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rfc:rfc6049

Internet Engineering Task Force (IETF) A. Morton Request for Comments: 6049 AT&T Labs Category: Standards Track E. Stephan ISSN: 2070-1721 France Telecom Orange

                                                          January 2011
                   Spatial Composition of Metrics

Abstract

 This memo utilizes IP performance metrics that are applicable to both
 complete paths and sub-paths, and it defines relationships to compose
 a complete path metric from the sub-path metrics with some accuracy
 with regard to the actual metrics.  This is called "spatial
 composition" in RFC 2330.  The memo refers to the framework for
 metric composition, and provides background and motivation for
 combining metrics to derive others.  The descriptions of several
 composed metrics and statistics follow.

Status of This Memo

 This is an Internet Standards Track document.
 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
 Internet Standards 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/rfc6049.

Morton & Stephan Standards Track [Page 1] RFC 6049 Spatial Composition January 2011

Copyright Notice

 Copyright (c) 2011 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
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 This document may contain material from IETF Documents or IETF
 Contributions published or made publicly available before November
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 material may not have granted the IETF Trust the right to allow
 modifications of such material outside the IETF Standards Process.
 Without obtaining an adequate license from the person(s) controlling
 the copyright in such materials, this document may not be modified
 outside the IETF Standards Process, and derivative works of it may
 not be created outside the IETF Standards Process, except to format
 it for publication as an RFC or to translate it into languages other
 than English.

Morton & Stephan Standards Track [Page 2] RFC 6049 Spatial Composition January 2011

Table of Contents

 1. Introduction ....................................................4
    1.1. Motivation .................................................6
    1.2. Requirements Language ......................................6
 2. Scope and Application ...........................................6
    2.1. Scope of Work ..............................................6
    2.2. Application ................................................7
    2.3. Incomplete Information .....................................7
 3. Common Specifications for Composed Metrics ......................8
    3.1. Name: Type-P ...............................................8
         3.1.1. Metric Parameters ...................................8
         3.1.2. Definition and Metric Units .........................9
         3.1.3. Discussion and Other Details ........................9
         3.1.4. Statistic ...........................................9
         3.1.5. Composition Function ................................9
         3.1.6. Statement of Conjecture and Assumptions ............10
         3.1.7. Justification of the Composition Function ..........10
         3.1.8. Sources of Deviation from the Ground Truth .........10
         3.1.9. Specific Cases where the Conjecture Might Fail .....11
         3.1.10. Application of Measurement Methodology ............12
 4. One-Way Delay Composed Metrics and Statistics ..................12
    4.1. Name: Type-P-Finite-One-way-Delay-<Sample>-Stream .........12
         4.1.1. Metric Parameters ..................................12
         4.1.2. Definition and Metric Units ........................12
         4.1.3. Discussion and Other Details .......................13
         4.1.4. Statistic ..........................................13
    4.2. Name: Type-P-Finite-Composite-One-way-Delay-Mean ..........13
         4.2.1. Metric Parameters ..................................13
         4.2.2. Definition and Metric Units of the Mean Statistic ..14
         4.2.3. Discussion and Other Details .......................14
         4.2.4. Statistic ..........................................14
         4.2.5. Composition Function: Sum of Means .................14
         4.2.6. Statement of Conjecture and Assumptions ............15
         4.2.7. Justification of the Composition Function ..........15
         4.2.8. Sources of Deviation from the Ground Truth .........15
         4.2.9. Specific Cases where the Conjecture Might Fail .....15
         4.2.10. Application of Measurement Methodology ............16
    4.3. Name: Type-P-Finite-Composite-One-way-Delay-Minimum .......16
         4.3.1. Metric Parameters ..................................16
         4.3.2. Definition and Metric Units of the Minimum
                Statistic ..........................................16
         4.3.3. Discussion and Other Details .......................16
         4.3.4. Statistic ..........................................16
         4.3.5. Composition Function: Sum of Minima ................16
         4.3.6. Statement of Conjecture and Assumptions ............17
         4.3.7. Justification of the Composition Function ..........17
         4.3.8. Sources of Deviation from the Ground Truth .........17

Morton & Stephan Standards Track [Page 3] RFC 6049 Spatial Composition January 2011

         4.3.9. Specific Cases where the Conjecture Might Fail .....17
         4.3.10. Application of Measurement Methodology ............17
 5. Loss Metrics and Statistics ....................................18
    5.1. Type-P-Composite-One-way-Packet-Loss-Empirical-Probability 18
         5.1.1. Metric Parameters ..................................18
         5.1.2. Definition and Metric Units ........................18
         5.1.3. Discussion and Other Details .......................18
         5.1.4. Statistic:
                Type-P-One-way-Packet-Loss-Empirical-Probability ...18
         5.1.5. Composition Function: Composition of
                Empirical Probabilities ............................18
         5.1.6. Statement of Conjecture and Assumptions ............19
         5.1.7. Justification of the Composition Function ..........19
         5.1.8. Sources of Deviation from the Ground Truth .........19
         5.1.9. Specific Cases where the Conjecture Might Fail .....19
         5.1.10. Application of Measurement Methodology ............19
 6. Delay Variation Metrics and Statistics .........................20
    6.1. Name: Type-P-One-way-pdv-refmin-<Sample>-Stream ...........20
         6.1.1. Metric Parameters ..................................20
         6.1.2. Definition and Metric Units ........................20
         6.1.3. Discussion and Other Details .......................21
         6.1.4. Statistics: Mean, Variance, Skewness, Quantile .....21
         6.1.5. Composition Functions ..............................22
         6.1.6. Statement of Conjecture and Assumptions ............23
         6.1.7. Justification of the Composition Function ..........23
         6.1.8. Sources of Deviation from the Ground Truth .........23
         6.1.9. Specific Cases where the Conjecture Might Fail .....24
         6.1.10. Application of Measurement Methodology ............24
 7. Security Considerations ........................................24
    7.1. Denial-of-Service Attacks .................................24
    7.2. User Data Confidentiality .................................24
    7.3. Interference with the Metrics .............................24
 8. IANA Considerations ............................................25
 9. Contributors and Acknowledgements ..............................27
 10. References ....................................................28
    10.1. Normative References .....................................28
    10.2. Informative References ...................................28

1. Introduction

 The IP Performance Metrics (IPPM) framework [RFC2330] describes two
 forms of metric composition: spatial and temporal.  The composition
 framework [RFC5835] expands and further qualifies these original
 forms into three categories.  This memo describes spatial
 composition, one of the categories of metrics under the umbrella of
 the composition framework.

Morton & Stephan Standards Track [Page 4] RFC 6049 Spatial Composition January 2011

 Spatial composition encompasses the definition of performance metrics
 that are applicable to a complete path, based on metrics collected on
 various sub-paths.
 The main purpose of this memo is to define the deterministic
 functions that yield the complete path metrics using metrics of the
 sub-paths.  The effectiveness of such metrics is dependent on their
 usefulness in analysis and applicability with practical measurement
 methods.
 The relationships may involve conjecture, and [RFC2330] lists four
 points that the metric definitions should include:
 o  the specific conjecture applied to the metric and assumptions of
    the statistical model of the process being measured (if any; see
    [RFC2330], Section 12),
 o  a justification of the practical utility of the composition in
    terms of making accurate measurements of the metric on the path,
 o  a justification of the usefulness of the composition in terms of
    making analysis of the path using A-frame concepts more effective,
    and
 o  an analysis of how the conjecture could be incorrect.
 Also, [RFC2330] gives an example using the conjecture that the delay
 of a path is very nearly the sum of the delays of the exchanges and
 clouds of the corresponding path digest.  This example is
 particularly relevant to those who wish to assess the performance of
 an inter-domain path without direct measurement, and the performance
 estimate of the complete path is related to the measured results for
 various sub-paths instead.
 Approximate functions between the sub-path and complete path metrics
 are useful, with knowledge of the circumstances where the
 relationships are/are not applicable.  For example, we would not
 expect that delay singletons from each sub-path would sum to produce
 an accurate estimate of a delay singleton for the complete path
 (unless all the delays were essentially constant -- very unlikely).
 However, other delay statistics (based on a reasonable sample size)
 may have a sufficiently large set of circumstances where they are
 applicable.

Morton & Stephan Standards Track [Page 5] RFC 6049 Spatial Composition January 2011

1.1. Motivation

 One-way metrics defined in other RFCs (such as [RFC2679] and
 [RFC2680]) all assume that the measurement can be practically carried
 out between the source and the destination of interest.  Sometimes
 there are reasons that the measurement cannot be executed from the
 source to the destination.  For instance, the measurement path may
 cross several independent domains that have conflicting policies,
 measurement tools and methods, and measurement time assignment.  The
 solution then may be the composition of several sub-path
 measurements.  This means each domain performs the one-way
 measurement on a sub-path between two nodes that are involved in the
 complete path, following its own policy, using its own measurement
 tools and methods, and using its own measurement timing.  Under the
 appropriate conditions, one can combine the sub-path one-way metric
 results to estimate the complete path one-way measurement metric with
 some degree of accuracy.

1.2. 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].
 In this memo, the characters "<=" should be read as "less than or
 equal to" and ">=" as "greater than or equal to".

2. Scope and Application

2.1. Scope of Work

 For the primary IP Performance Metrics RFCs for loss [RFC2680], delay
 [RFC2679], and delay variation [RFC3393], this memo gives a set of
 metrics that can be composed from the same or similar sub-path
 metrics.  This means that the composition function may utilize:
 o  the same metric for each sub-path;
 o  multiple metrics for each sub-path (possibly one that is the same
    as the complete path metric);
 o  a single sub-path metric that is different from the complete path
    metric;
 o  different measurement techniques like active [RFC2330], [RFC3432]
    and passive [RFC5474].

Morton & Stephan Standards Track [Page 6] RFC 6049 Spatial Composition January 2011

 We note a possibility: using a complete path metric and all but one
 sub-path metric to infer the performance of the missing sub-path,
 especially when the "last" sub-path metric is missing.  However, such
 de-composition calculations, and the corresponding set of issues they
 raise, are beyond the scope of this memo.

2.2. Application

 The composition framework [RFC5835] requires the specification of the
 applicable circumstances for each metric.  In particular, each
 section addresses whether the metric:
 o  Requires the same test packets to traverse all sub-paths or may
    use similar packets sent and collected separately in each
    sub-path.
 o  Requires homogeneity of measurement methodologies or can allow a
    degree of flexibility (e.g., active, active spatial division
    [RFC5644], or passive methods produce the "same" metric).  Also,
    the applicable sending streams will be specified, such as Poisson,
    Periodic, or both.
 o  Needs information or access that will only be available within an
    operator's domain, or is applicable to inter-domain composition.
 o  Requires synchronized measurement start and stop times in all
    sub-paths or largely overlapping measurement intervals, or no
    timing requirements.
 o  Requires the assumption of sub-path independence with regard to
    the metric being defined/composed or other assumptions.
 o  Has known sources of inaccuracy/error and identifies the sources.

2.3. Incomplete Information

 In practice, when measurements cannot be initiated on a sub-path (and
 perhaps the measurement system gives up during the test interval),
 then there will not be a value for the sub-path reported, and the
 entire test result SHOULD be recorded as "undefined".  This case
 should be distinguished from the case where the measurement system
 continued to send packets throughout the test interval, but all were
 declared lost.
 When a composed metric requires measurements from sub-paths A, B, and
 C, and one or more of the sub-path results are undefined, then the
 composed metric SHOULD also be recorded as undefined.

Morton & Stephan Standards Track [Page 7] RFC 6049 Spatial Composition January 2011

3. Common Specifications for Composed Metrics

 To reduce the redundant information presented in the detailed metrics
 sections that follow, this section presents the specifications that
 are common to two or more metrics.  The section is organized using
 the same subsections as the individual metrics, to simplify
 comparisons.
 Also, the index variables are represented as follows:
 o  m = index for packets sent.
 o  n = index for packets received.
 o  s = index for involved sub-paths.

3.1. Name: Type-P

 All metrics use the "Type-P" convention as described in [RFC2330].
 The rest of the name is unique to each metric.

3.1.1. Metric Parameters

 o  Src, the IP address of a host.
 o  Dst, the IP address of a host.
 o  T, a time (start of test interval).
 o  Tf, a time (end of test interval).
 o  lambda, a rate in reciprocal seconds (for Poisson Streams).
 o  incT, the nominal duration of inter-packet interval, first bit to
    first bit (for Periodic Streams).
 o  dT, the duration of the allowed interval for Periodic Stream
    sample start times.
 o  T0, a time that MUST be selected at random from the interval
    [T, T + dT] to start generating packets and taking measurements
    (for Periodic Streams).
 o  TstampSrc, the wire time of the packet as measured at MP(Src)
    (measurement point at the source).
 o  TstampDst, the wire time of the packet as measured at MP(Dst),
    assigned to packets that arrive within a "reasonable" time.

Morton & Stephan Standards Track [Page 8] RFC 6049 Spatial Composition January 2011

 o  Tmax, a maximum waiting time for packets at the destination, set
    sufficiently long to disambiguate packets with long delays from
    packets that are discarded (lost); thus, the distribution of delay
    is not truncated.
 o  M, the total number of packets sent between T0 and Tf.
 o  N, the total number of packets received at Dst (sent between T0
    and Tf).
 o  S, the number of sub-paths involved in the complete Src-Dst path.
 o  Type-P, as defined in [RFC2330], which includes any field that may
    affect a packet's treatment as it traverses the network.
 In metric names, the term "<Sample>" is intended to be replaced by
 the name of the method used to define a sample of values of parameter
 TstampSrc.  This can be done in several ways, including:
 1.  Poisson: a pseudo-random Poisson process of rate lambda, whose
     values fall between T and Tf.  The time interval between
     successive values of TstampSrc will then average 1/lambda, as per
     [RFC2330].
 2.  Periodic: a Periodic stream process with pseudo-random start time
     T0 between T and dT, and nominal inter-packet interval incT, as
     per [RFC3432].

3.1.2. Definition and Metric Units

 This section is unique for every metric.

3.1.3. Discussion and Other Details

 This section is unique for every metric.

3.1.4. Statistic

 This section is unique for every metric.

3.1.5. Composition Function

 This section is unique for every metric.

Morton & Stephan Standards Track [Page 9] RFC 6049 Spatial Composition January 2011

3.1.6. Statement of Conjecture and Assumptions

 This section is unique for each metric.  The term "ground truth" is
 frequently used in these sections and is defined in Section 4.7 of
 [RFC5835].

3.1.7. Justification of the Composition Function

 It is sometimes impractical to conduct active measurements between
 every Src-Dst pair.  Since the full mesh of N measurement points
 grows as N x N, the scope of measurement may be limited by testing
 resources.
 There may be varying limitations on active testing in different parts
 of the network.  For example, it may not be possible to collect the
 desired sample size in each test interval when access link speed is
 limited, because of the potential for measurement traffic to degrade
 the user traffic performance.  The conditions on a low-speed access
 link may be understood well enough to permit use of a small sample
 size/rate, while a larger sample size/rate may be used on other
 sub-paths.
 Also, since measurement operations have a real monetary cost, there
 is value in re-using measurements where they are applicable, rather
 than launching new measurements for every possible source-destination
 pair.

3.1.8. Sources of Deviation from the Ground Truth

3.1.8.1. Sub-Path List Differs from Complete Path

 The measurement packets, each having source and destination addresses
 intended for collection at edges of the sub-path, may take a
 different specific path through the network equipment and links when
 compared to packets with the source and destination addresses of the
 complete path.  Example sources of parallel paths include Equal Cost
 Multi-Path and parallel (or bundled) links.  Therefore, the
 performance estimated from the composition of sub-path measurements
 may differ from the performance experienced by packets on the
 complete path.  Multiple measurements employing sufficient sub-path
 address pairs might produce bounds on the extent of this error.
 We also note the possibility of re-routing during a measurement
 interval, as it may affect the correspondence between packets
 traversing the complete path and the sub-paths that were "involved"
 prior to the re-route.

Morton & Stephan Standards Track [Page 10] RFC 6049 Spatial Composition January 2011

3.1.8.2. Sub-Path Contains Extra Network Elements

 Related to the case of an alternate path described above is the case
 where elements in the measured path are unique to measurement system
 connectivity.  For example, a measurement system may use a dedicated
 link to a LAN switch, and packets on the complete path do not
 traverse that link.  The performance of such a dedicated link would
 be measured continuously, and its contribution to the sub-path
 metrics SHOULD be minimized as a source of error.

3.1.8.3. Sub-Paths Have Incomplete Coverage

 Measurements of sub-path performance may not cover all the network
 elements on the complete path.  For example, the network exchange
 points might be excluded unless a cooperative measurement is
 conducted.  In this example, test packets on the previous sub-path
 are received just before the exchange point, and test packets on the
 next sub-path are injected just after the same exchange point.
 Clearly, the set of sub-path measurements SHOULD cover all critical
 network elements in the complete path.

3.1.8.4. Absence of Route

 At a specific point in time, no viable route exists between the
 complete path source and destination.  The routes selected for one or
 more sub-paths therefore differ from the complete path.
 Consequently, spatial composition may produce finite estimation of a
 ground truth metric (see Section 4.7 of [RFC5835]) between a source
 and a destination, even when the route between them is undefined.

3.1.9. Specific Cases where the Conjecture Might Fail

 This section is unique for most metrics (see the metric-specific
 sections).
 For delay-related metrics, one-way delay always depends on packet
 size and link capacity, since it is measured in [RFC2679] from first
 bit to last bit.  If the size of an IP packet changes on its route
 (due to encapsulation), this can influence delay performance.
 However, the main error source may be the additional processing
 associated with encapsulation and encryption/decryption if not
 experienced or accounted for in sub-path measurements.
 Fragmentation is a major issue for composition accuracy, since all
 metrics require all fragments to arrive before proceeding, and
 fragmented complete path performance is likely to be different from
 performance with non-fragmented packets and composed metrics based on
 non-fragmented sub-path measurements.

Morton & Stephan Standards Track [Page 11] RFC 6049 Spatial Composition January 2011

 Highly manipulated routing can cause measurement error if not
 expected and compensated for.  For example, policy-based MPLS routing
 could modify the class of service for the sub-paths and complete
 path.

3.1.10. Application of Measurement Methodology

 o  The methodology SHOULD use similar packets sent and collected
    separately in each sub-path, where "similar" in this case means
    that Type-P contains as many equal attributes as possible, while
    recognizing that there will be differences.  Note that Type-P
    includes stream characteristics (e.g., Poisson, Periodic).
 o  The methodology allows a degree of flexibility regarding test
    stream generation (e.g., active or passive methods can produce an
    equivalent result, but the lack of control over the source,
    timing, and correlation of passive measurements is much more
    challenging).
 o  Poisson and/or Periodic streams are RECOMMENDED.
 o  The methodology applies to both inter-domain and intra-domain
    composition.
 o  The methodology SHOULD have synchronized measurement time
    intervals in all sub-paths, but largely overlapping intervals MAY
    suffice.
 o  Assumption of sub-path independence with regard to the metric
    being defined/composed is REQUIRED.

4. One-Way Delay Composed Metrics and Statistics

4.1. Name: Type-P-Finite-One-way-Delay-<Sample>-Stream

 This metric is a necessary element of delay composition metrics, and
 its definition does not formally exist elsewhere in IPPM literature.

4.1.1. Metric Parameters

 See the common parameters section (Section 3.1.1).

4.1.2. Definition and Metric Units

 Using the parameters above, we obtain the value of the Type-P-One-
 way-Delay singleton as per [RFC2679].

Morton & Stephan Standards Track [Page 12] RFC 6049 Spatial Composition January 2011

 For each packet "[i]" that has a finite one-way delay (in other
 words, excluding packets that have undefined one-way delay):
 Type-P-Finite-One-way-Delay-<Sample>-Stream[i] =
    FiniteDelay[i] = TstampDst - TstampSrc
 This metric is measured in units of time in seconds, expressed in
 sufficiently low resolution to convey meaningful quantitative
 information.  For example, resolution of microseconds is usually
 sufficient.

4.1.3. Discussion and Other Details

 The "Type-P-Finite-One-way-Delay" metric permits calculation of the
 sample mean statistic.  This resolves the problem of including lost
 packets in the sample (whose delay is undefined) and the issue with
 the informal assignment of infinite delay to lost packets (practical
 systems can only assign some very large value).
 The Finite-One-way-Delay approach handles the problem of lost packets
 by reducing the event space.  We consider conditional statistics, and
 estimate the mean one-way delay conditioned on the event that all
 packets in the sample arrive at the destination (within the specified
 waiting time, Tmax).  This offers a way to make some valid statements
 about one-way delay, at the same time avoiding events with undefined
 outcomes.  This approach is derived from the treatment of lost
 packets in [RFC3393], and is similar to [Y.1540].

4.1.4. Statistic

 All statistics defined in [RFC2679] are applicable to the finite one-
 way delay, and additional metrics are possible, such as the mean (see
 below).

4.2. Name: Type-P-Finite-Composite-One-way-Delay-Mean

 This section describes a statistic based on the Type-P-Finite-One-
 way-Delay-<Sample>-Stream metric.

4.2.1. Metric Parameters

 See the common parameters section (Section 3.1.1).

Morton & Stephan Standards Track [Page 13] RFC 6049 Spatial Composition January 2011

4.2.2. Definition and Metric Units of the Mean Statistic

 We define
 Type-P-Finite-One-way-Delay-Mean =
                                   N
                                  ---
                             1    \
                 MeanDelay = - *   >   (FiniteDelay [n])
                             N    /
                                  ---
                                 n = 1
 where all packets n = 1 through N have finite singleton delays.
 This metric is measured in units of time in seconds, expressed in
 sufficiently fine resolution to convey meaningful quantitative
 information.  For example, resolution of microseconds is usually
 sufficient.

4.2.3. Discussion and Other Details

 The Type-P-Finite-One-way-Delay-Mean metric requires the conditional
 delay distribution described in Section 4.1.3.

4.2.4. Statistic

 This metric, a mean, does not require additional statistics.

4.2.5. Composition Function: Sum of Means

 The Type-P-Finite-Composite-One-way-Delay-Mean, or CompMeanDelay, for
 the complete source to destination path can be calculated from the
 sum of the mean delays of all of its S constituent sub-paths.

Morton & Stephan Standards Track [Page 14] RFC 6049 Spatial Composition January 2011

 Then the
 Type-P-Finite-Composite-One-way-Delay-Mean =
                                    S
                                   ---
                                   \
                  CompMeanDelay =   >   (MeanDelay [s])
                                   /
                                   ---
                                  s = 1
 where sub-paths s = 1 to S are involved in the complete path.

4.2.6. Statement of Conjecture and Assumptions

 The mean of a sufficiently large stream of packets measured on each
 sub-path during the interval [T, Tf] will be representative of the
 ground truth mean of the delay distribution (and the distributions
 themselves are sufficiently independent), such that the means may be
 added to produce an estimate of the complete path mean delay.
 It is assumed that the one-way delay distributions of the sub-paths
 and the complete path are continuous.  The mean of multi-modal
 distributions has the unfortunate property that such a value may
 never occur.

4.2.7. Justification of the Composition Function

 See the common section (Section 3).

4.2.8. Sources of Deviation from the Ground Truth

 See the common section (Section 3).

4.2.9. Specific Cases where the Conjecture Might Fail

 If any of the sub-path distributions are multi-modal, then the
 measured means may not be stable, and in this case the mean will not
 be a particularly useful statistic when describing the delay
 distribution of the complete path.
 The mean may not be a sufficiently robust statistic to produce a
 reliable estimate, or to be useful even if it can be measured.
 If a link contributing non-negligible delay is erroneously included
 or excluded, the composition will be in error.

Morton & Stephan Standards Track [Page 15] RFC 6049 Spatial Composition January 2011

4.2.10. Application of Measurement Methodology

 The requirements of the common section (Section 3) apply here as
 well.

4.3. Name: Type-P-Finite-Composite-One-way-Delay-Minimum

 This section describes a statistic based on the Type-P-Finite-One-
 way-Delay-<Sample>-Stream metric, and the composed metric based on
 that statistic.

4.3.1. Metric Parameters

 See the common parameters section (Section 3.1.1).

4.3.2. Definition and Metric Units of the Minimum Statistic

 We define
 Type-P-Finite-One-way-Delay-Minimum =
             MinDelay = (FiniteDelay [j])
             such that for some index, j, where 1 <= j <= N
             FiniteDelay[j] <= FiniteDelay[n] for all n
 where all packets n = 1 through N have finite singleton delays.
 This metric is measured in units of time in seconds, expressed in
 sufficiently fine resolution to convey meaningful quantitative
 information.  For example, resolution of microseconds is usually
 sufficient.

4.3.3. Discussion and Other Details

 The Type-P-Finite-One-way-Delay-Minimum metric requires the
 conditional delay distribution described in Section 4.1.3.

4.3.4. Statistic

 This metric, a minimum, does not require additional statistics.

4.3.5. Composition Function: Sum of Minima

 The Type-P-Finite-Composite-One-way-Delay-Minimum, or CompMinDelay,
 for the complete source to destination path can be calculated from
 the sum of the minimum delays of all of its S constituent sub-paths.

Morton & Stephan Standards Track [Page 16] RFC 6049 Spatial Composition January 2011

 Then the
 Type-P-Finite-Composite-One-way-Delay-Minimum =
                                     S
                                    ---
                                    \
                   CompMinDelay =    >  (MinDelay [s])
                                    /
                                    ---
                                   s = 1

4.3.6. Statement of Conjecture and Assumptions

 The minimum of a sufficiently large stream of packets measured on
 each sub-path during the interval [T, Tf] will be representative of
 the ground truth minimum of the delay distribution (and the
 distributions themselves are sufficiently independent), such that the
 minima may be added to produce an estimate of the complete path
 minimum delay.
 It is assumed that the one-way delay distributions of the sub-paths
 and the complete path are continuous.

4.3.7. Justification of the Composition Function

 See the common section (Section 3).

4.3.8. Sources of Deviation from the Ground Truth

 See the common section (Section 3).

4.3.9. Specific Cases where the Conjecture Might Fail

 If the routing on any of the sub-paths is not stable, then the
 measured minimum may not be stable.  In this case the composite
 minimum would tend to produce an estimate for the complete path that
 may be too low for the current path.

4.3.10. Application of Measurement Methodology

 The requirements of the common section (Section 3) apply here as
 well.

Morton & Stephan Standards Track [Page 17] RFC 6049 Spatial Composition January 2011

5. Loss Metrics and Statistics

5.1. Type-P-Composite-One-way-Packet-Loss-Empirical-Probability

5.1.1. Metric Parameters

 See the common parameters section (Section 3.1.1).

5.1.2. Definition and Metric Units

 Using the parameters above, we obtain the value of the Type-P-One-
 way-Packet-Loss singleton and stream as per [RFC2680].
 We obtain a sequence of pairs with elements as follows:
 o  TstampSrc, as above.
 o  L, either zero or one, where L = 1 indicates loss and L = 0
    indicates arrival at the destination within TstampSrc + Tmax.

5.1.3. Discussion and Other Details

 None.

5.1.4. Statistic: Type-P-One-way-Packet-Loss-Empirical-Probability

 Given the stream parameter M, the number of packets sent, we can
 define the Empirical Probability of Loss Statistic (Ep), consistent
 with average loss in [RFC2680], as follows:
 Type-P-One-way-Packet-Loss-Empirical-Probability =
                                      M
                                     ---
                                1    \
                           Ep = - *   >  (L[m])
                                M    /
                                     ---
                                    m = 1
 where all packets m = 1 through M have a value for L.

5.1.5. Composition Function: Composition of Empirical Probabilities

 The Type-P-One-way-Composite-Packet-Loss-Empirical-Probability, or
 CompEp, for the complete source to destination path can be calculated
 by combining the Ep of all of its constituent sub-paths (Ep1, Ep2,
 Ep3, ...  Epn) as

Morton & Stephan Standards Track [Page 18] RFC 6049 Spatial Composition January 2011

 Type-P-Composite-One-way-Packet-Loss-Empirical-Probability =
   CompEp = 1 - {(1 - Ep1) x (1 - Ep2) x (1 - Ep3) x ... x (1 - EpS)}
 If any Eps is undefined in a particular measurement interval,
 possibly because a measurement system failed to report a value, then
 any CompEp that uses sub-path s for that measurement interval is
 undefined.

5.1.6. Statement of Conjecture and Assumptions

 The empirical probability of loss calculated on a sufficiently large
 stream of packets measured on each sub-path during the interval
 [T, Tf] will be representative of the ground truth empirical loss
 probability (and the probabilities themselves are sufficiently
 independent), such that the sub-path probabilities may be combined to
 produce an estimate of the complete path empirical loss probability.

5.1.7. Justification of the Composition Function

 See the common section (Section 3).

5.1.8. Sources of Deviation from the Ground Truth

 See the common section (Section 3).

5.1.9. Specific Cases where the Conjecture Might Fail

 A concern for loss measurements combined in this way is that root
 causes may be correlated to some degree.
 For example, if the links of different networks follow the same
 physical route, then a single catastrophic event like a fire in a
 tunnel could cause an outage or congestion on remaining paths in
 multiple networks.  Here it is important to ensure that measurements
 before the event and after the event are not combined to estimate the
 composite performance.
 Or, when traffic volumes rise due to the rapid spread of an email-
 borne worm, loss due to queue overflow in one network may help
 another network to carry its traffic without loss.

5.1.10. Application of Measurement Methodology

 See the common section (Section 3).

Morton & Stephan Standards Track [Page 19] RFC 6049 Spatial Composition January 2011

6. Delay Variation Metrics and Statistics

6.1. Name: Type-P-One-way-pdv-refmin-<Sample>-Stream

 This packet delay variation (PDV) metric is a necessary element of
 Composed Delay Variation metrics, and its definition does not
 formally exist elsewhere in IPPM literature (with the exception of
 [RFC5481]).

6.1.1. Metric Parameters

 In addition to the parameters of Section 3.1.1:
 o  TstampSrc[i], the wire time of packet[i] as measured at MP(Src)
    (measurement point at the source).
 o  TstampDst[i], the wire time of packet[i] as measured at MP(Dst),
    assigned to packets that arrive within a "reasonable" time.
 o  B, a packet length in bits.
 o  F, a selection function unambiguously defining the packets from
    the stream that are selected for the packet-pair computation of
    this metric.  F(current packet), the first packet of the pair,
    MUST have a valid Type-P-Finite-One-way-Delay less than Tmax (in
    other words, excluding packets that have undefined one-way delay)
    and MUST have been transmitted during the interval [T, Tf].  The
    second packet in the pair, F(min_delay packet) MUST be the packet
    with the minimum valid value of Type-P-Finite-One-way-Delay for
    the stream, in addition to the criteria for F(current packet).  If
    multiple packets have equal minimum Type-P-Finite-One-way-Delay
    values, then the value for the earliest arriving packet SHOULD be
    used.
 o  MinDelay, the Type-P-Finite-One-way-Delay value for F(min_delay
    packet) given above.
 o  N, the number of packets received at the destination that meet the
    F(current packet) criteria.

6.1.2. Definition and Metric Units

 Using the definition above in Section 5.1.2, we obtain the value of
 Type-P-Finite-One-way-Delay-<Sample>-Stream[n], the singleton for
 each packet[i] in the stream (a.k.a. FiniteDelay[i]).

Morton & Stephan Standards Track [Page 20] RFC 6049 Spatial Composition January 2011

 For each packet[n] that meets the F(first packet) criteria given
 above: Type-P-One-way-pdv-refmin-<Sample>-Stream[n] =
    PDV[n] = FiniteDelay[n] - MinDelay
 where PDV[i] is in units of time in seconds, expressed in
 sufficiently fine resolution to convey meaningful quantitative
 information.  For example, resolution of microseconds is usually
 sufficient.

6.1.3. Discussion and Other Details

 This metric produces a sample of delay variation normalized to the
 minimum delay of the sample.  The resulting delay variation
 distribution is independent of the sending sequence (although
 specific FiniteDelay values within the distribution may be
 correlated, depending on various stream parameters such as packet
 spacing).  This metric is equivalent to the IP Packet Delay Variation
 parameter defined in [Y.1540].

6.1.4. Statistics: Mean, Variance, Skewness, Quantile

 We define the mean PDV as follows (where all packets n = 1 through N
 have a value for PDV[n]):
 Type-P-One-way-pdv-refmin-Mean = MeanPDV =
                                 N
                                ---
                           1    \
                           - *   >   (PDV[n])
                           N    /
                                ---
                               n = 1
 We define the variance of PDV as follows:
 Type-P-One-way-pdv-refmin-Variance = VarPDV =
                             N
                            ---
                      1     \                      2
                   -------   >   (PDV[n] - MeanPDV)
                   (N - 1)  /
                            ---
                           n = 1

Morton & Stephan Standards Track [Page 21] RFC 6049 Spatial Composition January 2011

 We define the skewness of PDV as follows:
 Type-P-One-way-pdv-refmin-Skewness = SkewPDV =
                       N
                      ---                        3
                      \     /                  \
                       >   |  PDV[n] - MeanPDV  |
                      /     \                  /
                      ---
                     n = 1
                  -----------------------------------
                      /                         \
                     |                  ( 3/2 )  |
                      \ (N - 1) * VarPDV        /
 (See Appendix X of [Y.1541] for additional background information.)
 We define the quantile of the PDV sample as the value where the
 specified fraction of singletons is less than the given value.

6.1.5. Composition Functions

 This section gives two alternative composition functions.  The
 objective is to estimate a quantile of the complete path delay
 variation distribution.  The composed quantile will be estimated
 using information from the sub-path delay variation distributions.

6.1.5.1. Approximate Convolution

 The Type-P-Finite-One-way-Delay-<Sample>-Stream samples from each
 sub-path are summarized as a histogram with 1-ms bins representing
 the one-way delay distribution.
 From [STATS], the distribution of the sum of independent random
 variables can be derived using the relation:
 Type-P-Composite-One-way-pdv-refmin-quantile-a =
                     .  .
                    /  /
P(X + Y + Z <= a) = |  | P(X <= a - y - z) * P(Y = y) * P(Z = z) dy dz
                    /  /
                   `  `
                   z  y

Morton & Stephan Standards Track [Page 22] RFC 6049 Spatial Composition January 2011

 Note that dy and dz indicate partial integration above, and that y
 and z are the integration variables.  Also, the probability of an
 outcome is indicated by the symbol P(outcome), where X, Y, and Z are
 random variables representing the delay variation distributions of
 the sub-paths of the complete path (in this case, there are three
 sub-paths), and "a" is the quantile of interest.
 This relation can be used to compose a quantile of interest for the
 complete path from the sub-path delay distributions.  The histograms
 with 1-ms bins are discrete approximations of the delay
 distributions.

6.1.5.2. Normal Power Approximation (NPA)

 Type-P-One-way-Composite-pdv-refmin-NPA for the complete source to
 destination path can be calculated by combining the statistics of all
 the constituent sub-paths in the process described in [Y.1541],
 Clause 8 and Appendix X.

6.1.6. Statement of Conjecture and Assumptions

 The delay distribution of a sufficiently large stream of packets
 measured on each sub-path during the interval [T, Tf] will be
 sufficiently stationary, and the sub-path distributions themselves
 are sufficiently independent, so that summary information describing
 the sub-path distributions can be combined to estimate the delay
 distribution of the complete path.
 It is assumed that the one-way delay distributions of the sub-paths
 and the complete path are continuous.

6.1.7. Justification of the Composition Function

 See the common section (Section 3).

6.1.8. Sources of Deviation from the Ground Truth

 In addition to the common deviations, a few additional sources exist
 here.  For one, very tight distributions with ranges on the order of
 a few milliseconds are not accurately represented by a histogram with
 1-ms bins.  This size was chosen assuming an implicit requirement on
 accuracy: errors of a few milliseconds are acceptable when assessing
 a composed distribution quantile.
 Also, summary statistics cannot describe the subtleties of an
 empirical distribution exactly, especially when the distribution is
 very different from a classical form.  Any procedure that uses these
 statistics alone may incur error.

Morton & Stephan Standards Track [Page 23] RFC 6049 Spatial Composition January 2011

6.1.9. Specific Cases where the Conjecture Might Fail

 If the delay distributions of the sub-paths are somehow correlated,
 then neither of these composition functions will be reliable
 estimators of the complete path distribution.
 In practice, sub-path delay distributions with extreme outliers have
 increased the error of the composed metric estimate.

6.1.10. Application of Measurement Methodology

 See the common section (Section 3).

7. Security Considerations

7.1. Denial-of-Service Attacks

 This metric requires a stream of packets sent from one host (source)
 to another host (destination) through intervening networks.  This
 method could be abused for denial-of-service attacks directed at the
 destination and/or the intervening network(s).
 Administrators of source, destination, and intervening networks
 should establish bilateral or multilateral agreements regarding the
 timing, size, and frequency of collection of sample metrics.  Use of
 this method in excess of the terms agreed upon between the
 participants may be cause for immediate rejection or discarding of
 packets, or other escalation procedures defined between the affected
 parties.

7.2. User Data Confidentiality

 Active use of this method generates packets for a sample, rather than
 taking samples based on user data, and does not threaten user data
 confidentiality.  Passive measurement MUST restrict attention to the
 headers of interest.  Since user payloads may be temporarily stored
 for length analysis, suitable precautions MUST be taken to keep this
 information safe and confidential.  In most cases, a hashing function
 will produce a value suitable for payload comparisons.

7.3. Interference with the Metrics

 It may be possible to identify that a certain packet or stream of
 packets is part of a sample.  With that knowledge at the destination
 and/or the intervening networks, it is possible to change the

Morton & Stephan Standards Track [Page 24] RFC 6049 Spatial Composition January 2011

 processing of the packets (e.g., increasing or decreasing delay),
 which may distort the measured performance.  It may also be possible
 to generate additional packets that appear to be part of the sample
 metric.  These additional packets are likely to perturb the results
 of the sample measurement.
 To discourage the kind of interference mentioned above, packet
 interference checks, such as cryptographic hash, may be used.

8. IANA Considerations

 Metrics defined in the IETF are typically registered in the IANA IPPM
 Metrics Registry as described in the initial version of the registry
 [RFC4148].
 IANA has registered the following metrics in the
 IANA-IPPM-METRICS-REGISTRY-MIB:
    ietfFiniteOneWayDelayStream OBJECT-IDENTITY
       STATUS current
       DESCRIPTION
          "Type-P-Finite-One-way-Delay-Stream"
       REFERENCE "RFC 6049, Section 4.1."
       ::= { ianaIppmMetrics 71 }
    ietfFiniteOneWayDelayMean OBJECT-IDENTITY
       STATUS current
       DESCRIPTION
          "Type-P-Finite-One-way-Delay-Mean"
       REFERENCE "RFC 6049, Section 4.2."
       ::= { ianaIppmMetrics 72 }
    ietfCompositeOneWayDelayMean OBJECT-IDENTITY
       STATUS current
       DESCRIPTION
          "Type-P-Finite-Composite-One-way-Delay-Mean"
       REFERENCE "RFC 6049, Section 4.2.5."
       ::= { ianaIppmMetrics 73 }
    ietfFiniteOneWayDelayMinimum OBJECT-IDENTITY
       STATUS current
       DESCRIPTION
          "Type-P-Finite-One-way-Delay-Minimum"
       REFERENCE "RFC 6049, Section 4.3.2."
       ::= { ianaIppmMetrics 74 }

Morton & Stephan Standards Track [Page 25] RFC 6049 Spatial Composition January 2011

    ietfCompositeOneWayDelayMinimum OBJECT-IDENTITY
       STATUS current
       DESCRIPTION
          "Type-P-Finite-Composite-One-way-Delay-Minimum"
       REFERENCE "RFC 6049, Section 4.3."
       ::= { ianaIppmMetrics 75 }
    ietfOneWayPktLossEmpiricProb OBJECT-IDENTITY
       STATUS current
       DESCRIPTION
          "Type-P-One-way-Packet-Loss-Empirical-Probability"
       REFERENCE "RFC 6049, Section 5.1.4"
       ::= { ianaIppmMetrics 76 }
    ietfCompositeOneWayPktLossEmpiricProb OBJECT-IDENTITY
       STATUS current
       DESCRIPTION
          "Type-P-Composite-One-way-Packet-Loss-Empirical-Probability"
       REFERENCE "RFC 6049, Section 5.1."
       ::= { ianaIppmMetrics 77 }
    ietfOneWayPdvRefminStream OBJECT-IDENTITY
       STATUS current
       DESCRIPTION
          "Type-P-One-way-pdv-refmin-Stream"
       REFERENCE "RFC 6049, Section 6.1."
       ::= { ianaIppmMetrics 78 }
    ietfOneWayPdvRefminMean OBJECT-IDENTITY
       STATUS current
       DESCRIPTION
          "Type-P-One-way-pdv-refmin-Mean"
       REFERENCE "RFC 6049, Section 6.1.4."
       ::= { ianaIppmMetrics 79 }
    ietfOneWayPdvRefminVariance OBJECT-IDENTITY
       STATUS current
       DESCRIPTION
          "Type-P-One-way-pdv-refmin-Variance"
       REFERENCE "RFC 6049, Section 6.1.4."
       ::= { ianaIppmMetrics 80 }

Morton & Stephan Standards Track [Page 26] RFC 6049 Spatial Composition January 2011

    ietfOneWayPdvRefminSkewness OBJECT-IDENTITY
       STATUS current
       DESCRIPTION
          "Type-P-One-way-pdv-refmin-Skewness"
       REFERENCE "RFC 6049, Section 6.1.4."
       ::= { ianaIppmMetrics 81 }
    ietfCompositeOneWayPdvRefminQtil OBJECT-IDENTITY
       STATUS current
       DESCRIPTION
          "Type-P-Composite-One-way-pdv-refmin-quantile-a"
       REFERENCE "RFC 6049, Section 6.1.5.1."
       ::= { ianaIppmMetrics 82 }
    ietfCompositeOneWayPdvRefminNPA OBJECT-IDENTITY
       STATUS current
       DESCRIPTION
          "Type-P-One-way-Composite-pdv-refmin-NPA"
       REFERENCE "RFC 6049, Section 6.1.5.2."
       ::= { ianaIppmMetrics 83 }

9. Contributors and Acknowledgements

 The following people have contributed useful ideas, suggestions, or
 the text of sections that have been incorporated into this memo:
  1. Phil Chimento vze275m9@verizon.net
  1. Reza Fardid RFardid@cariden.com
  1. Dave Hoeflin dhoeflin@att.com
 A long time ago, in a galaxy far, far away (Minneapolis), Will Leland
 suggested the simple and elegant Type-P-Finite-One-way-Delay concept.
 Thanks Will.
 Yaakov Stein and Donald McLachlan also provided useful comments along
 the way.

Morton & Stephan Standards Track [Page 27] RFC 6049 Spatial Composition January 2011

10. References

10.1. Normative References

 [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.
 [RFC2679]  Almes, G., Kalidindi, S., and M. Zekauskas, "A One-way
            Delay Metric for IPPM", RFC 2679, September 1999.
 [RFC2680]  Almes, G., Kalidindi, S., and M. Zekauskas, "A One-way
            Packet Loss Metric for IPPM", RFC 2680, September 1999.
 [RFC3393]  Demichelis, C. and P. Chimento, "IP Packet Delay Variation
            Metric for IP Performance Metrics (IPPM)", RFC 3393,
            November 2002.
 [RFC3432]  Raisanen, V., Grotefeld, G., and A. Morton, "Network
            performance measurement with periodic streams", RFC 3432,
            November 2002.
 [RFC4148]  Stephan, E., "IP Performance Metrics (IPPM) Metrics
            Registry", BCP 108, RFC 4148, August 2005.
 [RFC5835]  Morton, A. and S. Van den Berghe, "Framework for Metric
            Composition", RFC 5835, April 2010.

10.2. Informative References

 [RFC5474]  Duffield, N., Chiou, D., Claise, B., Greenberg, A.,
            Grossglauser, M., and J. Rexford, "A Framework for Packet
            Selection and Reporting", RFC 5474, March 2009.
 [RFC5481]  Morton, A. and B. Claise, "Packet Delay Variation
            Applicability Statement", RFC 5481, March 2009.
 [RFC5644]  Stephan, E., Liang, L., and A. Morton, "IP Performance
            Metrics (IPPM): Spatial and Multicast", RFC 5644,
            October 2009.
 [STATS]    Mood, A., Graybill, F., and D. Boes, "Introduction to the
            Theory of Statistics, 3rd Edition", McGraw-Hill, New York,
            NY, 1974.

Morton & Stephan Standards Track [Page 28] RFC 6049 Spatial Composition January 2011

 [Y.1540]   ITU-T Recommendation Y.1540, "Internet protocol data
            communication service - IP packet transfer and
            availability performance parameters", November 2007.
 [Y.1541]   ITU-T Recommendation Y.1541, "Network Performance
            Objectives for IP-based Services", February 2006.

Authors' Addresses

 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/
 Stephan Emile
 France Telecom Orange
 2 avenue Pierre Marzin
 Lannion,   F-22307
 France
 EMail: emile.stephan@orange-ftgroup.com

Morton & Stephan Standards Track [Page 29]

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