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
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This memo provides information for the Internet community. This memo
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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