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Versions: 00                                                            
IP Performance Metrics Working Group        S. Ono, T. Miyata, H. Fukuda
Internet Draft                                                       NTT
Expiration Date: February 1999                                 Aug 1998


                 Loss Metrics of Grouped Packets for IPPM
                     <draft-ono-group-loss-00.txt>


1. Status of this Memo

   This document is an Internet Draft.  Internet Drafts are working doc-
   uments  of the Internet Engineering Task Force (IETF), its areas, and
   its working groups.  Note that other groups may also distribute work-
   ing documents as Internet Drafts.

   Internet  Drafts  are  draft  documents  valid  for  a maximum of six
   months, and may be updated, replaced, or obsoleted by other documents
   at any time.  It is inappropriate to use Internet Drafts as reference
   material or to cite them other than as ``work in progress''.

   To learn the current status of any Internet Draft, please  check  the
   ``1id-abstracts.txt'' listing contained in the Internet Drafts shadow
   directories  on  ftp.is.co.za   (Africa),   nic.nordu.net   (Europe),
   munnari.oz.au  (Pacific  Rim),  ftp.ietf.org  (US  East Coast), or
   ftp.isi.edu (US West Coast).

   This memo provides information for the Internet community.  This memo
   does  not  specify an Internet standard of any kind.  Distribution of
   this memo is unlimited.


2. Introduction

   This memo defines several metrics for loss of grouped packets across
   Internet paths. It builds on notions introduced and discussed in the
   IPPM Framework document (currently ''Framework for IP Performance
   Metrics'' <draft-ietf-ippm-framework-01.txt>); the reader is assumed
   to be familiar with that document.

   In addition, this memo is intended to be a generalization of the
   document  for  Packet  Loss [1] and to be very parallel in structure
   to  that document; the reader is also assumed to be familiar with
   that document.

   The structure of the memo is as follows:

 +    A 'singleton' analytic metric, called Type-P-One-way-Grouped-
      Packets-Loss-Pattern, will be introduced to measure a single
      observation, which is comprised of multiple related packet
      transmissions, each of the transmissions may be successful or
loss.
      (Section 3.1)

 +    A 'singleton' analytic metric, called Type-P-One-way-Grouped-
      Packets-LossTh, will be introduced to measure a single
observation,
      which can be derived from the metric Type-P-One-way-Grouped-
      Packets-Loss-Pattern. This metric is generalization of Type-P-
      One-way-Packet-Loss to a packet group such that a packet group is
      said to be 'lost' when s or more out of w packets in the group are
      lost, where w (0 < w) is a loss-window size, and s (0 < s <= w) is
      a loss threshold count. When both w and s are set to 1, it is the
same
      as the conventional metric Type-P-One-way-Packet-Loss.
      (Section 3.2)

 +    A 'singleton' analytic metric, called Type-P-One-way-Grouped-
      Packets-Loss, will be introduced to measure a single observation,
      which is the specialization of Type-P-One-way-Grouped-Packets-
      LossTh, such that the threshold value s is fixed to 1.
      Namely, a packet group is said to be lost only when all of w
      packets in the group are lost, where w (0 < w) is a loss-window
      size. Naturally,  when w is also set to 1, it is the same as
      Type-P-One-way-Packet-Loss.
      (Section 3.3)

 +    Using the singleton metric Type-P-One-way-Grouped-Packets-
      Loss-Pattern, a 'sample', called Type-P-One-way-Grouped-Packets-
      Loss-Pattern-Stream, will be introduced to measure a sequence of
      singleton loss patterns of grouped packets, where each measurement
      is started at times taken from a Poisson process.
      (Section 4.1)

 +    Using the singleton metric Type-P-One-way-Grouped-Packets-LossTh,
      a 'sample', called Type-P-One-way-Grouped-Packets-LossTh-Stream,
      will be introduced to measure a sequence of singleton losses
      of grouped packets, where each measurement is started at times
      taken from a Poisson process.
      (Section 4.2)

 +    Using the singleton metric Type-P-One-way-Grouped-Packets-Loss,
      a 'sample', called Type-P-One-way-Grouped-Packets-Loss-Stream,
      will be introduced to measure a sequence of singleton losses
      of grouped packets, where each measurement is started at times
      taken from a Poisson process.
      (Section 4.3)

 +    The relationships among these metrics are discussed.
      (Section 5)

 +    Using these samples, several 'statistics' of the samples will
      be defined and discussed.
      (Section 6)


   These progressions from singletons to samples to statistics, with
   clear separation among them, are important.

   Whenever a technical term from the IPPM Framework document  is  first
   used  in  this  memo,  it will be tagged with a trailing asterisk, as
   with >>term*<<.



2.1. Motivation:

   The work-in-progress document [2] defines the notion of one-way
   packet loss  of type-P packets transmitted from a source host* to a
destination
   host.  In that document, the unit of measurement is a packet.
   The main purpose of this document is to generalize it to a packet
   group, i.e. a sequence of multiple related packets.
   Such Generalization is useful for several reasons:

+     When successive packets are injected at short intervals,
      the loss probability of each packet may NOT be independent [3];
      For example, the loss probability of the next packet may become
      lower, if  the preceding packet has been  lost.
      Metrics using only one packet transmission for each sample
      cannot detect this kind of conditional loss probability.


+     The stream-oriented applications which handle audio and/or video
      are likely to inject packets at short intervals.

+     There exist applications that use FEC (forward error control/
      correction) technology. Typical FEC usage on packet networks is
      as follows. Application data are scattered among w different
      packets with redundancy so that the application data can be
      recovered if only s (0 < s < w) or  more packets out of the
      w packets are successfully received. In such recovered cases,
      losses of packets are invisible to the application.

+     Some applications take advantage of both FEC and hierarchical
      coding.  For example, low-quality audio data may be protected
      more thoroughly than high-quality audio, such that the low-quality
      data can be retrieved even if only one packet in a group is
received.
      In such a case, the application may provide degraded service to
      a user due to a heavily lossy network.

   It  is  outside  the scope of this document to define precisely how
   packet group loss metrics would be applied to specific problems.


2.2. General Issues Regarding Time

   Whenever a time (i.e., a moment in history) is mentioned here, it  is
   understood to be measured in seconds (and fractions) relative to UTC.

   Similar discussions are applicable as described in Section 2.2 of
   the document  for  Packet  Loss  (''A Packet Loss Metric
   for IPPM'' <draft-ietf-ippm-loss-xx.txt>).



3. Singleton Definitions for One-way Losses of Grouped Packets


3.1. A Singleton Definition for One-way Loss Pattern of Grouped Packets

3.1.1. Metric Name:

   Type-P-One-way-Grouped-Packets-Loss-Pattern


3.1.2. Metric Parameters:
 +    Src, the IP address of a host
 +    Dst, the IP address of a host
 +    T, a time
 +    n, a group size (GS) (positive integer number)
 +    TS, a monotonic sequence <t1, t2, ..., tn> of times with length n,
      where t1 should be always equal to T.
 +    Path, the path* from Src to Dst; in cases where there is only  one
      path from Src to Dst, this optional parameter can be omitted.
   {Comment:  For the discussion on path, please refer to Section 3.2 of
      the document  for  Packet  Loss  (''A Packet Loss Metric
      for IPPM'' <draft-ietf-ippm-loss-xx.txt>).


3.1.3. Metric Units:

   The value of a Type-P-One-way-Grouped-Packets-Loss-Pattern is a
   sequence of n (i.e. the group size) values, each of which is either
   a zero (signifying successful transmission of the packet) or
   a one (signifying loss).


3.1.4. Definition:

   >>The *Type-P-One-way-Grouped-Packets-Loss-Pattern* from Src to Dst
   at <t1, t2, ..., tn>  for GS n (1 <= n) [via path] is <L1, L2, ...,
   Ln> <<  means that for each i (1 <= i <= n), Src sent the first bit
   of the i-th type-P packet [via path] to Dst at wire-time ti, and that
   Dst received that packet correctly if Li is 0, and did not if Li is
1,
   respectively.


3.1.5. Discussion:

   It is a rather straight forward extension of
Type-P-One-way-Packet-Loss,
   such that grouped n packets are sent in sequence, and the loss
   is recorded packet-wise.

   As can be easily seen, when the group size n is set to 1, the Type-P-
   One-way-Grouped-Packets-Loss-Pattern is the same as
Type-P-One-way-Packet-
   Loss.

 +    Although no assumption is made for t1, ..., tn, the following are
      a partial list of possible alternatives.
        Stream type:
            Take ti such that ti = t1 + delta * (i - 1) where delta
            is a constant packet interval.
        Random type:
            ti is taken from a Poisson process.
        Burst type:
            Take ti as close as possible to the preceding packet.

 +    Similar discussions are applicable as described in Section 3.5 of
      the document  for  Packet  Loss  (''A Packet Loss Metric
      for IPPM'' <draft-ietf-ippm-loss-xx.txt>).


3.1.6. Methodologies:

   Similar discussions are applicable as described in Section 3.6 of
   the document  for  Packet  Loss  (''A Packet Loss Metric
   for IPPM'' <draft-ietf-ippm-loss-xx.txt>).



3.1.7. Errors and Uncertainties:

   Similar discussions are applicable as described in Section 3.7 of
   the document  for  Packet  Loss  (''A Packet Loss Metric
   for IPPM'' <draft-ietf-ippm-loss-xx.txt>).



3.2. A Singleton Definition for Window-based One-way Loss of
     Grouped Packets where Loss is Defined using Thresholds


3.2.1. Metric Name:

   Type-P-One-way-Grouped-Packets-LossTh


3.2.2. Metric Parameters:

 +    Src, the IP address of a host
 +    Dst, the IP address of a host
 +    T, a time
 +    n, a group size (GS) (positive integer number)
 +    TS, a monotonic sequence <t1, t2, ..., tn> of times with length n,
      where t1 should be always equal to T
 +    w, a loss window size (LWS) (positive integer number, and w <= n)
 +    s, a loss threshold count (LTC) (positive integer number and s <=
w)
 +    Path, the path from Src to Dst; in cases where there is only  one
      path from Src to Dst, this optional parameter can be omitted.


3.2.3. Metric Units:

   The value of a type-P-One-way-Grouped-Packets-LossTh is either a zero
   (signifying that at least s packets out of w packets have been
successfully
   transmitted) or a one (signifying that less than s packets out of
   w packets have been successfully transmitted, or all of the w packets
   have been lost).


3.2.4. Definition:

   >>The *Type-P-One-way-Grouped-Packets-LossTh* from Src to Dst at
   <t1, t2, ..., tn> for GS n (1 <= n), LWS w (1 <= w <= n), and LTC s
   (1 <= s <= w) [via path] is 0 << means that for each i (1 <= i <= n),
   Src sent the first bit of the i-th type-P packet Pi [via path] to Dst
   at wire-time ti, and that Dst received at least s packets among
   packets {P1, P2, ..., Pw}.

   >>The *Type-P-One-way-Grouped-Packets-LossTh* from Src to Dst at
   <t1, t2, ..., tn> for GS n (1 <= n), LWS w (1 <= w <= n), and LTC s
   (1 <= s <= w) [via path] is 1 << means that for each i (1 <= i <= n),
   Src sent the first bit of the i-th type-P packet Pi [via path] to Dst
   at wire-time ti, and that Dst received less than s or none of the
packets {P1, P2, ..., Pw}.


3.2.5. Discussion:

   This metric can be naturally derived from the Type-P-One-way-Grouped-
   Packets-Loss-Pattern.

   Assume a sample of Type-P-One-way-Grouped-Packets-Loss-Pattern of
   GS n (1 <= n) be
       < <t1, t2, ..., tn>, <L1, L2, ..., Ln> >.
   Then, Type-P-One-way-Grouped-Packets-LossTh of LWS w (1 <= w <= n)
   and LTC s (1 <= s <= w)  be
       < <t1, t2, ..., tn>, 0 > if  ((L1 + L2 + ... + Lw) <= w - s),
   and
       < <t1, t2, ..., tn>, 1 > if  ((L1 + L2 + ... + Lw) > w - s).

   When w < n, if we ignore the effect of packets Pw+1, ..., Pn to
   L1, ..., Lw, we can assume the group size to be equal to w.


3.2.6. Methodologies:

   See Section 3.1.6 of this document.


3.2.7. Errors and Uncertainties:

   See Section 3.1.7 of this document.



3.3. A Singleton Definition for Window-based One-way Loss of
     Grouped Packets

3.3.1. Metric Name:

   Type-P-One-way-Grouped-Packets-Loss


3.3.2. Metric Parameters:
 +    Src, the IP address of a host
 +    Dst, the IP address of a host
 +    T, a time
 +    n, a group size (GS) (positive integer number)
 +    TS, a monotonic sequence <t1, t2, ..., tn> of times with length n,
      where t1 should be always equals to T
 +    Path, the path from Src to Dst; in cases where there is only  one
      path from Src to Dst, this optional parameter can be omitted.
 +    w, a loss window size (LWS) (positive integer number, and w <= n)


3.3.3. Metric Units:

   The value of a type-P-One-way-Grouped-Packets-Loss is either a zero
   (signifying that at least one packet out of w packets has been
   successfully transmitted) or a one (signifying that none of
   w packets has been successfully transmitted.


3.3.4. Definition:

   >>The *Type-P-One-way-Grouped-Packets-Loss* from Src to Dst at
   <t1, t2, ..., tn> for GS n (1 <= n), LWS w (1 <= w <= n) [via path]
   is 0 << means that for each i (1 <= i <= n), Src sent the first bit
   of the i-th type-P packet Pi [via path] to Dst at wire-time ti,
   and that Dst received at least one of the packets
   {P1, P2, ..., Pw}.

   >>The *Type-P-One-way-Grouped-Packets-Loss* from Src to Dst at
   <t1, t2, ..., tn> for GS n (1 <= n), LWS w (1 <= w <= n) [via path]
   is 1 << means that for each i (1 <= i <= n), Src sent the first bit
   of the i-th type-P packet Pi [via path] to Dst at wire-time ti,
   and that Dst received none of the packets {P1, P2, ..., Pw}.


3.3.5. Discussion:

   This metric is specialization of
Type-P-One-way-Grouped-Packets-LossTh,
   such that LTC is always fixed to 1.

   In case w < n, if we ignore the effect of packets Pw+1, ..., Pn to
   L1, ..., Lw, we can assume the group size be equal to w.


3.3.6. Methodologies:

   See Section 3.1.6 of this document.


3.3.7. Errors and Uncertainties:

   See Section 3.1.7 of this document.



4. Definitions for Samples of One-way Losses of Grouped Packets

   Given the above-defined singleton metrics, namely
       (1) Type-P-One-way-Grouped-Packets-Loss-Pattern,
       (2) Type-P-One-way-Grouped-Packets-LossTh and
       (3) Type-P-One-way-Grouped-Packets-Loss,
   we now define one particular sample of such singletons. The idea
   of the sample is to select a particular  binding of the parameters
   Src, Dst, path, Type-P, GS n, LWS w (for metrics (2) and (3)) and
   LTC s (for the metric (2)), define a sample of values of parameter T.
   The way to  define the values of T is to select a beginning time
   T0, a final time Tf, and an average rate lambda, then define a
   pseudo-random Poisson arrival process of rate lambda, whose values
   fall between T0 and Tf. The time interval between successive values
   of T will then average 1/lambda.


4.1 A Definition for Samples of One-way Loss Pattern of Grouped Packets

4.1.1. Metric Name:

   Type-P-One-way-Grouped-Packets-Loss-Pattern-Stream


4.1.2. Metric Parameters:
 +    Src, the IP address of a host
 +    Dst, the IP address of a host
 +    Path, the path from Src to Dst; in cases where there is only  one
      path from Src to Dst, this optional parameter can be omitted
 +    n, a group size (GS) (positive integer number)
 +    T0, a time
 +    Tf, a time
 +    lambda, a rate in reciprocal seconds
 +    Tseq, a (possibly non-deterministic) mapping from a time to
      a monotonic sequence of times of length n. The first member of
      the resulting sequence is always the same as the input parameter.


4.1.3. Metric Units:

   A sequence of triplets; the elements of each triplets are:

 +    T, a time,
 +    TS, a monotonic sequence of times with length n, a valid instance
      of Tseq(T),
 +    LS, a sequence of values with length n, each of which is either
      a zero or a one.The values of T in the sequence increase
monotonically.
      Note that T and TS would be valid parameters for the
      Type-P-One-way-Grouped-Packets-Loss-Pattern, and that LS would be
a
      valid value of the Type-P-One-way-Grouped-Packets-Loss-Pattern.



4.1.4. Definition:

   Given  T0, Tf, and lambda, we compute a pseudo-random Poisson process
   beginning at or before T0, with an average arrival rate lambda, and
end-
   ing  at  or  after Tf.  Those time values greater than or equal to T0
   and less than or equal to Tf are then selected.  At each time T of
   the times in this process, we compute a time sequence TS = Tseq(T),
   and obtain the value of the
Type-P-One-way-Grouped-Packets-Loss-Pattern
   using this time sequence.  The value of the sample is the sequence
   made up of the resulting <time, TS, LS> triplets. If there are no
such
   triplets, the sequence is of length zero and the sample is said to be
   empty.


4.1.5. Discussion:

   Similar discussions are applicable as described in Section 4.5 of
   the document  for  Packet  Loss  (''A Packet Loss Metric
   for IPPM'' <draft-ietf-ippm-loss-xx.txt>).


4.1.6. Methodologies:

   Similar discussions are applicable as described in Section 4.6 of
   the document  for  Packet  Loss  (''A Packet Loss Metric
   for IPPM'' <draft-ietf-ippm-loss-xx.txt>).


4.1.7. Errors and Uncertainties:

   Similar discussions are applicable as described in Section 4.7 of
   the document  for  Packet  Loss  (''A Packet Loss Metric
   for IPPM'' <draft-ietf-ippm-loss-xx.txt>).

   Note also that errors could be caused by the algorithm of Tseq,
   and inaccuracy in the wire-time of the sending of the grouped
packets.



4.2. A Definition for Samples of Window-based One-way Loss of
     Grouped Packets where Loss is Defined using Thresholds


4.2.1. Metric Name:

   Type-P-One-way-Grouped-Packets-LossTh-Stream


4.2.2. Metric Parameters:
 +    Src, the IP address of a host
 +    Dst, the IP address of a host
 +    Path, the path from Src to Dst; in cases where there is only  one
      path from Src to Dst, this optional parameter can be omitted
 +    n, a group size (GS) (positive integer number)
 +    T0, a time
 +    Tf, a time
 +    lambda, a rate in reciprocal seconds
 +    Tseq, a (possibly non-deterministic) mapping from a time to
      a monotonic sequence of times of length n. The first member of
      the resulting sequence is always same as the input parameter.
 +    w, a loss window size (LWS) (positive integer number, and w <= n)
 +    s, a loss threshold count (LTC) (positive integer number and s <=
w)


4.2.3. Metric Units:

   A sequence of triplets; the elements of each triplets are:

 +    T, a time,
 +    TS, a monotonic sequence of times with length n, a valid instance
      of Tseq(T),
 +    L, either a zero or a one
   The values of T in the sequence increase monotonically.  Note that
   T and TS would be valid parameters for
Type-P-One-way-Grouped-Packets-
   LossTh, and that L would be a valid value of Type-P-One-way-
   Grouped-Packets-LossTh.


4.2.4. Definition:

   Given  T0, Tf, and lambda, we compute a pseudo-random Poisson process
   beginning at or before T0, with an average arrival rate lambda, and
end-
   ing  at  or  after Tf.  Those time values greater than or equal to T0
   and less than or equal to Tf are then selected.  At each time T of
   the times in this process, we compute a time sequence TS = Tseq(T),
   and obtain the value of Type-P-One-way-Grouped-Packets-LossTh
   using this time sequence.  The value of the sample is the sequence
   made up of the resulting <time, TS, L> triplets. If there are no such
   triplets, the sequence is of length zero and the sample is said to be
   empty.


4.2.5. Discussion:

   Similar discussions are applicable as described in Section 4.5 of
   the document  for  Packet  Loss  (''A Packet Loss Metric
   for IPPM'' <draft-ietf-ippm-loss-xx.txt>).


4.2.6. Methodologies:

   Similar discussions are applicable as described in Section 4.6 of
   the document  for  Packet  Loss  (''A Packet Loss Metric
   for IPPM'' <draft-ietf-ippm-loss-xx.txt>).


4.2.7. Errors and Uncertainties:

   Similar discussions are applicable as described in Section 4.7 of
   the document  for  Packet  Loss  (''A Packet Loss Metric
   for IPPM'' <draft-ietf-ippm-loss-xx.txt>).

   Note also that errors could be caused by the algorithm of Tseq,
   and inaccuracy in the wire-time of the sending of the grouped
packets.



4.3. A Definition for Samples of Window-based One-way Loss of
     Grouped Packets


4.3.1. Metric Name:

   Type-P-One-way-Grouped-Packets-Loss-Stream


4.3.2. Metric Parameters:

 +    Src, the IP address of a host
 +    Dst, the IP address of a host
 +    Path, the path from Src to Dst; in cases where there is only  one
      path from Src to Dst, this optional parameter can be omitted
 +    n, a group size (GS) (positive integer number)
 +    T0, a time
 +    Tf, a time
 +    lambda, a rate in reciprocal seconds
 +    Tseq, a (possibly non-deterministic) mapping from a time to
      a monotonic sequence of times of length n. The first member of
      the resulting sequence is always same as the input parameter.
 +    w, a loss window size (LWS) (positive integer number, and w <= n)


4.3.3. Metric Units:

   A sequence of triplets; the elements of each triplets are:

 +    T, a time,
 +    TS, a monotonic sequence of times with length n, a valid instance
      of Tseq(T),
 +    L, either a zero or a one
   The values of T in the sequence increase monotonically.  Note that
   T and TS would be valid parameters for Type-P-One-way-Grouped-Packets-
   Loss, and that L would be a valid value of Type-P-One-way-Grouped-
   Packets-Loss.


4.3.4. Definition:

   Given  T0, Tf, and lambda, we compute a pseudo-random Poisson process
   beginning at or before T0, with an average arrival rate lambda, and end-
   ing  at  or  after Tf.  Those time values greater than or equal to T0
   and less than or equal to Tf are then selected.  At each time T of
   the times in this process, we compute a time sequence TS = Tseq(T),
   and obtain the value of Type-P-One-way-Grouped-Packets-Loss
   using this time sequence.  The value of the sample is the sequence
   made up of the resulting <time, TS, L> triplets. If there are no such
   triplets, the sequence is of length zero and the sample is said to be
   empty.


4.3.5. Discussion:

   Similar discussions are applicable as described in Section 4.5 of
   the document  for  Packet  Loss  (''A Packet Loss Metric
   for IPPM'' <draft-ietf-ippm-loss-xx.txt>).


4.3.6. Methodologies:

   Similar discussions are applicable as described in Section 4.6 of
   the document  for  Packet  Loss  (''A Packet Loss Metric
   for IPPM'' <draft-ietf-ippm-loss-xx.txt>).


4.3.7. Errors and Uncertainties:

   Similar discussions are applicable as described in Section 4.7 of
   the document  for  Packet  Loss  (''A Packet Loss Metric
   for IPPM'' <draft-ietf-ippm-loss-xx.txt>).

   Note also that errors could be caused by the algorithm of Tseq,
   and inaccuracy in the wire-time of the sending of the grouped
   packets.


5. Metrics Conversion Examples for One-way Loss of Grouped Packets

   Among the three metrics defined in Section 3, the first one, i.e.
   Type-P-One-way-Grouped-Packets-Loss-Pattern, is the most general,
   and the other two can be computed from it.

   We will demonstrate this  using examples.
   Suppose that a sample of Type-P-One-way-Grouped-Packets-Loss-Pattern
   -Stream (group size 3) be as follows:

    Stream1 = <
                < t11, <t11, t12, t13>, <1, 0, 1> >
                < t21, <t21, t22, t23>, <0, 0, 0> >
                < t31, <t31, t32, t33>, <1, 1, 1> >
                < t41, <t41, t42, t43>, <1, 1, 0> >
                < t51, <t51, t52, t53>, <0, 1, 0> >
                < t61, <t61, t62, t63>, <1, 0, 0> >
              >

    From this sample, Type-P-One-way-Grouped-Packets-LossTh-Streams
    with various loss window sizes and loss threshold counts can be
    computed as follows:
    (Comments are added for readers' convenience. A comment starts with
    a mark '//' and ends at the end of the line.)


    (1) <LWS = 3, LTC = 2>
              <                                 // Threshold is 2.
                < t11, <t11, t12, t13>, 1 >     // <1,0,1> Only 1 packet arrived.
                < t21, <t21, t22, t23>, 0 >     // <0,0,0> 3 packets arrived.
                < t31, <t31, t32, t33>, 1 >     // <1,1,1> No packets arrived.
                < t41, <t41, t42, t43>, 1 >     // <1,1,0> Only 1 packet arrived.
                < t51, <t51, t52, t53>, 0 >     // <0,1,0> 2 packets arrived.
                < t61, <t61, t62, t63>, 0 >     // <1,0,0> 2 packets arrived.
              >


    (2) <LWS = 3, LTC = 1>
              <                                 // Threshold is 1.
                < t11, <t11, t12, t13>, 0 >
                < t21, <t21, t22, t23>, 0 >
                < t31, <t31, t32, t33>, 1 >     // <1,1,1> No packets arrived.
                < t41, <t41, t42, t43>, 0 >
                < t51, <t51, t52, t53>, 0 >
                < t61, <t61, t62, t63>, 0 >
              >


    (3) <LWS = 2, LTC = 1>
              <                         // Loss window size is 2.
                < t11, <t11, t12>, 0 >  // <1,0> One packet arrived.
                < t21, <t21, t22>, 0 >  // <0,0> All packets arrived.
                < t31, <t31, t32>, 1 >  // <1,1> No packets arrived.
                < t41, <t41, t42>, 1 >  // <1,1> No packets arrived.
                < t51, <t51, t52>, 0 >  // <0,1> One packet arrived.
                < t61, <t61, t62>, 0 >  // <1,0> One packet arrived.
              >
        (We set the group size to be LWS. See Section 3.2.5.)


    (4) <LWS = 1, LTC = 1>
              <                                 // Loss window size is 1.
                < t11, <t11>, 1 >               // <1> Lost.
                < t21, <t21>, 0 >               // <0> Arrived.
                < t31, <t31>, 1 >               // <1> Lost.
                < t41, <t41>, 1 >               // <1> Lost.
                < t51, <t51>, 0 >               // <0> Arrived.
                < t61, <t61>, 1 >               // <1> Lost.
              >
        (We set the group size to be LWS. See Section 3.2.5.)



6. Some Statistics Definitions for One-way Loss of Grouped Packets

   Given  sample  metrics such as Type-P-One-way-Grouped-Packets-Loss-
   Pattern defined in Section 4 above, we now offer several statistics
   of that sample. These statistics are offered mostly to be
illustrative
   of what could be done.


6.1. Type-P-One-way-Grouped-Packets-LossTh-Average

   Given a Type-P-One-way-Grouped-Packets-LossTh-Stream, the average
   of all the L values in the Stream.  In addition, the Type-P-One-way-
   Grouped-Packets-LossTh-Average is undefined if the sample is empty.

   Example: Suppose we take a sample and the results are:

    Stream2 = <
                < t11, <t11, t12, t13>, 1 >
                < t21, <t21, t22, t23>, 0 >
                < t31, <t31, t32, t33>, 1 >
                < t41, <t41, t42, t43>, 1 >
                < t51, <t51, t52, t53>, 0 >
                < t61, <t61, t62, t63>, 0 >
              >

   Then the average would be 0.5.


7. Security Considerations

   This memo raises no security issues.

8. Acknowledgments

   The authors are indebted to Jean-Chrysostome Bolot and Sacha
Fosse-Parisis of the Rodeo
   Project at inria for their valuable discussions regarding this draft.

   in addition, we are grateful to guy almes and sunil kalidindi for
their valuable internet
   drafts on metrics.

   Thanks are also due to Haruhisa Ichikawa of NTT Sofware Laboratories
for his
   helpful comments.

9. References

[1]   G. Almes and S. Kalidindi, "A One-way Loss Metric for IPPM",
Inter-
      net Draft <draft-ietf-ippm-loss-01.txt>, November 1997.

[2]   V. Paxson, G. Almes, J. Mahdavi, and M.  Mathis,  "Framework  for
IP
      Performance     Metrics",     Internet     Draft
<draft-ietf-ippm-
      framework-01.txt>, November 1997.

[3]   T. Miyata, H. Fukuda, S. Ono, "New network QoS measures for
      FEC-based audio application over the Internet", IEEE IPCCC'98,
      pp. 355 - 362, February 1998.

10. Authors' Addresses

   Satoshi Ono <ono@slab.ntt.co.jp>
   NTT Software Laboratories
   3-9-11, Midori-cho, Musashino-Shi,
   Tokyo, 180-8585
   JAPAN
   Phone: +81 422/59-3542

   Teruko Miyata <teruko@slab.ntt.co.jp>
   NTT Software Laboratories
   3-9-11, Midori-cho, Musashino-Shi,
   Tokyo, 180-8585
   JAPAN
   Phone: +81 422/59-2927

   Harumoto Fukuda <harumoto@slab.ntt.co.jp>
   NTT Software Laboratories
   3-9-11, Midori-cho, Musashino-Shi,
   Tokyo, 180-8585
   JAPAN
   Phone: +81 422/59-2882