A One-way Delay Metric for IPPM
RFC 2679

Network Working Group                                           G. Almes
Request for Comments: 2679                                  S. Kalidindi
Category: Standards Track                                   M. Zekauskas
                                             Advanced Network & Services
                                                          September 1999

                    A One-way Delay Metric for IPPM

1. 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) The Internet Society (1999).  All Rights Reserved.

2. Introduction

   This memo defines a metric for one-way delay of packets across
   Internet paths.  It builds on notions introduced and discussed in the
   IPPM Framework document, RFC 2330 [1]; the reader is assumed to be
   familiar with that document.

   This memo is intended to be parallel in structure to a companion
   document for Packet Loss ("A One-way Packet Loss Metric for IPPM")
   [2].

   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 [6].
   Although RFC 2119 was written with protocols in mind, the key words
   are used in this document for similar reasons.  They are used to
   ensure the results of measurements from two different implementations
   are comparable, and to note instances when an implementation could
   perturb the network.

   The structure of the memo is as follows:

   +  A 'singleton' analytic metric, called Type-P-One-way-Delay, will
      be introduced to measure a single observation of one-way delay.

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   +  Using this singleton metric, a 'sample', called Type-P-One-way-
      Delay-Poisson-Stream, will be introduced to measure a sequence of
      singleton delays measured at times taken from a Poisson process.

   +  Using this sample, several 'statistics' of the sample will be
      defined and discussed.

   This progression from singleton to sample to statistics, with clear
   separation among them, is important.

   Whenever a technical term from the IPPM Framework document is first
   used in this memo, it will be tagged with a trailing asterisk.  For
   example, "term*" indicates that "term" is defined in the Framework.

2.1. Motivation:

   One-way delay of a Type-P* packet from a source host* to a
   destination host is useful for several reasons:

   +  Some applications do not perform well (or at all) if end-to-end
      delay between hosts is large relative to some threshold value.

   +  Erratic variation in delay makes it difficult (or impossible) to
      support many real-time applications.

   +  The larger the value of delay, the more difficult it is for
      transport-layer protocols to sustain high bandwidths.

   +  The minimum value of this metric provides an indication of the
      delay due only to propagation and transmission delay.

   +  The minimum value of this metric provides an indication of the
      delay that will likely be experienced when the path* traversed is
      lightly loaded.

   +  Values of this metric above the minimum provide an indication of
      the congestion present in the path.

   The measurement of one-way delay instead of round-trip delay is
   motivated by the following factors:

   +  In today's Internet, the path from a source to a destination may
      be different than the path from the destination back to the source
      ("asymmetric paths"), such that different sequences of routers are
      used for the forward and reverse paths.  Therefore round-trip
      measurements actually measure the performance of two distinct
      paths together.  Measuring each path independently highlights the
      performance difference between the two paths which may traverse

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      different Internet service providers, and even radically different
      types of networks (for example, research versus commodity
      networks, or ATM versus packet-over-SONET).

   +  Even when the two paths are symmetric, they may have radically
      different performance characteristics due to asymmetric queueing.

   +  Performance of an application may depend mostly on the performance
      in one direction.  For example, a file transfer using TCP may
      depend more on the performance in the direction that data flows,
      rather than the direction in which acknowledgements travel.

   +  In quality-of-service (QoS) enabled networks, provisioning in one
      direction may be radically different than provisioning in the
      reverse direction, and thus the QoS guarantees differ.  Measuring
      the paths independently allows the verification of both
      guarantees.

   It is outside the scope of this document to say precisely how delay
   metrics would be applied to specific problems.

2.2. General Issues Regarding Time

   {Comment: the terminology below differs from that defined by ITU-T
   documents (e.g., G.810, "Definitions and terminology for
   synchronization networks" and I.356, "B-ISDN ATM layer cell transfer
   performance"), but is consistent with the IPPM Framework document.
   In general, these differences derive from the different backgrounds;
   the ITU-T documents historically have a telephony origin, while the
   authors of this document (and the Framework) have a computer systems
   background.  Although the terms defined below have no direct
   equivalent in the ITU-T definitions, after our definitions we will
   provide a rough mapping.  However, note one potential confusion: our
   definition of "clock" is the computer operating systems definition
   denoting a time-of-day clock, while the ITU-T definition of clock
   denotes a frequency reference.}

   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.

   As described more fully in the Framework document, there are four
   distinct, but related notions of clock uncertainty:

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   synchronization*

         measures the extent to which two clocks agree on what time it
         is.  For example, the clock on one host might be 5.4 msec ahead
         of the clock on a second host.  {Comment: A rough ITU-T
         equivalent is "time error".}

   accuracy*

         measures the extent to which a given clock agrees with UTC.
         For example, the clock on a host might be 27.1 msec behind UTC.
         {Comment: A rough ITU-T equivalent is "time error from UTC".}

   resolution*

         measures the precision of a given clock.  For example, the
         clock on an old Unix host might tick only once every 10 msec,
         and thus have a resolution of only 10 msec.  {Comment: A very
         rough ITU-T equivalent is "sampling period".}

   skew*

         measures the change of accuracy, or of synchronization, with
         time.  For example, the clock on a given host might gain 1.3
         msec per hour and thus be 27.1 msec behind UTC at one time and
         only 25.8 msec an hour later.  In this case, we say that the
         clock of the given host has a skew of 1.3 msec per hour
         relative to UTC, which threatens accuracy.  We might also speak
         of the skew of one clock relative to another clock, which
         threatens synchronization.  {Comment: A rough ITU-T equivalent
         is "time drift".}

3. A Singleton Definition for One-way Delay

3.1. Metric Name:

   Type-P-One-way-Delay

3.2. Metric Parameters:

   +  Src, the IP address of a host

   +  Dst, the IP address of a host

   +  T, a time

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3.3. Metric Units:

   The value of a Type-P-One-way-Delay is either a real number, or an
   undefined (informally, infinite) number of seconds.

3.4. Definition:

   For a real number dT, >>the *Type-P-One-way-Delay* from Src to Dst at
   T is dT<< means that Src sent the first bit of a Type-P packet to Dst
   at wire-time* T and that Dst received the last bit of that packet at
   wire-time T+dT.

   >>The *Type-P-One-way-Delay* from Src to Dst at T is undefined
   (informally, infinite)<< means that Src sent the first bit of a
   Type-P packet to Dst at wire-time T and that Dst did not receive that
   packet.

   Suggestions for what to report along with metric values appear in
   Section 3.8 after a discussion of the metric, methodologies for
   measuring the metric, and error analysis.

3.5. Discussion:

   Type-P-One-way-Delay is a relatively simple analytic metric, and one
   that we believe will afford effective methods of measurement.

   The following issues are likely to come up in practice:

   +  Real delay values will be positive.  Therefore, it does not make
      sense to report a negative value as a real delay.  However, an
      individual zero or negative delay value might be useful as part of
      a stream when trying to discover a distribution of a stream of
      delay values.

   +  Since delay values will often be as low as the 100 usec to 10 msec
      range, it will be important for Src and Dst to synchronize very
      closely.  GPS systems afford one way to achieve synchronization to
      within several 10s of usec.  Ordinary application of NTP may allow
      synchronization to within several msec, but this depends on the
      stability and symmetry of delay properties among those NTP agents
      used, and this delay is what we are trying to measure.  A
      combination of some GPS-based NTP servers and a conservatively
      designed and deployed set of other NTP servers should yield good
      results, but this is yet to be tested.

   +  A given methodology will have to include a way to determine
      whether a delay value is infinite or whether it is merely very
      large (and the packet is yet to arrive at Dst).  As noted by

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      Mahdavi and Paxson [4], simple upper bounds (such as the 255
      seconds theoretical upper bound on the lifetimes of IP packets
      [5]) could be used, but good engineering, including an
      understanding of packet lifetimes, will be needed in practice.
      {Comment: Note that, for many applications of these metrics, the
      harm in treating a large delay as infinite might be zero or very
      small.  A TCP data packet, for example, that arrives only after
      several multiples of the RTT may as well have been lost.}

   +  If the packet is duplicated along the path (or paths) so that
      multiple non-corrupt copies arrive at the destination, then the
      packet is counted as received, and the first copy to arrive
      determines the packet's one-way delay.

   +  If the packet is fragmented and if, for whatever reason,
      reassembly does not occur, then the packet will be deemed lost.

3.6. Methodologies:

   As with other Type-P-* metrics, the detailed methodology will depend
   on the Type-P (e.g., protocol number, UDP/TCP port number, size,
   precedence).

   Generally, for a given Type-P, the methodology would proceed as
   follows:

   +  Arrange that Src and Dst are synchronized; that is, that they have
      clocks that are very closely synchronized with each other and each
      fairly close to the actual time.

   +  At the Src host, select Src and Dst IP addresses, and form a test
      packet of Type-P with these addresses.  Any 'padding' portion of
      the packet needed only to make the test packet a given size should
      be filled with randomized bits to avoid a situation in which the
      measured delay is lower than it would otherwise be due to
      compression techniques along the path.

   +  At the Dst host, arrange to receive the packet.

   +  At the Src host, place a timestamp in the prepared Type-P packet,
      and send it towards Dst.

   +  If the packet arrives within a reasonable period of time, take a
      timestamp as soon as possible upon the receipt of the packet.  By
      subtracting the two timestamps, an estimate of one-way delay can
      be computed.  Error analysis of a given implementation of the
      method must take into account the closeness of synchronization
      between Src and Dst.  If the delay between Src's timestamp and the

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      actual sending of the packet is known, then the estimate could be
      adjusted by subtracting this amount; uncertainty in this value
      must be taken into account in error analysis.  Similarly, if the
      delay between the actual receipt of the packet and Dst's timestamp
      is known, then the estimate could be adjusted by subtracting this
      amount; uncertainty in this value must be taken into account in
      error analysis.  See the next section, "Errors and Uncertainties",
      for a more detailed discussion.

   +  If the packet fails to arrive within a reasonable period of time,
      the one-way delay is taken to be undefined (informally, infinite).
      Note that the threshold of 'reasonable' is a parameter of the
      methodology.

   Issues such as the packet format, the means by which Dst knows when
   to expect the test packet, and the means by which Src and Dst are
   synchronized are outside the scope of this document.  {Comment: We
   plan to document elsewhere our own work in describing such more
   detailed implementation techniques and we encourage others to as
   well.}

3.7. Errors and Uncertainties:

   The description of any specific measurement method should include an
   accounting and analysis of various sources of error or uncertainty.
   The Framework document provides general guidance on this point, but
   we note here the following specifics related to delay metrics:

   +  Errors or uncertainties due to uncertainties in the clocks of the
      Src and Dst hosts.

   +  Errors or uncertainties due to the difference between 'wire time'
      and 'host time'.

   In addition, the loss threshold may affect the results.  Each of
   these are discussed in more detail below, along with a section
   ("Calibration") on accounting for these errors and uncertainties.

3.7.1. Errors or uncertainties related to Clocks

   The uncertainty in a measurement of one-way delay is related, in
   part, to uncertainties in the clocks of the Src and Dst hosts.  In
   the following, we refer to the clock used to measure when the packet
   was sent from Src as the source clock, we refer to the clock used to
   measure when the packet was received by Dst as the destination clock,
   we refer to the observed time when the packet was sent by the source
   clock as Tsource, and the observed time when the packet was received
   by the destination clock as Tdest.  Alluding to the notions of

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   synchronization, accuracy, resolution, and skew mentioned in the
   Introduction, we note the following:

   +  Any error in the synchronization between the source clock and the
      destination clock will contribute to error in the delay
      measurement.  We say that the source clock and the destination
      clock have a synchronization error of Tsynch if the source clock
      is Tsynch ahead of the destination clock.  Thus, if we know the
      value of Tsynch exactly, we could correct for clock
      synchronization by adding Tsynch to the uncorrected value of
      Tdest-Tsource.

   +  The accuracy of a clock is important only in identifying the time
      at which a given delay was measured.  Accuracy, per se, has no
      importance to the accuracy of the measurement of delay.  When
      computing delays, we are interested only in the differences
      between clock values, not the values themselves.

   +  The resolution of a clock adds to uncertainty about any time
      measured with it.  Thus, if the source clock has a resolution of
      10 msec, then this adds 10 msec of uncertainty to any time value
      measured with it.  We will denote the resolution of the source
      clock and the destination clock as Rsource and Rdest,
      respectively.

   +  The skew of a clock is not so much an additional issue as it is a
      realization of the fact that Tsynch is itself a function of time.
      Thus, if we attempt to measure or to bound Tsynch, this needs to
      be done periodically.  Over some periods of time, this function
      can be approximated as a linear function plus some higher order
      terms; in these cases, one option is to use knowledge of the
      linear component to correct the clock.  Using this correction, the
      residual Tsynch is made smaller, but remains a source of
      uncertainty that must be accounted for.  We use the function
      Esynch(t) to denote an upper bound on the uncertainty in
      synchronization.  Thus, |Tsynch(t)| <= Esynch(t).

   Taking these items together, we note that naive computation Tdest-
   Tsource will be off by Tsynch(t) +/- (Rsource + Rdest).  Using the
   notion of Esynch(t), we note that these clock-related problems
   introduce a total uncertainty of Esynch(t)+ Rsource + Rdest.  This
   estimate of total clock-related uncertainty should be included in the
   error/uncertainty analysis of any measurement implementation.

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3.7.2. Errors or uncertainties related to Wire-time vs Host-time

   As we have defined one-way delay, we would like to measure the time
   between when the test packet leaves the network interface of Src and
   when it (completely) arrives at the network interface of Dst, and we
   refer to these as "wire times."  If the timings are themselves
   performed by software on Src and Dst, however, then this software can
   only directly measure the time between when Src grabs a timestamp
   just prior to sending the test packet and when Dst grabs a timestamp
   just after having received the test packet, and we refer to these two
   points as "host times".

   To the extent that the difference between wire time and host time is
   accurately known, this knowledge can be used to correct for host time
   measurements and the corrected value more accurately estimates the
   desired (wire time) metric.

   To the extent, however, that the difference between wire time and
   host time is uncertain, this uncertainty must be accounted for in an
   analysis of a given measurement method.  We denote by Hsource an
   upper bound on the uncertainty in the difference between wire time
   and host time on the Src host, and similarly define Hdest for the Dst
   host.  We then note that these problems introduce a total uncertainty
   of Hsource+Hdest.  This estimate of total wire-vs-host uncertainty
   should be included in the error/uncertainty analysis of any
   measurement implementation.

3.7.3. Calibration

   Generally, the measured values can be decomposed as follows:

      measured value = true value + systematic error + random error

   If the systematic error (the constant bias in measured values) can be
   determined, it can be compensated for in the reported results.

      reported value = measured value - systematic error

   therefore

      reported value = true value + random error

   The goal of calibration is to determine the systematic and random
   error generated by the instruments themselves in as much detail as
   possible.  At a minimum, a bound ("e") should be found such that the
   reported value is in the range (true value - e) to (true value + e)
   at least 95 percent of the time.  We call "e" the calibration error
   for the measurements.  It represents the degree to which the values

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   produced by the measurement instrument are repeatable; that is, how
   closely an actual delay of 30 ms is reported as 30 ms.  {Comment: 95
   percent was chosen because (1) some confidence level is desirable to
   be able to remove outliers, which will be found in measuring any
   physical property; (2) a particular confidence level should be
   specified so that the results of independent implementations can be
   compared; and (3) even with a prototype user-level implementation,
   95% was loose enough to exclude outliers.}

   From the discussion in the previous two sections, the error in
   measurements could be bounded by determining all the individual
   uncertainties, and adding them together to form

       Esynch(t) + Rsource + Rdest + Hsource + Hdest.

   However, reasonable bounds on both the clock-related uncertainty
   captured by the first three terms and the host-related uncertainty
   captured by the last two terms should be possible by careful design
   techniques and calibrating the instruments using a known, isolated,
   network in a lab.

   For example, the clock-related uncertainties are greatly reduced
   through the use of a GPS time source.  The sum of Esynch(t) + Rsource
   + Rdest is small, and is also bounded for the duration of the
   measurement because of the global time source.

   The host-related uncertainties, Hsource + Hdest, could be bounded by
   connecting two instruments back-to-back with a high-speed serial link
   or isolated LAN segment.  In this case, repeated measurements are
   measuring the same one-way delay.

   If the test packets are small, such a network connection has a
   minimal delay that may be approximated by zero.  The measured delay
   therefore contains only systematic and random error in the
   instrumentation.  The "average value" of repeated measurements is the
   systematic error, and the variation is the random error.

   One way to compute the systematic error, and the random error to a
   95% confidence is to repeat the experiment many times - at least
   hundreds of tests.  The systematic error would then be the median.
   The random error could then be found by removing the systematic error
   from the measured values.  The 95% confidence interval would be the
   range from the 2.5th percentile to the 97.5th percentile of these
   deviations from the true value.  The calibration error "e" could then
   be taken to be the largest absolute value of these two numbers, plus
   the clock-related uncertainty.  {Comment: as described, this bound is
   relatively loose since the uncertainties are added, and the absolute
   value of the largest deviation is used.  As long as the resulting

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   value is not a significant fraction of the measured values, it is a
   reasonable bound.  If the resulting value is a significant fraction
   of the measured values, then more exact methods will be needed to
   compute the calibration error.}

   Note that random error is a function of measurement load.  For
   example, if many paths will be measured by one instrument, this might
   increase interrupts, process scheduling, and disk I/O (for example,
   recording the measurements), all of which may increase the random
   error in measured singletons.  Therefore, in addition to minimal load
   measurements to find the systematic error, calibration measurements
   should be performed with the same measurement load that the
   instruments will see in the field.

   We wish to reiterate that this statistical treatment refers to the
   calibration of the instrument; it is used to "calibrate the meter
   stick" and say how well the meter stick reflects reality.

   In addition to calibrating the instruments for finite one-way delay,
   two checks should be made to ensure that packets reported as losses
   were really lost.  First, the threshold for loss should be verified.
   In particular, ensure the "reasonable" threshold is reasonable: that
   it is very unlikely a packet will arrive after the threshold value,
   and therefore the number of packets lost over an interval is not
   sensitive to the error bound on measurements.  Second, consider the
   possibility that a packet arrives at the network interface, but is
   lost due to congestion on that interface or to other resource
   exhaustion (e.g. buffers) in the instrument.

3.8. Reporting the metric:

   The calibration and context in which the metric is measured MUST be
   carefully considered, and SHOULD always be reported along with metric
   results.  We now present four items to consider: the Type-P of test
   packets, the threshold of infinite delay (if any), error calibration,
   and the path traversed by the test packets.  This list is not
   exhaustive; any additional information that could be useful in
   interpreting applications of the metrics should also be reported.

3.8.1. Type-P

   As noted in the Framework document [1], the value of the metric may
   depend on the type of IP packets used to make the measurement, or
   "type-P".  The value of Type-P-One-way-Delay could change if the
   protocol (UDP or TCP), port number, size, or arrangement for special
   treatment (e.g., IP precedence or RSVP) changes.  The exact Type-P
   used to make the measurements MUST be accurately reported.

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3.8.2. Loss threshold

   In addition, the threshold (or methodology to distinguish) between a
   large finite delay and loss MUST be reported.

3.8.3. Calibration results

   +  If the systematic error can be determined, it SHOULD be removed
      from the measured values.

   +  You SHOULD also report the calibration error, e, such that the
      true value is the reported value plus or minus e, with 95%
      confidence (see the last section.)

   +  If possible, the conditions under which a test packet with finite
      delay is reported as lost due to resource exhaustion on the
      measurement instrument SHOULD be reported.

3.8.4. Path

   Finally, the path traversed by the packet SHOULD be reported, if
   possible.  In general it is impractical to know the precise path a
   given packet takes through the network.  The precise path may be
   known for certain Type-P on short or stable paths.  If Type-P
   includes the record route (or loose-source route) option in the IP
   header, and the path is short enough, and all routers* on the path
   support record (or loose-source) route, then the path will be
   precisely recorded.  This is impractical because the route must be
   short enough, many routers do not support (or are not configured for)
   record route, and use of this feature would often artificially worsen
   the performance observed by removing the packet from common-case
   processing.  However, partial information is still valuable context.
   For example, if a host can choose between two links* (and hence two
   separate routes from Src to Dst), then the initial link used is
   valuable context.  {Comment: For example, with Merit's NetNow setup,
   a Src on one NAP can reach a Dst on another NAP by either of several
   different backbone networks.}

4. A Definition for Samples of One-way Delay

   Given the singleton metric Type-P-One-way-Delay, 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, and Type-P,
   then define a sample of values of parameter T.  The means for
   defining 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

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   Poisson process of rate lambda, whose values fall between T0 and Tf.
   The time interval between successive values of T will then average
   1/lambda.

   {Comment: Note that Poisson sampling is only one way of defining a
   sample.  Poisson has the advantage of limiting bias, but other
   methods of sampling might be appropriate for different situations.
   We encourage others who find such appropriate cases to use this
   general framework and submit their sampling method for
   standardization.}

4.1. Metric Name:

   Type-P-One-way-Delay-Poisson-Stream

4.2. Metric Parameters:

   +  Src, the IP address of a host

   +  Dst, the IP address of a host

   +  T0, a time

   +  Tf, a time

   +  lambda, a rate in reciprocal seconds

4.3. Metric Units:

   A sequence of pairs; the elements of each pair are:

   +  T, a time, and

   +  dT, either a real number or an undefined number of seconds.

   The values of T in the sequence are monotonic increasing.  Note that
   T would be a valid parameter to Type-P-One-way-Delay, and that dT
   would be a valid value of Type-P-One-way-Delay.

4.4. Definition:

   Given T0, Tf, and lambda, we compute a pseudo-random Poisson process
   beginning at or before T0, with average arrival rate lambda, and
   ending 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 of the times
   in this process, we obtain the value of Type-P-One-way-Delay at this
   time.  The value of the sample is the sequence made up of the
   resulting <time, delay> pairs.  If there are no such pairs, the

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   sequence is of length zero and the sample is said to be empty.

4.5. Discussion:

   The reader should be familiar with the in-depth discussion of Poisson
   sampling in the Framework document [1], which includes methods to
   compute and verify the pseudo-random Poisson process.

   We specifically do not constrain the value of lambda, except to note
   the extremes.  If the rate is too large, then the measurement traffic
   will perturb the network, and itself cause congestion.  If the rate
   is too small, then you might not capture interesting network
   behavior.  {Comment: We expect to document our experiences with, and
   suggestions for, lambda elsewhere, culminating in a "best current
   practices" document.}

   Since a pseudo-random number sequence is employed, the sequence of
   times, and hence the value of the sample, is not fully specified.
   Pseudo-random number generators of good quality will be needed to
   achieve the desired qualities.

   The sample is defined in terms of a Poisson process both to avoid the
   effects of self-synchronization and also capture a sample that is
   statistically as unbiased as possible.  {Comment: there is, of
   course, no claim that real Internet traffic arrives according to a
   Poisson arrival process.}  The Poisson process is used to schedule
   the delay measurements.  The test packets will generally not arrive
   at Dst according to a Poisson distribution, since they are influenced
   by the network.

   All the singleton Type-P-One-way-Delay metrics in the sequence will
   have the same values of Src, Dst, and Type-P.

   Note also that, given one sample that runs from T0 to Tf, and given
   new time values T0' and Tf' such that T0 <= T0' <= Tf' <= Tf, the
   subsequence of the given sample whose time values fall between T0'
   and Tf' are also a valid Type-P-One-way-Delay-Poisson-Stream sample.

4.6. Methodologies:

   The methodologies follow directly from:

   +  the selection of specific times, using the specified Poisson
      arrival process, and

   +  the methodologies discussion already given for the singleton
      Type-P-One-way-Delay metric.

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   Care must, of course, be given to correctly handle out-of-order
   arrival of test packets; it is possible that the Src could send one
   test packet at TS[i], then send a second one (later) at TS[i+1],
   while the Dst could receive the second test packet at TR[i+1], and
   then receive the first one (later) at TR[i].

4.7. Errors and Uncertainties:

   In addition to sources of errors and uncertainties associated with
   methods employed to measure the singleton values that make up the
   sample, care must be given to analyze the accuracy of the Poisson
   process with respect to the wire-times of the sending of the test
   packets.  Problems with this process could be caused by several
   things, including problems with the pseudo-random number techniques
   used to generate the Poisson arrival process, or with jitter in the
   value of Hsource (mentioned above as uncertainty in the singleton
   delay metric).  The Framework document shows how to use the
   Anderson-Darling test to verify the accuracy of a Poisson process
   over small time frames.  {Comment: The goal is to ensure that test
   packets are sent "close enough" to a Poisson schedule, and avoid
   periodic behavior.}

4.8. Reporting the metric:

   You MUST report the calibration and context for the underlying
   singletons along with the stream.  (See "Reporting the metric" for
   Type-P-One-way-Delay.)

5. Some Statistics Definitions for One-way Delay

   Given the sample metric Type-P-One-way-Delay-Poisson-Stream, we now
   offer several statistics of that sample.  These statistics are
   offered mostly to be illustrative of what could be done.

5.1. Type-P-One-way-Delay-Percentile

   Given a Type-P-One-way-Delay-Poisson-Stream and a percent X between
   0% and 100%, the Xth percentile of all the dT values in the Stream.
   In computing this percentile, undefined values are treated as
   infinitely large.  Note that this means that the percentile could
   thus be undefined (informally, infinite).  In addition, the Type-P-
   One-way-Delay-Percentile is undefined if the sample is empty.

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

      Stream1 = <
      <T1, 100 msec>
      <T2, 110 msec>
      <T3, undefined>
      <T4, 90 msec>
      <T5, 500 msec>
      >

   Then the 50th percentile would be 110 msec, since 90 msec and 100
   msec are smaller and 110 msec and 'undefined' are larger.

   Note that if the possibility that a packet with finite delay is
   reported as lost is significant, then a high percentile (90th or
   95th) might be reported as infinite instead of finite.

5.2. Type-P-One-way-Delay-Median

   Given a Type-P-One-way-Delay-Poisson-Stream, the median of all the dT
   values in the Stream.  In computing the median, undefined values are
   treated as infinitely large.  As with Type-P-One-way-Delay-
   Percentile, Type-P-One-way-Delay-Median is undefined if the sample is
   empty.

   As noted in the Framework document, the median differs from the 50th
   percentile only when the sample contains an even number of values, in
   which case the mean of the two central values is used.

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

   Stream2 = <
      <T1, 100 msec>
      <T2, 110 msec>
      <T3, undefined>
      <T4, 90 msec>
      >

   Then the median would be 105 msec, the mean of 100 msec and 110 msec,
   the two central values.

5.3. Type-P-One-way-Delay-Minimum

   Given a Type-P-One-way-Delay-Poisson-Stream, the minimum of all the
   dT values in the Stream.    In computing this, undefined values are
   treated as infinitely large.  Note that this means that the minimum
   could thus be undefined (informally, infinite) if all the dT values
   are undefined.  In addition, the Type-P-One-way-Delay-Minimum is

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   undefined if the sample is empty.

   In the above example, the minimum would be 90 msec.

5.4. Type-P-One-way-Delay-Inverse-Percentile

   Given a Type-P-One-way-Delay-Poisson-Stream and a time duration
   threshold, the fraction of all the dT values in the Stream less than
   or equal to the threshold.  The result could be as low as 0% (if all
   the dT values exceed threshold) or as high as 100%.  Type-P-One-way-
   Delay-Inverse-Percentile is undefined if the sample is empty.

   In the above example, the Inverse-Percentile of 103 msec would be
   50%.

6. Security Considerations

   Conducting Internet measurements raises both security and privacy
   concerns.  This memo does not specify an implementation of the
   metrics, so it does not directly affect the security of the Internet
   nor of applications which run on the Internet.  However,
   implementations of these metrics must be mindful of security and
   privacy concerns.

   There are two types of security concerns: potential harm caused by
   the measurements, and potential harm to the measurements.  The
   measurements could cause harm because they are active, and inject
   packets into the network.  The measurement parameters MUST be
   carefully selected so that the measurements inject trivial amounts of
   additional traffic into the networks they measure.  If they inject
   "too much" traffic, they can skew the results of the measurement, and
   in extreme cases cause congestion and denial of service.

   The measurements themselves could be harmed by routers giving
   measurement traffic a different priority than "normal" traffic, or by
   an attacker injecting artificial measurement traffic.  If routers can
   recognize measurement traffic and treat it separately, the
   measurements will not reflect actual user traffic.  If an attacker
   injects artificial traffic that is accepted as legitimate, the loss
   rate will be artificially lowered.  Therefore, the measurement
   methodologies SHOULD include appropriate techniques to reduce the
   probability measurement traffic can be distinguished from "normal"
   traffic.  Authentication techniques, such as digital signatures, may
   be used where appropriate to guard against injected traffic attacks.

   The privacy concerns of network measurement are limited by the active
   measurements described in this memo.  Unlike passive measurements,
   there can be no release of existing user data.

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7. Acknowledgements

   Special thanks are due to Vern Paxson of Lawrence Berkeley Labs for
   his helpful comments on issues of clock uncertainty and statistics.
   Thanks also to Garry Couch, Will Leland, Andy Scherrer, Sean Shapira,
   and Roland Wittig for several useful suggestions.

8. References

   [1]  Paxson, V., Almes, G., Mahdavi, J. and M. Mathis, "Framework for
        IP Performance Metrics", RFC 2330, May 1998.

   [2]  Almes, G., Kalidindi, S. and M. Zekauskas, "A One-way Packet
        Loss Metric for IPPM", RFC 2680, September 1999.

   [3]  Mills, D., "Network Time Protocol (v3)", RFC 1305, April 1992.

   [4]  Mahdavi J. and V. Paxson, "IPPM Metrics for Measuring
        Connectivity", RFC 2678, September 1999.

   [5]  Postel, J., "Internet Protocol", STD 5, RFC 791, September 1981.

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

   [7]  Bradner, S., "The Internet Standards Process -- Revision 3", BCP
        9, RFC 2026, October 1996.

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9. Authors' Addresses

   Guy Almes
   Advanced Network & Services, Inc.
   200 Business Park Drive
   Armonk, NY  10504
   USA

   Phone: +1 914 765 1120
   EMail: almes@advanced.org

   Sunil Kalidindi
   Advanced Network & Services, Inc.
   200 Business Park Drive
   Armonk, NY  10504
   USA

   Phone: +1 914 765 1128
   EMail: kalidindi@advanced.org

   Matthew J. Zekauskas
   Advanced Network & Services, Inc.
   200 Business Park Drive
   Armonk, NY 10504
   USA

   Phone: +1 914 765 1112
   EMail: matt@advanced.org

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10.  Full Copyright Statement

   Copyright (C) The Internet Society (1999).  All Rights Reserved.

   This document and translations of it may be copied and furnished to
   others, and derivative works that comment on or otherwise explain it
   or assist in its implementation may be prepared, copied, published
   and distributed, in whole or in part, without restriction of any
   kind, provided that the above copyright notice and this paragraph are
   included on all such copies and derivative works.  However, this
   document itself may not be modified in any way, such as by removing
   the copyright notice or references to the Internet Society or other
   Internet organizations, except as needed for the purpose of
   developing Internet standards in which case the procedures for
   copyrights defined in the Internet Standards process must be
   followed, or as required to translate it into languages other than
   English.

   The limited permissions granted above are perpetual and will not be
   revoked by the Internet Society or its successors or assigns.

   This document and the information contained herein is provided on an
   "AS IS" basis and THE INTERNET SOCIETY AND THE INTERNET ENGINEERING
   TASK FORCE DISCLAIMS ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING
   BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF THE INFORMATION
   HEREIN WILL NOT INFRINGE ANY RIGHTS OR ANY IMPLIED WARRANTIES OF
   MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE.

Acknowledgement

   Funding for the RFC Editor function is currently provided by the
   Internet Society.

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