RFC 2992               Analysis of ECMP Algorithm          November 2000   We now use the the concrete formulas for the sum of integers.  The   first summation is (K)(K-1)/2.  For the second summation notice that   we are summing the integers from 1 to N-K, thus it is (N-K)(N-K+1)/2.                             (K-1)(K) + (N-K)(N-K+1)                           = -----------------------                                   2(N)(N-1)   Considering the summations, one can see that the least disruption is   when K is as close to half way between 1 and N as possible.  This can   be proven by finding the minimum of the concrete formula for K   holding N constant.  First break apart the quantities and collect.                            2K*K - 2K - 2NK + N*N + N                          = -------------------------                                    2(N)(N-1)                             K*K - K - NK      N + 1                          = --------------  + -------                               (N)(N-1)        2(N-1)   Since we are minimizing for K the right side (N+1)/2(N-1) is constant   as is the denominator (N)(N-1) so we can drop them.  To minimize we   take the derivative.                             d                             -- (K*K - (N+1)K)                             dk                             = 2K - (N+1)   Which is zero when K is (N+1)/2.   The last thing to consider is that K must be an integer.  When N is   odd (N+1)/2 will yield an integer, however when N is even (N+1)/2   yields an integer + 1/2.  In the case, because of symmetry, we get   the least disruption when K is N/2 or N/2 + 1.   Now since the formula is quadratic with a global minimum half way   between 1 and N the maximum possible disruption must occur when edge   regions (1 and N) are removed.  If K is 1 or N the formula reduces to   1/2.   The minimum possible disruption is obtained by letting K=(N+1)/2.  In   this case the formula reduces to 1/4 + 1/(4*N).  So the range of   possible disruption is (1/4, 1/2].   To minimize disruption we recommend adding new regions to the center   rather than the ends.Hopps                        Informational                      [Page 5]RFC 2992               Analysis of ECMP Algorithm          November 20003.  Comparison to other algorithms   Other algorithms exist to decide which next-hop to use.  These   algorithms all have different performance and disruptive   characteristics.  Of these algorithms we will only consider ones that   are not disruptive by design (i.e., if no change to the set of next-   hops occurs the path a flow takes remains the same).  This will   exclude round-robin and random choice.  We will look at modulo-N and   highest random weight.   Modulo-N is a "simpler" form of hash-threshold.  Given N next-hops   the packet header fields which describe the flow are run through a   hash function.  A final modulo-N is applied to the output of the   hash.  This result then directly maps to one of the next-hops.   Modulo-N is the most disruptive of the algorithms; if a next-hop is   added or removed the disruption is (N-1)/N.  The performance of   Modulo-N is equivalent to hash-threshold.   Highest random weight (HRW) is a comparative method similar in some   ways to hash-threshold with non-fixed sized regions.  For each next-   hop, the router seeds a pseudo-random number generator with the   packet header fields which describe the flow and the next-hop to   obtain a weight.  The next-hop which receives the highest weight is   selected.  The advantage with using HRW is that it has minimal   disruption (i.e., disruption due to adding or removing a next-hop is   always 1/N.)  The disadvantage with HRW is that the next-hop   selection is more expensive than hash-threshold.  A description of   HRW along with comparisons to other methods can be found in [2].   Although not used for next-hop calculation an example usage of HRW   can be found in [3].   Since each of modulo-N, hash-threshold and HRW require a hash on the   packet header fields which define a flow, we can factor the   performance of the hash out of the comparison.  If the hash can not   be done inexpensively (e.g., in hardware) it too must be considered   when using any of the above methods.   The lookup performance for hash-threshold, like modulo-N is an   optimal O(1).  HRW's lookup performance is O(N).   Disruptive behavior is the opposite of performance.  HRW is best with   1/N.  Hash-threshold is between 1/4 and 1/2.  Finally Modulo-N is   (N-1)/N.   If the complexity of HRW's next-hop selection process is acceptable   we think it should be considered as an alternative to hash-threshold.   This could be the case when, for example, per-flow state is kept and   thus the next-hop choice is made infrequently.Hopps                        Informational                      [Page 6]RFC 2992               Analysis of ECMP Algorithm          November 2000   However, when HRW's next-hop selection is seen as too expensive the   obvious choice is hash-threshold as it performs as well as modulo-N   and is less disruptive.4.  Security Considerations   This document is an analysis of an algorithm used to implement an   ECMP routing decision.  This analysis does not directly affect the   security of the Internet Infrastructure.5.  References   [1]  Thaler, D. and C. Hopps, "Multipath Issues in Unicast and        Multicast", RFC 2991, November 2000.   [2]  Thaler, D. and C.V. Ravishankar, "Using Name-Based Mappings to        Increase Hit Rates", IEEE/ACM Transactions on Networking,        February 1998.   [3]  Estrin, D., Farinacci, D., Helmy, A., Thaler, D., Deering, S.,        Handley, M., Jacobson, V., Liu, C., Sharma, P. and L. Wei,        "Protocol Independent Multicast-Sparse Mode (PIM-SM): Protocol        Specification", RFC 2362, June 1998.6.  Author's Address   Christian E. Hopps   NextHop Technologies, Inc.   517 W. William Street   Ann Arbor, MI 48103-4943   U.S.A   Phone: +1 734 936 0291   EMail: chopps@nexthop.comHopps                        Informational                      [Page 7]RFC 2992               Analysis of ECMP Algorithm          November 20007.  Full Copyright Statement   Copyright (C) The Internet Society (2000).  All Rights Reserved.   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