| Rfc | 6576 |
|---|
| Title | IP Performance Metrics (IPPM) Standard Advancement Testing |
|---|
| Author | R. Geib,
Ed., A. Morton, R. Fardid, A. Steinmitz |
|---|
| Date | March 2012 |
|---|
| Format: | TXT,
HTML |
|---|
| Also | BCP0176 |
|---|
| Status: | BEST CURRENT PRACTICE |
|---|
|
Internet Engineering Task Force (IETF) R. Geib, Ed.
Request for Comments: 6576 Deutsche Telekom
BCP: 176 A. Morton
Category: Best Current Practice AT&T Labs
ISSN: 2070-1721 R. Fardid
Cariden Technologies
A. Steinmitz
Deutsche Telekom
March 2012
IP Performance Metrics (IPPM) Standard Advancement Testing
Abstract
This document specifies tests to determine if multiple independent
instantiations of a performance-metric RFC have implemented the
specifications in the same way. This is the performance-metric
equivalent of interoperability, required to advance RFCs along the
Standards Track. Results from different implementations of metric
RFCs will be collected under the same underlying network conditions
and compared using statistical methods. The goal is an evaluation of
the metric RFC itself to determine whether its definitions are clear
and unambiguous to implementors and therefore a candidate for
advancement on the IETF Standards Track. This document is an
Internet Best Current Practice.
Status of This Memo
This memo documents an Internet Best Current Practice.
This document is a product of the Internet Engineering Task Force
(IETF). It represents the consensus of the IETF community. It has
received public review and has been approved for publication by the
Internet Engineering Steering Group (IESG). Further information on
BCPs is available in Section 2 of RFC 5741.
Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
http://www.rfc-editor.org/info/rfc6576.
Copyright Notice
Copyright (c) 2012 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
(http://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents
carefully, as they describe your rights and restrictions with respect
to this document. Code Components extracted from this document must
include Simplified BSD License text as described in Section 4.e of
the Trust Legal Provisions and are provided without warranty as
described in the Simplified BSD License.
Table of Contents
1. Introduction ....................................................3
1.1. Requirements Language ......................................5
2. Basic Idea ......................................................5
3. Verification of Conformance to a Metric Specification ...........7
3.1. Tests of an Individual Implementation against a Metric
Specification ..............................................8
3.2. Test Setup Resulting in Identical Live Network
Testing Conditions .........................................9
3.3. Tests of Two or More Different Implementations
against a Metric Specification ............................15
3.4. Clock Synchronization .....................................16
3.5. Recommended Metric Verification Measurement Process .......17
3.6. Proposal to Determine an Equivalence Threshold for
Each Metric Evaluated .....................................20
4. Acknowledgements ...............................................21
5. Contributors ...................................................21
6. Security Considerations ........................................21
7. References .....................................................21
7.1. Normative References ......................................21
7.2. Informative References ....................................23
Appendix A. An Example on a One-Way Delay Metric Validation ......24
A.1. Compliance to Metric Specification Requirements ...........24
A.2. Examples Related to Statistical Tests for One-Way Delay ...25
Appendix B. Anderson-Darling K-sample Reference and 2 Sample
C++ Code .............................................27
Appendix C. Glossary .............................................36
1. Introduction
The Internet Standards Process as updated by RFC 6410 [RFC6410]
specifies that widespread deployment and use is sufficient to show
interoperability as a condition for advancement to Internet Standard.
The previous requirement of interoperability tests prior to advancing
an RFC to the Standard maturity level specified in RFC 2026 [RFC2026]
and RFC 5657 [RFC5657] has been removed. While the modified
requirement is applicable to protocols, wide deployment of different
measurement systems does not prove that the implementations measure
metrics in a standard way. Section 5.3 of RFC 5657 [RFC5657]
explicitly mentions the special case of Standards that are not "on-
the-wire" protocols. While this special case is not explicitly
mentioned by RFC 6410 [RFC6410], the four criteria in Section 2.2 of
RFC 6410 [RFC6410] are augmented by this document for RFCs that
specify performance metrics. This document takes the position that
flexible metric definitions can be proven to be clear and unambiguous
through tests that compare the results from independent
implementations. It describes tests that infer whether metric
specifications are sufficient using a definition of metric
"interoperability": measuring equivalent results (in a statistical
sense) under the same network conditions. The document expands on
this problem and its solution.
In the case of a protocol specification, the notion of
"interoperability" is reasonably intuitive -- the implementations
must successfully "talk to each other", while exercising all features
and options. To achieve interoperability, two implementors need to
interpret the protocol specifications in equivalent ways. In the
case of IP Performance Metrics (IPPM), this definition of
interoperability is only useful for test and control protocols like
the One-Way Active Measurement Protocol (OWAMP) [RFC4656] and the
Two-Way Active Measurement Protocol (TWAMP) [RFC5357].
A metric specification RFC describes one or more metric definitions,
methods of measurement, and a way to report the results of
measurement. One example would be a way to test and report the one-
way delay that data packets incur while being sent from one network
location to another, using the One-Way Delay Metric.
In the case of metric specifications, the conditions that satisfy the
"interoperability" requirement are less obvious, and there is a need
for IETF agreement on practices to judge metric specification
"interoperability" in the context of the IETF Standards Process.
This memo provides methods that should be suitable to evaluate metric
specifications for Standards Track advancement. The methods proposed
here MAY be generally applicable to metric specification RFCs beyond
those developed under the IPPM Framework [RFC2330].
Since many implementations of IP metrics are embedded in measurement
systems that do not interact with one another (they were built before
OWAMP and TWAMP), the interoperability evaluation called for in the
IETF Standards Process cannot be determined by observing that
independent implementations interact properly for various protocol
exchanges. Instead, verifying that different implementations give
statistically equivalent results under controlled measurement
conditions takes the place of interoperability observations. Even
when evaluating OWAMP and TWAMP RFCs for Standards Track advancement,
the methods described here are useful to evaluate the measurement
results because their validity would not be ascertained in protocol
interoperability testing.
The Standards advancement process aims at producing confidence that
the metric definitions and supporting material are clearly worded and
unambiguous, or reveals ways in which the metric definitions can be
revised to achieve clarity. The process also permits identification
of options that were not implemented, so that they can be removed
from the advancing specification. Thus, the product of this process
is information about the metric specification RFC itself:
determination of the specifications or definitions that are clear and
unambiguous and those that are not (as opposed to an evaluation of
the implementations that assist in the process).
This document defines a process to verify that implementations (or
practically, measurement systems) have interpreted the metric
specifications in equivalent ways and produce equivalent results.
Testing for statistical equivalence requires ensuring identical test
setups (or awareness of differences) to the best possible extent.
Thus, producing identical test conditions is a core goal of this
memo. Another important aspect of this process is to test individual
implementations against specific requirements in the metric
specifications using customized tests for each requirement. These
tests can distinguish equivalent interpretations of each specific
requirement.
Conclusions on equivalence are reached by two measures.
First, implementations are compared against individual metric
specifications to make sure that differences in implementation are
minimized or at least known.
Second, a test setup is proposed ensuring identical networking
conditions so that unknowns are minimized and comparisons are
simplified. The resulting separate data sets may be seen as samples
taken from the same underlying distribution. Using statistical
methods, the equivalence of the results is verified. To illustrate
application of the process and methods defined here, evaluation of
the One-Way Delay Metric [RFC2679] is provided in Appendix A. While
test setups will vary with the metrics to be validated, the general
methodology of determining equivalent results will not. Documents
defining test setups to evaluate other metrics should be developed
once the process proposed here has been agreed and approved.
The metric RFC advancement process begins with a request for protocol
action accompanied by a memo that documents the supporting tests and
results. The procedures of [RFC2026] are expanded in [RFC5657],
including sample implementation and interoperability reports.
[TESTPLAN] can serve as a template for a metric RFC report that
accompanies the protocol action request to the Area Director,
including a description of the test setup, procedures, results for
each implementation, and conclusions.
1.1. Requirements Language
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in RFC 2119 [RFC2119].
2. Basic Idea
The implementation of a standard compliant metric is expected to meet
the requirements of the related metric specification. So, before
comparing two metric implementations, each metric implementation is
individually compared against the metric specification.
Most metric specifications leave freedom to implementors on non-
fundamental aspects of an individual metric (or options). Comparing
different measurement results using a statistical test with the
assumption of identical test path and testing conditions requires
knowledge of all differences in the overall test setup. Metric
specification options chosen by implementors have to be documented.
It is RECOMMENDED to use identical metric options for any test
proposed here (an exception would be if a variable parameter of the
metric definition is not configurable in one or more
implementations). Calibrations specified by metric standards SHOULD
be performed to further identify (and possibly reduce) potential
sources of error in the test setup.
The IPPM Framework [RFC2330] expects that 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". This means an implementation is expected
to repeatedly measure a metric with consistent results (repeatability
with the same result). Small deviations in the test setup are
expected to lead to small deviations in results only. To
characterize statistical equivalence in the case of small deviations,
[RFC2330] and [RFC2679] suggest to apply a 95% confidence interval.
Quoting RFC 2679, "95 percent was chosen because ... a particular
confidence level should be specified so that the results of
independent implementations can be compared".
Two different implementations are expected to produce statistically
equivalent results if they both measure a metric under the same
networking conditions. Formulating in statistical terms: separate
metric implementations collect separate samples from the same
underlying statistical process (the same network conditions). The
statistical hypothesis to be tested is the expectation that both
samples do not expose statistically different properties. This
requires careful test design:
o The measurement test setup must be self-consistent to the largest
possible extent. To minimize the influence of the test and
measurement setup on the result, network conditions and paths MUST
be identical for the compared implementations to the largest
possible degree. This includes both the stability and non-
ambiguity of routes taken by the measurement packets. See
[RFC2330] for a discussion on self-consistency.
o To minimize the influence of implementation options on the result,
metric implementations SHOULD use identical options and parameters
for the metric under evaluation.
o The sample size must be large enough to minimize its influence on
the consistency of the test results. This consideration may be
especially important if two implementations measure with different
average packet transmission rates.
o The implementation with the lowest average packet transmission
rate determines the smallest temporal interval for which samples
can be compared.
o Repeat comparisons with several independent metric samples to
avoid random indications of compatibility (or the lack of it).
The metric specifications themselves are the primary focus of
evaluation, rather than the implementations of metrics. The
documentation produced by the advancement process should identify
which metric definitions and supporting material were found to be
clearly worded and unambiguous, OR it should identify ways in which
the metric specification text should be revised to achieve clarity
and unified interpretation.
The process should also permit identification of options that were
not implemented, so that they can be removed from the advancing
specification (this is an aspect more typical of protocol advancement
along the Standards Track).
Note that this document does not propose to base interoperability
indications of performance-metric implementations on comparisons of
individual singletons. Individual singletons may be impacted by many
statistical effects while they are measured. Comparing two
singletons of different implementations may result in failures with
higher probability than comparing samples.
3. Verification of Conformance to a Metric Specification
This section specifies how to verify compliance of two or more IPPM
implementations against a metric specification. This document only
proposes a general methodology. Compliance criteria to a specific
metric implementation need to be defined for each individual metric
specification. The only exception is the statistical test comparing
two metric implementations that are simultaneously tested. This test
is applicable without metric-specific decision criteria.
Several testing options exist to compare two or more implementations:
o Use a single test lab to compare the implementations and emulate
the Internet with an impairment generator.
o Use a single test lab to compare the implementations and measure
across the Internet.
o Use remotely separated test labs to compare the implementations
and emulate the Internet with two "identically" configured
impairment generators.
o Use remotely separated test labs to compare the implementations
and measure across the Internet.
o Use remotely separated test labs to compare the implementations,
measure across the Internet, and include a single impairment
generator to impact all measurement flows in a non-discriminatory
way.
The first two approaches work, but involve higher expenses than the
others (due to travel and/or shipping plus installation). For the
third option, ensuring two identically configured impairment
generators requires well-defined test cases and possibly identical
hardware and software.
As documented in a test report [TESTPLAN], the last option was
required to prove compatibility of two delay metric implementations.
An impairment generator is probably required when testing
compatibility of most other metrics, and it is therefore RECOMMENDED
to include an impairment generator in metric test setups.
3.1. Tests of an Individual Implementation against a Metric
Specification
A metric implementation is compliant with a metric specification if
it supports the requirements classified as "MUST" and "REQUIRED" in
the related metric specification. An implementation that implements
all requirements is fully compliant with the specification, and the
degree of compliance SHOULD be noted in the conclusions of the
report.
Further, supported options of a metric implementation SHOULD be
documented in sufficient detail to evaluate whether the specification
was correctly interpreted. The documentation of chosen options
should minimize (and recognize) differences in the test setup if two
metric implementations are compared. Further, this documentation is
used to validate or clarify the wording of the metric specification
option, to remove options that saw no implementation or that are
badly specified from the metric specification. This documentation
SHOULD be included for all implementation-relevant specifications of
a metric picked for a comparison, even those that are not explicitly
marked as "MUST" or "REQUIRED" in the RFC text. This applies for the
following sections of all metric specifications:
o Singleton Definition of the Metric.
o Sample Definition of the Metric.
o Statistics Definition of the Metric. As statistics are compared
by the test specified here, this documentation is required even in
the case that the metric specification does not contain a
Statistics Definition.
o Timing- and Synchronization-related specification (if relevant for
the Metric).
o Any other technical part present or missing in the metric
specification, which is relevant for the implementation of the
Metric.
[RFC2330] and [RFC2679] emphasize precision as an aim of IPPM metric
implementations. A single IPPM-conforming implementation should
under otherwise identical network conditions produce precise results
for repeated measurements of the same metric.
RFC 2330 prefers the "empirical distribution function" (EDF) to
describe collections of measurements. RFC 2330 determines, that
"unless otherwise stated, IPPM goodness-of-fit tests are done using
5% significance". The goodness-of-fit test determines by which
precision two or more samples of a metric implementation belong to
the same underlying distribution (of measured network performance
events). The goodness-of-fit test suggested for the metric test is
the Anderson-Darling K sample test (ADK sample test, K stands for the
number of samples to be compared) [ADK]. Please note that RFC 2330
and RFC 2679 apply an Anderson-Darling goodness-of-fit test, too.
The results of a repeated test with a single implementation MUST pass
an ADK sample test with a confidence level of 95%. The conditions
for which the ADK test has been passed with the specified confidence
level MUST be documented. To formulate this differently, the
requirement is to document the set of parameters with the smallest
deviation at which the results of the tested metric implementation
pass an ADK test with a confidence level of 95%. The minimum
resolution available in the reported results from each implementation
MUST be taken into account in the ADK test.
The test conditions to be documented for a passed metric test
include:
o The metric resolution at which a test was passed (e.g., the
resolution of timestamps).
o The parameters modified by an impairment generator.
o The impairment generator parameter settings.
3.2. Test Setup Resulting in Identical Live Network Testing Conditions
Two major issues complicate tests for metric compliance across live
networks under identical testing conditions. One is the general
point that metric definition implementations cannot be conveniently
examined in field measurement scenarios. The other one is more
broadly described as "parallelism in devices and networks", including
mechanisms like those that achieve load balancing (see [RFC4928]).
This section proposes two measures to deal with both issues.
Tunneling mechanisms can be used to avoid parallel processing of
different flows in the network. Measuring by separate parallel probe
flows results in repeated collection of data. If both measures are
combined, Wide Area Network (WAN) conditions are identical for a
number of independent measurement flows, no matter what the network
conditions are in detail.
Any measurement setup must be made to avoid the probing traffic
itself to impede the metric measurement. The created measurement
load must not result in congestion at the access link connecting the
measurement implementation to the WAN. The created measurement load
must not overload the measurement implementation itself, e.g., by
causing a high CPU load or by causing timestamp imprecision due to
unwanted queuing while transmitting or receiving test packets.
Tunneling multiple flows destined for a single physical port of a
network element allows transmission of all packets via the same path.
Applying tunnels to avoid undesired influence of standard routing for
measurement purposes is a concept known from literature, see e.g.,
GRE-encapsulated multicast probing [GU-Duffield]. An existing
IP-in-IP tunnel protocol can be applied to avoid Equal-Cost Multi-
Path (ECMP) routing of different measurement streams if it meets the
following criteria:
o Inner IP packets from different measurement implementations are
mapped into a single tunnel with a single outer IP origin and
destination address as well as origin and destination port numbers
that are identical for all packets.
o An easily accessible tunneling protocol allows for carrying out a
metric test from more test sites.
o A low operational overhead may enable a broader audience to set up
a metric test with the desired properties.
o The tunneling protocol should be reliable and stable in setup and
operation to avoid disturbances or influence on the test results.
o The tunneling protocol should not incur any extra cost for those
interested in setting up a metric test.
An illustration of a test setup with two layer 2 tunnels and two
flows between two linecards of one implementation is given in
Figure 1.
Implementation ,---. +--------+
+~~~~~~~~~~~/ \~~~~~~| Remote |
+------->-----F2->-| / \ |->---+ |
| +---------+ | Tunnel 1( ) | | |
| | transmit|-F1->-| ( ) |->+ | |
| | LC1 | +~~~~~~~~~| |~~~~| | | |
| | receive |-<--+ ( ) | F1 F2 |
| +---------+ | |Internet | | | | |
*-------<-----+ F2 | | | | | |
+---------+ | | +~~~~~~~~~| |~~~~| | | |
| transmit|-* *-| | | |--+<-* |
| LC2 | | Tunnel 2( ) | | |
| receive |-<-F1-| \ / |<-* |
+---------+ +~~~~~~~~~~~\ /~~~~~~| Router |
`-+-' +--------+
For simplicity, only two linecards of one implementation and two
flows F between them are shown.
Figure 1: Illustration of a Test Setup with Two Layer 2 Tunnels
Figure 2 shows the network elements required to set up layer 2
tunnels as shown by Figure 1.
Implementation
+-----+ ,---.
| LC1 | / \
+-----+ / \ +------+
| +-------+ ( ) +-------+ |Remote|
+--------+ | | | | | | | |
|Ethernet| | Tunnel| |Internet | | Tunnel| | |
|Switch |--| Head |--| |--| Head |--| |
+--------+ | Router| | | | Router| | |
| | | ( ) | | |Router|
+-----+ +-------+ \ / +-------+ +------+
| LC2 | \ /
+-----+ `-+-'
Figure 2: Illustration of a Hardware Setup to Realize the Test Setup
Illustrated by Figure 1 with Layer 2 Tunnels or Pseudowires
The test setup successfully used during a delay metric test
[TESTPLAN] is given as an example in Figure 3. Note that the shown
setup allows a metric test between two remote sites.
+----+ +----+ +----+ +----+
|LC10| |LC11| ,---. |LC20| |LC21|
+----+ +----+ / \ +-------+ +----+ +----+
| V10 | V11 / \ | Tunnel| | V20 | V21
| | ( ) | Head | | |
+--------+ +------+ | | | Router|__+----------+
|Ethernet| |Tunnel| |Internet | +---B---+ |Ethernet |
|Switch |--|Head |-| | | |Switch |
+-+--+---+ |Router| | | +---+---+ +--+--+----+
|__| +--A---+ ( )--|Option.| |__|
\ / |Impair.|
Bridge \ / |Gener. | Bridge
V20 to V21 `-+-? +-------+ V10 to V11
Figure 3: Example of Test Setup Successfully Used during a Delay
Metic Test
In Figure 3, LC10 identifies measurement clients / linecards. V10
and the others denote VLANs. All VLANs are using the same tunnel
from A to B and in the reverse direction. The remote site VLANs are
U-bridged at the local site Ethernet switch. The measurement packets
of site 1 travel tunnel A->B first, are U-bridged at site 2, and
travel tunnel B->A second. Measurement packets of site 2 travel
tunnel B->A first, are U-bridged at site 1, and travel tunnel A->B
second. So, all measurement packets pass the same tunnel segments,
but in different segment order.
If tunneling is applied, two tunnels MUST carry all test traffic in
between the test site and the remote site. For example, if 802.1Q
Virtual LANs (VLANs) are applied and the measurement streams are
carried in different VLANs, the IP tunnel or pseudowires respectively
are setup in physical port mode to avoid setup of pseudowires per
VLAN (which may see different paths due to ECMP routing); see
[RFC4448]. The remote router and the Ethernet switch shown in
Figure 3 have to support 802.1Q in this setup.
The IP packet size of the metric implementation SHOULD be chosen
small enough to avoid fragmentation due to the added Ethernet and
tunnel headers. Otherwise, the impact of tunnel overhead on
fragmentation and interface MTU size must be understood and taken
into account (see [RFC4459]).
An Ethernet port mode IP tunnel carrying several 802.1Q VLANs each
containing measurement traffic of a single measurement system was
successfully applied when testing compatibility of two metric
implementations [TESTPLAN]. Ethernet over Layer 2 Tunneling Protocol
Version 3 (L2TPv3) [RFC4719] was picked for this test.
The following headers may have to be accounted for when calculating
total packet length, if VLANs and Ethernet over L2TPv3 tunnels are
applied:
o Ethernet 802.1Q: 22 bytes.
o L2TPv3 Header: 4-16 bytes for L2TPv3 data messages over IP; 16-28
bytes for L2TPv3 data messages over UDP.
o IPv4 Header (outer IP header): 20 bytes.
o MPLS Labels may be added by a carrier. Each MPLS Label has a
length of 4 bytes. At the time of this writing, between 1 and 4
Labels seems to be a fair guess of what's expected.
The applicability of one or more of the following tunneling protocols
may be investigated by interested parties if Ethernet over L2TPv3 is
felt to be unsuitable: IP in IP [RFC2003] or Generic Routing
Encapsulation (GRE) [RFC2784]. RFC 4928 [RFC4928] proposes measures
how to avoid ECMP treatment in MPLS networks.
L2TP is a commodity tunneling protocol [RFC2661]. At the time of
this writing, L2TPv3 [RFC3931] is the latest version of L2TP. If
L2TPv3 is applied, software-based implementations of this protocol
are not suitable for the test setup, as such implementations may
cause incalculable delay shifts.
Ethernet pseudowires may also be set up on MPLS networks [RFC4448].
While there is no technical issue with this solution, MPLS interfaces
are mostly found in the network provider domain. Hence, not all of
the above criteria for selecting a tunneling protocol are met.
Note that setting up a metric test environment is not a plug-and-play
issue. Skilled networking engineers should be consulted and involved
if a setup between remote sites is preferred.
Passing or failing an ADK test with 2 samples could be a random
result (note that [RFC2330] defines a sample as a set of singleton
metric values produced by a measurement stream, and we continue to
use this terminology here). The error margin of a statistical test
is higher if the number of samples it is based on is low (the number
of samples taken influences the so-called "degree of freedom" of a
statistical test, and a higher degree of freedom produces more
reliable results). To pass an ADK with higher probability, the
number of samples collected per implementation under identical
networking conditions SHOULD be greater than 2. Hardware and load
constraints may enforce an upper limit on the number of simultaneous
measurement streams. The ADK test allows one to combine different
samples (see Section 9 of [ADK]) and then to run a 2-sample test
between combined samples. At least 4 samples per implementation
captured under identical networking conditions is RECOMMENDED when
comparing different metric implementations by a statistical test.
It is RECOMMENDED that tests be carried out by establishing N
different parallel measurement flows. Two or three linecards per
implementation serving to send or receive measurement flows should be
sufficient to create 4 or more parallel measurement flows. Other
options are to separate flows by DiffServ marks (without deploying
any Quality of Service (QoS) in the inner or outer tunnel) or to use
a single Constant Bitrate (CBR) flow and evaluate whether every n-th
singleton belongs to a specific measurement flow. Note that a
practical test indeed showed that ADK passed with 4 samples even if a
2-sample test failed [TESTPLAN].
Some additional guidelines to calculate and compare samples to
perform a metric test are:
o Comparing different probes of a common underlying distribution in
terms of metrics characterizing a communication network requires
respecting the temporal nature for which the assumption of a
common underlying distribution may hold. Any singletons or
samples to be compared must be captured within the same time
interval.
o If statistical events like rates are used to characterize measured
metrics of a time interval, a minimum of 5 singletons of a
relevant metric should be picked to ensure a minimum confidence
into the reported value. The error margin of the determined rate
depends on the number of singletons (refer to statistical
textbooks on student's t-test). As an example, any packet loss
measurement interval to be compared with the results of another
implementation contains at least five lost packets to have some
confidence that the observed loss rate wasn't caused by a small
number of random packet drops.
o The minimum number of singletons or samples to be compared by an
Anderson-Darling test should be 100 per tested metric
implementation. Note that the Anderson-Darling test detects small
differences in distributions fairly well and will fail for a high
number of compared results (RFC 2330 mentions an example with 8192
measurements where an Anderson-Darling test always failed).
o Generally, the Anderson-Darling test is sensitive to differences
in the accuracy or bias associated with varying implementations or
test conditions. These dissimilarities may result in differing
averages of samples to be compared. An example may be different
packet sizes, resulting in a constant delay difference between
compared samples. Therefore, samples to be compared by an
Anderson-Darling test MAY be calibrated by the difference of the
average values of the samples. Any calibration of this kind MUST
be documented in the test result.
3.3. Tests of Two or More Different Implementations against a Metric
Specification
[RFC2330] expects that "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". A small variation in conditions in the context of the metric
test proposed here can be seen as different implementations measuring
the same metric along the same path.
IPPM metric specifications, however, allow for implementor options to
the largest possible degree. It cannot be expected that two
implementors allow 100% identical options in their implementations.
Testers SHOULD pick the same metric measurement configurations for
their systems when comparing their implementations by a metric test.
In some cases, a goodness-of-fit test may not be possible or show
disappointing results. To clarify the difficulties arising from
different metric implementation options, the individual options
picked for every compared metric implementation should be documented
as specified in Section 3.5. If the cause of the failure is a lack
of specification clarity or multiple legitimate interpretations of
the definition text, the text should be modified and the resulting
memo proposed for consensus and (possible) advancement to Internet
Standard.
The same statistical test as applicable to quantify precision of a
single metric implementation must be used to compare metric result
equivalence for different implementations. To document
compatibility, the smallest measurement resolution at which the
compared implementations passed the ADK sample test must be
documented.
For different implementations of the same metric, "variations in
conditions" are reasonably expected. The ADK test comparing samples
of the different implementations may result in a lower precision than
the test for precision in the same-implementation comparison.
3.4. Clock Synchronization
Clock synchronization effects require special attention. Accuracy of
one-way active delay measurements for any metric implementation
depends on clock synchronization between the source and destination
of tests. Ideally, one-way active delay measurement [RFC2679] test
endpoints either have direct access to independent GPS or CDMA-based
time sources or indirect access to nearby NTP primary (stratum 1)
time sources, equipped with GPS receivers. Access to these time
sources may not be available at all test locations associated with
different Internet paths, for a variety of reasons out of scope of
this document.
When secondary (stratum 2 and above) time sources are used with NTP
running across the same network, whose metrics are subject to
comparative implementation tests, network impairments can affect
clock synchronization and distort sample one-way values and their
interval statistics. Discarding sample one-way delay values for any
implementation is recommended when one of the following reliability
conditions is met:
o Delay is measured and is finite in one direction but not the
other.
o Absolute value of the difference between the sum of one-way
measurements in both directions and the round-trip measurement is
greater than X% of the latter value.
Examination of the second condition requires round-trip time (RTT)
measurement for reference, e.g., based on TWAMP [RFC5357] in
conjunction with one-way delay measurement.
Specification of X% to strike a balance between identification of
unreliable one-way delay samples and misidentification of reliable
samples under a wide range of Internet path RTTs requires further
study.
An IPPM-compliant metric implementation of an RFC that requires
synchronized clocks is expected to provide precise measurement
results.
IF an implementation publishes a specification of its precision, such
as "a precision of 1 ms (+/- 500 us) with a confidence of 95%", then
the specification should be met over a useful measurement duration.
For example, if the metric is measured along an Internet path that is
stable and not congested, then the precision specification should be
met over durations of an hour or more.
3.5. Recommended Metric Verification Measurement Process
In order to meet their obligations under the IETF Standards Process,
the IESG must be convinced that each metric specification advanced to
Internet Standard status is clearly written, that there are a
sufficient number of verified equivalent implementations, and that
options that have been implemented are documented.
In the context of this document, metrics are designed to measure some
characteristic of a data network. An aim of any metric definition
should be that it is specified in a way that can reliably measure the
specific characteristic in a repeatable way across multiple
independent implementations.
Each metric, statistic, or option of those to be validated MUST be
compared against a reference measurement or another implementation as
specified in this document.
Finally, the metric definitions, embodied in the text of the RFCs,
are the objects that require evaluation and possible revision in
order to advance to Internet Standard.
IF two (or more) implementations do not measure an equivalent metric
as specified by this document,
AND sources of measurement error do not adequately explain the lack
of agreement,
THEN the details of each implementation should be audited along with
the exact definition text to determine if there is a lack of clarity
that has caused the implementations to vary in a way that affects the
correspondence of the results.
IF there was a lack of clarity or multiple legitimate interpretations
of the definition text,
THEN the text should be modified and the resulting memo proposed for
consensus and (possible) advancement along the Standards Track.
Finally, all the findings MUST be documented in a report that can
support advancement to Internet Standard, as described here (similar
to the reports described in [RFC5657]). The list of measurement
devices used in testing satisfies the implementation requirement,
while the test results provide information on the quality of each
specification in the metric RFC (the surrogate for feature
interoperability).
The complete process of advancing a metric specification to a
Standard as defined by this document is illustrated in Figure 4.
,---.
/ \
( Start )
\ / Implementations
`-+-' +-------+
| /| 1 `.
+---+----+ / +-------+ `.-----------+ ,-------.
| RFC | / |Check for | ,' was RFC `. YES
| | / |Equivalence.... clause x ------+
| |/ +-------+ |under | `. clear? ,' |
| Metric \.....| 2 ....relevant | `---+---' +----+-----+
| Metric |\ +-------+ |identical | No | |Report |
| Metric | \ |network | +--+----+ |results + |
| ... | \ |conditions | |Modify | |Advance |
| | \ +-------+ | | |Spec +--+RFC |
+--------+ \| n |.'+-----------+ +-------+ |request |
+-------+ +----------+
Figure 4: Illustration of the Metric Standardization Process
Any recommendation for the advancement of a metric specification MUST
be accompanied by an implementation report. The implementation
report needs to include the tests performed, the applied test setup,
the specific metrics in the RFC, and reports of the tests performed
with two or more implementations. The test plan needs to specify the
precision reached for each measured metric and thus define the
meaning of "statistically equivalent" for the specific metrics being
tested.
Ideally, the test plan would co-evolve with the development of the
metric, since that's when participants have the clearest context in
their minds regarding the different subtleties that can arise.
In particular, the implementation report MUST include the following
at minimum:
o The metric compared and the RFC specifying it. This includes
statements as required by Section 3.1 ("Tests of an Individual
Implementation against a Metric Specification") of this document.
o The measurement configuration and setup.
o A complete specification of the measurement stream (mean rate,
statistical distribution of packets, packet size or mean packet
size, and their distribution), Differentiated Services Code Point
(DSCP), and any other measurement stream properties that could
result in deviating results. Deviations in results can also be
caused if chosen IP addresses and ports of different
implementations result in different layer 2 or layer 3 paths due
to operation of Equal Cost Multi-Path routing in an operational
network.
o The duration of each measurement to be used for a metric
validation, the number of measurement points collected for each
metric during each measurement interval (i.e., the probe size),
and the level of confidence derived from this probe size for each
measurement interval.
o The result of the statistical tests performed for each metric
validation as required by Section 3.3 ("Tests of Two or More
Different Implementations against a Metric Specification") of this
document.
o A parameterization of laboratory conditions and applied traffic
and network conditions allowing reproduction of these laboratory
conditions for readers of the implementation report.
o The documentation helping to improve metric specifications defined
by this section.
All of the tests for each set SHOULD be run in a test setup as
specified in Section 3.2 ("Test Setup Resulting in Identical Live
Network Testing Conditions".
If a different test setup is chosen, it is recommended to avoid
effects falsifying results of validation measurements caused by real
data networks (like parallelism in devices and networks). Data
networks may forward packets differently in the case of:
o Different packet sizes chosen for different metric
implementations. A proposed countermeasure is selecting the same
packet size when validating results of two samples or a sample
against an original distribution.
o Selection of differing IP addresses and ports used by different
metric implementations during metric validation tests. If ECMP is
applied on the IP or MPLS level, different paths can result (note
that it may be impossible to detect an MPLS ECMP path from an IP
endpoint). A proposed countermeasure is to connect the
measurement equipment to be compared by a NAT device or establish
a single tunnel to transport all measurement traffic. The aim is
to have the same IP addresses and port for all measurement packets
or to avoid ECMP-based local routing diversion by using a layer 2
tunnel.
o Different IP options.
o Different DSCP.
o If the N measurements are captured using sequential measurements
instead of simultaneous ones, then the following factors come into
play: time varying paths and load conditions.
3.6. Proposal to Determine an Equivalence Threshold for Each Metric
Evaluated
This section describes a proposal for maximum error of equivalence,
based on performance comparison of identical implementations. This
comparison may be useful for both ADK and non-ADK comparisons.
Each metric is tested by two or more implementations (cross-
implementation testing).
Each metric is also tested twice simultaneously by the *same*
implementation, using different Src/Dst Address pairs and other
differences such that the connectivity differences of the cross-
implementation tests are also experienced and measured by the same
implementation.
Comparative results for the same implementation represent a bound on
cross-implementation equivalence. This should be particularly useful
when the metric does *not* produce a continuous distribution of
singleton values, such as with a loss metric or a duplication metric.
Appendix A indicates how the ADK will work for one-way delay and
should be likewise applicable to distributions of delay variation.
Appendix B discusses two possible ways to perform the ADK analysis:
the R statistical language [Rtool] with ADK package [Radk] and C++
code.
Conclusion: the implementation with the largest difference in
homogeneous comparison results is the lower bound on the equivalence
threshold, noting that there may be other systematic errors to
account for when comparing implementations.
Thus, when evaluating equivalence in cross-implementation results:
Maximum_Error = Same_Implementation_Error + Systematic_Error
and only the systematic error need be decided beforehand.
In the case of ADK comparison, the largest same-implementation
resolution of distribution equivalence can be used as a limit on
cross-implementation resolutions (at the same confidence level).
4. Acknowledgements
Gerhard Hasslinger commented a first draft version of this document;
he suggested statistical tests and the evaluation of time series
information. Matthias Wieser's thesis on a metric test resulted in
new input for this document. Henk Uijterwaal and Lars Eggert have
encouraged and helped to organize this work. Mike Hamilton, Scott
Bradner, David Mcdysan, and Emile Stephan commented on this document.
Carol Davids reviewed a version of the document before it became a WG
item.
5. Contributors
Scott Bradner, Vern Paxson, and Allison Mankin drafted [METRICTEST],
and major parts of it are included in this document.
6. Security Considerations
This memo does not raise any specific security issues.
7. References
7.1. Normative References
[RFC2003] Perkins, C., "IP Encapsulation within IP", RFC 2003,
October 1996.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[RFC2330] Paxson, V., Almes, G., Mahdavi, J., and M. Mathis,
"Framework for IP Performance Metrics", RFC 2330,
May 1998.
[RFC2661] Townsley, W., Valencia, A., Rubens, A., Pall, G.,
Zorn, G., and B. Palter, "Layer Two Tunneling Protocol
"L2TP"", RFC 2661, August 1999.
[RFC2679] Almes, G., Kalidindi, S., and M. Zekauskas, "A One-way
Delay Metric for IPPM", RFC 2679, September 1999.
[RFC2784] Farinacci, D., Li, T., Hanks, S., Meyer, D., and P.
Traina, "Generic Routing Encapsulation (GRE)",
RFC 2784, March 2000.
[RFC3931] Lau, J., Townsley, M., and I. Goyret, "Layer Two
Tunneling Protocol - Version 3 (L2TPv3)", RFC 3931,
March 2005.
[RFC4448] Martini, L., Rosen, E., El-Aawar, N., and G. Heron,
"Encapsulation Methods for Transport of Ethernet over
MPLS Networks", RFC 4448, April 2006.
[RFC4656] Shalunov, S., Teitelbaum, B., Karp, A., Boote, J., and
M. Zekauskas, "A One-way Active Measurement Protocol
(OWAMP)", RFC 4656, September 2006.
[RFC4719] Aggarwal, R., Townsley, M., and M. Dos Santos,
"Transport of Ethernet Frames over Layer 2 Tunneling
Protocol Version 3 (L2TPv3)", RFC 4719, November 2006.
[RFC4928] Swallow, G., Bryant, S., and L. Andersson, "Avoiding
Equal Cost Multipath Treatment in MPLS Networks",
BCP 128, RFC 4928, June 2007.
[RFC5657] Dusseault, L. and R. Sparks, "Guidance on
Interoperation and Implementation Reports for
Advancement to Draft Standard", BCP 9, RFC 5657,
September 2009.
[RFC6410] Housley, R., Crocker, D., and E. Burger, "Reducing the
Standards Track to Two Maturity Levels", BCP 9,
RFC 6410, October 2011.
7.2. Informative References
[ADK] Scholz, F. and M. Stephens, "K-sample Anderson-Darling
Tests of Fit, for Continuous and Discrete Cases",
University of Washington, Technical Report No. 81,
May 1986.
[GU-Duffield] Gu, Y., Duffield, N., Breslau, L., and S. Sen, "GRE
Encapsulated Multicast Probing: A Scalable Technique
for Measuring One-Way Loss", SIGMETRICS'07 San Diego,
California, USA, June 2007.
[METRICTEST] Bradner, S. and V. Paxson, "Advancement of metrics
specifications on the IETF Standards Track", Work
in Progress, August 2007.
[RFC2026] Bradner, S., "The Internet Standards Process --
Revision 3", BCP 9, RFC 2026, October 1996.
[RFC4459] Savola, P., "MTU and Fragmentation Issues with In-the-
Network Tunneling", RFC 4459, April 2006.
[RFC5357] Hedayat, K., Krzanowski, R., Morton, A., Yum, K., and
J. Babiarz, "A Two-Way Active Measurement Protocol
(TWAMP)", RFC 5357, October 2008.
[Radk] Scholz, F., "adk: Anderson-Darling K-Sample Test and
Combinations of Such Tests. R package version 1.0",
2008.
[Rtool] R Development Core Team, "R: A language and
environment for statistical computing. R Foundation
for Statistical Computing, Vienna, Austria. ISBN
3-900051-07-0", 2011, <http://www.R-project.org/>.
[TESTPLAN] Ciavattone, L., Geib, R., Morton, A., and M. Wieser,
"Test Plan and Results for Advancing RFC 2679 on the
Standards Track", Work in Progress, March 2012.
Appendix A. An Example on a One-Way Delay Metric Validation
The text of this appendix is not binding. It is an example of what
parts of a One-Way Delay Metric test could look like.
A.1. Compliance to Metric Specification Requirements
One-Way Delay, Loss Threshold, RFC 2679
This test determines if implementations use the same configured
maximum waiting time delay from one measurement to another under
different delay conditions and correctly declare packets arriving in
excess of the waiting time threshold as lost. See Sections 3.5 (3rd
bullet point) and 3.8.2 of [RFC2679].
(1) Configure a path with 1-second one-way constant delay.
(2) Measure one-way delay with 2 or more implementations, using
identical waiting time thresholds for loss set at 2 seconds.
(3) Configure the path with 3-second one-way delay.
(4) Repeat measurements.
(5) Observe that the increase measured in step 4 caused all packets
to be declared lost and that all packets that arrive
successfully in step 2 are assigned a valid one-way delay.
One-Way Delay, First Bit to Last Bit, RFC 2679
This test determines if implementations register the same relative
increase in delay from one measurement to another under different
delay conditions. This test tends to cancel the sources of error
that may be present in an implementation. See Section 3.7.2 of
[RFC2679] and Section 10.2 of [RFC2330].
(1) Configure a path with X ms one-way constant delay and ideally
include a low-speed link.
(2) Measure one-way delay with 2 or more implementations, using
identical options and equal size small packets (e.g., 100 octet
IP payload).
(3) Maintain the same path with X ms one-way delay.
(4) Measure one-way delay with 2 or more implementations, using
identical options and equal size large packets (e.g., 1500 octet
IP payload).
(5) Observe that the increase measured in steps 2 and 4 is
equivalent to the increase in ms expected due to the larger
serialization time for each implementation. Most of the
measurement errors in each system should cancel, if they are
stationary.
One-Way Delay, RFC 2679
This test determines if implementations register the same relative
increase in delay from one measurement to another under different
delay conditions. This test tends to cancel the sources of error
that may be present in an implementation. This test is intended to
evaluate measurements in Sections 3 and 4 of [RFC2679].
(1) Configure a path with X ms one-way constant delay.
(2) Measure one-way delay with 2 or more implementations, using
identical options.
(3) Configure the path with X+Y ms one-way delay.
(4) Repeat measurements.
(5) Observe that the increase measured in steps 2 and 4 is ~Y ms for
each implementation. Most of the measurement errors in each
system should cancel, if they are stationary.
Error Calibration, RFC 2679
This is a simple check to determine if an implementation reports the
error calibration as required in Section 4.8 of [RFC2679]. Note that
the context (Type-P) must also be reported.
A.2. Examples Related to Statistical Tests for One-Way Delay
A one-way delay measurement may pass an ADK test with a timestamp
result of 1 ms. The same test may fail if timestamps with a
resolution of 100 microseconds are evaluated. The implementation is
then conforming to the metric specification up to a timestamp
resolution of 1 ms.
Let's assume another one-way delay measurement comparison between
implementation 1 probing with a frequency of 2 probes per second and
implementation 2 probing at a rate of 2 probes every 3 minutes. To
ensure reasonable confidence in results, sample metrics are
calculated from at least 5 singletons per compared time interval.
This means that sample delay values are calculated for each system
for identical 6-minute intervals for the duration of the whole test.
Per 6-minute interval, the sample metric is calculated from 720
singletons for implementation 1 and from 6 singletons for
implementation 2. Note that if outliers are not filtered, moving
averages are an option for an evaluation too. The minimum move of an
averaging interval is three minutes in this example.
The data in Table 1 may result from measuring one-way delay with
implementation 1 (see column Implemnt_1) and implementation 2 (see
column Implemnt_2). Each data point in the table represents a
(rounded) average of the sampled delay values per interval. The
resolution of the clock is one micro-second. The difference in the
delay values may result, e.g., from different probe packet sizes.
+------------+------------+-----------------------------+
| Implemnt_1 | Implemnt_2 | Implemnt_2 - Delta_Averages |
+------------+------------+-----------------------------+
| 5000 | 6549 | 4997 |
| 5008 | 6555 | 5003 |
| 5012 | 6564 | 5012 |
| 5015 | 6565 | 5013 |
| 5019 | 6568 | 5016 |
| 5022 | 6570 | 5018 |
| 5024 | 6573 | 5021 |
| 5026 | 6575 | 5023 |
| 5027 | 6577 | 5025 |
| 5029 | 6580 | 5028 |
| 5030 | 6585 | 5033 |
| 5032 | 6586 | 5034 |
| 5034 | 6587 | 5035 |
| 5036 | 6588 | 5036 |
| 5038 | 6589 | 5037 |
| 5039 | 6591 | 5039 |
| 5041 | 6592 | 5040 |
| 5043 | 6599 | 5047 |
| 5046 | 6606 | 5054 |
| 5054 | 6612 | 5060 |
+------------+------------+-----------------------------+
Table 1
Average values of sample metrics captured during identical time
intervals are compared. This excludes random differences caused by
differing probing intervals or differing temporal distance of
singletons resulting from their Poisson-distributed sending times.
In the example, 20 values have been picked (note that at least 100
values are recommended for a single run of a real test). Data must
be ordered by ascending rank. The data of Implemnt_1 and Implemnt_2
as shown in the first two columns of Table 1 clearly fails an ADK
test with 95% confidence.
The results of Implemnt_2 are now reduced by the difference of the
averages of column 2 (rounded to 6581 us) and column 1 (rounded to
5029 us), which is 1552 us. The result may be found in column 3 of
Table 1. Comparing column 1 and column 3 of the table by an ADK test
shows that the data contained in these columns passes an ADK test
with 95% confidence.
Comment: Extensive averaging was used in this example because of the
vastly different sampling frequencies. As a result, the
distributions compared do not exactly align with a metric in
[RFC2679] but illustrate the ADK process adequately.
Appendix B. Anderson-Darling K-sample Reference and 2 Sample C++ Code
There are many statistical tools available, and this appendix
describes two that are familiar to the authors.
The "R tool" is a language and command-line environment for
statistical computing and plotting [Rtool]. With the optional "adk"
package installed [Radk], it can perform individual and combined
sample ADK computations. The user must consult the package
documentation and the original paper [ADK] to interpret the results,
but this is as it should be.
The C++ code below will perform an AD2-sample comparison when
compiled and presented with two column vectors in a file (using white
space as separation). This version contains modifications made by
Wes Eddy in Sept 2011 to use the vectors and run as a stand-alone
module. The status of the comparison can be checked on the command
line with "$ echo $?" or the last line can be replaced with a printf
statement for adk_result instead.
/*
Copyright (c) 2012 IETF Trust and the persons identified
as authors of the code. All rights reserved.
Redistribution and use in source and binary forms, with
or without modification, is permitted pursuant to, and subject
to the license terms contained in, the Simplified BSD License
set forth in Section 4.c of the IETF Trust's Legal Provisions
Relating to IETF Documents (http://trustee.ietf.org/license-info).
*/
/* Routines for computing the Anderson-Darling 2 sample
* test statistic.
*
* Implemented based on the description in
* "Anderson-Darling K Sample Test" Heckert, Alan and
* Filliben, James, editors, Dataplot Reference Manual,
* Chapter 15 Auxiliary, NIST, 2004.
* Official Reference by 2010
* Heckert, N. A. (2001). Dataplot website at the
* National Institute of Standards and Technology:
* http://www.itl.nist.gov/div898/software/dataplot.html/
* June 2001.
*/
#include <iostream>
#include <fstream>
#include <vector>
#include <sstream>
using namespace std;
int main() {
vector<double> vec1, vec2;
double adk_result;
static int k, val_st_z_samp1, val_st_z_samp2,
val_eq_z_samp1, val_eq_z_samp2,
j, n_total, n_sample1, n_sample2, L,
max_number_samples, line, maxnumber_z;
static int column_1, column_2;
static double adk, n_value, z, sum_adk_samp1,
sum_adk_samp2, z_aux;
static double H_j, F1j, hj, F2j, denom_1_aux, denom_2_aux;
static bool next_z_sample2, equal_z_both_samples;
static int stop_loop1, stop_loop2, stop_loop3,old_eq_line2,
old_eq_line1;
static double adk_criterium = 1.993;
/* vec1 and vec2 to be initialized with sample 1 and
* sample 2 values in ascending order */
while (!cin.eof()) {
double f1, f2;
cin >> f1;
cin >> f2;
vec1.push_back(f1);
vec2.push_back(f2);
}
k = 2;
n_sample1 = vec1.size() - 1;
n_sample2 = vec2.size() - 1;
// -1 because vec[0] is a dummy value
n_total = n_sample1 + n_sample2;
/* value equal to the line with a value = zj in sample 1.
* Here j=1, so the line is 1.
*/
val_eq_z_samp1 = 1;
/* value equal to the line with a value = zj in sample 2.
* Here j=1, so the line is 1.
*/
val_eq_z_samp2 = 1;
/* value equal to the last line with a value < zj
* in sample 1. Here j=1, so the line is 0.
*/
val_st_z_samp1 = 0;
/* value equal to the last line with a value < zj
* in sample 1. Here j=1, so the line is 0.
*/
val_st_z_samp2 = 0;
sum_adk_samp1 = 0;
sum_adk_samp2 = 0;
j = 1;
// as mentioned above, j=1
equal_z_both_samples = false;
next_z_sample2 = false;
//assuming the next z to be of sample 1
stop_loop1 = n_sample1 + 1;
// + 1 because vec[0] is a dummy, see n_sample1 declaration
stop_loop2 = n_sample2 + 1;
stop_loop3 = n_total + 1;
/* The required z values are calculated until all values
* of both samples have been taken into account. See the
* lines above for the stoploop values. Construct required
* to avoid a mathematical operation in the while condition.
*/
while (((stop_loop1 > val_eq_z_samp1)
|| (stop_loop2 > val_eq_z_samp2)) && stop_loop3 > j)
{
if(val_eq_z_samp1 < n_sample1+1)
{
/* here, a preliminary zj value is set.
* See below how to calculate the actual zj.
*/
z = vec1[val_eq_z_samp1];
/* this while sequence calculates the number of values
* equal to z.
*/
while ((val_eq_z_samp1+1 < n_sample1)
&& z == vec1[val_eq_z_samp1+1] )
{
val_eq_z_samp1++;
}
}
else
{
val_eq_z_samp1 = 0;
val_st_z_samp1 = n_sample1;
// this should be val_eq_z_samp1 - 1 = n_sample1
}
if(val_eq_z_samp2 < n_sample2+1)
{
z_aux = vec2[val_eq_z_samp2];;
/* this while sequence calculates the number of values
* equal to z_aux
*/
while ((val_eq_z_samp2+1 < n_sample2)
&& z_aux == vec2[val_eq_z_samp2+1] )
{
val_eq_z_samp2++;
}
/* the smaller of the two actual data values is picked
* as the next zj.
*/
if(z > z_aux)
{
z = z_aux;
next_z_sample2 = true;
}
else
{
if (z == z_aux)
{
equal_z_both_samples = true;
}
/* This is the case if the last value of column1 is
* smaller than the remaining values of column2.
*/
if (val_eq_z_samp1 == 0)
{
z = z_aux;
next_z_sample2 = true;
}
}
}
else
{
val_eq_z_samp2 = 0;
val_st_z_samp2 = n_sample2;
// this should be val_eq_z_samp2 - 1 = n_sample2
}
/* in the following, sum j = 1 to L is calculated for
* sample 1 and sample 2.
*/
if (equal_z_both_samples)
{
/* hj is the number of values in the combined sample
* equal to zj
*/
hj = val_eq_z_samp1 - val_st_z_samp1
+ val_eq_z_samp2 - val_st_z_samp2;
/* H_j is the number of values in the combined sample
* smaller than zj plus one half the number of
* values in the combined sample equal to zj
* (that's hj/2).
*/
H_j = val_st_z_samp1 + val_st_z_samp2
+ hj / 2;
/* F1j is the number of values in the 1st sample
* that are less than zj plus one half the number
* of values in this sample that are equal to zj.
*/
F1j = val_st_z_samp1 + (double)
(val_eq_z_samp1 - val_st_z_samp1) / 2;
/* F2j is the number of values in the 1st sample
* that are less than zj plus one half the number
* of values in this sample that are equal to zj.
*/
F2j = val_st_z_samp2 + (double)
(val_eq_z_samp2 - val_st_z_samp2) / 2;
/* set the line of values equal to zj to the
* actual line of the last value picked for zj.
*/
val_st_z_samp1 = val_eq_z_samp1;
/* Set the line of values equal to zj to the actual
* line of the last value picked for zj of each
* sample. This is required as data smaller than zj
* is accounted differently than values equal to zj.
*/
val_st_z_samp2 = val_eq_z_samp2;
/* next the lines of the next values z, i.e., zj+1
* are addressed.
*/
val_eq_z_samp1++;
/* next the lines of the next values z, i.e.,
* zj+1 are addressed
*/
val_eq_z_samp2++;
}
else
{
/* the smaller z value was contained in sample 2;
* hence, this value is the zj to base the following
* calculations on.
*/
if (next_z_sample2)
{
/* hj is the number of values in the combined
* sample equal to zj; in this case, these are
* within sample 2 only.
*/
hj = val_eq_z_samp2 - val_st_z_samp2;
/* H_j is the number of values in the combined sample
* smaller than zj plus one half the number of
* values in the combined sample equal to zj
* (that's hj/2).
*/
H_j = val_st_z_samp1 + val_st_z_samp2
+ hj / 2;
/* F1j is the number of values in the 1st sample that
* are less than zj plus one half the number of values in
* this sample that are equal to zj.
* As val_eq_z_samp2 < val_eq_z_samp1, these are the
* val_st_z_samp1 only.
*/
F1j = val_st_z_samp1;
/* F2j is the number of values in the 1st sample that
* are less than zj plus one half the number of values in
* this sample that are equal to zj. The latter are from
* sample 2 only in this case.
*/
F2j = val_st_z_samp2 + (double)
(val_eq_z_samp2 - val_st_z_samp2) / 2;
/* Set the line of values equal to zj to the actual line
* of the last value picked for zj of sample 2 only in
* this case.
*/
val_st_z_samp2 = val_eq_z_samp2;
/* next the line of the next value z, i.e., zj+1 is
* addressed. Here, only sample 2 must be addressed.
*/
val_eq_z_samp2++;
if (val_eq_z_samp1 == 0)
{
val_eq_z_samp1 = stop_loop1;
}
}
/* the smaller z value was contained in sample 2;
* hence, this value is the zj to base the following
* calculations on.
*/
else
{
/* hj is the number of values in the combined
* sample equal to zj; in this case, these are
* within sample 1 only.
*/
hj = val_eq_z_samp1 - val_st_z_samp1;
/* H_j is the number of values in the combined
* sample smaller than zj plus one half the number
* of values in the combined sample equal to zj
* (that's hj/2).
*/
H_j = val_st_z_samp1 + val_st_z_samp2
+ hj / 2;
/* F1j is the number of values in the 1st sample that
* are less than zj plus; in this case, these are within
* sample 1 only one half the number of values in this
* sample that are equal to zj. The latter are from
* sample 1 only in this case.
*/
F1j = val_st_z_samp1 + (double)
(val_eq_z_samp1 - val_st_z_samp1) / 2;
/* F2j is the number of values in the 1st sample that
* are less than zj plus one half the number of values
* in this sample that are equal to zj. As
* val_eq_z_samp1 < val_eq_z_samp2, these are the
* val_st_z_samp2 only.
*/
F2j = val_st_z_samp2;
/* Set the line of values equal to zj to the actual line
* of the last value picked for zj of sample 1 only in
* this case.
*/
val_st_z_samp1 = val_eq_z_samp1;
/* next the line of the next value z, i.e., zj+1 is
* addressed. Here, only sample 1 must be addressed.
*/
val_eq_z_samp1++;
if (val_eq_z_samp2 == 0)
{
val_eq_z_samp2 = stop_loop2;
}
}
}
denom_1_aux = n_total * F1j - n_sample1 * H_j;
denom_2_aux = n_total * F2j - n_sample2 * H_j;
sum_adk_samp1 = sum_adk_samp1 + hj
* (denom_1_aux * denom_1_aux) /
(H_j * (n_total - H_j)
- n_total * hj / 4);
sum_adk_samp2 = sum_adk_samp2 + hj
* (denom_2_aux * denom_2_aux) /
(H_j * (n_total - H_j)
- n_total * hj / 4);
next_z_sample2 = false;
equal_z_both_samples = false;
/* index to count the z. It is only required to prevent
* the while slope to execute endless
*/
j++;
}
// calculating the adk value is the final step.
adk_result = (double) (n_total - 1) / (n_total
* n_total * (k - 1))
* (sum_adk_samp1 / n_sample1
+ sum_adk_samp2 / n_sample2);
/* if(adk_result <= adk_criterium)
* adk_2_sample test is passed
*/
return adk_result <= adk_criterium;
}
Appendix C. Glossary
+-------------+-----------------------------------------------------+
| ADK | Anderson-Darling K-Sample test, a test used to |
| | check whether two samples have the same statistical |
| | distribution. |
| ECMP | Equal Cost Multipath, a load-balancing mechanism |
| | evaluating MPLS Labels stacks, IP addresses, and |
| | ports. |
| EDF | The "empirical distribution function" 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 smaller than or equal to x. |
| Metric | A measured quantity related to the performance and |
| | reliability of the Internet, expressed by a value. |
| | This could be a singleton (single value), a sample |
| | of single values, or a statistic based on a sample |
| | of singletons. |
| OWAMP | One-Way Active Measurement Protocol, a protocol for |
| | communication between IPPM measurement systems |
| | specified by IPPM. |
| OWD | One-Way Delay, a performance metric specified by |
| | IPPM. |
| Sample | A sample metric is derived from a given singleton |
| metric | metric by evaluating a number of distinct instances |
| | together. |
| Singleton | A singleton metric is, in a sense, one atomic |
| metric | measurement of this metric. |
| Statistical | A 'statistical' metric is derived from a given |
| metric | sample metric by computing some statistic of the |
| | values defined by the singleton metric on the |
| | sample. |
| TWAMP | Two-way Active Measurement Protocol, a protocol for |
| | communication between IPPM measurement systems |
| | specified by IPPM. |
+-------------+-----------------------------------------------------+
Authors' Addresses
Ruediger Geib (editor)
Deutsche Telekom
Heinrich Hertz Str. 3-7
Darmstadt 64295
Germany
Phone: +49 6151 58 12747
EMail: Ruediger.Geib@telekom.de
Al Morton
AT&T Labs
200 Laurel Avenue South
Middletown, NJ 07748
USA
Phone: +1 732 420 1571
Fax: +1 732 368 1192
EMail: acmorton@att.com
URI: http://home.comcast.net/~acmacm/
Reza Fardid
Cariden Technologies
888 Villa Street, Suite 500
Mountain View, CA 94041
USA
Phone:
EMail: rfardid@cariden.com
Alexander Steinmitz
Deutsche Telekom
Memmelsdorfer Str. 209b
Bamberg 96052
Germany
Phone:
EMail: Alexander.Steinmitz@telekom.de