Network Working Group                                         Alex Zinin
Internet Draft                                                   Alcatel
Expiration Date: January June 2006                                  July                                  October 2005
File name: draft-ietf-rtgwg-microloop-analysis-00.txt draft-ietf-rtgwg-microloop-analysis-01.txt

               Analysis and Minimization of Microloops in
                      Link-state Routing Protocols

               draft-ietf-rtgwg-microloop-analysis-00.txt

               draft-ietf-rtgwg-microloop-analysis-01.txt

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Abstract

   Link-state routing protocols (e.g. OSPF or IS-IS) are known to
   converge to a loop-free state within a finite period of time after a
   change in the topology. It is normal, however, to observe short-term
   loops during the period of topology update propagation, route
   recalculation, and forwarding table update, due to the asynchronous
   nature of link-state protocol operation. This document provides an
   analysis of formation of such microloops and suggests simple
   mechanisms to minimize them.

1 Introduction

   Link-state routing protocols, such as [OSPF] and [ISIS] converge to a
   loop-free state within a finite period of time after a topology
   change. Additional changes postpone the convergence, but do not get
   in its way.

   During the period of convergence, however, link-state protocols
   exhibit short-term routing table inconsistencies caused by the
   protocol's asynchronous nature.  These incornsistencies may cause
   short-term packet loops, also known as microloops. For example, see a
   sample network in Figure 1.

                         +--+    1    +--+
                         |A |---------|B |
                         +--+         +--+
                          |  \  10      |
                         5|   ------    |1
                          |         \   |
                         +--+   10   \+--+
                         |E |---------|C |
                         +--+         +--+
                             \_      /
                            5  \    /1 (failure)
                                +--+
                                |D |
                                +--+

                        Figure 1. Microloop example

   We are interested in routers A and B and their best paths towards D.
   Before failure, B's best path to D is B-C-D with cost 2, and A's best
   path is A-B-C-D with cost 3.  When link C-D fails, both C and D
   announce their link state information with link C-D missing. Within a
   finite period of time, both A and B shall receive the topology
   updates and converge on them, installing new best paths: A-E-D (10)
   for A, and B-A-E-D (11) for B.  However, if, due to the timing
   differences, B calculates and installs its new best path through A
   before A has a chance to switch from B to E, a microloop will form
   between A and B for the duration of time required for A to complete
   its routing table update.

   Similar microloops may form when other topological changes happen in
   the network, for example, when a new link or a node is added, a link
   cost is changed, etc. In summary, whenever a topological change in
   the network results in changes of the shortest path three (SPT) for
   more than one node, it is possible for the network to exhibit
   temporary loops.

   This document provides an analysis of microloop formation.
   Specifically, we categorize different types of reconvergence
   scenarios, and explore their properties. We then show that in certain
   scenaiors
   scenarios microloops do not form, in others they can be eliminated
   using simple techniques described in this document, and define
   scenarios where more sophisticated loop avoidance mechanisms may be
   necessary.

   It is useful to understand the relationship between [IPFRR] and the
   technique described here. The two mechanisms play complimentary roles
   to each other: deploying [IPFRR] without micro-loop prevention only
   partially addresses the goal of minimizing packet loss during network
   reconvergence, since packets will be lost due to microloops. On the
   other hand, micro-loop prevention described in this document relies
   on [IPFRR] local failure protection, as routers will keep forwarding
   traffic down the old path until the new next-hops are known to be
   safe.

2 Analysis

   To analyse the behavior of a network during reconvergence, we look at
   a given router and its neighbors before failure and during the
   transition to the new routes. More specifically, we analyse whether
   switching to the new routing information can result in loop formation
   or not.

2.1 Terminology

   The following terms are used in the draft.

     Dopt(X,Y)
          Integer function defined as the cumulative cost of the least-
          cost path from node X to node Y in a topology graph. Normally
          calculated by link-state routing protocols using Dijkstra
          algorithm as part of regular route calculation procedures.
          This is the same as "Distance_opt(A,B)" defined in [IPFRR-FW]

     Downstream neighbor
          Neighbor N of router S is considered S's downstream neighbor
          for destination D, if Dopt(N, D) < Dopt(S, D)

     Primary neighbor
          Neighbor N of router S is considered S's primary neighbor for
          destination D, if the path via N is such that D(S, D) is mini-
          mized, i.e. N provides the a shortest path to D according to the
          SPF calculation.

     Loop-free neighbor
          Neighbor N of router S is considered S's loop-free neighbor
          for destination D, if Dopt(N, D) < Dopt(N, S) + Dopt(S, D).
          Note that a loop-free neighbor may be, for example, router's
          primary before or and/or after failure.

 2.2 Next hop safety condition

   We start the analysis of single-hop loops with the following observation:

     When router X learns about observa-
   tion:

     After a topology change and starts using
     neighbor Y as its new primary neighbor for a given destination, change, there are precisely two situations when a
     microloop between routers X and Y can only form if the topology before
     failure or topology after failure are such that Y uses form:

     a)   Before failure, X uses Y as its
     primary neighbor for the same destination.

     Indeed, if the topologies before and after failure are such that next-hop. After failure, Y
     does not use
          uses X as it's next hop, then there is no moment in time
     before its next-hop. Y learned about updates its routes based on the failure or after it learned about it
     when it would forward traffic to new
          topology before X. Hence, at least one

     b)   Exact opposite of the two
     topologies must be such that previous case. Before failure, Y uses X
          as its next hop for a
     microloop between next-hop.  After failure, X and uses Y to form.

   Based as its next-hop. X
          updates its routes based on the above, we new topology before Y.

   Formulating this for a given calculating router S (either X or Y in
   the above example) switching to a new primary Pn, a microloop may
   occur between S and Pn only if Pn was forwarding through S before
   failure.

   Based on the above, we can define a general safety condition for any
   neighbor Y N (whether new primary or not) of router X S that has just
   learned about a topology change. Note that the condition must satisfy
   the topological criteria above, and be non-recursive, i.e. not lead
   to loops if both X S and Y N follow it.

     Next-hop safety condition:

          For networks with symmetric link costs, after

          After a topology change, it is safe for router X S to switch to
          neigbor Y N as its next-hop for a specific destination if the
          path through Y sat-
          isfies N satisfies both of the following criteria:

          1.   X   S considered Y N as its loop-free neighbor based on the
               topology before change AND

          2:   X

          2.   S considers Y N as its downstream neighbor based on the
               topology after change.

          The first requirement ensures that Y N has not been forwarding
          traffic to X S before the change occured and both X S and Y N used
          old topology. The second requirement makes sure Y N does not
          forward traffic to X S when Y N learns the new topology. Note
          again, that N is S's any neighbor, and may or may not be used
          by S as its new primary or a temporary safe neighbor.

          The difference in the conditions before and after failure is
          there to make sure that X S and Y N do not recursively consider
          each other as safe next-hops when they learn about the fail-
          ure.

   For networks with asymmetric link costs, the safety condition is mod-
   ified as follows:

          Y is X's downstream neighbor based on the topology both before
          AND after the change.

   Whether a given router uses a the safety condition for symmetric or
   assymetric link costs will affect micro-loop coverage. Generally, the
   stricter condition for asymmetric link costs will result in poorer
   coverage, however using the less strict (symmetric-link) condition in
   networks with asymmetric link costs may result in transformation of
   single-hop loops into multi-hop ones rather than their removal.

   Routers SHOULD use the symmetric-link safety condition by default,
   MAY attempt to dynamically determine the method that needs to be
   applied based on the topological information from the routing
   protocol, and SHOULD provide the administrator an opportunity to man-
   ually override this setting.

 2.3 Transition types

   Here, we analyse different types of scenarios that a given router may
   find itself in after learning about a topology change.

   For each destination affected by a topological change, the network
   will have three major types of nodes categorized by the degree of
   safety of their old primary, new primary, and other neighbors. (Note
   that we do not yet consider ECMP, which will be discussed in section
   3.2.)

     Type A

          Routers whose new primary next-hops after the topology change
          are safe and transition to them will not create a microloop.
          Two subtypes are recognized:

          A1:  Routers whose primaries haven't changed as a result of
               the topology change

          A2:  Routers whose new primary satisfies the safety condition

     Type B

          Routers whose new primary next-hops after the topology change
          do not satisfy the safety condition, but that have at least
          one other neighbor that does. Note that such a neighbor can be
          the router's old primary (type B1) or a neighbor that is nei-
          ther old nor new primary (type B2).

     Type C

          Routers that have no neighbor that satisfies the safety condi-
          tion.

   It is clear that nothing special needs to be done for type-A routers
   as they either do not need to modify their routes or can immediately
   switch to their the new primary next hops once they are calculated after the topology change. hops.

   It can also be shown that if type-B routers do not immediately switch
   to their new primaries, but use their safe next-hops for some time,
   switching to the new primaries later will not create loops, provided
   that later will not create loops, provided
   that their downstream routers have also switched to the safe hops or
   have already switched to the new primaries.

     NOTE: The above analysis applies to single-hop loops. Multi-hop
     loops, possible in networks with asymmetric link costs could be
     prevented by using a tighter safety condition.  However, as shown
     by simulations on real-life network topologies, doing so would
     decrease micro-loop coverage and thus result in increased number of
     unprevented single-hop loops.

   The following section formally defines the mechanism.

3. Loop prevention mechanism

 3.1 Basic procedures

   The essense of the mechanism defined here, also known as "path lock-
   ing via safe neighbors" (PLSN), can be informally summarized as fol-
   lows. Upon a topology change, for each destination:

     -    Each router in the network assesses safety of its new primary
          next-hops.

     -    If the new primaries are safe, they are used immediately, oth-
          erwise, partial
           ordering of updates is introduced:

          o    If non-primary safe neighbors are found, they are used
               for a period of time, thus locking traffic to a safe path
               while the new primaries complete their downstream routers have also switched transition to the
               new routes

          o    If no safe hops or
   have already switched neighbors are found, the forwarding path is
               locked on the old next-hop for some time to give the new primaries.

   The following section formally defines the mechanism.

3. Loop prevention mechanism

 3.1 Basic procedures
               primary enough time to complete route updates.

   For a description of several architectural constants used in this
   document (named as "DELAY_xxx"), refer to section 3.4.

   On receiving a topology update, the router delays its SPF calculation
   by DELAY_SPF time in order to collect the remaining updates that
   relate to the same topological event (e.g. update from the router
   connected to the second end of a point-to-point link). link in case of a
   link failure, or updates from other neighbors of a failed node).

   Upon expiration of DELAY_SPF, the router calculates the new SPT, the
   new routes, checks the safety status of each neighbor relative to
   each affected destination using the con-
   ditions conditions in section 3.1, and
   applies the following logic for each route depending on the type of
   role it finds itself in:

     Type A:
          The route SHALL be updated with the new primary next-hops
          without an additional delay.

     Type B:
          Without an additional delay, the
          The route SHALL be updated with one or more temporary next-hops next-
          hops that satisfy the safety condi-
          tion. condition without an additional
          delay. These temporary next-hops SHALL be used for the duration dura-
          tion of DELAY_TYPEB. After DELAY_TYPEB, the route SHALL be
          updated with the new primary next-hops.

     Type C:
          The route's old (primary) next-hops SHALL continue to be used
          for DELAY_TYPEC.  After DELAY_TYPEC, the route SHALL be
          updated with the new primary next-hops.

   If, after expiration of DELAY_SPF, the router receives a topology
   update sooner than DELAY_STABLE after the previous one, the router
   MUST fall back to the regular convergence mechanisms (immediate
   installation of the new primary next-hops) aborting any transition
   processes initiated as part of procedures described here (i.e., if by prematurely
   expiring DELAY_TYPEB or DELAY_TYPEC timers if they are still running), running
   (thus causing immediate installation of the new primary next-hops),
   MUST recalcu-
   late recalculate its routing table as soon as practical, and MUST
   refrain from using the mechanisms described here until it has seen no
   topological updates for at least DELAY_STABLE. This is a safeguard
   mechanism to ensure that procedures described here are applied only
   when a single failure is experienced and that the network converges
   in a situation where multiple topological events or network instabilities instabil-
   ities are expe-
   rienced. experienced.

   [ISIS] includes the concept of an Overload bit (OL) that indicates a
   node in the network that shouldn't be used as transit. A similar
   notion is introduced in OSPF by [STUB] using LSInfinity link costs.
   Honoring these conventions, implementations of this document MUST NOT
   use neighbors with the OL bit set in IS-IS or announcing links to the
   calculating router with LSInfinity cost in OSPF as temporary safe
   neighbors.

     NOTE: In OSPF, if S's neighbor N is a stub router, the S->N link,
     visited first by the SPF algorithm, will normally have a real link
     cost, and it is the backwards link N->S announced by N that will
     have its cost set to LSInfinity. Implementations have to account
     for this details when satisfying the above requirement.

3.2 Equal Cost Multipath Considerations

   In situations where more than one primary next-hop is available after
   the topology change, there are several possible combination of their
   safety properties:

     1)   All new next-hops satisfy the safery condition (a pure type-A
          situation)

     2)   Some of the new next-hops satisfy the safety condition, some
          of them do not (a combination of type-A and type-B, or type-A
          and type-C) type-B)

     3)   None of the new next-hops satisfy the safety condition, how-
          ever, there's at least one other neighbor that satisfies it (a
          type-B situation)
          safe non-primary next-hop, causing new primaries to be type-B)

     4)   None of the new next-hops satisfy the safety condition, and
          there is no other neighbor that satisfies it (a pure type-C
          situation).

   For situations 1, 3, and 4 above, the implementation merely follows
   the basic procedures described in section 3.1

   For situation 2 (an A/B or an A/C combination), the implementation:

     1)   SHALL update the route with the new next-hops that satisfy the
          safety condition without an additional delay

     2)   SHALL add the remaining new next-hops after DELAY_TYPEB.

3.3 IP Fast Reroute Considerations

   If the router implements [IPFRR] and performs local failure repair,
   procedures describes

   Note that one could potentially use temporary safe neighbors in situ-
   ation 2 above, however this specification does not recommend this document still need to be applied in
   order to prevent micro-loops while reconverging on the new topology.
   avoid unnessesary traffic rerouting and hence packet reordering.

3.3 Local Failure and IP Fast Reroute Considerations

   After detecting a local failure and initiating the local repair, repair
   process if IPFRR is supported, the router directly attached to the
   point of failure follows the procedures described in this docu-
   ment--it delays its SPF calculation to collect updates from other
   routers, calculates new routes, and classifies the next-hops.

   The

   For routers implementing IPFRR, the difference with routers that
   learn about the failure from the routing protocol updates, is that
   one or more of the repairing router's old next-hops has become
   unavailable, and hence cannot be considered as the temporary safe
   next-hops for type-B operation. Also, if the router was able to
   locally repair the failure, and the new primary next-hops do not satisfy sat-
   isfy the safety condition, the router should consider itself in the
   middle of type-B operation with the temporary safe neighbor engaged
   as part of IP Fast Reroute operation.

   Another difference distinct situation is when the router does not support IPFRR
   or could not repair the failure, the new primary next-hops do not
   satisfy the safety condition, and there's no other neighbor that
   does, i.e. a type-C situation. Unlike other routers in the network,
   the router directly connected to the network does not have the old
   next-hop any more, and cannot continue using it. Immediately switch-
   ing to the new next-hops, on the other hand, may result in a micro-
   loop. In this situation, the router MUST revert to discard traffic forwarded
   along the regular
   convergence procedures, affected route for the duration of DELAY_TYPEC, and then
   update the route with routes. Implementations MAY have a configuration option to
   allow switching immediately to the new next-hops
   with no additional delay. for situations where
   this type of a micro-loop is not a concern. If implemented, this
   option MUST be disabled by default.

   As a result, there are the following possible scenarios:

     1)   If the new primary next-hops satisfy the safery condition, the
          router updates the routes without an additional delay.

     2)   Otherwise, if the failure could be repaired locally by IP Fast
          Reroute, the router continues to use the repair path for
          DELAY_TYPEB and updates the routes with the new primary next-
          hops after it expires.

     3)   Otherwise (new next-hops are not safe, and IPFRR is not sup-
          ported or the failure couldn't be repaired), the router reverts to the regular procedures dis-
          cards traffic for DELAY_TYPEC and updates the route routes with the
          new primary next-hops without an additional
          delay. after its expiration.

3.4 Architectural Constants

   The following architectural constants have been used in the descrip-
   tion of the algorithm above:

     DELAY_SPF
          The delay between the moment the router receives a the first
          topology update after a period of stability and the moment it
          starts its routing table recalculation.  This delay is
          necessary to collect multiple updates originated by different
          routers that relate to the same topological event.

     DELAY_STABLE
          Period of time, during which the network topology is consid-
          ered to be stable if the router receives no topological
          updates. When the first update after DELAY_STABLE is received,
          all other updates that fit within DELAY_SPF are considered as
          related to a single topological event.

     DELAY_TYPEB and DELAY_TYPEC
          Periods of time used by the router to delay installation of
          new primary next-hops after a topology change when the router
          has (type-B) or has not (type-C) a safe neighbor to temporary
          divert the traffic to in the meantime.

   While correctness and effectiveness of the algorithm described here
   does not depend on the actual values assigned to the architectural
   constants, it does depend on the relationship between them, and the
   assumption that all routers in the same network use the same values.

   To satisfy these constrains, and yet allow these delays to be
   decreased as implementations continue to improve towards faster con-
   vergence, this document defines the architectural constants as con-
   figurable, specifies the required relationship between the values,
   and the default values that should be used by the implementations.

   The following equations define the relationship between the constants
   that needs to be maintained in order for the mechanism described here
   to provide desireable results:

    DELAY_SPF > update-propagation-time

    DELAY_STABLE > DELAY_TYPEB > DELAY_TYPEC > fault-propagation-time

   where:

     o    update-propagation-time is the time it is expected to take
          routers in the network to detect the failure, and originate
          and propagate new link-state information.

     o    fault-propagation-time is update-propagation plus the time it
          is expected to take routers in the network to calculate the
          new SPT, check the safety condition of the neighbors, and
          install required FIB entries.

   Because fault-propagation-time includes update-propagation-time, and
   DELAY_SPF (since every router will delay its SPF according to this
   document):

    fault-propagation-time > DELAY_SPF + update-propagation-time

   and hence the equations above can be converted to one:

    DELAY_STABLE > DELAY_TYPEB > DELAY_TYPEC > (DELAY_SPF + update-prop-
   agation-time)

   The implementations SHOULD use the following default values for the
   architectural constants:

        Constant                   Default val
       ----------------------------------------
        DELAY_SPF                   500 msec
        DELAY_TYPEC                   2 sec
        DELAY_TYPEB                   4 sec
        DELAY_STABLE                 10 sec

4 Coverage analysis

   The above algorithm minimizes the probability of loop formation. More
   specifically, loops will only be possible when two neighboring
   routers both experience the type C condition after the topology
   change. Appendix A shows that transitions between A-A, A-B, A-C, and
   B-C routers are loop-free.

   While this mechanism does not remove all possible micro-loops, it
   addresses the majority of them in topologies with a reasonable level
   of physical redundancy.  Topologically, micro-loop coverage provided
   by this algorithm is very similar to that provided by [IPFRR]. This
   is due to the fact that similar construct are used by both mecha-
   nisms.

5 Backwards Compatibility Analysis

   Effectiveness of the mechanism described here relies on the assump-
   tion that all routers in the network support it.

   In a situation where some routers do not support the describer mecha-
   nism, the network will continue to converge properly fundamentals of
   the routing system are not changed.  When a topology change event
   occurs in such a network, Type-A and Type-B routers will not substan-
   tially change the convergence patterns, as they will switch to
   routers that are guaranteed to forward traffic correctly after
   DELAY_SPF.  (Note that routers today already implement a delay
   similar to DELAY_SPF.)  Type-C routers, when mixed with routers not
   supporting this mechanism, may induce longer than usual micro-loops
   (up to DELAY_TYPEC), however this delay is in the same order of mag-
   nitude as in most deployed networks today.

6 Security Considerations

   The mechanism described in this document does not modify any routing
   protocol messages, and hence no new threats related to packet modifi-
   cations or replay attacks are introduced. The mechanism changes cer-
   tain delays used in node-local algorithms and introduces partial
   event ordering after a topology change has occured. This, however,
   does not introduce new security risks. For type-B situations, traffic
   to certain destinations can be temporarily routed via next-hop
   routers that would not be used with the same topology change if this
   mechanism wasn't employed. However, these next-hop routers can be
   used anyway when a different topological change occurs, and hence
   this can't be viewed as a new security threat.

7 Acknowledgements

   The author would like to thank Don Fedyk, Chris Martin, Mike Shand, Alex Audu,
   Olivier Bonaventure, Stefano Previdi, and other members of the IETF
   RTGWG for their useful comments. Special thanks go to Alia
   Atlas who, among other things, was Atlas,
   Mike Shand, and Steward Bryant, who were instrumental in development
   of this mechanism, such as fine-tuning the safety condition. condition, simulat-
   ing the mechanism, proof-reading the document, and without whom this
   work wouldn't be possible.

8 References

 8.1 Normative References

   [OSPF] J. Moy. OSPF version 2. Technical Report RFC 2328, Internet
          Engineering Task Force, 1998.

   [ISIS] ISO, "Intermediate system to Intermediate system routeing
          information exchange protocol for use in conjunction with the
          Protocol for providing the Connectionless-mode Network Service
          (ISO 8473)," ISO/IEC 10589:1992.

   [IPFRR] Atlas, A., "Basic Specification for IP Fast-Reroute:
          Loop-free Alternates", Internet Engineering Task Force, Work
          in Progress, draft-ietf-rtgwg-ipfrr-spec-base-03.txt

 8.2 Informative References

   [IPFRR-FW] Shand, M., S. Bryant, "IP Fast Reroute Framework",
          Internet Engineering Task Force, Work in Progress, draft-ietf-
          rtgwg-ipfrr-framework-04.txt

   [STUB] Retana, A., et al, OSPF Stub Router Advertisement, RFC 3137,
          Internet Engineering Task Force, 2001.

Author's Address

    Alex Zinin
    Alcatel
    701 E Middlefield Rd
    Mountain View, CA 94043
    E-mail: zinin@psg.com

Appendix A. Loop formation analysis

   S is the calculating router discovering the failure through a link-state
   update. P is the old primary, NP is the new primary.

    BF:
                        <------
            [P]----------------[S]----------------[NP]
               ...>?

    AF:

                                  ------>
            [P]----------------[S]----------------[NP]
                                              ?<...

   To analyze possible loop formation, we need to check the following:

     1)   if it is possible for P to start forwarding packets to S
          before S switches to NP

     2)   if it is possible for NP to be forwarding packets back to S
          before or after S starts using it

   Assumptions are that type-As switch-over to NP immediately, and type-
   Bs and type-Cs wait certain amount of time so that:

      DELAY_TYPEB > DELAY_TYPEC > fault-propagation-time

   1. S is type A:

   BF analysis:

     1.1 If P is another type-A, then S cannot be its new primary, since
     S has not been P's LFA before (since it's been fwd'ing through P).
     Hence, P will not route through S AF, and the will be no loops
     between P and S.

     1.2 If P is a type-B, then S hasn't been P's LF neighbor BF, and P
     will not forward through S at least for DELAY_TYPEB, which gives S
     enough time to switch to NP. After DELAY_TYPEB P may start using S
     as it's new primary.

     1.3 If P is a type-C, then it hasn't been forwarding traffic to S
     BF, and will not use S as its new primary at least for DELAY_TYPEC,
     which should give S enough time to switch to NP.

     1.4 Consequently, no loops will form between a type-A node and it's
     old primary before the type-A nodes switches to its new primary.

   AF analysis:

     1.5 Regardless of its type, NP has not been forwarding packets to S
     BF and will not do so AF by definition of type-A.

     1.6 Consequently, no loops will form between a type-A node and it's
     new primary before or after the type-A nodes switches to it.

   2. S is type B:

   BF analysis:

     2.1 If P is a type-A, then similarly to 1.1 above, there will be no
     routes between P and S.

     2.2 If P is another type-B, then similarly to 1.2, S will not be
     used by P for at least DELAY_TYPEB, and S will have enough time to
     switch to its safe hops or NP.

     2.3 If P is a type-C, then similarly to 1.3, S hasn't been receiv-
     ing traffic from P BF, and will not AF for at least DELAY_TYPEC,
     which should give S enough time to switch to its safe hops or NP.

     2.4 Consequently, no loops will form between a type-B node and it's
     old primary before the type-B nodes switches to its new primary.

   AF analysis:

     2.5 If NP is a type-A, then because of the DELAY_TYPEB NP must have
     had enough time to switch to its new NP, which cannot be S by defi-
     nition of SPT considering that NP is S's new nexthop in the SPT AF.

     2.6 If NP is another type-B, then because of DELAY_TYPEB, NP must
     have had enough time to switch from its old primary and can equally
     likely be routing through either its safe hops, or its new primary.
     Neither of the two can be S by definition of a downstream node (for
     safe hops) and SPT (for new primary).

     2.7 If NP is a type-C, then because DELAY_TYPEB > DELAY_TYPEC, NP
     must have had enough time to switch to its new primary, which can't
     be S by definition of SPT and considering that NP is S's nexthop in
     the SPT AF.

     2.8 Consequently, no loops will form between a type-B node and it's
     new primary before or after the type-A nodes switches to it.

   3. S is type C:

   BF analysis:

     3.1 If P is a type-A, then similarly to 1.1 before, S has not been
     P's LF neighbor before and hence won't be its new primary, so no
     loops will form between P and S.

     3.2 If P is a type-B, then similarly to 1.2, S will not be used by
     P for at least DELAY_TYPEB, and because DELAY_TYPEB > DELAY_TYPEC,
     S will have enough time to switch to NP.

     3.3 If P is another type-C, then it hasn't been using S as its pri-
     mary BF, but it is possible for P to consider S as its new primary
     AF and to install routes before S after their DELAY_TYPEC expires.
     Hence, a microloop is possible between P and S.

     3.4 Consequently, a microloop between a type-C node and its old
     primary is possible only if the old primary is also a type-C node
     and it considers S as its new primary AF. Note that DELAY_TYPEC
     only delays probably loop formation, but does not increase its
     duration, as both neighboring routers are using the same delay.

   AF analysis:

     3.5 If NP is a type-A, then because of the DELAY_TYPEC NP must have
     had enough time to switch to its new NP, which cannot be S by defi-
     nition of SPT considering that NP is S's new nexthop in the SPT AF.

     3.6 If NP is a type-B, then because of DELAY_TYPEC, NP must have
     had enough time to switch to its safe hops, which can't be S by
     definition of a downstream node and considering that NP is S's new
     SPT next-hop.

     3.7 If NP is another type-C, a loop is possible if S's DELAY_TYPEC
     expires before that on NP and NP has been using S as its primary
     BF.

     3.8 Consequently, a microloop between a type-C node and its new
     primary is possible only if the new primary is also a type-C node
     and S was NP's primary BF.

   4. Given the above analysis, it can be noted that, for a given fail-
   ure, presence of single type-C nodes in the network does not create
   microloops.
    It is the C-C combination that introduces this potential.

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