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Internet Engineering Task Force
INTERNET-DRAFT                                          Anura Jayasumana
draft-jayasumana-reorder-density-01.txt               Nischal M. Piratla
                                                         Abhijit A. Bare
                                                             Tarun Banka
                                               Colorado State University
                                                               July 2003
                                                  Expires: December 2003

   Reorder Density Function - Metric for packet reordering measurement


Status of this memo

   This document is an Internet-Draft and is subject to all provisions
   of Section 10 of RFC2026.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF), its areas, and its working groups.  Note that
   other groups may also distribute working 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."

   The list of current Internet-Drafts can be accessed at
   http://www.ietf.org/ietf/1id-abstracts.txt

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   http://www.ietf.org/shadow.html

   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.

Abstract

   Out-of-order arrival of packets can significantly degrade the
   performance of many TCP-based, VoIP-based and Video-based
   applications. There is a need for a metric that can meaningfully,
   accurately and unambiguously characterize reordering. This memo
   proposes a new metric called Reorder Density function (RD), which can
   give an in-depth view of the reordering present in any packet
   sequence. This well-defined metric can also be used to evaluate
   effects of protocol and hardware implementations on packet
   reordering. The memo also provides an algorithm to compute the
   reorder density function followed by some illustrative examples.


Anura Jayasumana                                                [Page 1]

Internet Draft  <draft-jayasumana-reorder-density-01.txt>      July 2003


1. Introduction and Motivation

   Out-of-order arrival of packets is a common phenomenon on the
   Internet. Major cause of reordering of packets is the local
   parallelism present in network routers and switches. This parallelism
   is caused due to different load balancing algorithms used in routers
   and switches. Packets can also be reordered due to different queuing
   schemes within the networking equipment itself. Packet reordering
   leads to degradation of the performance of the applications. For
   example, perceived quality of voice degrades if a VoIP application
   receives packets out of order. Once we are able to quantify the
   degree of reordering in arriving packet streams, it is possible to
   predict the effects of reordering on applications that are sensitive
   to reordering, and perhaps even compensate for reordering. This can
   further help us in evaluating network protocols with respect to
   packet reordering.

   Until now, the percentage of out-of-order packets has been used as a
   metric for characterizing reordering.  However, this metric is vague
   and lacks in detail. There is also no uniform definition for the
   the degree of reordering of an arrived packet. For example, consider
   two packet sequences (1,3,4,2,5) and (1,4,3,2,5). It is possible to
   interpret the reordering of packets differently in this case,
   for example [1],

   (i) Packets 2, 3 and 4 are out ûof order in both cases.
   (ii) Only packet 2 is out of order in the first sequence, while
   packets 2 and 3 are out of order in the second.
   (iii) Packets 3 and 4 are out of order in both the sequences.
   (iv) Packets 2, 3 and 4 are out of order in the first sequence, while
   packets 4 and 2 are out ûof order in the second sequence.

   In essence, the percentage of out-of-order packets is subject to
   interpretation and it cannot capture the reordering unambiguously
   and, hence, accurately. Thus, there is a need for a more precise and
   complete definition.

   Taking any of the above sequences, if buffers are available to store
   the packets 3 and 4 while waiting for packet 2, it is possible to
   recover from the reordering. However, there may be cases where the
   application requirement is such that the arrival of packet 2 after
   this delay renders it useless. While one can argue that a good packet
   reordering measurement scheme should capture such effects, a counter
   argument can also be made that packet reordering should be measured
   strictly with respect to the order of delivery and should be
   application independent.

   In this memo, we define a metric called Reorder Density function
   (RD). RD is the normalized form of a histogram of the occupancy of a
   hypothetical buffer that would allow the recovery from out-of-order
   delivery of packets.


Anura Jayasumana                                                [Page 2]

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   In addition to providing  a consistent percentage reordering measure,
   RD can  also be used to compute the percentages corresponding to
   different degrees of reordering. Next sections explain the concept of
   the reorder density function (RD).

2. Definitions of terms used

   Some important terms are defined, which will help us describe the
   Reorder Density Function (RD).

2.1 Out-of-order packet:

   When a packet other than the expected packet arrives, it is
   considered as an out-of-order packet, provided it is not a duplicate
   of an already received packet.

2.2 Buffer Occupancy (D):

    An arrived packet with a sequence number greater than the expected
    packet is considered to be stored in a buffer. Note that this is
    only a hypothetical buffer that we use to define RD. At any packet
    arrival instant, the buffer occupancy (assuming one buffer per
    packet) is therefore the number of such out-of-order packets.
    If the newly arrived packet is out of order, it will occupy the
    buffer as well. For example, for the sequence of packets
    (1,2,4,5,3), the buffer occupancy value, when the packet with the
    sequence number 4 arrives is 1 because it arrived when 3 was
    expected. Similarly, the buffer occupancy becomes 2 when the packet
    with the sequence number 5 arrives, as both packets 3 and 4 have to
    be held in the buffer. When packet 3 arrives, the occupancy becomes
    zero as it is no longer necessary to hold packets 4 and 5 to recover
    from reordering. (The term buffer occupancy was called displacement
    in [1].)

2.3 Occupancy Threshold (DT):

   This parameter defines the tolerance of the application to the
   maximum allowed displacement of a packet. It can also be viewed as
   the maximum size of the hypothetical buffer.  If an out-of-order
   packet needs to be stored in the hypothetical buffer already filled
   to the value of occupancy threshold, the currently expected packet is
   considered to be delayed more than the tolerance and hence, is
   assumed to be lost. The threshold may be chosen such that even if the
   packet ultimately arrives after the threshold, it is of no use to the
   application. Many factors influence the selection of the occupancy
   threshold value, for example, the transport layer protocol (UDP or
   TCP), the amount of redundant information sent to recover from
   losses, and whether the sequence of packets belong to a
   time-sensitive application or not.


Anura Jayasumana                                                [Page 3]

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   In case of a VoIP application, for example, with  a bit-rate of 128
   kbps and packet size of 200 bytes, DT value can be determined as
   follows. Assume that the application can wait at most 50 ms for an
   expected packet, and that the packets arrive at constant rate. That
   means within 50 ms, the application can receive
   (128*1000*0.05)/(200*8) i.e. 4 packets. Therefore, the occupancy
   threshold should be kept at 4.

   In case of TCP, a lost or delayed packet will be retransmitted and
   will reach the destination. So the value of the DT should be kept at
   least equal to the size of the receiving window on the receiver side.

2.4 Frequency of Occurences (F)

   At the arrival of each packet the buffer occupancy may take any value
   'i' ranging from 0 to DT. The frequency of occurrence F[i] is the
   number of times the occupancy takes the value of 'i'.

2.5 Definition of RD

   RD is defined as the distribution of all frequencies of occurrence,
   F[i], normalized with respect to the total number of occurrences,
   i.e, sum(F[i]) for all i in [0, DT].

3. Algorithm to compute reorder density (RD) function

   This section describes an algorithm to compute the reorder density
   function. Without loss of generality, the description assumes that
   the sequence numbers start at 1 and increment by 1 for each in-order
   packet.

   ---------------------------------------------------------------------
   # E : Next expected sequence number.
   # S : Sequence number of the packet just arrived.
   # D : Current buffer occupancy.
   # DT : Occupancy threshold.
   # F[i] : Frequency of occurrence of D = i.
   # RD[i] : Normalized frequency for D = i.
   # in_buffer(N) : True if the packet with sequence number N is
     already stored in the buffer.
   =====================================================================

   1.  Initialize E = 1,D = 0 and F[i] = 0 for all values of i (0 <= i
       <= DT).

   2.  Do the following for each arrived packet.

          If (in_buffer(S) || S < E) /*Do nothing*/;
          /* Case a: S is a duplicate or delayed packet. Discard the
            packet.*/


Anura Jayasumana                                                [Page 4]

Internet Draft  <draft-jayasumana-reorder-density-01.txt>      July 2003


          ElseIf (S == E)
          /* Case b: Expected packet has arrived.*/
          {
             E = E + 1;
             While (in_buffer(E))
             {
                D = D - 1; /* Free buffer occupied by E.*/
                E = E + 1; /* Expect next packet.*/
             }
             F[D] = F[D] + 1; /* Update frequency for buffer occupancy
                                 D.*/
          } /* End of ElseIf (S == E)*/

          ElseIf (S > E)
          /*Case c: Arrived packet has a sequence number higher
            than expected.*/
          {
             If (D < DT)
             /* Store the arrived packet in a buffer.*/
                D = D + 1;
             Else
             /* Expected packet is delayed beyond the DT. Treat it as
                lost.*/
             {
                 Repeat
                 {
                    E = E + 1;
                 }
                 Until (in_buffer(E) || E == S);

                 While (in_buffer(E) || E == S)
                 {
                       D = D - 1;
                       E = E + 1;
                 }
             }
             F[D] = F[D] + 1; /* Update frequency for buffer occupancy
                                 D.*/
          } /* End of ElseIf (S > E)*/

   3. Normalize F[i] to get RD[i] for all values of i (0 <= i <= DT)
      using
                            F[i]
      RD[i] = ----------------------------------
                  Sum(F[j] for 0 <= j <= DT)
   ---------------------------------------------------------------------

   The algorithm starts with the initialization of D to 0 and E to 1.
   Let S be the sequence number of an arrived packet.



Anura Jayasumana                                                [Page 5]

Internet Draft  <draft-jayasumana-reorder-density-01.txt>      July 2003


   If S has been received previously or delayed subject to the
   occupancy threshold condition (case a), it is discarded.

   If S is the expected packet (case b), E is incremented by 1 (i.e. the
   next packet in the sequence is now expected). If the packet with new
   E, i.e., the next packet in the sequence has already arrived, it need
   not be held in the buffer any more (it can be used by the
   application). So the buffer occupancy value is reduced by 1 and E is
   incremented by 1. This is repeated till all the in-sequence waiting
   packets are removed.

   If the received packet with the sequence number S is not the expected
   packet, two cases are possible. First case is when S is higher than E
   (case c), i.e., received packet is an out-of-order packet. If the
   buffer occupancy is less than the occupancy threshold, the packet
   with the sequence number E can still be expected. The value of the
   buffer occupancy is incremented, because the newly arrived packet
   needs to be stored in the hypothetical buffer. On the other hand, if
   the buffer occupancy is equal to the occupancy threshold, the
   currently expected packet E is assumed to be lost and E is
   incremented repeatedly till E reaches the sequence number of a packet
   that has been already received. This packet can now be removed from
   the hypothetical buffer giving space to the newly arrived packet. E
   is incremented further to check if there are any packets with higher
   sequence numbers already arrived and waiting, similar to what is done
   in the S=E case (in case b).

   The frequency value for the new value of the buffer occupancy is
   incremented as shown in the algorithm.

   Once the algorithm deals with all the packets and the frequency F[D]
   is computed, for all the values of D, the F[D] values are normalized
   to get the density with respect to D. This function is called the
   Reorder Density function.


4. Examples

   We consider a few different sequences to exemplify the above
   algorithm.

   a. Case of no packet loss:

   Consider a sequence of 5 packets (1,4,2,5,3) with DT = 10.

   Table 1 and 2 show the computation steps when RD algorithm is applied
   to above sequence.





Anura Jayasumana                                                [Page 6]

Internet Draft  <draft-jayasumana-reorder-density-01.txt>      July 2003


   -------------------------------------------------
   Table 1: Reorder Histogram computation steps
   -------------------------------------------------
   E     1     2     2     3     3
   S     1     4     2     5     3
   D     0     1     1     2     0
   F[D]  1     1     2     1     2
   -------------------------------------------------
   (E,S,D,F[D] as described in section 3)
   -------------------------------------------------

   The last row (F[D]) represents the current frequency of occurrence of
   the buffer occupancy D. The final set of values for F[D] are shown in
   table 2.

   When the first packet with the sequence number S=1 arrives, it is
   same as the expected sequence number E=1, resulting in the
   buffer occupancy D=0. Next, when the packet S=4 arrives instead of
   the expected packet E=2, the buffer occupancy D becomes 1. After
   receiving the packet with the sequence number 2, the buffer occupancy
   D is still 1, since the packet 3 that is expected now is not yet
   received. Packet 4 continues to occupy a buffer. Only one buffer is
   needed and hence D = 1. On receiving the packet with the sequence
   number 5, the buffer occupancy D becomes 2. Finally, when we receive
   the packet with sequence number 3, all the packets up to the sequence
   number 5 have been received. Thus the buffers can be released and
   hence the buffer occupancy D becomes 0. The reorder density function
   (RD) is derived by normalizing reorder histogram in Table 1 as
   follows:

   -------------------------------------------------
   Table 2: Reorder Density Function (RD)
   -------------------------------------------------
   D          0     1     2
   F[D]       2     2     1
   RD[D]      0.4   0.4   0.2
   -------------------------------------------------
   (D,F[D],RD[D] as described in section 3)
   -------------------------------------------------













Anura Jayasumana                                                [Page 7]

Internet Draft  <draft-jayasumana-reorder-density-01.txt>      July 2003


   Graphical representation of above RD is as follows:

                 ^
                 |
             0.4 |_____
         ^       |  |  |
         |       |  |  |
             0.2 |  |  |__
       RD[D]     |  |  |  |
                 |  |  |  |
               0 +--+--+--+----------->
                   0  1  2
                     D  -->

   b. Case of packet loss:

   Consider a sequence of 6 packets (1,2,4,5,6,7) with DT = 3.

   Tables 3 and 4 show the computation steps, when the RD algorithm is
   applied to the above sequence.

   -------------------------------------------------
   Table 3: Reorder Histogram computation steps
   -------------------------------------------------
   E     1     2     3     3     3     3
   S     1     2     4     5     6     7
   D     0     0     1     2     3     0
   F[D]  1     2     1     1     1     3
   -------------------------------------------------
   (E,S,D,F[D] as described in section 3)
   -------------------------------------------------

   When a packet with the sequence number 4 is received, the expected
   packet E is 3. So the buffer occupancy D increases by 1. When the
   packets with the sequence numbers 5 and 6 arrive, D increases to 2
   and then to 3 respectively. The buffer occupancy is now equal to the
   occupancy threshold DT=3. Therefore, when the packet 7 is received,
   we no longer expect the packet with the sequence number 3 to arrive
   and assume that it is lost. We can now use all the waiting packets
   (4,5,6 and 7), reducing the buffer occupancy to 0. The reorder
   density function (RD) is derived by normalizing the reorder histogram
   in Table 3 as follows:

   -------------------------------------------------
   Table 4: Reorder Density Function (RD)
   -------------------------------------------------
   D          0     1     2     3
   F[D]       3     1     1     1
   RD[D]      0.5   0.17  0.17  0.17
   -------------------------------------------------
   (D,F[D],RD[D] as described in section 3)
   -------------------------------------------------


Anura Jayasumana                                                [Page 8]

Internet Draft  <draft-jayasumana-reorder-density-01.txt>      July 2003


   Graphical representation of above RD is as follows.


                 ^
             0.5 |__
         ^       |  |
         |       |  |
                 |  |
       RD[D] 0.17|  |________
                 |  |  |  |  |
               0 +--+--+--+--+--------->
                   0  1  2  3
                     D  -->

   c.  Case of Duplicate packets:

   Consider a sequence of 6 packets (1,3,2,3,4,5) with DT = 5.

   Tables 5 and 6 show the computation steps when the RD algorithm is
   applied to the above sequence.

   -------------------------------------------------
   Table 5: Reorder Histogram computation steps
   -------------------------------------------------
   E     1     2     2     4     4     5
   S     1     3     2     3     4     5
   D     0     1     0     -     0     0
   F[D]  1     1     2     -     3     4
   -------------------------------------------------
   (E,S,D,F[D] as described in section 3)
   -------------------------------------------------

   In the above sequence, duplicate packets are received by the
   destination. The RD algorithm ignores the arrivals of the duplicate
   packets.

   The reorder density function (RD) is derived by normalizing reorder
   histogram in Table 5 as follows:

   -------------------------------------------------
   Table 6: Reorder Density Function (RD)
   -------------------------------------------------
   D          0     1
   F[D]       4     1
   RD[D]      0.80  0.20
   -------------------------------------------------
   (D,F[D],RD[D] as described in section 3)
   -------------------------------------------------




Anura Jayasumana                                                [Page 9]

Internet Draft  <draft-jayasumana-reorder-density-01.txt>      July 2003


   Graphical Representation of RD is as follows:

                 ^
                 |
            0.80 |__
                 |  |
                 |  |
         ^       |  |
         |       |  |
                 |  |
       RD[D] 0.20|  |__
                 |  |  |
               0 +--+--+----->
                   0  1
                     D  --->


5. Simple metrics derived from RD

   While the reorder density can provide a detailed picture of the
   degree of reordering present in a sequence of packets, there may be
   instances, where a simpler metric is needed to compare sequences.
   The following parameters derived from the reorder density may be used
   as simpler  metrics for packet reordering. Reference [1] shows that
   these simpler metrics can effectively capture the relative degrees of
   reordering of packet in sequences effectively.

5.1 90th percentile of RD

    This parameter is the buffer occupancy value, such that 90 % of the
    arrived packets have buffer occupancy less than this value.

5.2 Mean and Standard Deviation of RD

    Mean and standard deviation of the buffer occupancy values of the
    arrived packets may be used as simple metrics.


6. Current Schemes

   Currently, the percentage of  out-of-order packets is the most
   commonly used packet reordering metric. With the percentage reorder
   metric, the information provided by the metric is purely for
   information only. For example, consider two sequences at the receiver
   end (2,3,4,5,1) and (2,1,3,4,5). Taking the definition of late
   arrival as reordered packet [2], in both the cases the percentage
   reordering is 20. However, it is obvious that the reordering in the
   second sequence is more acceptable than the first one as the recovery
   from the packet reordering is much easier in the former case. This
   metric is a significant simplification and is not useful in the
   recovery from reordering.


Anura Jayasumana                                               [Page 10]

Internet Draft  <draft-jayasumana-reorder-density-01.txt>      July 2003


   N-reordering [3] is a metric where an expected packet is 1-reordered,
   2-reordered and so on till it arrives. If a packet arrives after 40
   positions from its expected position then it is 40-reordered. Two
   examples are listed in Appendix A to show the difference between
   reorder density and N-reordering. These examples how that
   N-reordering is much more susceptible to delayed packets as it cannot
   treat them as lost when their useful life is over, whereas with RD
   this is taken care of using threshold.

   Reordering offset[4] is another metric to measure reordering. In this
   metric the packet is not reordered until it arrives. However, a
   duplicate packet is considered as a reordered packet. Unlike RD, this
   metric is not orthogonal to duplication of packets. Appendix B uses a
   few few example sequences to compare Reordering offset and RD.


7. Security Considerations

   This document does not define any protocol.  The metric definition
   per se is believed to have no security implications.


8. IANA Considerations

   This document requires nothing from the IANA.


9. References

   1. T. Banka, A. A. Bare, A. P. Jayasumana, "Metrics for Degree of
   Reordering in Packet Sequences", Proc. 27th IEEE Conference on Local
   Computer Networks, Tampa, FL, Nov. 2002.

   2. V.Paxson, "Measurements and Analysis of End-to-End Internet
   Dynamics," Ph.D. dissertation, U.C. Berkeley, 1997,
   ftp://ftp.ee.lbl.gov/papers/vp-thesis/dis.ps.gz.

   3. S. Shalunov, "Definition of IP Packet Reordering Metric",
   Internet Draft, <draft-shalunov-reordering-definition-02.txt>,
   March 2003.

   4. A. Morton, L. Ciavattone, G. Ramachandran, S.Shalunov and J.
   Perser, "Packet Reordering Metric for IPPM", Internet Draft,
   <draft-ietf-ippm-reordering-03.txt>, June 2003.






Anura Jayasumana                                               [Page 11]

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10. Authors' Addresses

   Anura Jayasumana <anura@engr.colostate.edu>
   Nischal M. Piratla <nischal@engr.colostate.edu>
   Abhijit A. Bare <abhijit@cs.colostate.edu>
   Tarun Banka <tarunb@cs.colostate.edu>



   Computer Networking Research Laboratory,
   Department of Electrical and Computer Engineering,
   Colorado State University,
   Fort Collins, CO 80523.

   Expiration Date: December 2003


11. Appendix A


   Example 1:For the sequence <1,2,3,4,5,2,1>

   RD output:
   -------------------------------------------------
   D          0     1     2     3
   F[D]       5     0     0     0
   RD[D]      1.00  0.00  0.00  0.00
   -------------------------------------------------

   N-reordering output:
   1-reordering = 33.333333%
   2-reordering = 40.000000%
   3-reordering = 50.000000%
   4-reordering = 33.333333%
   5-reordering = 50.000000%
   no 6-reordering

   1-reordering = 2
   2-reordering = 2
   3-reordering = 2
   4-reordering = 1
   5-reordering = 1
   no 6-reordering

   In this example, the N-reordering algo shows that there is
   5-reordering. If you look at the sequence there are two duplicate
   packets namely, seq#s 2 & 1. In RD, the F[D] does not exist for D>0
   i.e., there is no reordering. As one can see, the sequence has no
   reordering.


Anura Jayasumana                                               [Page 12]

Internet Draft  <draft-jayasumana-reorder-density-01.txt>      July 2003


   Example 2: For Sequence:
   <1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
   27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,2>

   RD output with DT = 5:
   -------------------------------------------------
   D          0     1     2     3     4     5
   F[D]       35    1     1     1     1     1
   RD[D]      0.875 0.025 0.025 0.025 0.025 0.025
   -------------------------------------------------

   N-reordering output:

   1-reordering = 2.500000%
   2-reordering = 2.564103%
   3-reordering = 2.631579%
   4-reordering = 2.702703%
   5-reordering = 2.777778%
   6-reordering = 2.857143%
   7-reordering = 2.941176%
   8-reordering = 3.030303%
   9-reordering = 3.125000%
   10-reordering = 3.225806%
   11-reordering = 3.333333%
   12-reordering = 3.448276%
   13-reordering = 3.571429%
   14-reordering = 3.703704%
   15-reordering = 3.846154%
   16-reordering = 4.000000%
   17-reordering = 4.166667%
   18-reordering = 4.347826%
   19-reordering = 4.545455%
   20-reordering = 4.761905%
   21-reordering = 5.000000%
   22-reordering = 5.263158%
   23-reordering = 5.555556%
   24-reordering = 5.882353%
   25-reordering = 6.250000%
   26-reordering = 6.666667%
   27-reordering = 7.142857%
   28-reordering = 7.692308%
   29-reordering = 8.333333%
   30-reordering = 9.090909%
   31-reordering = 10.000000%
   32-reordering = 11.111111%
   33-reordering = 12.500000%
   34-reordering = 14.285714%
   35-reordering = 16.666667%


Anura Jayasumana                                               [Page 13]

Internet Draft  <draft-jayasumana-reorder-density-01.txt>      July 2003


   36-reordering = 20.000000%
   37-reordering = 25.000000%
   38-reordering = 33.333333%
   39-reordering = 50.000000%
   no 40-reordering

   This example clearly shows that N-reordering is much more susceptible
   to delayed packets as it cannot treat them as lost when their useful
   life is over, whereas with RD this is taken care of.


12. Appendix B

   From <draft-ietf-ippm-reordering-01.txt>

   "...Table 1 Example with Packet 4 Reordered,

   Sending order(SrcNum@Src): 1,2,3,4,5,6,7,8,9,10

   SrcNum       Src     Dst                     Dst     Posit.  Late
   @Dst NextExp Time    Time    Delay   IPDV    Order   Offset  Time
   1     1       0      68      68              1
    2     2      20      88      68       0      2
    3     3      40     108      68       0      3
    5     4      80     148      68     -82      4
    6     6     100     168      68       0      5
    7     7     120     188      68       0      6
    8     8     140     208      68       0      7
    4     9      60     210     150      82      8      4       62
    9     9     160     228      68       0      9
   10    10     180     248      68       0     10

   Each column gives the following information:

   SrcNum   Packet sequence number at the Source.
   NextExp   The value of NextExp when the packet arrived(before
   update).
   SrcTime  Packet time stamp at the Source, ms.
   DstTime  Packet time stamp at the Destination, ms.
   Delay    1-way delay of the packet, ms.
   IPDV     IP Packet Delay Variation, ms
            IPDV = Delay(SrcNum)-Delay(SrcNum-1)..."

   Reorder Density for the above example:





Anura Jayasumana                                               [Page 14]

Internet Draft  <draft-jayasumana-reorder-density-01.txt>      July 2003


   SrcNum
   @Dst NextExp  Buffer occupancy  Frequency
    1     1          0              F[0] = 1
    2     2          0              F[0]++
    3     3          0              F[0]++
    5     4          1              F[1] = 1
    6     4          2              F[2] = 1
    7     4          3              F[3] = 1
    8     4          4              F[4] = 1
    4     4          0              F[0]++
    9     9          0              F[0]++
   10    10          0              F[0]++

   Normalized F[i] for all i: F[0] = 0.6, F[1] = 0.1, F[2] = 0.1, F[3] =
   0.1, F[4] = 0.1

   In this case, if we can set DT = 3, then the table will look like
   this:

   SrcNum
   @Dst Expected  Buffer occupancy Frequency
   1     1          0              F[0] = 1
   2     2          0              F[0]++
   3     3          0              F[0]++
   5     4          1              F[1] = 1
   6     4          2              F[2] = 1
   7     4          3              F[3] = 1
   8     4          0              F[0]++    {after the current packet's
                                              arrival, packet '4' is
                                              rendered useless!}
   4     9          0              -         {discarded pkt.}
   9     9          0              F[0]++
   10    10         0              F[0]++

   Normalized F[i] for all i: F[0] = 5/9, F[1] = 1/9, F[2] = 1/9, F[3] =
   1/9


   Other examples in current metrics are almost similar. However, there
   are no examples with packet duplication. Here is one for the metric
   proposed. (Note: Packet '5' is duplicated.)








Anura Jayasumana                                               [Page 15]

Internet Draft  <draft-jayasumana-reorder-density-01.txt>      July 2003


   SrcNum
   @Dst NextExp  Buffer Occupancy  Frequency
   1     1          0              F[0] = 1
   2     2          0              F[0]++
   3     3          0              F[0]++
   5     4          1              F[1] = 1
   6     4          2              F[2] = 1
   7     4          3              F[3] = 1
   8     4          4              F[4] = 1
   4     4          0              F[0]++
   5     9          0              -   {duplicate packet}
   9     9          0              F[0]++


   Normalized F[i] for all i: F[0] = 5/9, F[1] = 1/9, F[2] = 1/9, F[3] =
   1/9, F[4] = 1/9.

   At the arrival of a duplicate packet the buffer occupancy is held the
   same as the duplicate packet will be discarded and the contents of
   the buffer remain.































Anura Jayasumana                                              [Page 16]


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