "length

of B octets", then 0 <= B <= 65535; and if B is the payload length in

octets, then B <= (65535-IP header size in octets).

So, for example, one might imagine defining an IP connectivity metric

as "IP-type-P-connectivity for standard-formed packets with the IP

TOS field set to 0", or, more succinctly, "IP-type-P-connectivity

with the IP TOS field set to 0", since standard-formed is already

implied by convention.

A particular type of standard-formed packet often useful to consider

is the "minimal IP packet from A to B" - this is an IP packet with

the following properties:

+ It is standard-formed.

+ Its data payload is 0 octets.

+ It contains no options.

(Note that we do not define its protocol field, as different values

may lead to different treatment by the network.)

When defining IP metrics we keep in mind that no packet smaller or

simpler than this can be transmitted over a correctly operating IP

network.

16. Acknowledgements

The comments of Brian Carpenter, Bill Cerveny, Padma Krishnaswamy

Jeff Sedayao and Howard Stanislevic are appreciated.

17. Security Considerations

This document concerns definitions and concepts related to Internet

measurement. We discuss measurement procedures only in high-level

terms, regarding principles that lend themselves to sound

measurement. As such, the topics discussed do not affect the

security of the Internet or of applications which run on it.

That said, it should be recognized that conducting Internet

measurements can raise both security and privacy concerns. Active

techniques, in which traffic is injected into the network, can be

abused for denial-of-service attacks disguised as legitimate

measurement activity. Passive techniques, in which existing traffic

is recorded and analyzed, can expose the contents of Internet traffic

to unintended recipients. Consequently, the definition of each

metric and methodology must include a corresponding discussion of

security considerations.

18. Appendix

Below we give routines written in C for computing the Anderson-

Darling test statistic (A2) for determining whether a set of values

is consistent with a given statistical distribution. Externally, the

two main routines of interest are:

double exp_A2_known_mean(double x[], int n, double mean)

double unif_A2_known_range(double x[], int n,

double min_val, double max_val)

Both take as their first argument, x, the array of n values to be

tested. (Upon return, the elements of x are sorted.) The remaining

parameters characterize the distribution to be used: either the mean

(1/lambda), for an exponential distribution, or the lower and upper

bounds, for a uniform distribution. The names of the routines stress

that these values must be known in advance, and *not* estimated from

the data (for example, by computing its sample mean). Estimating the

parameters from the data *changes* the significance level of the test

statistic. While [DS86] gives alternate significance tables for some

instances in which the parameters are estimated from the data, for

our purposes we expect that we should indeed know the parameters in

advance, since what we will be testing are generally values such as

packet sending times that we wish to verify follow a known

distribution.

Both routines return a significance level, as described earlier. This

is a value between 0 and 1. The correct use of the routines is to

pick in advance the threshold for the significance level to test;

generally, this will be 0.05, corresponding to 5%, as also described

above. Subsequently, if the routines return a value strictly less

than this threshold, then the data are deemed to be inconsistent with

the presumed distribution, *subject to an error corresponding to the

significance level*. That is, for a significance level of 5%, 5% of

the time data that is indeed drawn from the presumed distribution

will be erroneously deemed inconsistent.

Thus, it is important to bear in mind that if these routines are used

frequently, then one will indeed encounter occasional failures, even

if the data is unblemished.

Another important point concerning significance levels is that it is

unsound to compare them in order to determine which of two sets of

values is a "better" fit to a presumed distribution. Such testing

should instead be done using "closeness-of-fit metrics" such as the

lambda^2 metric described in [Pa94].

While the routines provided are for exponential and uniform

distributions with known parameters, it is generally straight-forward

to write comparable routines for any distribution with known

parameters. The heart of the A2 tests lies in a statistic computed

for testing whether a set of values is consistent with a uniform

distribution between 0 and 1, which we term Unif(0, 1). If we wish

to test whether a set of values, X, is consistent with a given

distribution G(x), we first compute

Y = G_inverse(X)

If X is indeed distributed according to G(x), then Y will be

distributed according to Unif(0, 1); so by testing Y for consistency

with Unif(0, 1), we also test X for consistency with G(x).

We note, however, that the process of computing Y above might yield

values of Y outside the range (0..1). Such values should not occur

if X is indeed distributed according to G(x), but easily can occur if

it is not. In the latter case, we need to avoid computing the

central A2 statistic, since floating-point exceptions may occur if

any of the values lie outside (0..1). Accordingly, the routines

check for this possibility, and if encountered, return a raw A2

statistic of -1. The routine that converts the raw A2 statistic to a

significance level likewise propagates this value, returning a

significance level of -1. So, any use of these routines must be

prepared for a possible negative significance level.

The last important point regarding use of A2 statistic concerns n,

the number of values being tested. If n < 5 then the test is not

meaningful, and in this case a significance level of -1 is returned.

On the other hand, for "real" data the test *gains* power as n

becomes larger. It is well known in the statistics community that

real data almost never exactly matches a theoretical distribution,

even in cases such as rolling dice a great many times (see [Pa94] for

a brief discussion and references). The A2 test is sensitive enough

that, for sufficiently large sets of real data, the test will almost

always fail, because it will manage to detect slight imperfections in

the fit of the data to the distribution.

For example, we have found that when testing 8,192 measured wire

times for packets sent at Poisson intervals, the measurements almost

always fail the A2 test. On the other hand, testing 128 measurements

failed at 5% significance only about 5% of the time, as expected.

Thus, in general, when the test fails, care must be taken to

understand why it failed.

The remainder of this appendix gives C code for the routines

mentioned above.

/* Routines for computing the Anderson-Darling A2 test statistic.

*

* Implemented based on the description in "Goodness-of-Fit

* Techniques," R. D'Agostino and M. Stephens, editors,

* Marcel Dekker, Inc., 1986.

*/

#include <stdio.h>

#include <stdlib.h>

#include <math.h>

/* Returns the raw A^2 test statistic for n sorted samples

* z[0] .. z[n-1], for z ~ Unif(0,1).

*/

extern double compute_A2(double z[], int n);

/* Returns the significance level associated with a A^2 test

* statistic value of A2, assuming no parameters of the tested

* distribution were estimated from the data.

*/

extern double A2_significance(double A2);

/* Returns the A^2 significance level for testing n observations

* x[0] .. x[n-1] against an exponential distribution with the

* given mean.

*

* SIDE EFFECT: the x[0..n-1] are sorted upon return.

*/

extern double exp_A2_known_mean(double x[], int n, double mean);

/* Returns the A^2 significance level for testing n observations

* x[0] .. x[n-1] against the uniform distribution [min_val, max_val].

*

* SIDE EFFECT: the x[0..n-1] are sorted upon return.

*/

extern double unif_A2_known_range(double x[], int n,

double min_val, double max_val);

/* Returns a pseudo-random number distributed according to an

* exponential distribution with the given mean.

*/

extern double random_exponential(double mean);

/* Helper function used by qsort() to sort double-precision

* floating-point values.

*/

static int

compare_double(const void *v1, const void *v2)

{

double d1 = *(double *) v1;

double d2 = *(double *) v2;

if (d1 < d2)

return -1;

else if (d1 > d2)

return 1;

else

return 0;

}

double

compute_A2(double z[], int n)

{

int i;

double sum = 0.0;

if ( n < 5 )

/* Too few values. */

return -1.0;

/* If any of the values are outside the range (0, 1) then

* fail immediately (and avoid a possible floating point

* exception in the code below).

*/

for (i = 0; i < n; ++i)

if ( z[i] <= 0.0 z[i] >= 1.0 )

return -1.0;

/* Page 101 of D'Agostino and Stephens. */

for (i = 1; i <= n; ++i) {

sum += (2 * i - 1) * log(z[i-1]);

sum += (2 * n + 1 - 2 * i) * log(1.0 - z[i-1]);

}

return -n - (1.0 / n) * sum;

}

double

A2_significance(double A2)

{

/* Page 105 of D'Agostino and Stephens. */

if (A2 < 0.0)

return A2; /* Bogus A2 value - propagate it. */

/* Check for possibly doctored values. */

if (A2 <= 0.201)

return 0.99;

else if (A2 <= 0.240)

return 0.975;

else if (A2 <= 0.283)

return 0.95;

else if (A2 <= 0.346)

return 0.90;

else if (A2 <= 0.399)

return 0.85;

/* Now check for possible inconsistency. */

if (A2 <= 1.248)

return 0.25;

else if (A2 <= 1.610)

return 0.15;

else if (A2 <= 1.933)

return 0.10;

else if (A2 <= 2.492)

return 0.05;

else if (A2 <= 3.070)

return 0.025;

else if (A2 <= 3.880)

return 0.01;

else if (A2 <= 4.500)

return 0.005;

else if (A2 <= 6.000)

return 0.001;

else

return 0.0;

}

double

exp_A2_known_mean(double x[], int n, double mean)

{

int i;

double A2;

/* Sort the first n values. */

qsort(x, n, sizeof(x[0]), compare_double);

/* Assuming they match an exponential distribution, transform

* them to Unif(0,1).

*/

for (i = 0; i < n; ++i) {

x[i] = 1.0 - exp(-x[i] / mean);

}

/* Now make the A^2 test to see if they're truly uniform. */

A2 = compute_A2(x, n);

return A2_significance(A2);

}

double

unif_A2_known_range(double x[], int n, double min_val, double max_val)

{

int i;

double A2;

double range = max_val - min_val;

/* Sort the first n values. */

qsort(x, n, sizeof(x[0]), compare_double);

/* Transform Unif(min_val, max_val) to Unif(0,1). */

for (i = 0; i < n; ++i)

x[i] = (x[i] - min_val) / range;

/* Now make the A^2 test to see if they're truly uniform. */

A2 = compute_A2(x, n);

return A2_significance(A2);

}

double

random_exponential(double mean)

{

return -mean * log1p(-drand48());

}

19. References

[AK97] G. Almes and S. Kalidindi, "A One-way Delay Metric for IPPM",

Work in Progress, November 1997.

[BM92] I. Bilinskis and A. Mikelsons, Randomized Signal Processing,

Prentice Hall International, 1992.

[DS86] R. D'Agostino and M. Stephens, editors, Goodness-of-Fit

Techniques, Marcel Dekker, Inc., 1986.

[CPB93] K. Claffy, G. Polyzos, and H-W. Braun, "Application of

Sampling Methodologies to Network Traffic Characterization," Proc.

SIGCOMM '93, pp. 194-203, San Francisco, September 1993.

[FJ94] S. Floyd and V. Jacobson, "The Synchronization of Periodic

Routing Messages," IEEE/ACM Transactions on Networking, 2(2), pp.

122-136, April 1994.

[Mi92] Mills, D., "Network Time Protocol (Version 3) Specification,

Implementation and Analysis", RFC1305, March 1992.

[Pa94] V. Paxson, "Empirically-Derived Analytic Models of Wide-Area

TCP Connections," IEEE/ACM Transactions on Networking, 2(4), pp.

316-336, August 1994.

[Pa96] V. Paxson, "Towards a Framework for Defining Internet

Performance Metrics," Proceedings of INET '96,

FTP://ftp.ee.lbl.gov/papers/metrics-framework-INET96.ps.Z

[Pa97] V. Paxson, "Measurements and Analysis of End-to-End Internet

Dynamics," Ph.D. dissertation, U.C. Berkeley, 1997,

ftp://ftp.ee.lbl.gov/papers/vp-thesis/dis.ps.gz.

20. Authors' Addresses

Vern Paxson

MS 50B/2239

Lawrence Berkeley National Laboratory

University of California

Berkeley, CA 94720

USA

Phone: +1 510/486-7504

EMail: vern@ee.lbl.gov

Guy Almes

Advanced Network & Services, Inc.

200 Business Park Drive

Armonk, NY 10504

USA

Phone: +1 914/765-1120

EMail: almes@advanced.org

Jamshid Mahdavi

Pittsburgh Supercomputing Center

4400 5th Avenue

Pittsburgh, PA 15213

USA

Phone: +1 412/268-6282

EMail: mahdavi@psc.edu

Matt Mathis

Pittsburgh Supercomputing Center

4400 5th Avenue

Pittsburgh, PA 15213

USA

Phone: +1 412/268-3319

EMail: mathis@psc.edu

21. Full Copyright Statement

Copyright (C) The Internet Society (1998). All Rights Reserved.

This document and translations of it may be copied and furnished to

others, and derivative works that comment on or otherwise explain it

or assist in its implementation may be prepared, copied, published

and distributed, in whole or in part, without restriction of any

kind, provided that the above copyright notice and this paragraph are

included on all such copies and derivative works. However, this

document itself may not be modified in any way, such as by removing

the copyright notice or references to the Internet Society or other

Internet organizations, except as needed for the purpose of

developing Internet standards in which case the procedures for

copyrights defined in the Internet Standards process must be

followed, or as required to translate it into languages other than

English.

The limited permissions granted above are perpetual and will not be

revoked by the Internet Society or its successors or assigns.

This document and the information contained herein is provided on an

"AS IS" basis and THE INTERNET SOCIETY AND THE INTERNET ENGINEERING

TASK FORCE DISCLAIMS ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING

BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE INFORMATION

HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED WARRANTIES OF

MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.

 
 
 
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