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

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