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Internet Engineering Task Force                              Sally Floyd
INTERNET DRAFT                                                      ICSI
draft-floyd-tcp-highspeed-00.txt                              June, 2002



               HighSpeed TCP for Large Congestion Windows



                          Status of this Memo


   This document is an Internet-Draft and is in full conformance with
   all provisions of Section 10 of RFC2026.

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Abstract

   This document proposes HighSpeed TCP, a modification to TCP's
   congestion control mechanism for use with TCP connections with large
   congestion windows.  The congestion control mechanisms of the
   current, Standard TCP constrains the congestion windows that can be
   achieved by TCP in realistic environments.  For example, for a
   Standard TCP connection with 1500-byte packets and a 100 ms round-
   trip time, achieving a steady-state throughput of 10 Gbps would
   require an average congestion window of 83,333 segments, and a packet
   drop rate of at most one congestion event every 5,000,000,000 packet
   (or equivalently, at most one congestion event every 1 2/3 hours).
   We do not consider this a realistic condition.  To address this
   limitation of TCP, this document proposes HighSpeed TCP, and solicits
   experimentation and feedback from the wider community.




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1.  Introduction.

   This document proposes HighSpeed TCP, a modification to TCP's
   congestion control mechanism for use with TCP connections with large
   congestion windows.  In a steady-state environment, with a packet
   loss rate p, the current Standard TCP's average congestion window is
   roughly 1.2/sqrt(p) segments.  This places a serious constraint on
   the congestion windows that can be achieved by TCP in realistic
   environments.  For example, for a Standard TCP connection with
   1500-byte packets and a 100 ms round-trip time, achieving a steady-
   state throughput of 10 Gbps would require an average congestion
   window of 83,333 segments, and a packet drop rate of at most one
   congestion event every 5,000,000,000 packet (or equivalently, at most
   one congestion event every 1 2/3 hours).  To address this limitation
   of TCP, this document proposes HighSpeed TCP, and solicits
   experimentation and feedback from the wider community.

2.  The Problem Description.

   This section describes the number of round-trip times between
   congestion events required for a Standard TCP flow to achieve an
   average throughput of B bps, given packets of D bytes and a round-
   trip time of R seconds.  A congestion event refers to a window of
   data with one or more dropped or ECN-marked packets.

   From Appendix A, achieving an average TCP throughput of B bps
   requires a loss event at most every BR/(12D) round-trip times.  This
   is illustrated in Table 1, for R = 0.1 seconds and D = 1500 bytes.
   The table also gives the average congestion window W of BR/(8D), and
   the steady-state packet drop rate P of 1.5/W^2.

    TCP Throughput (Mbps)   RTTs Between Losses       W       P
    ---------------------   -------------------   ------    -----
                1                    5.5             8.3    0.02
               10                   55.5            83.3    0.0002
              100                  555.5           833.3    0.000002
             1000                 5555.5          8333.3    0.00000002
            10000                55555.5         83333.3    0.0000000002

   Table 1: RTTs Between Congestion Events for Standard TCP, for
   1500-Byte Packets and a Round-Trip Time of 0.1 Seconds.

   This document proposes HighSpeed TCP, a minimal modification to TCP's
   increase and decrease parameters, for TCP connections with larger
   congestion windows, to allow TCP to achieve high throughput with more
   realistic requirements for the steady-state packet drop rate.
   Equivalently, HighSpeed TCP has more realistic requirements for the
   number of round-trip times between loss events.



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3.  Design Guidelines.

   Our proposal for HighSpeed TCP is motivated by the following
   requirements:

   * Achieve high per-connection throughput without requiring
   unrealistically low packet loss rates.

   * Reach high throughput reasonably quickly when in slow-start.

   * Reach high throughput without overly long delays when recovering
   from multiple retransmit timeouts, or when ramping-up from a period
   with small congestion windows.

   * No additional feedback or support required from routers:

   For example, the goal is for acceptable performance in both ECN-
   capable and non-ECN-capable environments, and with Drop-Tail as well
   as with Active Queue Management such as RED in the routers.

   * No additional feedback required from TCP receivers.

   * TCP-compatible performance in environments with moderate or high
   congestion:

   Equivalently, the requirement is that there be no additional load on
   the network (in terms of increased packet drop rates) in environments
   with moderate or high congestion.

   * Performance at least as good as Standard TCP in environments with
   moderate or high congestion.

   * Acceptable transient performance, in terms of increases in the
   congestion window in one round-trip time, responses to severe
   congestion, and convergence times to fairness.

   Currently, users wishing to achieve throughputs of 1Gbps or more
   typically open up multiple TCP connections in parallel, or use MulTCP
   [GRK99], which behaves roughly like the aggregate of N virtual TCP
   connections.  While this approach suffices for the occasional user on
   well-provisioned links, it leaves the parameter N to be determined by
   the user, and results in more aggressive performance and higher
   steady-state packet drop rates if used in environments with periods
   of moderate or high congestion.  We believe that a new approach is
   needed that offers more flexibility, more effectively scales to a
   wide range of available bandwidths, and competes more fairly with
   Standard TCP in congested environments.




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4.  Non-Goals.

   The following are explicitly *not* goals of our work:

   * Non-goal: TCP-compatible performance in environments with very low
   packet drop rates.

   We note that our proposal does not require, or deliver, TCP-
   compatible performance in environments with very low packet drop
   rates, e.g., with packet loss rates of 10^-5 or 10^-6.  As we discuss
   later in this document, we assume that Standard TCP is unable to make
   effective use of the available bandwidth in environments with loss
   rates of 10^-6 in any case, so that it is acceptable and appropriate
   for HighSpeed TCP to perform more aggressively than Standard TCP is
   such an environment.

   * Non-goal: Ramping-up more quickly than allowed by slow-start.

   It is our belief that ramping-up more quickly than allowed by slow-
   start would necessitate more explicit feedback from routers along the
   path.  The proposal for HighSpeed TCP is focused on changes to TCP
   that could be effectively deployed in the current Internet
   environment.

   * Non-goal: Avoiding oscillations in environments with only one-way,
   long-lived flows all with the same round-trip times.

   While we agree that attention to oscillatory behavior is useful,
   avoiding oscillations in aggregate throughput has not been our
   primary consideration, particularly for simplified environments
   limited to one-way, long-lived flows all with the same, large round-
   trip times.  Our assessment is that some oscillatory behavior in
   these extreme environments is an acceptable price to pay for the
   other benefits of HighSpeed TCP.

5.  Modifying the TCP Response Function.

   The TCP response function, w = 1.2/sqrt(p), gives TCP's average
   congestion window w in MSS-sized segments, as a function of the
   steady-state packet drop rate p [FF98].  This TCP response function
   is a direct consequence of TCP's Additive Increase Multiplicative
   Decrease (AIMD) mechanisms of increasing the congestion window by
   roughly one segment per round-trip time in the absence of congestion,
   and halving the congestion window in response to a round-trip time
   with a congestion event.  This response function for Standard TCP is
   reflected in the table below.  In this proposal we restrict our
   attention to TCP performance in environments with packet loss rates
   of at most 10^-2, and so we can ignore the more complex response



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   functions that are required to model TCP performance in more
   congested environments with retransmit timeouts.  From Appendix A, an
   average congestion window of W corresponds to an average of W/1.5
   round-trip times between loss events for Standard TCP.

       Packet Drop Rate P   Congestion Window W    RTTs Between Losses
       ------------------   -------------------    -------------------
              10^-2                     12                8
              10^-3                     38               25
              10^-4                    120               80
              10^-5                    379              252
              10^-6                   1200              800
              10^-7                   3795             2530
              10^-8                  12000             8000
              10^-9                  37948            25298
              10^-10                120000            80000

   Table 2: TCP Response Function for Standard TCP.  The average
   congestion window W in MSS-sized segments is given as a function of
   the packet drop rate P.

   To specify a modified response function for HighSpeed TCP, we use
   three parameters, Low_Window, High_Window, and High_P.  To ensure TCP
   compatibility, the HighSpeed response function uses the same response
   function as Standard TCP when the current congestion window is at
   most Low_Window, and uses the HighSpeed response function when the
   current congestion window is greater than Low_Window.  In this
   document we set Low_Window to 38 MSS-sized segments, corresponding to
   a packet drop rate of 10^-3 for TCP.

   To specify the upper end of the HighSpeed response function, we
   specify the packet drop rate needed in the HighSpeed response
   function to achieve an average congestion window of 83000 segments.
   This is roughly the window needed to sustain 10Gbps throughput, for a
   TCP connection with the default packet size and round-trip time used
   earlier in this document.  For High_Window set to 83000, we specify
   High_P of 10^-7; that is, with HighSpeed TCP a packet drop rate of
   10^-7 allows the HighSpeed TCP connection to achieve an average
   congestion window of 83000 segments.  We believe that this loss rate
   sets an achieveable target for high-speed environments, while still
   allowing acceptable fairness for the HighSpeed response function when
   competing with Standard TCP in environments with packet drop rates of
   10^-4 or 10^5.

   For simplicity, for the HighSpeed response function we maintain the
   property that the response function gives a straight line on a log-
   log scale (as does the response function for Standard TCP, for low to
   moderate congestion).  This results in the following response



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   function, for values of the average congestion window W greater than
   Low_Window:

     W = (p/Low_P)^S Low_Window,

   for Low_P the packet drop rate corresponding to Low_Window, and for S
   as following constant [FRS02]:

     S = (log High_Window - log Low_Window)/(log High_P - log Low_P).

   For example, for Low_Window set to 38, we have Low_P of 10^-3 (for
   compatibility with Standard TCP).  Thus, for High_Window set to 83000
   and High_P set to 10^-7, we get the following response function:

     W = 0.12/p^0.835.                                    (1)

   This HighSpeed response function is illustrated in Table 3 below.
   For HighSpeed TCP, the number of round-trip times between losses,
   1/(pW), equals 12.7 W^0.2, for W > 38 segments.

       Packet Drop Rate P   Congestion Window W    RTTs Between Losses
       ------------------   -------------------    -------------------
              10^-2                    12                   8
              10^-3                    38                  25
              10^-4                   263                  38
              10^-5                  1795                  57
              10^-6                 12279                  83
              10^-7                 83981                 123
              10^-8                574356                 180
              10^-9               3928088                 264
              10^-10             26864653                 388

   Table 3: TCP Response Function for HighSpeed TCP.  The average
   congestion window W in MSS-sized segments is given as a function of
   the packet drop rate P.

   It has been suggested that a less "ad-hoc" guideline for a response
   function for HighSpeed TCP would be to specify a constant value for
   the number of round-trip times between congestion events.  However,
   this seems inadvisable to us, as it gives less clearly-differentiated
   feedback in an environment with steady-state background losses at
   fixed intervals for all flows (as might occur with a wireless link
   with occasional short error bursts, giving losses for all flows every
   N seconds regardless of their sending rate).  While it is not a goal
   to have perfect fairness in an environment with synchronized losses,
   it would be good to have moderately acceptable performance in this
   regime.  This goal argues against a response function with a constant
   number of round-trip times between congestion events.



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   We believe that the problem of backward compatibility with Standard
   TCP requires a response function that is quite close to that of
   Standard TCP for loss rates of 10^-1, 10^-2, or 10^-3.  We believe,
   however, that such stringent TCP-compatibility is not required for
   smaller loss rates, and that an appropriate response function is one
   that gives a plausible packet drop rate for a connection throughput
   of 10Gbps.  This also gives a slowly increasing number of round-trip
   times between loss events as a function of a decreasing packet drop
   rate.

   We do not in this document attempt to seriously evaluate the
   HighSpeed response function for per-connection bandwidths greater
   than 10Gbps.  We believe that we will learn more about the
   requirements for sustaining the throughput of best-effort connections
   in that range as we gain more experience with HighSpeed TCP at 1 Gbps
   and 10 Gbps ranges.  There also might be limitations to the per-
   connection throughput that can be realistically achieved for best-
   effort traffic in the absence of additional support or feedback from
   the routers along the path.

6.  Fairness Implications of the HighSpeed Response Function.

   The Standard and Highspeed Response Functions can be used directly to
   infer the relative fairness between flows using the two response
   functions.  For example, given a packet drop rate P, assume that
   Standard TCP has an average congestion window of W_Standard, and
   HighSpeed TCP has a higher average congestion window of W_HighSpeed.
   In this case, a single HighSpeed TCP connection is receiving
   W_HighSpeed/W_Standard times the throughput of a single Standard TCP
   connection competing in the same environment.

   This relative fairness is illustrated below in Table 4, for the
   parameters used for the Highspeed response function in the section
   above.

















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       Packet Drop Rate P   Relative Fairness
       ------------------   -----------------
              10^-2               1.0
              10^-3               1.0
              10^-4               2.2
              10^-5               4.7
              10^-6              10.2
              10^-7              22.1
              10^-8              47.9
              10^-9             103.5
              10^-10            223.9

   Table 4: Relative Fairness between the HighSpeed and Standard
   Response Functions.

   Thus, for packet drop rates of 10^-4, a flow with the HighSpeed
   response function can expect to receive 2.2 times the throughput of a
   flow using the Standard response function, given the same round-trip
   times and packet sizes.  With packet drop rates of 10^-6 (or 10^-7),
   the unfairness is more severe, and we have entered the regime where a
   Standard TCP connection requires a congestion event at most every 800
   (or 2530) round-trip times in order to make use of the available
   bandwidth.  Our judgement would be that there are not a lot of TCP
   connections effectively operating in this regime today, with
   congestion windows of thousands of packets, and that therefore the
   benefits of the HighSpeed response function would outweigh the
   unfairness that would be experienced by Standard TCP in this regime.
   However, one purpose of this document is to solicit feedback on this
   issue.  The parameter Low_Window determines directly the point of
   divergence between the Standard and HighSpeed Response Functions.

7.  Translating the HighSpeed Response Function into Congestion Control
Parameters.

   For equation-based congestion control such as TFRC, the HighSpeed
   Response Function above could be used directly by the TFRC congestion
   control mechanism.  However, for TCP the HighSpeed response function
   would have to be translated into additive increase and multiplicative
   decrease parameters.  The HighSpeed response function cannot be
   achieved by TCP with an additive increase of one segment per round-
   trip time and a multiplicative decrease of halving the current
   congestion window; HighSpeed TCP will have to modify either the
   increase or the decrease parameter, or both.  We have concluded that
   HighSpeed TCP is most likely to achieve an acceptable compromise
   between moderate increases and timely decreases by modifying both the
   increase and the decrease parameter.

   That is, for HighSpeed TCP let the congestion window increase by a(w)



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   segments per round-trip time in the absence of congestion, and let
   the congestion window decrease to w(1-b(w)) segments in response to a
   round-trip time with one or more loss events.  Thus, in response to a
   single acknowledgement HighSpeed TCP increases its congestion window
   in segments as follows:

    w <- w + a(w)/w.

   In response to a congestion event, HighSpeed TCP decreases as
   follows:

    w <- (1-b(w))w.

   For Standard TCP, a(w) = 1 and b(w) = 1/2, regardless of the value of
   w.  HighSpeed TCP uses the same values of a(w) and b(w) for w <=
   Low_Window.  This section specifies a(w) and b(w) for HighSpeed TCP
   for larger values of w.

   For w = High_Window, we have specified a loss rate of High_P.  From
   [FRS02], or from elementary calculations, this requires the following
   relationship between a(w) and b(w) for w = High_Window:

    a(w) = High_Window^2 * High_P * 2 * b(w)/(2-b(w).     (2)

   We use the parameter High_Decrease to specify the decrease parameter
   b(w) for w = High_Window, and use Equation (2) to derive the increase
   parameter a(w) for w = High_Window.  Along with High_P = 10^-7 and
   High_Window = 83000, for example, we specify High_Decrease = 0.1,
   specifying that b(83000) = 0.1, giving a decrease of 10% after a
   congestion event.  Equation (2) then gives a(83000) = 72, for an
   increase of 72 segments, or just under 0.1%, within a round-trip
   time, for w = 83000.

   This moderate decrease strikes us as acceptable, particularly when
   coupled with the role of TCP's ACK-clocking in limiting the sending
   rate in response to more severe congestion [BBFS01].  A more severe
   decrease would require a more aggressive increase in the congestion
   window for a round-trip time without congestion.  In particular, a
   decrease factor High_Decrease of 0.5, as in Standard TCP, would
   require an increase of 459 segments per round-trip time when w =
   83000.

   Given decrease parameters of b(w) = 1/2 for w = Low_Window, and b(w)
   = High_Decrease for w = High_Window, we are left to specify the value
   of b(w) for other values of w > Low_Window.  From [FRS02], we let
   b(w) vary linearly as the log of w, as follows:

    b(w) = (High_Decrease - 0.5) (log(w)-log(W)) / (log(W_1)-log(W)) +



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   0.5.

   The increase parameter a(w) can then be computed as follows:

    a(w) = w^2 * p(w) * 2 * b(w)/(2-b(w)),

   for p(w) the packet drop rate for congestion window w.  From
   inverting Equation (1), we get p(w) as follows:

    p(w) = 0.078/w^1.2.

   We assume that experimental implementations of HighSpeed TCP for
   further investigation will use a pre-computed look-up table for
   finding a(w) and b(w).  In the appendix we give such a table for our
   default values of Low_Window = 38, High_Window = 83,000, High_P =
   10^-7, and High_Decrease = 0.1.  These are also the default values in
   the NS simulator; example simulations in NS can be run with the
   command "./test-all-tcpHighspeed" in the directory tcl/test.

8.  Slow-Start.

   An companion internet-draft on "Limited Slow-Start for TCP with Large
   Congestion Windows" [F02b] proposes a modification to TCP's slow-
   start procedure that can significantly improve the performance of TCP
   connections slow-starting up to large congestion windows.  For TCP
   connections that are able to use congestion windows of thousands (or
   tens of thousands) of MSS-sized segments (for MSS the sender's
   MAXIMUM SEGMENT SIZE), the current slow-start procedure can result in
   increasing the congestion window by thousands of segments in a single
   round-trip time.  Such an increase can easily result in thousands of
   packets being dropped in one round-trip time.  This is often counter-
   productive for the TCP flow itself, and is also hard on the rest of
   the traffic sharing the congested link.

   [F02b] proposes Limited Slow-Start, limiting the number of segments
   by which the congestion window is increased for one window of data
   during slow-start, in order to improve performance for TCP
   connections with large congestion windows.  We have separated out
   Limited Slow-Start to a separate draft because it can be used both
   with Standard or with HighSpeed TCP.

   Limited Slow-Start is illustrated in the NS simulator, for snapshots
   after May 1, 2002, in the tests "./test-all-tcpHighspeed tcp1A" and
   "./test-all-tcpHighspeed tcpHighspeed1" in the subdirectory
   "tcl/lib".

   In order for best-effort flows to safely start-up faster than slow-
   start, e.g., in future high-bandwidth networks, we believe that it



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   would be necessary for the flow to have explicit feedback from the
   routers along the path.  There are a number of proposals for this,
   ranging from a minimal proposal for an IP option that allows TCP SYN
   packets to collect information from routers along the path about the
   allowed initial window [J02], to proposals with more power that
   require more fine-tuned and continuous feedback from routers.  These
   proposals all are somewhat longer-term proposals that the HighSpeed
   TCP proposal in this document, requiring longer lead times and more
   coordination for deployment, and will be discussed in later
   documents.

9.  Related Work in HighSpeed TCP.

   HighSpeed TCP has been separately investigated in simulations by
   Sylvia Ratnasamy and by Evandro de Souza, and reports of some of
   these simulations should be available shortly.  The simulations by
   Evandro verify the fairness properties of HighSpeed TCP when sharing
   a link with Standard TCP.

   These simulations explore the relative fairness of HighSpeed TCP
   flows when competing with Standard TCP.  The simulation environment
   include background forward and reverse-path TCP traffic limited by
   the TCP receive window, along with a small amount of forward and
   reverse-path traffic from the web traffic generator.  Most of the
   simulations so far explore performance on a simple dumbbell topology
   with a 1Gbps link with a propagation delay of 50 ms.  Simulations
   have been run both the Adaptive RED and with DropTail queue
   management.

   Future work to explore in more detail includes convergence times
   after new flows start-up; recovery time after a transient outage; the
   response to sudden severe congestion, and investigations of the
   potential for oscillations.  Additional future work includes
   evaluating more fully the choices of parameters for HighSpeed TCP.
   We invite contributions from others in this work.

   Suggestions to other citations of related work would also be welcome.

10.  Relationship to other Work.

   Our assumption is that HighSpeed TCP will be used along with the TCP
   SACK option, and also with the increased Initial Window of three or
   four segments, as allowed by [AFP02].  For paths that have
   substantial reordering, TCP performance would be greatly improved by
   some of the mechanisms still in the research stages for robust
   performance in the presence of reordered packets.





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11.  Conclusions.

   This is an initial proposal, and we are asking from feedback from the
   wider community.  We have explored this proposal in simulations,
   though we have not yet finished our reports on these simulations.  We
   would welcome additional analysis, simulations, and particularly,
   experimentation.

   There are three parameters that determine the HighSpeed Response
   Function, and an additional parameter that determines HighSpeed TCP's
   tradeoffs between increases and decreases using that response
   function.  We solicit feedback on our setting of these parameters as
   well as on other issues.

   We also have not exhaustively explored the transient dynamics of
   HighSpeed TCP.

12.  Acknowledgements

   The HighSpeed TCP proposal is from joint work with Sylvia Ratnasamy
   and Scott Shenker.  Additional investigations of HighSpeed TCP were
   joint work with Evandro de Souza and Deb Agarwal.  We are grateful to
   the End-to-End Research Group and to members of the IPAM program in
   Large Scale Communication Networks for feedback, and to contributions
   and feedback from the following individuals: Tom Kelly, Jitendra
   Padhye, Brian Tierney.

13.  Normative References

   [RFC2581] M. Allman and V. Paxson, "TCP Congestion Control", RFC
   2581, April 1999.

14.  Informative References

   [BBFS01] Deepak Bansal, Hari Balakrishnan, Sally Floyd, and Scott
   Shenker, "Dynamic Behavior of Slowly-Responsive Congestion Control
   Algorithms", SIGCOMM 2001, August 2001.

   [FF98] Floyd, S., and Fall, K., "Promoting the Use of End-to-End
   Congestion Control in the Internet", IEEE/ACM Transactions on
   Networking, August 1999.

   [FRS02] Sally Floyd, Sylvia Ratnasamy, and Scott Shenker, "Modifying
   TCP's Congestion Control for High Speeds", May 2002.  URL
   "http://www.icir.org/floyd/notes.html".

   [J02] Amit Jain, "Initial Congestion Window Discovery", rough draft,
   work in progress, 2002.  Citation for acknowledgement purposes only.



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15.  Security Considerations

   This proposal makes no changes to the underlying security of TCP.

16.  IANA Considerations

   There are no IANA considerations regarding this document.

A.  TCP's Loss Event Rate in Steady-State

   This section gives the number of round-trip times between congestion
   events for a TCP flow with D-byte packets, for D=1500, as a function
   of the connection's average throughput B in bps.  To achieve this
   average throughput B, a TCP connection with round-trip time R in
   seconds requires an average congestion window w of BR/(8D) segments.

   In steady-state, TCP's average congestion window w is roughly
   1.2/sqrt(p) segments.  This is equivalent to a lost event at most
   once every 1/p packets, or at most once every 1/(pw) = w/1.5 round-
   trip times.  Substituting for w, this is a loss event at most every
   (BR)/12D)round-trip times.

   An an example, for R = 0.1 seconds and D = 1500 bytes, this gives
   B/180000 round-trip times between loss events.

B.  A table for a(w) and b(w).

   This section gives a table for the increase and decrease parameters
   a(w) and b(w) for HighSpeed TCP, for the default values of Low_Window
   = 38, High_Window = 83000, High_P = 10^-7, and High_Decrease = 0.1.





















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        w  a(w)  b(w)
     ----  ----  ----
       38     1  0.50
      118     2  0.44
      221     3  0.41
      347     4  0.38
      495     5  0.37
      663     6  0.35
      851     7  0.34
     1058     8  0.33
     1284     9  0.32
     1529    10  0.31
     1793    11  0.30
     2076    12  0.29
     2378    13  0.28
     2699    14  0.28
     3039    15  0.27
     3399    16  0.27
     3778    17  0.26
     4177    18  0.26
     4596    19  0.25
     5036    20  0.25
     5497    21  0.24
     5979    22  0.24
     6483    23  0.23
     7009    24  0.23
     7558    25  0.22
     8130    26  0.22
     8726    27  0.22
     9346    28  0.21
     9991    29  0.21
    10661    30  0.21
    11358    31  0.20
    12082    32  0.20
    12834    33  0.20
    13614    34  0.19
    14424    35  0.19
    15265    36  0.19
    16137    37  0.19
    17042    38  0.18
    17981    39  0.18
    18955    40  0.18
    19965    41  0.17
    21013    42  0.17
    22101    43  0.17
    23230    44  0.17
    24402    45  0.16
    25618    46  0.16



Floyd                         Experimental                     [Page 14]


draft-floyd-tcp-highspeed-00                                   June 2002


    26881    47  0.16
    28193    48  0.16
    29557    49  0.15
    30975    50  0.15
    32450    51  0.15
    33986    52  0.15
    35586    53  0.14
    37253    54  0.14
    38992    55  0.14
    40808    56  0.14
    42707    57  0.13
    44694    58  0.13
    46776    59  0.13
    48961    60  0.13
    51258    61  0.13
    53677    62  0.12
    56230    63  0.12
    58932    64  0.12
    61799    65  0.12
    64851    66  0.11
    68113    67  0.11
    71617    68  0.11
    75401    69  0.10
    79517    70  0.10
    84035    71  0.10
    89053    72  0.10
    94717    73  0.09

   Table 4: Parameters for HighSpeed TCP.

   This table was computed with the following Perl program:




















Floyd                         Experimental                     [Page 15]


draft-floyd-tcp-highspeed-00                                   June 2002


    $top = 100000;
    $num = 38;
    if ($num == 38) {
      print "     w  a(w)  b(w)0;
      print "  ----  ----  ----0;
      print "    38     1  0.500;
      $oldb = 0.50;
      $olda = 1;
    }
    while ($num < $top) {
      $bw = (0.1 -0.5)*(log($num)-log(38))/(log(83000)-log(38))+0.5;
      $aw = ($num**2*2.0*$bw) / ((2.0-$bw)*$num**1.2*12.8);
      if ($aw > $olda + 1) {
         printf "%6d %5d  %3.2f0, $num, $aw, $bw;
         $olda = $aw;
      }
      $num ++;
    }

   Table 5: Perl Program for computing parameters for HighSpeed TCP.

   AUTHORS' ADDRESSES


      Sally Floyd
      Phone: +1 (510) 666-2989
      ICIR (ICSI Center for Internet Research)
      Email: floyd@icir.org
      URL: http://www.icir.org/floyd/

      This draft was created in June 2002.




















Floyd                         Experimental                     [Page 16]


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