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rfc:rfc5644

Network Working Group E. Stephan Request for Comments: 5644 France Telecom Category: Standards Track L. Liang

                                                  University of Surrey
                                                             A. Morton
                                                             AT&T Labs
                                                          October 2009
        IP Performance Metrics (IPPM): Spatial and Multicast

Abstract

 The IETF has standardized IP Performance Metrics (IPPM) for measuring
 end-to-end performance between two points.  This memo defines two new
 categories of metrics that extend the coverage to multiple
 measurement points.  It defines spatial metrics for measuring the
 performance of segments of a source to destination path, and metrics
 for measuring the performance between a source and many destinations
 in multiparty communications (e.g., a multicast tree).

Status of This Memo

 This document specifies an Internet standards track protocol for the
 Internet community, and requests discussion and suggestions for
 improvements.  Please refer to the current edition of the "Internet
 Official Protocol Standards" (STD 1) for the standardization state
 and status of this protocol.  Distribution of this memo is unlimited.

Copyright Notice

 Copyright (c) 2009 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 BSD License.
 This document may contain material from IETF Documents or IETF
 Contributions published or made publicly available before November
 10, 2008.  The person(s) controlling the copyright in some of this
 material may not have granted the IETF Trust the right to allow

Stephan, et al. Standards Track [Page 1] RFC 5644 Spatial and Multicast Metrics October 2009

 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.

Table of Contents

 1. Introduction and Scope ..........................................3
 2. Terminology .....................................................4
 3. Brief Metric Descriptions .......................................7
 4. Motivations ....................................................10
 5. Spatial Vector Metrics Definitions .............................12
 6. Spatial Segment Metrics Definitions ............................19
 7. One-to-Group Metrics Definitions ...............................27
 8. One-to-Group Sample Statistics .................................30
 9. Measurement Methods: Scalability and Reporting .................40
 10. Manageability Considerations ..................................44
 11. Security Considerations .......................................49
 12. Acknowledgments ...............................................50
 13. IANA Considerations ...........................................50
 14. References ....................................................56
    14.1. Normative References .....................................56
    14.2. Informative References ...................................57

Stephan, et al. Standards Track [Page 2] RFC 5644 Spatial and Multicast Metrics October 2009

1. Introduction and Scope

 IETF has standardized IP Performance Metrics (IPPM) for measuring
 end-to-end performance between two points.  This memo defines two new
 categories of metrics that extend the coverage to multiple
 measurement points.  It defines spatial metrics for measuring the
 performance of segments of a source to destination path, and metrics
 for measuring the performance between a source and many destinations
 in multiparty communications (e.g., a multicast tree).
 The purpose of this memo is to define metrics to fulfill the new
 requirements of measurement involving multiple measurement points.
 Spatial metrics measure the performance of each segment along a path.
 One-to-group metrics measure the performance for a group of users.
 These metrics are derived from one-way end-to-end metrics, all of
 which follow the IPPM framework [RFC2330].
 This memo is organized as follows: Section 2 introduces new terms
 that extend the original IPPM framework [RFC2330].  Section 3 briefly
 introduces the new metrics, and Section 4 motivates each metric
 category.  Sections 5 through 8 develop each category of metrics with
 definitions and statistics.  Then the memo discusses the impact of
 the measurement methods on the scalability and proposes an
 information model for reporting the measurements.  Finally, the memo
 discusses security aspects related to measurement and registers the
 metrics in the IANA IP Performance Metrics Registry [RFC4148].
 The scope of this memo is limited to metrics using a single source
 packet or stream, and observations of corresponding packets along the
 path (spatial), at one or more destinations (one-to-group), or both.
 Note that all the metrics defined herein are based on observations of
 packets dedicated to testing, a process that is called active
 measurement.  Passive measurement (for example, a spatial metric
 based on the observation of user traffic) is beyond the scope of this
 memo.

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].

Stephan, et al. Standards Track [Page 3] RFC 5644 Spatial and Multicast Metrics October 2009

2. Terminology

2.1. Naming of the Metrics

 The names of the metrics, including capitalized letters, are as close
 as possible of the names of the one-way end-to-end metrics they are
 derived from.

2.2. Terms Defined Elsewhere

 host: section 5 of RFC 2330
 router: section 5 of RFC 2330
 loss threshold: section 2.8.2 of RFC 2680
 path: section 5 of RFC 2330
 sample: section 11 of RFC 2330
 singleton: section 11 of RFC 2330

2.3. Routers Digest

 The list of the routers on the path from the source to the
 destination that act as points of interest, also referred to as the
 routers digest.

2.4. Multiparty Metric

 A metric is said to be multiparty if the topology involves more than
 one measurement collection point.  All multiparty metrics designate a
 set of hosts as "points of interest", where one host is the source
 and other hosts are the measurement collection points.  For example,
 if the set of points of interest is < ha, hb, hc, ..., hn >, where ha
 is the source and < hb, hc, ..., hn > are the destinations, then
 measurements may be conducted between < ha, hb>, < ha, hc>, ..., <ha,
 hn >.
 For the purposes of this memo (reflecting the scope of a single
 source), the only multiparty metrics are one-to-group metrics.

2.5. Spatial Metric

 A metric is said to be spatial if one of the hosts (measurement
 collection points) involved is neither the source nor a destination
 of the measured packet(s).  Such measurement hosts will usually be
 routers that are members of the routers digest.

Stephan, et al. Standards Track [Page 4] RFC 5644 Spatial and Multicast Metrics October 2009

2.6. One-to-Group Metric

 A metric is said to be one-to-group if the measured packet is sent by
 one source and (potentially) received by more than one destination.
 Thus, the topology of the communication group can be viewed as a
 center-distributed or server-client topology with the source as the
 center/server in the topology.

2.7. Points of Interest

 Points of interest are the hosts (as per the RFC 2330 definition,
 "hosts" include routing nodes) that are measurement collection
 points, which are a sub-set of the set of hosts involved in the
 delivery of the packets (in addition to the source itself).
 For spatial metrics, points of interest are a (possibly arbitrary)
 sub-set of all the routers involved in the path.
 Points of interest of one-to-group metrics are the intended
 destination hosts for packets from the source (in addition to the
 source itself).
                       Src                   Dst
                       `.          ,-.
                         `.      ,'   `...... 1
                           `.   ;       :
                             `. ;       :
                               ;         :... 2
                               |         |
                               :         ;
                                :       ;.... 3
                                :       ;
                                 `.   ,'
                                   `-'....... I
               Figure 1: One-to-Group Points of Interest
 A candidate point of interest for spatial metrics is a router from
 the set of routers involved in the delivery of the packets from
 source to destination.

Stephan, et al. Standards Track [Page 5] RFC 5644 Spatial and Multicast Metrics October 2009

                       Src ------.           Hosts
                                  \
                                   `---X   --- 1
                                       \
                                        x
                                       /
                            .---------X   ---- 2
                          /
                         x
                          ...
                          `---X           ---- ...
                                 \
                                  \
                                   \
                                    X     ---- J
                                     \
                                      \
                                       \
                                        `---- Dst
              Note: 'X' are nodes that are points of interest,
                    'x' are nodes that are not points of interest
                 Figure 2: Spatial Points of Interest

2.8. Reference Point

 A reference point is defined as the server where the statistical
 calculations will be carried out.  It is usually a centralized server
 in the measurement architecture that is controlled by a network
 operator, where measurement data can be collected for further
 processing.  The reference point is distinctly different from hosts
 at measurement collection points, where the actual measurements are
 carried out (e.g., points of interest).

2.9. Vector

 A vector is a set of singletons (single atomic results) comprised of
 observations corresponding to a single source packet at different
 hosts in a network.  For instance, if the one-way delay singletons
 observed at N receivers for Packet P sent by the source Src are dT1,
 dT2,..., dTN, then a vector V with N elements can be organized as
 {dT1, dT2,..., dTN}.  The element dT1 is distinct from all others as
 the singleton at receiver 1 in response to a packet sent from the
 source at a specific time.  The complete vector gives information
 over the dimension of space, a set of N receivers in this example.

Stephan, et al. Standards Track [Page 6] RFC 5644 Spatial and Multicast Metrics October 2009

 The singleton elements of any vector are distinctly different from
 each other in terms of their measurement collection point.  Different
 vectors for common measurement points of interest are distinguished
 by the source packet sending time.

2.10. Matrix

 Several vectors form a matrix, which contains results observed over a
 sampling interval at different places in a network at different
 times.  For example, the one-way delay vectors V1={dT11, dT12,...,
 dT1N}, V2={dT21, dT22,..., dT2N},..., Vm={dTm1, dTm2,..., dTmN} for
 Packet P1, P2,...,Pm, form a one-way delay Matrix {V1, V2,...,Vm}.
 The matrix organizes the vector information to present network
 performance in both space and time.
 A one-dimensional matrix (row) corresponds to a sample in simple
 point-to-point measurement.
 The relationship among singleton, sample, vector, and matrix is
 illustrated in Figure 3.
               points of        singleton
               interest           /       samples(time)
                ,----.    ^      /
               /   R1.....|  / R1dT1   R1dT2   R1dT3 ... R3dTk \
              /         \ | |                                   |
             ;  R2........| |  R2dT1   R2dT2   R2dT3 ... R3dTk  |
        Src  |           || |                                   |
             |      R3....| |  R3dT1   R3dT2   R3dT3 ... R3dTk  |
             |           || |                                   |
             :           ;| |                                   |
              \         / | |                                   |
               \  Rn......|  \ RndT1   RndT2   RndT3 ... RndTk /
                `-----'   +-------------------------------------> time
                              vector           matrix
                             (space)      (time and space)
   Figure 3: Relationship between Singletons, Samples, Vectors, and
                                Matrix

3. Brief Metric Descriptions

 The metrics for spatial and one-to-group measurement are based on the
 source-to-destination, or end-to-end metrics defined by IETF in
 [RFC2679], [RFC2680], [RFC3393], and [RFC3432].

Stephan, et al. Standards Track [Page 7] RFC 5644 Spatial and Multicast Metrics October 2009

 This memo defines seven new spatial metrics using the [RFC2330]
 framework of parameters, units of measure, and measurement
 methodologies.  Each definition includes a section that describes
 measurement constraints and issues, and provides guidance to increase
 the accuracy of the results.
 The spatial metrics are:
 o  Type-P-Spatial-One-way-Delay-Vector divides the end-to-end Type-P-
    One-way-Delay [RFC2679] into a spatial vector of one-way delay
    singletons.
 o  Type-P-Spatial-One-way-Packet-Loss-Vector divides an end-to-end
    Type-P-One-way-Packet-Loss [RFC2680] into a spatial vector of
    packet loss singletons.
 o  Type-P-Spatial-One-way-ipdv-Vector divides an end-to-end Type-P-
    One-way-ipdv into a spatial vector of ipdv (IP Packet Delay
    Variation) singletons.
 o  Using elements of the Type-P-Spatial-One-way-Delay-Vector metric,
    a sample called Type-P-Segment-One-way-Delay-Stream collects one-
    way delay metrics between two points of interest on the path over
    time.
 o  Likewise, using elements of the Type-P-Spatial-Packet-Loss-Vector
    metric, a sample called Type-P-Segment-Packet-Loss-Stream collects
    one-way delay metrics between two points of interest on the path
    over time.
 o  Using the Type-P-Spatial-One-way-Delay-Vector metric, a sample
    called Type-P-Segment-ipdv-prev-Stream will be introduced to
    compute ipdv metrics (using the previous packet selection
    function) between two points of interest on the path over time.
 o  Again using the Type-P-Spatial-One-way-Delay-Vector metric, a
    sample called Type-P-Segment-ipdv-min-Stream will define another
    set of ipdv metrics (using the minimum delay packet selection
    function) between two points of interest on the path over time.
 The memo also defines three one-to-group metrics to measure the one-
 way performance between a source and a group of receivers.  They are:
 o  Type-P-One-to-group-Delay-Vector which collects the set of Type-P-
    One-way-Delay singletons between one sender and N receivers;

Stephan, et al. Standards Track [Page 8] RFC 5644 Spatial and Multicast Metrics October 2009

 o  Type-P-One-to-group-Packet-Loss-Vector which collects the set of
    Type-P-One-way-Packet-Loss singletons between one sender and N
    receivers; and
 o  Type-P-One-to-group-ipdv-Vector which collects the set of Type-P-
    One-way-ipdv singletons between one sender and N receivers.
 Finally, based on the one-to-group vector metrics listed above,
 statistics are defined to capture single receiver performance, group
 performance, and the relative performance for a multiparty
 communication:
 o  Using the Type-P-One-to-group-Delay-Vector, a metric called Type-
    P-One-to-group-Receiver-n-Mean-Delay, or RnMD, presents the mean
    of delays between one sender and a single receiver 'n'.  From this
    metric, three additional metrics are defined to characterize the
    mean delay over the entire group of receivers during the same time
    interval:
  • Type-P-One-to-group-Mean-Delay, or GMD, presents the mean of

delays;

  • Type-P-One-to-group-Range-Mean-Delay, or GRMD, presents the

range of mean delays; and

  • Type-P-One-to-group-Max-Mean-Delay, or GMMD, presents the

maximum of mean delays.

 o  Using the Type-P-One-to-group-Packet-Loss-Vector, a metric called
    Type-P-One-to-group-Receiver-n-Loss-Ratio, or RnLR, captures the
    packet loss ratio between one sender and a single receiver 'n'.
    Based on this definition, two more metrics are defined to
    characterize packet loss over the entire group during the same
    time interval:
  • Type-P-One-to-group-Loss-Ratio, or GLR, captures the overall

packet loss ratio for the entire group of receivers; and

  • Type-P-One-to-group-Range-Loss-Ratio, or GRLR, presents the

comparative packet loss ratio during the test interval between

       one sender and N receivers.
 o  Using the Type-P-One-to-group-Packet-Loss-Vector, a metric called
    Type-P-One-to-group-Receiver-n-Comp-Loss-Ratio, or RnCLR, computes
    a packet loss ratio using the maximum number of packets received
    at any receiver.

Stephan, et al. Standards Track [Page 9] RFC 5644 Spatial and Multicast Metrics October 2009

 o  Using Type-P-One-to-group-ipdv-Vector, a metric called Type-P-One-
    to-group-Range-Delay-Variation, or GRDV, presents the range of
    delay variation between one sender and a group of receivers.

4. Motivations

 All existing IPPM metrics are defined for end-to-end (source-to-
 destination) measurement of point-to-point paths.  It is logical to
 extend them to multiparty situations such as one-to-one trajectory
 metrics and one-to-multipoint metrics.

4.1. Motivations for Spatial Metrics

 Spatial metrics are needed for:
 o  Decomposing the performance of an inter-domain path to quantify
    the per-AS (Autonomous System) contribution to the end-to-end
    performance.
 o  Traffic engineering and troubleshooting, which benefit from
    spatial views of one-way delay and ipdv consumption, or
    identification of the path segment where packets were lost.
 o  Monitoring the decomposed performance of a multicast tree based on
    MPLS point-to-multipoint communications.
 o  Dividing end-to-end metrics, so that some segment measurements can
    be re-used and help measurement systems reach large-scale
    coverage.  Spatial measures could characterize the performance of
    an intra-domain segment and provide an elementary piece of
    information needed to estimate inter-domain performance to another
    destination using Spatial Composition metrics [SPATIAL].

4.2. Motivations for One-to-group Metrics

 While the node-to-node-based spatial measures can provide very useful
 data in the view of each connection, we also need measures to present
 the performance of a multiparty communication topology.  A simple
 point-to-point metric cannot completely describe the multiparty
 situation.  New one-to-group metrics assess performance of the
 multiple paths for further statistical analysis.  The new metrics are
 named one-to-group performance metrics, and they are based on the
 unicast metrics defined in IPPM RFCs.  One-to-group metrics are one-
 way metrics from one source to a group of destinations or receivers.
 The metrics are helpful for judging the overall performance of a
 multiparty communications network and for describing the performance
 variation across a group of destinations.

Stephan, et al. Standards Track [Page 10] RFC 5644 Spatial and Multicast Metrics October 2009

 One-to-group performance metrics are needed for:
 o  Designing and engineering multicast trees and MPLS point-to-
    multipoint Label Switched Paths (LSPs).
 o  Evaluating and controlling the quality of multicast services,
    including inter-domain multicast.
 o  Presenting and evaluating the performance requirements for
    multiparty communications and overlay multicast.
 To understand the packet transfer performance between one source and
 any one receiver in the multiparty communication group, we need to
 collect instantaneous end-to-end metrics, or singletons.  This gives
 a very detailed view into the performance of each branch of the
 multicast tree, and can provide clear and helpful information for
 engineers to identify the branch with problems in a complex
 multiparty routing tree.
 The one-to-group metrics described in this memo introduce the
 multiparty topology into the IPPM framework, and they describe the
 performance delivered to a group receiving packets from the same
 source.  The concept extends the "path" of the point-to-point
 measurement to "path tree" to cover one-to-many topologies.  If
 applied to one-to-one topology, the one-to-group metrics provide
 exactly the same results as the corresponding one-to-one metrics.

4.3. Discussion on Group-to-One and Group-to-Group Metrics

 We note that points of interest can also be selected to define
 measurements on group-to-one and group-to-group topologies.  These
 topologies are beyond the scope of this memo, because they would
 involve multiple packets launched from different sources.  However,
 this section gives some insights on these two cases.
 The measurements for group-to-one topology can be easily derived from
 the one-to-group measurement.  The measurement point is the host that
 is acting as a receiver while all other hosts act as sources in this
 case.
 The group-to-group communication topology has no obvious focal point:
 the sources and the measurement collection points can be anywhere.
 However, it is possible to organize the problem by applying
 measurements in one-to-group or group-to-one topologies for each host
 in a uniform way (without taking account of how the real

Stephan, et al. Standards Track [Page 11] RFC 5644 Spatial and Multicast Metrics October 2009

 communication might be carried out).  For example, one group of hosts
 < ha, hb, hc, ..., hn > might act as sources to send data to another
 group of hosts < Ha, Hb, Hc, ..., Hm >, and they can be organized
 into n sets of points of interest for one-to-group communications:
 < ha, Ha, Hb, Hc, ..., Hm >, < hb, Ha, Hb, Hc, ..., Hm >, <hc, Ha,
 Hb, Hc, ..., Hm >, ..., < hn, Ha, Hb, Hc, ..., Hm >.

5. Spatial Vector Metrics Definitions

 This section defines vectors for the spatial decomposition of end-to-
 end singleton metrics over a path.
 Spatial vector metrics are based on the decomposition of standard
 end-to-end metrics defined by the IPPM WG in [RFC2679], [RFC2680],
 [RFC3393], and [RFC3432].
 The spatial vector definitions are coupled with the corresponding
 end-to-end metrics.  Measurement methodology aspects are common to
 all the vectors defined and are consequently discussed in a common
 section.

5.1. A Definition for Spatial One-Way Delay Vector

 This section is coupled with the definition of Type-P-One-way-Delay
 in section 3 of [RFC2679].  When a parameter from the definition in
 [RFC2679] is re-used in this section, the first instance will be
 tagged with a trailing asterisk.
 Sections 3.5 to 3.8 of [RFC2679] give requirements and applicability
 statements for end-to-end one-way delay measurements.  They are
 applicable to each point of interest, Hi, involved in the measure.
 Spatial one-way delay measurements MUST respect them, especially
 those related to methodology, clock, uncertainties, and reporting.

5.1.1. Metric Name

 Type-P-Spatial-One-way-Delay-Vector

5.1.2. Metric Parameters

 o  Src*, the IP address of the sender.
 o  Dst*, the IP address of the receiver.
 o  i, an integer in the ordered list <1,2,...,n> of routers in the
    path.

Stephan, et al. Standards Track [Page 12] RFC 5644 Spatial and Multicast Metrics October 2009

 o  Hi, a router of the routers digest.
 o  T*, a time, the sending (or initial observation) time for a
    measured packet.
 o  dT*, a delay, the one-way delay for a measured packet.
 o  dTi, a delay, the one-way delay for a measured packet from the
    source to router Hi.
 o  <dT1,... dTi,... dTn> a list of n delay singletons.
 o  Type-P*, the specification of the packet type.
 o  <H1, H2,..., Hn> the routers digest.

5.1.3. Metric Units

 The value of Type-P-Spatial-One-way-Delay-Vector is a sequence of
 times (a real number in the dimension of seconds with sufficient
 resolution to convey the results).

5.1.4. Definition

 Given a Type-P packet sent by the Src at wire-time (first bit) T to
 the receiver Dst on the path <H1, H2,..., Hn>.  There is a sequence
 of values <T+dT1,T+dT2,...,T+dTn,T+dT> such that dT is the Type-P-
 One-way-Delay from Src to Dst, and for each Hi of the path, T+dTi is
 either a real number corresponding to the wire-time the packet passes
 (last bit received) Hi, or undefined if the packet does not pass Hi
 within a specified loss threshold* time.
 Type-P-Spatial-One-way-Delay-Vector metric is defined for the path
 <Src, H1, H2,..., Hn, Dst> as the sequence of values
 <T,dT1,dT2,...,dTn,dT>.

5.1.5. Discussion

 Some specific issues that may occur are as follows:
 o  the delay singletons "appear" to decrease: dTi > dTi+1.  This may
    occur despite being physically impossible with the definition
    used.

Stephan, et al. Standards Track [Page 13] RFC 5644 Spatial and Multicast Metrics October 2009

  • This is frequently due to a measurement clock synchronization

issue. This point is discussed in section 3.7.1 "Errors or

       uncertainties related to Clocks" of [RFC2679].  Consequently,
       the values of delays measured at multiple routers may not match
       the order of those routers on the path.
  • The actual order of routers on the path may change due to

reconvergence (e.g., recovery from a link failure).

  • The location of the measurement collection point in the device

influences the result. If the packet is not observed directly

       on the input interface, the delay includes buffering time and
       consequently an uncertainty due to the difference between
       'wire-time' and 'host time'.

5.2. A Definition for Spatial Packet Loss Vector

 This section is coupled with the definition of Type-P-One-way-Packet-
 Loss.  When a parameter from section 2 of [RFC2680] is used in this
 section, the first instance will be tagged with a trailing asterisk.
 Sections 2.5 to 2.8 of [RFC2680] give requirements and applicability
 statements for end-to-end one-way packet loss measurements.  They are
 applicable to each point of interest, Hi, involved in the measure.
 Spatial packet loss measurement MUST respect them, especially those
 related to methodology, clock, uncertainties, and reporting.
 The following sections define the spatial loss vector, adapt some of
 the points above, and introduce points specific to spatial loss
 measurement.

5.2.1. Metric Name

 Type-P-Spatial-Packet-Loss-Vector

5.2.2. Metric Parameters

 o  Src*, the IP address of the sender.
 o  Dst*, the IP address of the receiver.
 o  i, an integer in the ordered list <1,2,...,n> of routers in the
    path.
 o  Hi, a router of the routers digest.
 o  T*, a time, the sending time for a measured packet.

Stephan, et al. Standards Track [Page 14] RFC 5644 Spatial and Multicast Metrics October 2009

 o  dTi, a delay, the one-way delay for a measured packet from the
    source to host Hi.
 o  <dT1,..., dTn>, list of n delay singletons.
 o  Type-P*, the specification of packet type.
 o  <H1, H2,..., Hn>, the routers digest.
 o  <L1, L2, ...,Ln>, a list of Boolean values.

5.2.3. Metric Units

 The value of Type-P-Spatial-Packet-Loss-Vector is a sequence of
 Boolean values.

5.2.4. Definition

 Given a Type-P packet sent by the Src at time T to the receiver Dst
 on the path <H1, H2, ..., Hn>.  For the sequence of times <T+dT1,T+
 dT2,..., T+dTi, ...,T+dTn> the packet passes in <H1, H2, ..., Hi,
 ..., Hn>, define the Type-P-Packet-Loss-Vector metric as the sequence
 of values <T, L1, L2, ..., Ln> such that for each Hi of the path, a
 value of 0 for Li means that dTi is a finite value, and a value of 1
 means that dTi is undefined.

5.2.5. Discussion

 Some specific issues that may occur are as follows:
 o  The result might include the sequence of values 1,0.  Although
    this appears physically impossible (a packet is lost, then re-
    appears later on the path):
  • The actual routers on the path may change due to reconvergence

(e.g., recovery from a link failure).

  • The order of routers on the path may change due to

reconvergence.

  • A packet may not be observed in a router due to some buffer or

CPU overflow at the measurement collection point.

5.3. A Definition for Spatial One-Way ipdv Vector

 When a parameter from section 2 of [RFC3393] (the definition of Type-
 P-One-way-ipdv) is used in this section, the first instance will be
 tagged with a trailing asterisk.

Stephan, et al. Standards Track [Page 15] RFC 5644 Spatial and Multicast Metrics October 2009

 The following sections define the spatial ipdv vector, adapt some of
 the points above, and introduce points specific to spatial ipdv
 measurement.

5.3.1. Metric Name

 Type-P-Spatial-One-way-ipdv-Vector

5.3.2. Metric Parameters

 o  Src*, the IP address of the sender.
 o  Dst*, the IP address of the receiver.
 o  i, an integer in the ordered list <1,2,...,n> of routers in the
    path.
 o  Hi, a router of the routers digest.
 o  T1*, a time, the sending time for a first measured packet.
 o  T2*, a time, the sending time for a second measured packet.
 o  dT*, a delay, the one-way delay for a measured packet.
 o  dTi, a delay, the one-way delay for a measured packet from the
    source to router Hi.
 o  Type-P*, the specification of the packet type.
 o  P1, the first packet sent at time T1.
 o  P2, the second packet sent at time T2.
 o  <H1, H2,..., Hn>, the routers digest.
 o  <T1,dT1.1, dT1.2,..., dT1.n,dT1>, the Type-P-Spatial-One-way-
    Delay-Vector for a packet sent at time T1.
 o  <T2,dT2.1, dT2.2,..., dT2.n,dT2>, the Type-P-Spatial-One-way-
    Delay-Vector for a packet sent at time T2.
 o  L*, a packet length in bits.  The packets of a Type-P packet
    stream from which the Type-P-Spatial-One-way-Delay-Vector metric
    is taken MUST all be of the same length.

Stephan, et al. Standards Track [Page 16] RFC 5644 Spatial and Multicast Metrics October 2009

5.3.3. Metric Units

 The value of Type-P-Spatial-One-way-ipdv-Vector is a sequence of
 times (a real number in the dimension of seconds with sufficient
 resolution to convey the results).

5.3.4. Definition

 Given P1 the Type-P packet sent by the sender Src at wire-time (first
 bit) T1 to the receiver Dst. Given <T1, dT1.1, dT1.2,..., dT1.n, dT1>
 the Type-P-Spatial-One-way-Delay-Vector of P1 over the sequence of
 routers <H1, H2,..., Hn>.
 Given P2 the Type-P packet sent by the sender Src at wire-time (first
 bit) T2 to the receiver Dst. Given <T2, dT2.1, dT2.2,..., dT2.n, dT2>
 the Type-P-Spatial-One-way-Delay-Vector of P2 over the same path.
 The Type-P-Spatial-One-way-ipdv-Vector metric is defined as the
 sequence of values <T1, T2, dT2.1-dT1.1, dT2.2-dT1.2 ,..., dT2.n-
 dT1.n, dT2-dT1> such that for each Hi of the sequence of routers <H1,
 H2,..., Hn>, dT2.i-dT1.i is either a real number if the packets P1
 and P2 pass Hi at wire-time (last bit) dT1.i and dT2.i respectively,
 or undefined if at least one of them never passes Hi (and the
 respective one-way delay is undefined).  The T1,T2* pair indicates
 the inter-packet emission interval and dT2-dT1 is ddT* the Type-P-
 One-way-ipdv.

5.4. Spatial Methodology

 The methodology, reporting specifications, and uncertainties
 specified in section 3 of [RFC2679] apply to each point of interest
 (or measurement collection point), Hi, measuring an element of a
 spatial delay vector.
 Likewise, the methodology, reporting specifications, and
 uncertainties specified in section 2 of [RFC2680] apply to each point
 of interest, Hi, measuring an element of a spatial packet loss
 vector.
 Sections 3.5 to 3.7 of [RFC3393] give requirements and applicability
 statements for end-to-end One-way ipdv measurements.  They are
 applicable to each point of interest, Hi, involved in the measure.
 Spatial One-way ipdv measurement MUST respect the methodology, clock,
 uncertainties, and reporting aspects given there.

Stephan, et al. Standards Track [Page 17] RFC 5644 Spatial and Multicast Metrics October 2009

 Generally, for a given Type-P packet of length L at a specific Hi,
 the methodology for spatial vector metrics may proceed as follows:
 o  At each Hi, points of interest/measurement collection points
    prepare to capture the packet sent at time T, record a timestamp
    Ti', and determine the internal delay correction dTi' (see section
    3.7.1.  "Errors or uncertainties related to Clocks" of [RFC2679]);
 o  Each Hi extracts the path ordering information from the packet
    (e.g., time-to-live (TTL));
 o  Each Hi computes the corrected wire-time from Src to Hi: Ti = Ti'
    - dTi'.  This arrival time is undefined if the packet is not
    detected after the 'loss threshold' duration;
 o  Each Hi extracts the timestamp T from the packet;
 o  Each Hi computes the one-way delay from Src to Hi: dTi = Ti - T;
 o  The reference point gathers the result of each Hi and arranges
    them according to the path ordering information received to build
    the Type-P spatial one-way vector (e.g., Type-P-Spatial-One-way-
    Delay-Vector metric <T, dT1, dT2,..., dTn, dT>) over the path
    <Src, H1, H2,..., Hn, Dst> at time T.

5.4.1. Packet Loss Detection

 In a pure end-to-end measurement, packet losses are detected by the
 receiver only.  A packet is lost when Type-P-One-way-Delay is
 undefined or very large (see sections 2.4 and 2.5 of [RFC2680] and
 section 3.5 of [RFC2680]).  A packet is deemed lost by the receiver
 after a duration that starts at the time the packet is sent.  This
 timeout value is chosen by a measurement process.  It determines the
 threshold between recording a long packet transfer time as a finite
 value or an undefined value.
 In a spatial measurement, packet losses may be detected at several
 measurement collection points.  Depending on the consistency of the
 packet loss detections among the points of interest, a packet may be
 considered as lost at one point despite having a finite delay at
 another, or it may be observed by the last measurement collection
 point of the path but considered lost by Dst.
 There is a risk of misinterpreting such results: has the path
 changed?  Did the packet arrive at the destination or was it lost on
 the very last link?

Stephan, et al. Standards Track [Page 18] RFC 5644 Spatial and Multicast Metrics October 2009

 The same concern applies to one-way delay measures: a delay measured
 may be computed as infinite by one observation point but as a real
 value by another one, or may be measured as a real value by the last
 observation point of the path but designated as undefined by Dst.
 The observation/measurement collection points and the destination
 SHOULD use consistent methods to detect packets losses.  The methods
 and parameters must be systematically reported to permit careful
 comparison and to avoid introducing any confounding factors in the
 analysis.

5.4.2. Routers Digest

 The methodology given above relies on knowing the order of the
 router/measurement collection points on the path [RFC2330].
 Path instability might cause a test packet to be observed more than
 once by the same router, resulting in the repetition of one or more
 routers in the routers digest.
 For example, repeated observations may occur during rerouting phases
 that introduce temporary micro loops.  During such an event, the
 routers digest for a packet crossing Ha and Hb may include the
 pattern <Hb, Ha, Hb, Ha, Hb>, meaning that Ha ended the computation
 of the new path before Hb and that the initial path was from Ha to
 Hb, and that the new path is from Hb to Ha.
 Consequently, duplication of routers in the routers digest of a
 vector MUST be identified before computation of statistics to avoid
 producing corrupted information.

6. Spatial Segment Metrics Definitions

 This section defines samples to measure the performance of a segment
 of a path over time.  The definitions rely on the matrix of the
 spatial vector metrics defined above.
 First, this section defines a sample of one-way delay, Type-P-
 Segment-One-way-Delay-Stream, and a sample of packet loss, Type-P-
 Segment-Packet-Loss-Stream.
 Then, it defines two different samples of ipdv: Type-P-Segment-ipdv-
 prev-Stream uses the current and previous packets as the selection
 function, and Type-P-Segment-ipdv-min-Stream uses the minimum delay
 as one of the selected packets in every pair.

Stephan, et al. Standards Track [Page 19] RFC 5644 Spatial and Multicast Metrics October 2009

6.1. A Definition of a Sample of One-Way Delay of a Segment of the Path

 This metric defines a sample of one-way delays over time between a
 pair of routers on a path.  Since it is very close semantically to
 the metric Type-P-One-way-Delay-Poisson-Stream defined in section 4
 of [RFC2679], sections 4.5 to 4.8 of [RFC2679] are integral parts of
 the definition text below.

6.1.1. Metric Name

 Type-P-Segment-One-way-Delay-Stream

6.1.2. Metric Parameters

 o  Src, the IP address of the sender.
 o  Dst, the IP address of the receiver.
 o  Type-P, the specification of the packet type.
 o  i, an integer in the ordered list <1,2,...,n> of routers in the
    path.
 o  k, an integer that orders the packets sent.
 o  a and b, two integers where b > a.
 o  Hi, a router of the routers digest.
 o  <H1,..., Ha, ..., Hb, ...., Hn>, the routers digest.
 o  <T1, T2, ..., Tm>, a list of times.

6.1.3. Metric Units

 The value of a Type-P-Segment-One-way-Delay-Stream is a pair of:
    A list of times <T1, T2, ..., Tm>; and
    A sequence of delays.

6.1.4. Definition

 Given two routers, Ha and Hb, of the path <H1, H2,..., Ha, ..., Hb,
 ..., Hn>, and the matrix of Type-P-Spatial-One-way-Delay-Vector for
 the packets sent from Src to Dst at times <T1, T2, ..., Tm-1, Tm> :

Stephan, et al. Standards Track [Page 20] RFC 5644 Spatial and Multicast Metrics October 2009

    <T1, dT1.1, dT1.2, ..., dT1.a, ..., dT1.b,..., dT1.n, dT1>;
    <T2, dT2.1, dT2.2, ..., dT2.a, ..., dT2.b,..., dT2.n, dT2>;
    ...
    <Tm, dTm.1, dTm.2, ..., dTm.a, ..., dTm.b,..., dTm.n, dTm>.
 We define the sample Type-P-Segment-One-way-Delay-Stream as the
 sequence <dT1.ab, dT2.ab, ..., dTk.ab, ..., dTm.ab> such that for
 each time Tk, 'dTk.ab' is either the real number 'dTk.b - dTk.a', if
 the packet sent at the time Tk passes Ha and Hb, or is undefined if
 this packet never passes Ha or (inclusive) never passes Hb.

6.1.5. Discussion

 Some specific issues that may occur are as follows:
 o  the delay singletons "appear" to decrease: dTi > DTi+1, and is
    discussed in section 5.1.5.
  • This could also occur when the clock resolution of one

measurement collection point is larger than the minimum delay

       of a path.  For example, the minimum delay of a 500 km path
       through optical fiber facilities is 2.5 ms, but the measurement
       collection point has a clock resolution of 8 ms.
 The metric SHALL be invalid for times < T1 , T2, ..., Tm-1, Tm> if
 the following conditions occur:
 o  Ha or Hb disappears from the path due to some routing change.
 o  The order of Ha and Hb changes in the path.

6.2. A Definition of a Sample of Packet Loss of a Segment of the Path

 This metric defines a sample of packet loss over time between a pair
 of routers of a path.  Since it is very close semantically to the
 metric Type-P-Packet-loss-Stream defined in section 3 of [RFC2680],
 sections 3.5 to 3.8 of [RFC2680] are integral parts of the definition
 text below.

6.2.1. Metric Name

 Type-P-Segment-Packet-Loss-Stream

Stephan, et al. Standards Track [Page 21] RFC 5644 Spatial and Multicast Metrics October 2009

6.2.2. Metric Parameters

 o  Src, the IP address of the sender.
 o  Dst, the IP address of the receiver.
 o  Type-P, the specification of the packet type.
 o  k, an integer that orders the packets sent.
 o  n, an integer that orders the routers on the path.
 o  a and b, two integers where b > a.
 o  <H1, H2, ..., Ha, ..., Hb, ...,Hn>, the routers digest.
 o  Hi, a router of the routers digest.
 o  <T1, T2, ..., Tm>, a list of times.
 o  <L1, L2, ..., Ln>, a list of Boolean values.

6.2.3. Metric Units

 The value of a Type-P-Segment-Packet-Loss-Stream is a pair of:
    The list of times <T1, T2, ..., Tm>; and
    A sequence of Boolean values.

6.2.4. Definition

 Given two routers, Ha and Hb, of the path <H1, H2,..., Ha, ..., Hb,
 ..., Hn> and the matrix of Type-P-Spatial-Packet-Loss-Vector for the
 packets sent from Src to Dst at times <T1, T2, ..., Tm-1, Tm> :
    <T1, L1.1, L1.2,..., L1.a, ..., L1.b, ..., L1.n, L>,
    <T2, L2.1, L2.2,..., L2.a, ..., L2.b, ..., L2.n, L>,
    ...,
    <Tm, Lm.1, Lm.2,..., Lma, ..., Lm.b, ..., Lm.n, L>.
 We define the value of the sample Type-P-Segment-Packet-Loss-Stream
 from Ha to Hb as the sequence of Booleans <L1.ab, L2.ab,..., Lk.ab,
 ..., Lm.ab> such that for each Tk:

Stephan, et al. Standards Track [Page 22] RFC 5644 Spatial and Multicast Metrics October 2009

 o  A value of Lk of 0 means that Ha and Hb observed the packet sent
    at time Tk (both Lk.a and Lk.b have a value of 0).
 o  A value of Lk of 1 means that Ha observed the packet sent at time
    Tk (Lk.a has a value of 0) and that Hb did not observe the packet
    sent at time Tk (Lk.b has a value of 1).
 o  The value of Lk is undefined when neither Ha nor Hb observed the
    packet (both Lk.a and Lk.b have a value of 1).

6.2.5. Discussion

 Unlike Type-P-Packet-loss-Stream, Type-P-Segment-Packet-Loss-Stream
 relies on the stability of the routers digest.  The metric SHALL be
 invalid for times < T1 , T2, ..., Tm-1, Tm> if the following
 conditions occur:
 o  Ha or Hb disappears from the path due to some routing change.
 o  The order of Ha and Hb changes in the path.
 o  Lk.a or Lk.b is undefined.
 o  Lk.a has the value 1 (not observed) and Lk.b has the value 0
    (observed).
 o  L has the value 0 (the packet was received by Dst) and Lk.ab has
    the value 1 (the packet was lost between Ha and Hb).

6.3. A Definition of a Sample of ipdv of a Segment Using the Previous

    Packet Selection Function
 This metric defines a sample of ipdv [RFC3393] over time between a
 pair of routers using the previous packet as the selection function.

6.3.1. Metric Name

 Type-P-Segment-ipdv-prev-Stream

6.3.2. Metric Parameters

 o  Src, the IP address of the sender.
 o  Dst, the IP address of the receiver.
 o  Type-P, the specification of the packet type.
 o  k, an integer that orders the packets sent.

Stephan, et al. Standards Track [Page 23] RFC 5644 Spatial and Multicast Metrics October 2009

 o  n, an integer that orders the routers on the path.
 o  a and b, two integers where b > a.
 o  <H1, H2, ..., Ha, ..., Hb, ...,Hn>, the routers digest.
 o  <T1, T2, ..., Tm-1, Tm>, a list of times.
 o  <Tk, dTk.1, dTk.2, ..., dTk.a, ..., dTk.b,..., dTk.n, dTk>, a
    Type-P-Spatial-One-way-Delay-Vector.

6.3.3. Metric Units

 The value of a Type-P-Segment-ipdv-prev-Stream is a pair of:
    The list of <T1, T2, ..., Tm-1, Tm>; and
    A list of pairs of interval of times and delays;

6.3.4. Definition

 Given two routers, Ha and Hb, of the path <H1, H2,..., Ha, ..., Hb,
 ..., Hn> and the matrix of Type-P-Spatial-One-way-Delay-Vector for
 the packets sent from Src to Dst at times <T1, T2, ..., Tm-1, Tm> :
    <T1, dT1.1, dT1.2, ..., dT1.a, ..., dT1.b,..., dT1.n, dT1>,
    <T2, dT2.1, dT2.2, ..., dT2.a, ..., dT2.b,..., dT2.n, dT2>,
    ...
    <Tm, dTm.1, dTm.2, ..., dTm.a, ..., dTm.b,..., dTm.n, dTm>.
 We define the Type-P-Segment-ipdv-prev-Stream as the sequence of
 packet time pairs and delay variations
 <(T1, T2 , dT2.ab - dT1.ab) ,...,
 (Tk-1, Tk, dTk.ab - dTk-1.ab), ...,
 (Tm-1, Tm, dTm.ab - dTm-1.ab)>
 For any pair, Tk, Tk-1 in k=1 through m, the difference dTk.ab - dTk-
 1.ab is undefined if:
 o  the delay dTk.a or the delay dTk-1.a is undefined, OR
 o  the delay dTk.b or the delay dTk-1.b is undefined.

Stephan, et al. Standards Track [Page 24] RFC 5644 Spatial and Multicast Metrics October 2009

6.3.5. Discussion

 This metric belongs to the family of inter-packet delay variation
 metrics (IPDV in uppercase) whose results are extremely sensitive to
 the inter-packet interval in practice.
 The inter-packet interval of an end-to-end IPDV metric is under the
 control of the source (ingress point of interest).  In contrast, the
 inter-packet interval of a segment IPDV metric is not under the
 control the ingress point of interest of the measure, Ha.  The
 interval will certainly vary if there is delay variation between the
 Source and Ha.  Therefore, the ingress inter-packet interval must be
 known at Ha in order to fully comprehend the delay variation between
 Ha and Hb.

6.4. A Definition of a Sample of ipdv of a Segment Using the Minimum

    Delay Selection Function
 This metric defines a sample of ipdv [RFC3393] over time between a
 pair of routers on a path using the minimum delay as one of the
 selected packets in every pair.

6.4.1. Metric Name

 Type-P-Segment-One-way-ipdv-min-Stream

6.4.2. Metric Parameters

 o  Src, the IP address of the sender.
 o  Dst, the IP address of the receiver.
 o  Type-P, the specification of the packet type.
 o  k, an integer that orders the packets sent.
 o  i, an integer that identifies a packet sent.
 o  n, an integer that orders the routers on the path.
 o  a and b, two integers where b > a.
 o  <H1, H2, ..., Ha, ..., Hb, ...,Hn>, the routers digest.
 o  <T1, T2, ..., Tm-1, Tm>, a list of times.
 o  <Tk, dTk.1, dTk.2, ..., dTk.a, ..., dTk.b,..., dTk.n, dTk>, a
    Type-P-Spatial-One-way-Delay-Vector.

Stephan, et al. Standards Track [Page 25] RFC 5644 Spatial and Multicast Metrics October 2009

6.4.3. Metric Units

 The value of a Type-P-Segment-One-way-ipdv-min-Stream is a pair of:
    The list of <T1, T2, ..., Tm-1, Tm>; and
    A list of times.

6.4.4. Definition

 Given two routers, Ha and Hb, of the path <H1, H2,..., Ha, ..., Hb,
 ..., Hn> and the matrix of Type-P-Spatial-One-way-Delay-Vector for
 the packets sent from Src to Dst at times <T1, T2, ..., Tm-1, Tm> :
    <T1, dT1.1, dT1.2, ..., dT1.a, ..., dT1.b,..., dT1.n, dT1>,
    <T2, dT2.1, dT2.2, ..., dT2.a, ..., dT2.b,..., dT2.n, dT2>,
    ...
    <Tm, dTm.1, dTm.2, ..., dTm.a, ..., dTm.b,..., dTm.n, dTm>.
 We define the Type-P-Segment-One-way-ipdv-min-Stream as the sequence
 of times <dT1.ab - min(dTi.ab) ,..., dTk.ab - min(dTi.ab), ...,
 dTm.ab - min(dTi.ab)> where:
 o  min(dTi.ab) is the minimum value of the tuples (dTk.b - dTk.a);
 o  for each time Tk, dTk.ab is undefined if dTk.a or (inclusive)
    dTk.b is undefined, or the real number (dTk.b - dTk.a) is
    undefined.

6.4.5. Discussion

 This metric belongs to the family of packet delay variation metrics
 (PDV).  PDV distributions have less sensitivity to inter-packet
 interval variations than IPDV values, as discussed above.
 In principle, the PDV distribution reflects the variation over many
 different inter-packet intervals, from the smallest inter-packet
 interval, up to the length of the evaluation interval, Tm - T1.
 Therefore, when delay variation occurs and disturbs the packet
 spacing observed at Ha, the PDV results will likely compare favorably
 to a PDV measurement where the source is Ha and the destination is
 Hb, because a wide range of spacings are reflected in any PDV
 distribution.

Stephan, et al. Standards Track [Page 26] RFC 5644 Spatial and Multicast Metrics October 2009

7. One-to-Group Metrics Definitions

 This section defines performance metrics between a source and a group
 of receivers.

7.1. A Definition for One-to-Group Delay

 This section defines a metric for one-way delay between a source and
 a group of receivers.

7.1.1. Metric Name

 Type-P-One-to-group-Delay-Vector

7.1.2. Metric Parameters

 o  Src, the IP address of a host acting as the source.
 o  Recv1,..., RecvN, the IP addresses of the N hosts acting as
    receivers.
 o  T, a time.
 o  dT1,...,dTn a list of times.
 o  Type-P, the specification of the packet type.
 o  Gr, the receiving group identifier.  The parameter Gr is the
    multicast group address if the measured packets are transmitted
    over IP multicast.  This parameter is to differentiate the
    measured traffic from other unicast and multicast traffic.  It is
    OPTIONAL for this metric to avoid losing any generality, i.e., to
    make the metric also applicable to unicast measurement where there
    is only one receiver.

7.1.3. Metric Units

 The value of a Type-P-One-to-group-Delay-Vector is a set of Type-P-
 One-way-Delay singletons [RFC2679], that is a sequence of times (a
 real number in the dimension of seconds with sufficient resolution to
 convey the results).

7.1.4. Definition

 Given a Type-P packet sent by the source Src at time T, and the N
 hosts { Recv1,...,RecvN } which receive the packet at the time {
 T+dT1,...,T+dTn }, or the packet does not pass a receiver within a
 specified loss threshold time, then the Type-P-One-to-group-Delay-

Stephan, et al. Standards Track [Page 27] RFC 5644 Spatial and Multicast Metrics October 2009

 Vector is defined as the set of the Type-P-One-way-Delay singletons
 between Src and each receiver with value of { dT1, dT2,...,dTn },
 where any of the singletons may be undefined if the packet did not
 pass the corresponding receiver within a specified loss threshold
 time.

7.2. A Definition for One-to-Group Packet Loss

7.2.1. Metric Name

 Type-P-One-to-group-Packet-Loss-Vector

7.2.2. Metric Parameters

 o  Src, the IP address of a host acting as the source.
 o  Recv1,..., RecvN, the IP addresses of the N hosts acting as
    receivers.
 o  T, a time.
 o  Type-P, the specification of the packet type.
 o  Gr, the receiving group identifier, OPTIONAL.

7.2.3. Metric Units

 The value of a Type-P-One-to-group-Packet-Loss-Vector is a set of
 Type-P-One-way-Packet-Loss singletons [RFC2680].
 o  T, time the source packet was sent.
 o  L1,...,LN a list of Boolean values.

7.2.4. Definition

 Given a Type-P packet sent by the source Src at T and the N hosts,
 Recv1,...,RecvN, the Type-P-One-to-group-Packet-Loss-Vector is
 defined as a set of the Type-P-One-way-Packet-Loss singletons between
 Src and each of the receivers:
 {T, <L1=0|1>,<L2=0|1>,..., <LN=0|1>},
 where the Boolean value 0|1 depends on receiving the packet at a
 particular receiver within a loss threshold time.

Stephan, et al. Standards Track [Page 28] RFC 5644 Spatial and Multicast Metrics October 2009

7.3. A Definition for One-to-Group ipdv

7.3.1. Metric Name

 Type-P-One-to-group-ipdv-Vector

7.3.2. Metric Parameters

 o  Src, the IP address of a host acting as the source.
 o  Recv1,..., RecvN, the IP addresses of the N hosts acting as
    receivers.
 o  T1, a time.
 o  T2, a time.
 o  ddT1, ...,ddTn, a list of times.
 o  Type-P, the specification of the packet type.
 o  F, a selection function non-ambiguously defining the two packets
    from the stream selected for the metric.
 o  Gr, the receiving group identifier.  The parameter Gr is the
    multicast group address if the measured packets are transmitted
    over IP multicast.  This parameter is to differentiate the
    measured traffic from other unicast and multicast traffic.  It is
    OPTIONAL in the metric to avoid losing any generality, i.e., to
    make the metric also applicable to unicast measurement where there
    is only one receiver.

7.3.3. Metric Units

 The value of a Type-P-One-to-group-ipdv-Vector is a set of Type-P-
 One-way-ipdv singletons [RFC3393].

7.3.4. Definition

 Given a Type-P packet stream, Type-P-One-to-group-ipdv-Vector is
 defined for two packets transferred from the source Src to the N
 hosts {Recv1,...,RecvN }, which are selected by the selection
 function F as the difference between the value of the Type-P-One-to-
 group-Delay-Vector from Src to { Recv1,..., RecvN } at time T1 and
 the value of the Type-P-One-to-group-Delay-Vector from Src to {
 Recv1,...,RecvN } at time T2.  T1 is the wire-time at which Src sent

Stephan, et al. Standards Track [Page 29] RFC 5644 Spatial and Multicast Metrics October 2009

 the first bit of the first packet, and T2 is the wire-time at which
 Src sent the first bit of the second packet.  This metric is derived
 from the Type-P-One-to-group-Delay-Vector metric.
 For a set of real numbers {ddT1,...,ddTn}, the Type-P-One-to-group-
 ipdv-Vector from Src to { Recv1,...,RecvN } at T1, T2 is
 {ddT1,...,ddTn} means that Src sent two packets, the first at wire-
 time T1 (first bit), and the second at wire-time T2 (first bit) and
 the packets were received by { Recv1,...,RecvN } at wire-time {dT1+
 T1,...,dTn+T1}(last bit of the first packet), and at wire-time {dT'1+
 T2,...,dT'n+T2} (last bit of the second packet), and that {dT'1-
 dT1,...,dT'n-dTn} ={ddT1,...,ddTn}.
 For any pair of selected packets, the difference dT'n-dTn is
 undefined if:
 o  the delay dTn to Receiver n is undefined, OR
 o  the delay dT'n to Receiver n is undefined.

8. One-to-Group Sample Statistics

 The one-to-group metrics defined above are directly achieved by
 collecting relevant unicast one-way metrics measurements results and
 by gathering them per group of receivers.  They produce network
 performance information that guides engineers toward potential
 problems that may have happened on any branch of a multicast routing
 tree.
 The results of these metrics are not directly usable to present the
 performance of a group because each result is made of a huge number
 of singletons that are difficult to read and analyze.  As an example,
 delays are not comparable because the distance between receiver and
 sender differs.  Furthermore, they don't capture relative performance
 situations in a multiparty communication.
 From the performance point of view, the multiparty communication
 services not only require the support of absolute performance
 information but also information on "relative performance".
 "Relative performance" means the difference between absolute
 performance of all users.  Directly using the one-way metrics cannot
 present the relative performance situation.  However, if we use the
 variations of all users' one-way parameters, we can have new metrics
 to measure the difference of the absolute performance and hence
 provide the threshold value of relative performance that a multiparty
 service might demand.  A very good example of the high relative
 performance requirement is online gaming.  A very small difference in
 delay might result in failure in the game.  We have to use multicast-

Stephan, et al. Standards Track [Page 30] RFC 5644 Spatial and Multicast Metrics October 2009

 specific statistic metrics to define the relative delay required by
 online gaming.  There are many other services, e.g., online biding,
 online stock market, etc., that require multicast metrics in order to
 evaluate the network against their requirements.  Therefore, we can
 see the importance of new, multicast specific, statistic metrics to
 feed this need.
 We might also use some one-to-group statistic conceptions to present
 and report the group performance and relative performance to save the
 report transmission bandwidth.  Statistics have been defined for One-
 way metrics in corresponding RFCs.  They provide the foundation of
 definition for performance statistics.  For instance, there are
 definitions for minimum and maximum one-way delay in [RFC2679].
 However, there is a dramatic difference between the statistics for
 one-to-one communications and for one-to-many communications.  The
 former one only has statistics over the time dimension while the
 later one can have statistics over both time and space dimensions.
 This space dimension is introduced by the Matrix concept as
 illustrated in Figure 4.  For a Matrix M, each row is a set of one-
 way singletons spreading over the time dimension and each column is
 another set of One-way singletons spreading over the space dimension.
          Receivers
           Space
             ^
           1 |    / R1dT1   R1dT2     R1dT3 ... R1dTk \
             |   |                                     |
           2 |   |  R2dT1   R2dT2     R2dT3 ... R2dTk  |
             |   |                                     |
           3 |   |  R3dT1   R3dT2     R3dT3 ... R3dTk  |
           . |   |                                     |
           . |   |                                     |
           . |   |                                     |
           n |    \ RndT1   RndT2     RndT3 ... RndTk /
             +--------------------------------------------> time
            T0
                       Figure 4: Matrix M (n*m)
 In Matrix M, each element is a one-way delay singleton.  Each column
 is a delay vector.  It contains the one-way delays of the same packet
 observed at n points of interest.  It implies the geographical factor
 of the performance within a group.  Each row is a set of one-way
 delays observed during a sampling interval at one of the points of
 interest.  It presents the delay performance at a receiver over the
 time dimension.

Stephan, et al. Standards Track [Page 31] RFC 5644 Spatial and Multicast Metrics October 2009

 Therefore, one can either calculate statistics by rows over the space
 dimension or by columns over the time dimension.  It's up to the
 operators or service providers in which dimension they are
 interested.  For example, a TV broadcast service provider might want
 to know the statistical performance of each user in a long-term run
 to make sure their services are acceptable and stable.  While for an
 online gaming service provider, he might be more interested in
 knowing if all users are served fairly by calculating the statistics
 over the space dimension.  This memo does not intend to recommend
 which of the statistics are better than the others.
 To save the report transmission bandwidth, each point of interest can
 send statistics in a pre-defined time interval to the reference point
 rather than sending every one-way singleton it observed.  As long as
 an appropriate time interval is decided, appropriate statistics can
 represent the performance in a certain accurate scale.  How to decide
 the time interval and how to bootstrap all points of interest and the
 reference point depend on applications.  For instance, applications
 with a lower transmission rate can have the time interval be longer,
 and ones with higher transmission rate can have the time interval be
 shorter.  However, this is out of the scope of this memo.
 Moreover, after knowing the statistics over the time dimension, one
 might want to know how these statistics are distributed over the
 space dimension.  For instance, a TV broadcast service provider had
 the performance Matrix M and calculated the one-way delay mean over
 the time dimension to obtain a delay Vector as {V1,V2,..., VN}.  He
 then calculated the mean of all the elements in the Vector to see
 what level of delay he has served to all N users.  This new delay
 mean gives information on how well the service has been delivered to
 a group of users during a sampling interval in terms of delay.  It
 requires twice as much calculation to have this statistic over both
 time and space dimensions.  These kinds of statistics are referred to
 as 2-level statistics to distinguish them from 1-level statistics
 calculated over either space or time dimension.  It can be easily
 proven that no matter over which dimension a 2-level statistic is
 calculated first, the results are the same.  That is, one can
 calculate the 2-level delay mean using the Matrix M by having the
 1-level delay mean over the time dimension first and then calculate
 the mean of the obtained vector to find out the 2-level delay mean.
 Or, he can do the 1-level statistic calculation over the space
 dimension first and then have the 2-level delay mean.  Both results
 will be exactly the same.  Therefore, when defining a 2-level
 statistic, there is no need to specify the order in which the
 calculation is executed.

Stephan, et al. Standards Track [Page 32] RFC 5644 Spatial and Multicast Metrics October 2009

 Many statistics can be defined for the proposed one-to-group metrics
 over the space dimension, the time dimension, or both.  This memo
 treats the case where a stream of packets from the Source results in
 a sample at each of the Receivers in the Group, and these samples are
 each summarized with the usual statistics employed in one-to-one
 communication.  New statistic definitions are presented, which
 summarize the one-to-one statistics over all the Receivers in the
 Group.

8.1. Discussion on the Impact of Packet Loss on Statistics

 Packet loss does have effects on one-way metrics and their
 statistics.  For example, a lost packet can result in an infinite
 one-way delay.  It is easy to handle the problem by simply ignoring
 the infinite value in the metrics and in the calculation of the
 corresponding statistics.  However, the packet loss has such a strong
 impact on the statistics calculation for the one-to-group metrics
 that it can not be solved by the same method used for one-way
 metrics.  This is due to the complexity of building a matrix, which
 is needed for calculation of the statistics proposed in this memo.
 The situation is that measurement results obtained by different end
 users might have different packet loss pattern.  For example, for
 User1, packet A was observed to be lost.  And for User2, packet A was
 successfully received, but packet B was lost.  If the method to
 overcome the packet loss for one-way metrics is applied, the two
 singleton sets reported by User1 and User2 will be different in terms
 of the transmitted packets.  Moreover, if User1 and User2 have a
 different number of lost packets, the size of the results will be
 different.  Therefore, for the centralized calculation, the reference
 point will not be able to use these two results to build up the group
 Matrix and cannot calculate the statistics.  The extreme situation
 being the case when no packets arrive at any user.  One of the
 possible solutions is to replace the infinite/undefined delay value
 by the average of the two adjacent values.  For example, if the
 result reported by User1 is { R1dT1 R1dT2 R1dT3 ...  R1dTK-1 UNDEF
 R1dTK+1...  R1MD } where "UNDEF" is an undefined value, the reference
 point can replace it by R1dTK = {(R1dTK-1)+( R1dTK+1)}/2.  Therefore,
 this result can be used to build up the group Matrix with an
 estimated value R1dTK.  There are other possible solutions, such as
 using the overall mean of the whole result to replace the infinite/
 undefined value, and so on.  However, this is out of the scope of
 this memo.
 For the distributed calculation, the reported statistics might have
 different "weight" to present the group performance, which is
 especially true for delay and ipdv relevant metrics.  For example,

Stephan, et al. Standards Track [Page 33] RFC 5644 Spatial and Multicast Metrics October 2009

 User1 calculates the Type-P-Finite-One-way-Delay-Mean R1MD as shown
 in Figure 7 without any packet loss, and User2 calculates the R2MD
 with N-2 packet loss.  The R1MD and R2MD should not be treated with
 equal weight because R2MD was calculated only based on two delay
 values in the whole sample interval.  One possible solution is to use
 a weight factor to mark every statistic value sent by users and use
 this factor for further statistic calculation.

8.2. General Metric Parameters

 o  Src, the IP address of a host.
 o  G, the receiving group identifier.
 o  N, the number of Receivers (Recv1, Recv2, ...  RecvN).
 o  T, a time (start of test interval).
 o  Tf, a time (end of test interval).
 o  K, the number of packets sent from the source during the test
    interval.
 o  J[n], the number of packets received at a particular Receiver, n,
    where 1<=n<=N.
 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  T0, a time that MUST be selected at random from the interval [T,
    T+I] to start generating packets and taking measurements (for
    Periodic Streams).
 o  TstampSrc, the wire-time of the packet as measured at MP(Src) (the
    Source Measurement Point).
 o  TstampRecv, the wire-time of the packet as measured at MP(Recv),
    assigned to packets that arrive within a "reasonable" time.
 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  dT, shorthand notation for a one-way delay singleton value.

Stephan, et al. Standards Track [Page 34] RFC 5644 Spatial and Multicast Metrics October 2009

 o  L, shorthand notation for a one-way loss singleton value, either
    zero or one, where L=1 indicates loss and L=0 indicates arrival at
    the destination within TstampSrc + Tmax, may be indexed over n
    Receivers.
 o  DV, shorthand notation for a one-way delay variation singleton
    value.

8.3. One-to-Group Delay Statistics

 This section defines the overall one-way delay statistics for a
 receiver and for an entire group as illustrated by the matrix below.
    Recv    /----------- Sample -------------\   Stats      Group Stat
     1      R1dT1   R1dT2     R1dT3 ... R1dTk    R1MD  \
                                                        |
     2      R2dT1   R2dT2     R2dT3 ... R2dTk    R2MD   |
                                                        |
     3      R3dT1   R3dT2     R3dT3 ... R3dTk    R3MD    > Group Delay
     .                                                  |
     .                                                  |
     .                                                  |
     n      RndT1   RndT2     RndT3 ... RndTk    RnMD  /
                                               Receiver-n
                                                 Delay
                   Figure 5: One-to-Group Mean Delay
 Statistics are computed on the finite one-way delays of the matrix
 above.
 All one-to-group delay statistics are expressed in seconds with
 sufficient resolution to convey three significant digits.

8.3.1. Type-P-One-to-group-Receiver-n-Mean-Delay

 This section defines Type-P-One-to-group-Receiver-n-Mean-Delay, the
 Delay Mean, at each Receiver N, also named RnMD.
 We obtain the value of Type-P-One-way-Delay singleton for all packets
 sent during the test interval at each Receiver (Destination), as per
 [RFC2679].  For each packet that arrives within Tmax of its sending
 time, TstampSrc, the one-way delay singleton (dT) will be the finite
 value TstampRecv[i] - TstampSrc[i] in units of seconds.  Otherwise,
 the value of the singleton is Undefined.

Stephan, et al. Standards Track [Page 35] RFC 5644 Spatial and Multicast Metrics October 2009

                         J[n]
                         ---
                    1    \
         RnMD =    --- *  >  TstampRecv[i] - TstampSrc[i]
                   J[n]  /
                         ---
                         i = 1
         Note:  RnMD value is Undefined when J[n] = 0 for all n.
          Figure 6: Type-P-One-to-group-Receiver-n-Mean-Delay
 where all packets i= 1 through J[n] have finite singleton delays.

8.3.2. Type-P-One-to-group-Mean-Delay

 This section defines Type-P-One-to-group-Mean-Delay, the Mean one-way
 Delay calculated over the entire Group, also named GMD.
                                       N
                                      ---
                                 1    \
                          GMD =  - *   >   RnMD
                                 N    /
                                      ---
                                      n = 1
               Figure 7: Type-P-One-to-group-Mean-Delay
 Note that the Group Mean Delay can also be calculated by summing the
 finite one-way delay singletons in the matrix, and dividing by the
 number of finite one-way delay singletons.

8.3.3. Type-P-One-to-group-Range-Mean-Delay

 This section defines a metric for the Range of Mean Delays over all N
 receivers in the Group (R1MD, R2MD...RnMD).
 Type-P-One-to-group-Range-Mean-Delay = GRMD = max(RnMD) - min(RnMD)

8.3.4. Type-P-One-to-group-Max-Mean-Delay

 This section defines a metric for the Maximum of Mean Delays over all
 N receivers in the Group (R1MD, R2MD,...RnMD).
 Type-P-One-to-group-Max-Mean-Delay = GMMD = max(RnMD)

Stephan, et al. Standards Track [Page 36] RFC 5644 Spatial and Multicast Metrics October 2009

8.4. One-to-Group Packet Loss Statistics

 This section defines the overall one-way loss statistics for a
 receiver and for an entire group as illustrated by the matrix below.
  Recv    /----------- Sample ----------\   Stats     Group Stat
    1      R1L1   R1L2     R1L3 ... R1Lk     R1LR \
                                                   |
    2      R2L1   R2L2     R2L3 ... R2Lk     R2LR  |
                                                   |
    3      R3L1   R3L2     R3L3 ... R3Lk     R3LR   > Group Loss Ratio
    .                                              |
    .                                              |
    .                                              |
    n      RnL1   RnL2     RnL3 ... RnLk     RnLR /
                                         Receiver-n
                                         Loss Ratio
                   Figure 8: One-to-Group Loss Ratio
 Statistics are computed on the sample of Type-P-One-way-Packet-Loss
 [RFC2680] of the matrix above.
 All loss ratios are expressed in units of packets lost to total
 packets sent.

8.4.1. Type-P-One-to-group-Receiver-n-Loss-Ratio

 Given a Matrix of loss singletons as illustrated above, determine the
 Type-P-One-way-Packet-Loss-Average for the sample at each receiver,
 according to the definitions and method of [RFC2680].  The Type-P-
 One-way-Packet-Loss-Average and the Type-P-One-to-group-Receiver-n-
 Loss-Ratio, also named RnLR, are equivalent metrics.  In terms of the
 parameters used here, these metrics definitions can be expressed as
                                         K
                                        ---
                                   1    \
                           RnLR =  - *   >   RnLk
                                   K    /
                                        ---
                                       k = 1
          Figure 9: Type-P-One-to-group-Receiver-n-Loss-Ratio

Stephan, et al. Standards Track [Page 37] RFC 5644 Spatial and Multicast Metrics October 2009

8.4.2. Type-P-One-to-group-Receiver-n-Comp-Loss-Ratio

 Usually, the number of packets sent is used in the denominator of
 packet loss ratio metrics.  For the comparative metrics defined here,
 the denominator is the maximum number of packets received at any
 receiver for the sample and test interval of interest.  The numerator
 is the sum of the losses at receiver n.
 The Comparative Loss Ratio, also named, RnCLR, is defined as
                                K
                               ---
                               \
                                >   Ln(k)
                               /
                               ---
                               k=1
          RnCLR =  -----------------------------
                            /    K         \
                            |   ---        |
                            |   \          |
                    K - Min |    >   Ln(k) |
                            |   /          |
                            |   ---        |
                            \   k=1        / N
          Note: Ln is a set of one-way loss values at receiver n.
                There is one value for each of the K packets sent.
       Figure 10: Type-P-One-to-group-Receiver-n-Comp-Loss-Ratio

8.4.3. Type-P-One-to-group-Loss-Ratio

 Type-P-One-to-group-Loss-Ratio, the overall Group Loss Ratio, also
 named GLR, is defined as:
                                       K,N
                                       ---
                                 1     \
                          GLR = --- *   >   Ln(k)
                                K*N    /
                                       ---
                                      k,n = 1
               Figure 11: Type-P-One-to-group-Loss-Ratio

Stephan, et al. Standards Track [Page 38] RFC 5644 Spatial and Multicast Metrics October 2009

 Where the sum includes all of the Loss singletons, Ln(k), over the N
 receivers and K packets sent, in a ratio with the total packets over
 all receivers.

8.4.4. Type-P-One-to-group-Range-Loss-Ratio

 The One-to-group Loss Ratio Range is defined as:
 Type-P-One-to-group-Range-Loss-Ratio = max(RnLR) - min(RnLR)
 It is most effective to indicate the range by giving both the maximum
 and minimum loss ratios for the Group, rather than only reporting the
 difference between them.

8.5. One-to-group Delay Variation Statistics

 This section defines one-way delay variation (DV) statistics for an
 entire group as illustrated by the matrix below.
  Recv    /------------- Sample --------------\   Stats
   1      R1ddT1   R1ddT2     R1ddT3 ... R1ddTk   R1DV  \
                                                         |
   2      R2ddT1   R2ddT2     R2ddT3 ... R2ddTk   R2DV   |
                                                         |
   3      R3ddT1   R3ddT2     R3ddT3 ... R3ddTk   R3DV    > Group Stat
   .                                                     |
   .                                                     |
   .                                                     |
   n      RnddT1   RnddT2     RnddT3 ... RnddTk   RnDV  /
         Figure 12: One-to-group Delay Variation Matrix (DVMa)
 Statistics are computed on the sample of Type-P-One-way-ipdv
 singletons of the group delay variation matrix above where RnddTk is
 the Type-P-One-way-ipdv singleton evaluated at Receiver n for the
 packet k and where RnDV is the point-to-point one-way packet delay
 variation for Receiver n.
 All One-to-group delay variation statistics are expressed in seconds
 with sufficient resolution to convey three significant digits.

8.5.1. Type-P-One-to-group-Range-Delay-Variation

 This section defines a metric for the Range of Delay Variation over
 all N receivers in the Group.

Stephan, et al. Standards Track [Page 39] RFC 5644 Spatial and Multicast Metrics October 2009

 Maximum DV and minimum DV over all receivers summarize the
 performance over the Group (where DV is a point-to-point metric).
 For each receiver, the DV is usually expressed as the 1-10^(-3)
 quantile of one-way delay minus the minimum one-way delay.
 Type-P-One-to-group-Range-Delay-Variation = GRDV =
 = max(RnDV) - min(RnDV) for all n receivers
 This range is determined from the minimum and maximum values of the
 point-to-point one-way IP Packet Delay Variation for the set of
 Destinations in the group and a population of interest, using the
 Packet Delay Variation expressed as the 1-10^-3 quantile of one-way
 delay minus the minimum one-way delay.  If a more demanding service
 is considered, one alternative is to use the 1-10^-5 quantile, and in
 either case, the quantile used should be recorded with the results.
 Both the minimum and the maximum delay variation are recorded, and
 both values are given to indicate the location of the range.

9. Measurement Methods: Scalability and Reporting

 Virtually all the guidance on measurement processes supplied by the
 earlier IPPM RFCs (such as [RFC2679] and [RFC2680]) for one-to-one
 scenarios is applicable here in the spatial and multiparty
 measurement scenario.  The main difference is that the spatial and
 multiparty configurations require multiple points of interest where a
 stream of singletons will be collected.  The amount of information
 requiring storage grows with both the number of metrics and the
 points of interest, so the scale of the measurement architecture
 multiplies the number of singleton results that must be collected and
 processed.
 It is possible that the architecture for results collection involves
 a single reference point with connectivity to all the points of
 interest.  In this case, the number of points of interest determines
 both storage capacity and packet transfer capacity of the host acting
 as the reference point.  However, both the storage and transfer
 capacity can be reduced if the points of interest are capable of
 computing the summary statistics that describe each measurement
 interval.  This is consistent with many operational monitoring
 architectures today, where even the individual singletons may not be
 stored at each point of interest.
 In recognition of the likely need to minimize the form of the results
 for storage and communication, the Group metrics above have been
 constructed to allow some computations on a per-Receiver basis.  This

Stephan, et al. Standards Track [Page 40] RFC 5644 Spatial and Multicast Metrics October 2009

 means that each Receiver's statistics would normally have an equal
 weight with all other Receivers in the Group (regardless of the
 number of packets received).

9.1. Computation Methods

 The scalability issue can be raised when there are thousands of
 points of interest in a group who are trying to send back the
 measurement results to the reference point for further processing and
 analysis.  The points of interest can send either the whole measured
 sample or only the calculated statistics.  The former is a
 centralized statistic calculation method and the latter is a
 distributed statistic calculation method.  The sample should include
 all metrics parameters, the values, and the corresponding sequence
 numbers.  The transmission of the whole sample can cost much more
 bandwidth than the transmission of the statistics that should include
 all statistic parameters specified by policies and the additional
 information about the whole sample, such as the size of the sample,
 the group address, the address of the point of interest, the ID of
 the sample session, and so on.  Apparently, the centralized
 calculation method can require much more bandwidth than the
 distributed calculation method when the sample size is big.  This is
 especially true when the measurement has a very large number of the
 points of interest.  It can lead to a scalability issue at the
 reference point by overloading the network resources.
 The distributed calculation method can save much more bandwidth and
 mitigate issues arising from scalability at the reference point side.
 However, it may result in a loss of information.  As not all measured
 singletons are available for building up the group matrix, the real
 performance over time can be hidden from the result.  For example,
 the loss pattern can be missed by simply accepting the loss ratio.
 This tradeoff between bandwidth consumption and information
 acquisition has to be taken into account when designing the
 measurement approach.
 One possible solution could be to transmit the statistic parameters
 to the reference point first to obtain the general information of the
 group performance.  If detailed results are required, the reference
 point should send the requests to the points of interest, which could
 be particular ones or the whole group.  This procedure can happen in
 the off peak time and can be well scheduled to avoid delivery of too
 many points of interest at the same time.  Compression techniques can
 also be used to minimize the bandwidth required by the transmission.
 This could be a measurement protocol to report the measurement
 results.  However, this is out of the scope of this memo.

Stephan, et al. Standards Track [Page 41] RFC 5644 Spatial and Multicast Metrics October 2009

9.2. Measurement

 To prevent any bias in the result, the configuration of a one-to-many
 measure must take into consideration that more packets will be routed
 than sent (copies of a packet sent are expected to arrive at many
 destination points) and select a test packet rate that will not
 impact the network performance.

9.3. Effect of Time and Space Aggregation Order on Stats

 This section presents the impact of the aggregation order on the
 scalability of the reporting and of the computation.  It makes the
 hypothesis that receivers are not co-located and that results are
 gathered in a point of reference for further usages.
 Multimetric samples are represented in a matrix as illustrated below
    Point of
    Interest
      1      R1S1   R1S1     R1S1 ... R1Sk    \
                                               |
      2      R2S1   R2S2     R2S3 ... R2Sk     |
                                               |
      3      R3S1   R3S2     R3S3 ... R3Sk      >  Sample over Space
      .                                        |
      .                                        |
      .                                        |
      n      RnS1   RnS2     RnS3 ... RnSk    /
             S1M    S2M      S3M  ... SnM     Stats over Space
             \-------------  ------------/
                           \/
               Stats over Space and Time
     Figure 13: Impact of Space Aggregation on Multimetrics Stats
 Two methods are available to compute statistics on a matrix:
 o  Method 1: The statistic metric is computed over time and then over
    space; or
 o  Method 2: The statistic metric is computed over space and then
    over time.

Stephan, et al. Standards Track [Page 42] RFC 5644 Spatial and Multicast Metrics October 2009

 These two methods differ only by the order of the aggregation.  The
 order does not impact the computation resources required.  It does
 not change the value of the result.  However, it impacts severely the
 minimal volume of data to report:
 o  Method 1: Each point of interest periodically computes statistics
    over time to lower the volume of data to report.  They are
    reported to the reference point for subsequent computations over
    the spatial dimension.  This volume no longer depends on the
    number of samples.  It is only proportional to the computation
    period.
 o  Method 2: The volume of data to report is proportional to the
    number of samples.  Each sample, RiSi, must be reported to the
    reference point for computing statistic over space and statistic
    over time.  The volume increases with the number of samples.  It
    is proportional to the number of test packets;
 Method 2 has severe drawbacks in terms of security and dimensioning:
 o  Increasing the rate of the test packets may result in a Denial of
    Service (DoS) toward the points of reference;
 o  The dimensioning of a measurement system is quite impossible to
    validate because any increase of the rate of the test packets will
    increase the bandwidth requested to collect the raw results.
 The computation period over time period (commonly named the
 aggregation period) provides the reporting side with a control of
 various collecting aspects such as bandwidth, computation, and
 storage capacities.  So this document defines metrics based on method
 1.

9.3.1. Impact on Spatial Statistics

 Two methods are available to compute spatial statistics:
 o  Method 1: Spatial segment metrics and statistics are preferably
    computed over time for each points of interest;
 o  Method 2: Vectors metrics are intrinsically instantaneous space
    metrics, which must be reported using Method 2 whenever
    instantaneous metrics information is needed.

Stephan, et al. Standards Track [Page 43] RFC 5644 Spatial and Multicast Metrics October 2009

9.3.2. Impact on One-to-Group Statistics

 Two methods are available to compute group statistics:
 o  Method 1: Figure 5 and Figure 8 illustrate the method.  The one-
    to-one statistic is computed per interval of time before the
    computation of the mean over the group of receivers.
 o  Method 2: Figure 13 presents the second method.  The metric is
    computed over space and then over time.

10. Manageability Considerations

 This section defines the reporting of all the metrics introduced in
 the document.
 Information models of spatial metrics and of one-to-group metrics are
 similar except that points of interests of spatial vectors MUST be
 ordered.
 The complexity of the reporting relies on the number of points of
 interest.

10.1. Reporting Spatial Metric

 The reporting of spatial metrics shares a lot of aspects with RFC
 2679 and RFC 2680.  New ones are common to all the definitions and
 are mostly related to the reporting of the path and of methodology
 parameters that may bias raw results analysis.  This section presents
 these specific parameters and then lists exhaustively the parameters
 that SHOULD be reported.

10.1.1. Path

 End-to-end metrics can't determine the path of the measure despite
 the fact that IPPM RFCs recommend it be reported (see section 3.8.4
 of [RFC2679]).  Spatial metrics vectors provide this path.  The
 report of a spatial vector MUST include the points of interests
 involved: the sub-set of the routers of the path participating to the
 instantaneous measure.

10.1.2. Host Order

 A spatial vector MUST order the points of interest according to their
 order in the path.  The ordering MAY be based on information from the
 TTL in IPv4, the Hop Limit in IPv6, or the corresponding information
 in MPLS.

Stephan, et al. Standards Track [Page 44] RFC 5644 Spatial and Multicast Metrics October 2009

 The report of a spatial vector MUST include the ordered list of the
 hosts involved in the instantaneous measure.

10.1.3. Timestamping Bias

 The location of the point of interest inside a node influences the
 timestamping skew and accuracy.  As an example, consider that some
 internal machinery delays the timestamping up to three milliseconds;
 then the minimal uncertainty reported be 3 ms if the internal delay
 is unknown at the time of the timestamping.
 The report of a spatial vector MUST include the uncertainty of the
 timestamping compared to wire-time.

10.1.4. Reporting Spatial One-Way Delay

 The reporting includes information to report for one-way delay as
 section 3.6 of [RFC2679].  The same applies for packet loss and ipdv.

10.2. Reporting One-to-Group Metric

 All reporting rules described in [RFC2679] and [RFC2680] apply to the
 corresponding One-to-group metrics.  The following are specific
 parameters that SHOULD be reported.

10.2.1. Path

 As suggested by [RFC2679] and [RFC2680], the path traversed by the
 packet SHOULD be reported, if possible.  For One-to-group metrics,
 the path tree between the source and the destinations or the set of
 paths between the source and each destination SHOULD be reported.
 The path tree might not be as valuable as individual paths because an
 incomplete path might be difficult to identify in the path tree.  For
 example, how many points of interest are reached by a packet
 traveling along an incomplete path?

10.2.2. Group Size

 The group size SHOULD be reported as one of the critical management
 parameters.  One-to-group metrics, unlike spatial metrics, don't
 require the ordering of the points of interests because group members
 receive the packets in parallel.

10.2.3. Timestamping Bias

 It is the same as described in section 10.1.3.

Stephan, et al. Standards Track [Page 45] RFC 5644 Spatial and Multicast Metrics October 2009

10.2.4. Reporting One-to-group One-way Delay

 It is the same as described in section 10.1.4.

10.2.5. Measurement Method

 As explained in section 9, the measurement method will have impact on
 the analysis of the measurement result.  Therefore, it SHOULD be
 reported.

10.3. Metric Identification

 IANA assigns each metric defined by the IPPM WG a unique identifier
 as per [RFC4148] in the IANA-IPPM-METRICS-REGISTRY-MIB.

10.4. Information Model

 This section presents the elements of information and the usage of
 the information reported for network performance analysis.  It is out
 of the scope of this section to define how the information is
 reported.
 The information model is built with pieces of information introduced
 and explained in the sections of [RFC2679] , [RFC2680] , [RFC3393],
 and [RFC3432] that define the IPPM metrics and from any of the
 sections named "Reporting the metric" , "Methodology", and "Errors
 and Uncertainties" whenever they exist in these documents.
 The following are the elements of information taken from end-to-end
 metrics definitions referred to in this memo and from spatial and
 multicast metrics it defines:
 o  Packet_type, the Type-P of test packets (Type-P).
 o  Packet_length, a packet length in bits (L).
 o  Src_host, the IP address of the sender.
 o  Dst_host, the IP address of the receiver.
 o  Hosts_series: <H1, H2,..., Hn>, a list of points of interest
    participating in the instantaneous measure.  They are routers in
    the case of spatial metrics or receivers in the case of one-to-
    group metrics.
 o  Loss_threshold, the threshold of infinite delay.

Stephan, et al. Standards Track [Page 46] RFC 5644 Spatial and Multicast Metrics October 2009

 o  Systematic_error, constant delay between wire-time and
    timestamping.
 o  Calibration_error, maximal uncertainty.
 o  Src_time, the sending time for a measured packet.
 o  Dst_time, the receiving time for a measured packet.
 o  Result_status, an indicator of usability of a result 'Resource
    exhaustion' 'infinite', 'lost'.
 o  Delays_series, <dT1,..., dTn>, a list of delays.
 o  Losses_series, <B1, B2, ..., Bi, ..., Bn>, a list of Boolean
    values (spatial) or a set of Boolean values (one-to-group).
 o  Result_status_series, a list of results status.
 o  dT, a delay.
 o  Singleton_number, a number of singletons.
 o  Observation_duration, an observation duration.
 o  metric_identifier.
 The following is the information of each vector that SHOULD be
 available to compute samples:
 o  Packet_type;
 o  Packet_length;
 o  Src_host, the sender of the packet;
 o  Dst_host, the receiver of the packet, apply only for spatial
    vectors;
 o  Hosts_series, not ordered for one-to-group;
 o  Src_time, the sending time for the measured packet;
 o  dT, the end-to-end one-way delay for the measured packet, apply
    only for spatial vectors;
 o  Delays_series, apply only for delays and ipdv vector, not ordered
    for one-to-group;

Stephan, et al. Standards Track [Page 47] RFC 5644 Spatial and Multicast Metrics October 2009

 o  Losses_series, apply only for packets loss vector, not ordered for
    one-to-group;
 o  Result_status_series;
 o  Observation_duration, the difference between the time of the last
    singleton and the time of the first singleton.
 Following is the context information (measure, points of interests)
 that SHOULD be available to compute samples:
 o  Loss threshold;
 o  Systematic error, constant delay between wire-time and
    timestamping;
 o  Calibration error, maximal uncertainty.
 A spatial or a one-to-group sample is a collection of singletons
 giving the performance from the sender to a single point of interest.
 The following is the information that SHOULD be available for each
 sample to compute statistics:
 o  Packet_type;
 o  Packet_length;
 o  Src_host, the sender of the packet;
 o  Dst_host, the receiver of the packet;
 o  Start_time, the sending time of the first packet;
 o  Delays_series, apply only for delays and ipdv samples;
 o  Losses_series, apply only for packets loss samples;
 o  Result_status_series;
 o  Observation_duration, the difference between the time of the last
    singleton of the last sample and the time of the first singleton
    of the first sample.
 The following is the context information (measure, points of
 interests) that SHOULD be available to compute statistics:
 o  Loss threshold;

Stephan, et al. Standards Track [Page 48] RFC 5644 Spatial and Multicast Metrics October 2009

 o  Systematic error, constant delay between wire-time and
    timestamping;
 o  Calibration error, maximal uncertainty;
 The following is the information of each statistic that SHOULD be
 reported:
 o  Result;
 o  Start_time;
 o  Duration;
 o  Result_status;
 o  Singleton_number, the number of singletons on which the statistic
    is computed;

11. Security Considerations

 Spatial and one-to-group metrics are defined on the top of end-to-end
 metrics.  Security considerations discussed in the one-way delay
 metrics definitions of [RFC2679], in packet loss metrics definitions
 of [RFC2680] and in IPDV metrics definitions of [RFC3393] and
 [RFC3432] apply to metrics defined in this memo.
 Someone may spoof the identity of a point of interest identity and
 intentionally send corrupt results in order to remotely orient the
 traffic engineering decisions.
 A point of interest could intentionally corrupt its results in order
 to remotely orient the traffic engineering decisions.

11.1. Spatial Metrics

 Malicious generation of packets that systematically match the hash
 function used to detect the packets may lead to a DoS attack toward
 the point of reference.
 Spatial measurement results carry the performance of individual
 segments of the path and the identity of nodes.  An attacker may
 infer from this information the points of weakness of a network
 (e.g., congested node) that would require the least amount of
 additional attacking traffic to exploit.  Therefore, monitoring
 information should be carried in a way that prevents unintended

Stephan, et al. Standards Track [Page 49] RFC 5644 Spatial and Multicast Metrics October 2009

 recipients from inspecting the measurement reports.  A
 straightforward solution is to restrict access to the reports using
 encrypted sessions or secured networks.

11.2. One-to-Group Metrics

 Reporting of measurement results from a huge number of probes may
 overload reference point resources (network, network interfaces,
 computation capacities, etc.).
 The configuration of a measurement must take into consideration that
 implicitly more packets will be routed than sent and select a test
 packet rate accordingly.  Collecting statistics from a huge number of
 probes may overload any combination of the network to which the
 measurement controller is attached, measurement controller network
 interfaces, and measurement controller computation capacities.
 One-to-group metric measurements should consider using source
 authentication protocols, standardized in the MSEC group, to avoid
 fraud packet in the sampling interval.  The test packet rate could be
 negotiated before any measurement session to avoid denial-of-service
 attacks.
 A point of interest could intentionally degrade its results in order
 to remotely increase the quality of the network on the branches of
 the multicast tree to which it is connected.

12. Acknowledgments

 Lei would like to acknowledge Professor Zhili Sun from CCSR,
 University of Surrey, for his instruction and helpful comments on
 this work.

13. IANA Considerations

 Metrics defined in this memo have been 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:
 ietfSpatialOneWayDelayVector OBJECT-IDENTITY
    STATUS current
    DESCRIPTION

Stephan, et al. Standards Track [Page 50] RFC 5644 Spatial and Multicast Metrics October 2009

       "Type-P-Spatial-One-way-Delay-Vector"
    REFERENCE
       "RFC 5644, section 5.1."
    := { ianaIppmMetrics 52 }
 ietfSpatialPacketLossVector OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-Spatial-Packet-Loss-Vector"
    REFERENCE
       "RFC 5644, section 5.2."
    := { ianaIppmMetrics 53 }
 ietfSpatialOneWayIpdvVector OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-Spatial-One-way-ipdv-Vector"
    REFERENCE
       "RFC 5644, section 5.3."
    := { ianaIppmMetrics 54 }
 ietfSegmentOneWayDelayStream OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-Segment-One-way-Delay-Stream"
    REFERENCE
       "RFC 5644, section 6.1."

Stephan, et al. Standards Track [Page 51] RFC 5644 Spatial and Multicast Metrics October 2009

    := { ianaIppmMetrics 55 }
 ietfSegmentPacketLossStream OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-Segment-Packet-Loss-Stream"
    REFERENCE
       "RFC 5644, section 6.2."
    := { ianaIppmMetrics 56 }
 ietfSegmentIpdvPrevStream OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-Segment-ipdv-prev-Stream"
    REFERENCE
       "RFC 5644, section 6.3."
    := { ianaIppmMetrics 57 }
 ietfSegmentIpdvMinStream OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-Segment-ipdv-min-Stream"
    REFERENCE
       "RFC 5644, section 6.4."
    := { ianaIppmMetrics 58 }
  1. - One-to-group metrics
 ietfOneToGroupDelayVector OBJECT-IDENTITY

Stephan, et al. Standards Track [Page 52] RFC 5644 Spatial and Multicast Metrics October 2009

    STATUS current
    DESCRIPTION
       "Type-P-One-to-group-Delay-Vector"
    REFERENCE
       "RFC 5644, section 7.1."
    := { ianaIppmMetrics 59 }
 ietfOneToGroupPacketLossVector OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-One-to-group-Packet-Loss-Vector"
    REFERENCE
       "RFC 5644, section 7.2."
    := { ianaIppmMetrics 60 }
 ietfOneToGroupIpdvVector OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-One-to-group-ipdv-Vector"
    REFERENCE
       "RFC 5644, section 7.3."
    := { ianaIppmMetrics 61 }
  1. - One to group statistics
  1. -
 ietfOnetoGroupReceiverNMeanDelay OBJECT-IDENTITY
    STATUS current

Stephan, et al. Standards Track [Page 53] RFC 5644 Spatial and Multicast Metrics October 2009

    DESCRIPTION
       "Type-P-One-to-group-Receiver-n-Mean-Delay"
    REFERENCE
       "RFC 5644, section 8.3.1."
    := { ianaIppmMetrics 62 }
 ietfOneToGroupMeanDelay OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-One-to-group-Mean-Delay"
    REFERENCE
       "RFC 5644, section 8.3.2."
    := { ianaIppmMetrics 63 }
 ietfOneToGroupRangeMeanDelay OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-One-to-group-Range-Mean-Delay"
    REFERENCE
       "RFC 5644, section 8.3.3."
    := { ianaIppmMetrics 64 }
 ietfOneToGroupMaxMeanDelay OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-One-to-group-Max-Mean-Delay"
    REFERENCE

Stephan, et al. Standards Track [Page 54] RFC 5644 Spatial and Multicast Metrics October 2009

       "RFC 5644, section 8.3.4."
    := { ianaIppmMetrics 65 }
 ietfOneToGroupReceiverNLossRatio OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-One-to-group-Receiver-n-Loss-Ratio"
    REFERENCE
       "RFC 5644, section 8.4.1."
    := { ianaIppmMetrics 66 }
  1. -
 ietfOneToGroupReceiverNCompLossRatio OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-One-to-group-Receiver-n-Comp-Loss-Ratio"
    REFERENCE
       "RFC 5644, section 8.4.2."
    := { ianaIppmMetrics 67 }
 ietfOneToGroupLossRatio OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-One-to-group-Loss-Ratio"
    REFERENCE
       "RFC 5644, section 8.4.3."
    := { ianaIppmMetrics 68 }

Stephan, et al. Standards Track [Page 55] RFC 5644 Spatial and Multicast Metrics October 2009

  1. -
 ietfOneToGroupRangeLossRatio OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-One-to-group-Range-Loss-Ratio"
    REFERENCE
       "RFC 5644, section 8.4.4."
    := { ianaIppmMetrics 69 }
 ietfOneToGroupRangeDelayVariation OBJECT-IDENTITY
    STATUS current
    DESCRIPTION
       "Type-P-One-to-group-Range-Delay-Variation"
    REFERENCE
       "RFC 5644, section 8.5.1."
    := { ianaIppmMetrics 70 }
  1. -

14. References

14.1. Normative References

 [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
            Requirement Levels", BCP 14, RFC 2119, March 1997.
 [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.

Stephan, et al. Standards Track [Page 56] RFC 5644 Spatial and Multicast Metrics October 2009

 [RFC4148]  Stephan, E., "IP Performance Metrics (IPPM) Metrics
            Registry", BCP 108, RFC 4148, August 2005.

14.2. Informative References

 [RFC2330]  Paxson, V., Almes, G., Mahdavi, J., and M. Mathis,
            "Framework for IP Performance Metrics", RFC 2330,
            May 1998.
 [RFC3432]  Raisanen, V., Grotefeld, G., and A. Morton, "Network
            performance measurement with periodic streams", RFC 3432,
            November 2002.
 [SPATIAL]  Morton, A. and E. Stephan, "Spatial Composition of
            Metrics", Work in Progress, June 2009.

Authors' Addresses

 Stephan Emile
 France Telecom Division R&D
 2 avenue Pierre Marzin
 Lannion  F-22307
 France
 Fax:   +33 2 96 05 18 52
 EMail: emile.stephan@orange-ftgroup.com
 Lei Liang
 CCSR, University of Surrey
 Guildford
 Surrey  GU2 7XH
 UK
 Fax:   +44 1483 683641
 EMail: L.Liang@surrey.ac.uk
 Al Morton
 200 Laurel Ave. South
 Middletown, NJ  07748
 USA
 Phone: +1 732 420 1571
 EMail: acmorton@att.com

Stephan, et al. Standards Track [Page 57]

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