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+Network Working Group V. Fuller
+Request for Comments: 1519 BARRNet
+Obsoletes: 1338 T. Li
+Category: Standards Track cisco
+ J. Yu
+ MERIT
+ K. Varadhan
+ OARnet
+ September 1993
+
+
+ Classless Inter-Domain Routing (CIDR):
+ an Address Assignment and Aggregation Strategy
+
+Status of this Memo
+
+ This RFC specifies an Internet standards track protocol for the
+ Internet community, and requests discussion and suggestions for
+ improvements. Please refer to the current edition of the "Internet
+ Official Protocol Standards" for the standardization state and status
+ of this protocol. Distribution of this memo is unlimited.
+
+Abstract
+
+ This memo discusses strategies for address assignment of the existing
+ IP address space with a view to conserve the address space and stem
+ the explosive growth of routing tables in default-route-free routers.
+
+Table of Contents
+
+ Acknowledgements ................................................. 2
+ 1. Problem, Goal, and Motivation ................................ 2
+ 2. CIDR address allocation ...................................... 3
+ 2.1 Aggregation and its limitations ............................. 3
+ 2.2 Distributed network number allocation ....................... 5
+ 3. Cost-benefit analysis ........................................ 6
+ 3.1 Present allocation figures .................................. 7
+ 3.2 Historic growth rates ....................................... 8
+ 3.3 Detailed analysis ........................................... 8
+ 3.3.1 Benefits of new addressing plan ........................... 9
+ 3.3.2 Growth rate projections ................................... 9
+ 4. Changes to inter-domain routing protocols and practices ...... 11
+ 4.1 Protocol-independent semantic changes ....................... 11
+ 4.2 Rules for route advertisement ............................... 11
+ 4.3 How the rules work .......................................... 13
+ 4.4 Responsibility for and configuration of aggregation ......... 14
+ 4.5 Intra-domain protocol considerations ........................ 15
+ 5. Example of new allocation and routing ........................ 15
+
+
+
+Fuller, Li, Yu & Varadhan [Page 1]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ 5.1 Address allocation .......................................... 15
+ 5.2 Routing advertisements ...................................... 17
+ 6. Extending CIDR to class A addresses .......................... 18
+ 7. Domain Naming Service considerations ......................... 20
+ 7.1 Procedural changes for class-C "supernets" ................... 20
+ 7.2 Procedural changes for class-A subnetting .................... 21
+ 8. Transitioning to a long term solution ........................ 22
+ 9. Conclusions .................................................. 22
+ 10. Recommendations ............................................. 22
+ 11. References .................................................. 23
+ 12. Security Considerations ..................................... 23
+ 13. Authors' Addresses .......................................... 24
+
+Acknowledgements
+
+ The authors wish to express their appreciation to the members of the
+ ROAD group with whom many of the ideas contained in this document
+ were inspired and developed.
+
+1. Problem, Goal, and Motivation
+
+ As the Internet has evolved and grown over in recent years, it has
+ become evident that it is soon to face several serious scaling
+ problems. These include:
+
+ 1. Exhaustion of the class B network address space. One
+ fundamental cause of this problem is the lack of a network
+ class of a size which is appropriate for mid-sized
+ organization; class C, with a maximum of 254 host
+ addresses, is too small, while class B, which allows up to
+ 65534 addresses, is too large for most organizations.
+
+ 2. Growth of routing tables in Internet routers beyond the
+ ability of current software, hardware, and people to
+ effectively manage.
+
+ 3. Eventual exhaustion of the 32-bit IP address space.
+
+ It has become clear that the first two of these problems are likely
+ to become critical within the next one to three years. This memo
+ attempts to deal with these problems by proposing a mechanism to slow
+ the growth of the routing table and the need for allocating new IP
+ network numbers. It does not attempt to solve the third problem,
+ which is of a more long-term nature, but instead endeavors to ease
+ enough of the short to mid-term difficulties to allow the Internet to
+ continue to function efficiently while progress is made on a longer-
+ term solution.
+
+
+
+
+Fuller, Li, Yu & Varadhan [Page 2]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ The proposed solution is to topologically allocate future IP address
+ assignment, by allocating segments of the IP address space to the
+ transit routing domains.
+
+ This plan for allocating IP addresses should be undertaken as soon as
+ possible. We believe that this will suffice as a short term
+ strategy, to fill the gap between now and the time when a viable long
+ term plan can be put into place and deployed effectively. This plan
+ should be viable for at least three (3) years, after which time,
+ deployment of a suitable long term solution is expected to occur.
+
+ This plan is primarily directed at the first two problems listed
+ above. We believe that the judicious use of variable-length
+ subnetting techniques should help defer the onset of the last problem
+ problem, the exhaustion of the 32-bit address space. Note also that
+ improved tools for performing address allocation in a "supernetted"
+ and variably-subnetted world would greatly help the user community in
+ accepting these sometimes confusing techniques. Efforts to create
+ some simple tools for this purpose should be encouraged by the
+ Internet community.
+
+ Note that this plan neither requires nor assumes that already
+ assigned addresses will be reassigned, though if doing so were
+ possible, it would further reduce routing table sizes. It is assumed
+ that routing technology will be capable of dealing with the current
+ routing table size and with some reasonably small rate of growth.
+ The emphasis of this plan is on significantly slowing the rate of
+ this growth.
+
+ Note that this plan does not require domains to renumber if they
+ change their attached transit routing domain. Domains are encouraged
+ to renumber so that their individual address allocations do not need
+ to be advertised.
+
+ This plan will not affect the deployment of any specific long term
+ plan, and therefore, this document will not discuss any long term
+ plans for routing and address architectures.
+
+2. CIDR address allocation
+
+ There are two basic components of this addressing and routing plan:
+ one, to distribute the allocation of Internet address space and two,
+ to provide a mechanism for the aggregation of routing information.
+
+ 2.1 Aggregation and its limitations
+
+ One major goal of this addressing plan is to allocate Internet
+ address space in such a manner as to allow aggregation of routing
+
+
+
+Fuller, Li, Yu & Varadhan [Page 3]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ information along topological lines. For simple, single-homed
+ clients, the allocation of their address space out of a transit
+ routing domain's space will accomplish this automatically - rather
+ than advertise a separate route for each such client, the transit
+ domain may advertise a single aggregate route which describes all of
+ the destinations connected to it. Unfortunately, not all sites are
+ singly-connected to the network, so some loss of ability to aggregate
+ is realized for the non-trivial cases.
+
+ There are two situations that cause a loss of aggregation efficiency.
+
+ o Organizations which are multi-homed. Because multi-homed
+ organizations must be advertised into the system by each of
+ their service providers, it is often not feasible to
+ aggregate their routing information into the address space
+ any one of those providers. Note that they still may receive
+ their address allocation out of a transit domain's address
+ space (which has other advantages), but their routing
+ information must still be explicitly advertised by most of
+ their service providers (the exception being that if the
+ site's allocation comes out of its least-preferable service
+ provider, then that service provider need not advertise the
+ explicit route - longest-match will insure that its
+ aggregated route is used to get to the site on a backup
+ basis). For this reason, the routing cost for these
+ organizations will typically be about the same as it is
+ today.
+
+ o Organizations which change service provider but do not
+ renumber. This has the effect of "punching a hole" in the
+ aggregation of the original service provider's advertisement.
+ This plan will handle the situation by requiring the newer
+ service provider to advertise a specific advertisement for
+ the new client, which is preferred by virtue of being the
+ longest match. To maintain efficiency of aggregation, it is
+ recommended that organizations which do change service
+ providers plan to eventually migrate their address
+ assignments from the old provider's space to that of the new
+ provider. To this end, it is recommended that mechanisms to
+ facilitate such migration, including improved protocols and
+ procedures for dynamic host address assignment, be developed.
+
+ Note that some aggregation efficiency gain can still be had for
+ multi-homed sites (and, in general, for any site composed of
+ multiple, logical IP network numbers) - by allocating a contiguous
+ power-of-two block of network numbers to the client (as opposed to
+ multiple, independent network numbers) the client's routing
+ information may be aggregated into a single (net, mask) pair. Also,
+
+
+
+Fuller, Li, Yu & Varadhan [Page 4]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ since the routing cost associated with assigning a multi-homed site
+ out of a service provider's address space is no greater than the
+ current method of a random allocation by a central authority, it
+ makes sense to allocate all address space out of blocks assigned to
+ service providers.
+
+ It is also worthwhile to mention that since aggregation may occur at
+ multiple levels in the system, it may still be possible to aggregate
+ these anomalous routes at higher levels of whatever hierarchy may be
+ present. For example, if a site is multi-homed to two NSFNET regional
+ networks both of whom obtain their address space from the NSFNET,
+ then aggregation by the NSFNET of routes from the regionals will
+ include all routes to the multi-homed site.
+
+ Finally, it should also be noted that deployment of the new
+ addressing plan described in this document may (and should) begin
+ almost immediately but effective use of the plan to aggregate routing
+ information will require changes to some Inter-Domain routing
+ protocols. Likewise, deploying classless Inter-Domain protocols
+ without deployment of the new address plan will not allow useful
+ aggregation to occur (in other words, the addressing plan and routing
+ protocol changes are both required for supernetting, and its
+ resulting reduction in table growth, to be effective.) Note,
+ however, that during the period of time between deployment of the
+ addressing plan and deployment of the new protocols, the size of
+ routing tables may temporarily grow very rapidly. This must be
+ considered when planning the deployment of the two plans.
+
+ Note: in the discussion and examples which follow, the network and
+ mask notation is used to represent routing destinations. This is used
+ for illustration only and does not require that routing protocols use
+ this representation in their updates.
+
+ 2.2 Distributed allocation of address space
+
+ The basic idea of the plan is to allocate one or more blocks of Class
+ C network numbers to each network service provider. Organizations
+ using the network service provider for Internet connectivity are
+ allocated bitmask-oriented subsets of the provider's address space as
+ required.
+
+ It is also worthwhile to mention that once inter-domain protocols
+ which support classless network destinations are widely deployed, the
+ rules described by this plan generalize to permit arbitrary
+ super/subnetting of the remaining class A and class B address space
+ (the assumption being that classless inter-domain protocols will
+ either allow for non-contiguous subnets to exist in the system or
+ that all components of a sub-allocated class A/B will be contained
+
+
+
+Fuller, Li, Yu & Varadhan [Page 5]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ within a single routing domain). This will allow this plan to
+ continue to be used in the event that the class C space is exhausted
+ before implementation of a long-term solution is deployed. This
+ alternative is discussed further below in section 6.
+
+ Hierarchical sub-allocation of addresses in this manner implies that
+ clients with addresses allocated out of a given service provider are,
+ for routing purposes, part of that service provider and will be
+ routed via its infrastructure. This implies that routing information
+ about multi-homed organizations, i.e., organizations connected to
+ more than one network service provider, will still need to be known
+ by higher levels in the hierarchy.
+
+ The advantages of hierarchical assignment in this fashion are
+
+ a) It is expected to be easier for a relatively small number of
+ service providers to obtain addresses from the central
+ authority, rather than a much larger, and monotonically
+ increasing, number of individual clients. This is not to be
+ considered as a loss of part of the service providers' address
+ space.
+
+ b) Given the current growth of the Internet, a scalable and
+ delegatable method of future allocation of network numbers has
+ to be achieved.
+
+ For these reasons, and in the interest of providing a consistent
+ procedure for obtaining Internet addresses, it is recommended that
+ most, if not all, network numbers be distributed through service
+ providers. These issues are discussed in much greater length in [2].
+
+3. Cost-benefit analysis
+
+ This new method of assigning address through service providers can be
+ put into effect immediately and will, from the start, have the
+ benefit of distributing the currently centralized process of
+ assigning new addresses. Unfortunately, before the benefit of
+ reducing the size of globally-known routing destinations can be
+ achieved, it will be necessary to deploy an Inter-Domain routing
+ protocol capable of handling arbitrary network and mask pairs. Only
+ then will it be possible to aggregate individual class C networks
+ into larger blocks represented by single routing table entries.
+
+ This means that upon introduction, the new addressing allocation plan
+ will not in and of itself help solve the routing table size problem.
+ Once the new Inter-Domain routing protocol is deployed, however, an
+ immediate drop in the number of destinations which clients of the new
+ protocol must carry will occur. A detailed analysis of the magnitude
+
+
+
+Fuller, Li, Yu & Varadhan [Page 6]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ of this expected drop and the permanent reduction in rate of growth
+ is given in the next section.
+
+ In should also be noted that the present method of flat address
+ allocations imposes a large bureaucratic cost on the central address
+ allocation authority. For scaling reasons unrelated to address space
+ exhaustion or routing table overflow, this should be changed. Using
+ the mechanism proposed in this paper will have the fortunate side
+ effect of distributing the address allocation procedure, greatly
+ reducing the load on the central authority.
+
+ 3.1 Present Allocation Figures
+
+ An informal analysis of "network-contacts.txt" (available from the
+ DDN NIC) indicates that as of 2/25/92, 46 of 126 class A network
+ numbers have been allocated (leaving 81) and 5467 of 16382 class B
+ numbers have been allocated, leaving 10915. Assuming that recent
+ trends continue, the number of allocated class B's will continue to
+ double approximately once a year. At this rate of growth, all class
+ B's will be exhausted within about 15 months. As of 1/13/93, 52
+ class A network numbers have been allocated and 7133 class B's have
+ been allocated. We suggest that the change in the class B allocation
+ rate is due to the initial deployment of this address allocation
+ plan.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Fuller, Li, Yu & Varadhan [Page 7]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ 3.2 Historic growth rates
+
+ MM/YY ROUTES MM/YY ROUTES
+ ADVERTISED ADVERTISED
+ ------------------------ -----------------------
+ Dec-92 8561 Sep-90 1988
+ Nov-92 7854 Aug-90 1894
+ Oct-92 7354 Jul-90 1727
+ Sep-92 6640 Jun-90 1639
+ Aug-92 6385 May-90 1580
+ Jul-92 6031 Apr-90 1525
+ Jun-92 5739 Mar-90 1038
+ May-92 5515 Feb-90 997
+ Apr-92 5291 Jan-90 927
+ Mar-92 4976 Dec-89 897
+ Feb-92 4740 Nov-89 837
+ Jan-92 4526 Oct-89 809
+ Dec-91 4305 Sep-89 745
+ Nov-91 3751 Aug-89 650
+ Oct-91 3556 Jul-89 603
+ Sep-91 3389 Jun-89 564
+ Aug-91 3258 May-89 516
+ Jul-91 3086 Apr-89 467
+ Jun-91 2982 Mar-89 410
+ May-91 2763 Feb-89 384
+ Apr-91 2622 Jan-89 346
+ Mar-91 2501 Dec-88 334
+ Feb-91 2417 Nov-88 313
+ Jan-91 2338 Oct-88 291
+ Dec-90 2190 Sep-88 244
+ Nov-90 2125 Aug-88 217
+ Oct-90 2063 Jul-88 173
+
+ Table I : Growth in routing table size, total numbers
+ Source for the routing table size data is MERIT
+
+ 3.3 Detailed Analysis
+
+ There is a small technical cost and minimal administrative cost
+ associated with deployment of the new address assignment plan. The
+ administrative cost is basically that of convincing the NIC, the
+ IANA, and the network service providers to agree to this plan, which
+ is not expected to be too difficult. In addition, administrative
+ cost for the central numbering authorities (the NIC and the IANA)
+ will be greatly decreased by the deployment of this plan. To take
+ advantage of aggregation of routing information, however, it is
+ necessary that the capability to represent routes as arbitrary
+ network and mask fields (as opposed to the current class A/B/C
+
+
+
+Fuller, Li, Yu & Varadhan [Page 8]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ distinction) be added to the common Internet inter-domain routing
+ protocol(s). Thus, the technical cost is in the implementation of
+ classless interdomain routing protocols.
+
+ 3.3.1 Benefits of the new addressing plan
+
+ There are two benefits to be had by deploying this plan:
+
+ o The current problem with depletion of the available class B
+ address space can be ameliorated by assigning more-
+ appropriately sized blocks of class C's to mid-sized
+ organizations (in the 200-4000 host range).
+
+ o When the improved inter-domain routing protocol is deployed,
+ an immediate decrease in the number routing table entries
+ should occur, followed by a significant reduction in the rate
+ growth of routing table size (for default-free routers).
+
+ 3.3.2 Growth rate projections
+
+ As of Jan '92, a default-free routing table (for example, the routing
+ tables maintained by the routers in the NSFNET backbone) contained
+ approximately 4700 entries. This number reflects the current size of
+ the NSFNET routing database. Historic data shows that this number, on
+ average, has doubled every 10 months between 1988 and 1991. Assuming
+ that this growth rate is going to persist in the foreseeable future
+ (and there is no reason to assume otherwise), we expect the number of
+ entries in a default-free routing table to grow to approximately
+ 30000 in two years time. In the following analysis, we assume that
+ the growth of the Internet has been, and will continue to be,
+ exponential.
+
+ It should be stressed that these projections do not consider that the
+ current shortage of class B network numbers may increase the number
+ of instances where many class C's are used rather than a class B.
+ Using an assumption that new organizations which formerly obtained
+ class B's will now obtain somewhere between 4 and 16 class C's, the
+ rate of routing table growth can conservatively be expected to at
+ least double and probably quadruple. This means the number of entries
+ in a default-free routing table may well exceed 10,000 entries within
+ six months and 20,000 entries in less than a year.
+
+ As of Dec '92, the routing table contains 8500 routes. The original
+ growth curves would predict over 9400 routes. At this time, it is
+ not clear if this would indicate a significant change in the rate of
+ growth.
+
+ Under the proposed plan, growth of the routing table in a default-
+
+
+
+Fuller, Li, Yu & Varadhan [Page 9]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ free router is greatly reduced since most new address assignment will
+ come from one of the large blocks allocated to the service providers.
+ For the sake of this analysis, we assume prompt implementation of
+ this proposal and deployment of the revised routing protocols. We
+ make the initial assumption that any initial block given to a
+ provider is sufficient to satisfy its needs for two years.
+
+ Since under this plan, multi-homed networks must continue to be
+ explicitly advertised throughout the system (according to Rule #1
+ described in section 4.2), the number multi-homed routes is expected
+ to be the dominant factor in future growth of routing table size,
+ once the supernetting plan is applied.
+
+ Presently, it is estimated that there are fewer than 100 multi-homed
+ organizations connected to the Internet. Each such organization's
+ network is comprised of one or more network numbers. In many cases
+ (and in all future cases under this plan), the network numbers used
+ by an organization are consecutive, meaning that aggregation of those
+ networks during route advertisement may be possible. This means that
+ the number of routes advertised within the Internet for multi-homed
+ networks may be approximated as the total number of multi-homed
+ organizations. Assuming that the number of multi-homed organization
+ will double every year (which may be a over-estimation, given that
+ every connection costs money), the number of routes for multi-homed
+ networks would be expected to grow to approximately 800 in three
+ years.
+
+ If we further assume that there are approximately 100 service
+ providers, then each service provider will also need to advertise its
+ block of addresses. However, due to aggregation, these
+ advertisements will be reduced to only 100 additional routes. We
+ assume that after the initial two years, new service providers
+ combined with additional requests from existing providers will
+ require an additional 50 routes per year. Thus, the total is 4700 +
+ 800 + 150 = 5650. This represents an annual growth rate of
+ approximately 6%. This is in clear contrast to the current annual
+ growth of 130%. This analysis also assumes an immediate deployment
+ of this plan with full compliance. Note that this analysis assumes
+ only a single level of route aggregation in the current Internet -
+ intelligent address allocation should significantly improve this.
+
+ Clearly, this is not a very conservative assumption in the Internet
+ environment nor can 100% adoption of this proposal be expected.
+ Still, with only a 90% participation in this proposal by service
+ providers, at the end of the target three years, global routing table
+ size will be "only" 4700 + 800 + 145 + 7500 = 13145 routes -- without
+ any action, the routing table will grow to approximately 75000 routes
+ during that time period.
+
+
+
+Fuller, Li, Yu & Varadhan [Page 10]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+4. Changes to inter-domain routing protocols and practices
+
+ In order to support supernetting efficiently, it is clear that some
+ changes will need to be made to both routing protocols themselves and
+ to the way in which routing information is interpreted. In the case
+ of "new" inter-domain protocols, the actual protocol syntax changes
+ should be relatively minor. This mechanism will not work with older
+ inter-domain protocols such as EGP2; the only ways to interoperate
+ with old systems using such protocols are either to use existing
+ mechanisms for providing "default" routes or b) require that new
+ routers talking to old routers "explode" supernet information into
+ individual network numbers. Since the first of these is trivial
+ while the latter is cumbersome (at best -- consider the memory
+ requirements it imposes on the receiver of the exploded information),
+ it is recommended that the first approach be used -- that older
+ systems to continue to the mechanisms they currently employ for
+ default handling.
+
+ Note that a basic assumption of this plan is that those organizations
+ which need to import "supernet" information into their routing
+ systems must run IGPs (such as OSPF [1]) which support classless
+ routes. Systems running older IGPs may still advertise and receive
+ "supernet" information, but they will not be able to propagate such
+ information through their routing domains.
+
+ 4.1 Protocol-independent semantic changes
+
+ There are two fundamental changes which must be applied to Inter-
+ Domain routing protocols in order for this plan to work. First, the
+ concept of network "class" needs to be deprecated - this plan assumes
+ that routing destinations are represented by network and mask pairs
+ and that routing is done on a longest-match basis (i.e., for a given
+ destination which matches multiple network+mask pairs, the match with
+ the longest mask is used). Second, current inter-domain protocols
+ generally do not support the concept of route aggregation, so the new
+ semantics need to be implemented in a new set of inter-domain
+ protocols. In particular, when doing aggregation, dealing with
+ multi-homed sites or destinations which change service providers is
+ difficult. Fortunately, it is possible to define several fairly
+ simple rules for dealing with such cases.
+
+ 4.2. Rules for route advertisement
+
+ 1. Routing to all destinations must be done on a longest-match
+ basis only. This implies that destinations which are multi-
+ homed relative to a routing domain must always be explicitly
+ announced into that routing domain - they cannot be summarized
+ (this makes intuitive sense - if a network is multi-homed, all
+
+
+
+Fuller, Li, Yu & Varadhan [Page 11]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ of its paths into a routing domain which is "higher" in the
+ hierarchy of networks must be known to the "higher" network).
+
+ 2. A routing domain which performs summarization of multiple
+ routes must discard packets which match the summarization but
+ do not match any of the explicit routes which makes up the
+ summarization. This is necessary to prevent routing loops in
+ the presence of less-specific information (such as a default
+ route). Implementation note - one simple way to implement
+ this rule would be for the border router to maintain a "sink"
+ route for each of its aggregations. By the rule of longest
+ match, this would cause all traffic destined to components of
+ the aggregation which are not explicitly known to be
+ discarded.
+
+ Note that during failures, partial routing of traffic to a site which
+ takes its address space from one service provider but which is
+ actually reachable only through another (i.e., the case of a site
+ which has change service providers) may occur because such traffic
+ will be routed along the path advertised by the aggregated route.
+ Rule #2 will prevent any real problem from occurring by forcing such
+ traffic to be discarded by the advertiser of the aggregated route,
+ but the output of "traceroute" and other similar tools will suggest
+ that a problem exists within the service provider advertising the
+ aggregate, which may be confusing to network operators (see the
+ example in section 5.2 for details). Solutions to this problem appear
+ to be challenging and not likely to be implementable by current
+ Inter-Domain protocols within the time-frame suggested by this
+ document. This decision may need to be revisited as Inter-Domain
+ protocols evolve.
+
+ An implementation following these rules should also be generalized,
+ so that an arbitrary network number and mask are accepted for all
+ routing destinations. The only outstanding constraint is that the
+ mask must be left contiguous. Note that the degenerate route 0.0.0.0
+ mask 0.0.0.0 is used as a default route and MUST be accepted by all
+ implementations. Further, to protect against accidental
+ advertisements of this route via the inter-domain protocol, this
+ route should never be advertised unless there is specific
+ configuration information indicating to do so.
+
+ Systems which process route announcements must also be able to verify
+ that information which they receive is correct. Thus, implementations
+ of this plan which filter route advertisements must also allow masks
+ in the filter elements. To simplify administration, it would be
+ useful if filter elements automatically allowed more specific network
+ numbers and masks to pass in filter elements given for a more general
+ mask. Thus, filter elements which looked like:
+
+
+
+Fuller, Li, Yu & Varadhan [Page 12]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ accept 128.32.0.0
+ accept 128.120.0.0
+ accept 134.139.0.0
+ deny 36.2.0.0
+ accept 36.0.0.0
+
+ would look something like:
+
+ accept 128.32.0.0 255.255.0.0
+ accept 128.120.0.0 255.255.0.0
+ accept 134.139.0.0 255.255.0.0
+ deny 36.2.0.0 255.255.0.0
+ accept 36.0.0.0 255.0.0.0
+
+ This is merely making explicit the network mask which was implied by
+ the class A/B/C classification of network numbers.
+
+ 4.3. How the rules work
+
+ Rule #1 guarantees that the routing algorithm used is consistent
+ across implementations and consistent with other routing protocols,
+ such as OSPF. Multi-homed networks are always explicitly advertised
+ by every service provider through which they are routed even if they
+ are a specific subset of one service provider's aggregate (if they
+ are not, they clearly must be explicitly advertised). It may seem as
+ if the "primary" service provider could advertise the multi-homed
+ site implicitly as part of its aggregate, but the assumption that
+ longest-match routing is always done causes this not to work.
+
+ Rule #2 guarantees that no routing loops form due to aggregation.
+ Consider a mid-level network which has been allocated the 2048 class
+ C networks starting with 192.24.0.0 (see the example in section 5 for
+ more on this). The mid-level advertises to a "backbone"
+ 192.24.0.0/255.248.0.0. Assume that the "backbone", in turn, has been
+ allocated the block of networks 192.0.0.0/255.0.0.0. The backbone
+ will then advertise this aggregate route to the mid-level. Now, if
+ the mid-level loses internal connectivity to the network
+ 192.24.1.0/255.255.255.0 (which is part of its aggregate), traffic
+ from the "backbone" to the mid-level to destination 192.24.1.1 will
+ follow the mid-level's advertised route. When that traffic gets to
+ the mid-level, however, the mid-level *must not* follow the route
+ 192.0.0.0/255.0.0.0 it learned from the backbone, since that would
+ result in a routing loop. Rule #2 says that the mid-level may not
+ follow a less-specific route for a destination which matches one of
+ its own aggregated routes. Note that handling of the "default" route
+ (0.0.0.0/0.0.0.0) is a special case of this rule - a network must not
+ follow the default to destinations which are part of one of it's
+ aggregated advertisements.
+
+
+
+Fuller, Li, Yu & Varadhan [Page 13]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ 4.4. Responsibility for and configuration of aggregation
+
+ The domain which has been allocated a range of addresses has the sole
+ authority for aggregation of its address space. In the usual case,
+ the AS will install manual configuration commands in its border
+ routers to aggregate some portion of its address space. An domain
+ can also delegate aggregation authority to another domain. In this
+ case, aggregation is done in the other domain by one of its border
+ routers.
+
+ When an inter-domain border router performs route aggregation, it
+ needs to know the range of the block of IP addresses to be
+ aggregated. The basic principle is that it should aggregate as much
+ as possible but not to aggregate those routes which cannot be treated
+ as part of a single unit due to multi-homing, policy, or other
+ constraints.
+
+ One mechanism is to do aggregation solely based on dynamically
+ learned routing information. This has the danger of not specifying a
+ precise enough range since when a route is not present, it is not
+ always possible to distinguish whether it is temporarily unreachable
+ or that it does not belong in the aggregate. Purely dynamic routing
+ also does not allow the flexibility of defining what to aggregate
+ within a range. The other mechanism is to do all aggregation based on
+ ranges of blocks of IP addresses preconfigured in the router. It is
+ recommended that preconfiguration be used, since it more flexible and
+ allows precise specification of the range of destinations to
+ aggregate.
+
+ Preconfiguration does require some manually-maintained configuration
+ information, but not excessively more so than what router
+ administrators already maintain today. As an addition to the amount
+ of information that must be typed in and maintained by a human,
+ preconfiguration is just a line or two defining the range of the
+ block of IP addresses to aggregate. In terms of gathering the
+ information, if the advertising router is doing the aggregation, its
+ administrator knows the information because the aggregation ranges
+ are assigned to its domain. If the receiving domain has been granted
+ the authority to and task of performing aggregation, the information
+ would be known as part of the agreement to delegate aggregation.
+ Given that it is common practice that a network administrator learns
+ from its neighbor which routes it should be willing to accept,
+ preconfiguration of aggregation information does not introduce
+ additional administrative overhead.
+
+ Implementation note: aggregates which encompass the class D address
+ space (multicast addresses) are currently not well understood. At
+ present, it appears that the optimal strategy is to consider
+
+
+
+Fuller, Li, Yu & Varadhan [Page 14]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ aggregates to never encompass class D space, even if they do so
+ numerically.
+
+ 4.5 Intra-domain protocol considerations
+
+ While no changes need be made to internal routing protocols to
+ support the advertisement of aggregated routing information between
+ autonomous systems, it is often the case that external routing
+ information is propagated within interior protocols for policy
+ reasons or to aid in the propagation of information through a transit
+ network. At the point when aggregated routing information starts to
+ appear in the new exterior protocols, this practice of importing
+ external information will have to be modified. A transit network
+ which imports external information will have to do one of:
+
+ a) use an interior protocol which supports aggregated routing
+
+ b) find some other method of propagating external information
+ which does not involve flooding it through the interior
+ protocol (i.e., by the use of internal BGP, for example).
+
+ c) stop the importation of external information and flood a
+ "default" route through the internal protocol for discovery
+ of paths to external destinations.
+
+ For case (a), the modifications necessary to a routing protocol to
+ allow it to support aggregated information may not be simple. For
+ protocols such as OSPF and IS-IS, which represent routing information
+ as either a destination+mask (OSPF) or as a prefix+prefix-length
+ (IS-IS) changes to support aggregated information are conceptually
+ fairly simple; for protocols which are dependent on the class-A/B/C
+ nature of networks or which support only fixed-sized subnets, the
+ changes are of a more fundamental nature. Even in the "conceptually
+ simple" cases of OSPF and IS-IS, an implementation may need to be
+ modified to support supernets in the database or in the forwarding
+ table.
+
+5. Example of new allocation and routing
+
+ 5.1 Address allocation
+
+ Consider the block of 2048 class C network numbers beginning with
+ 192.24.0.0 (0xC0180000 and ending with 192.31.255.0 (0xC01FFF00)
+ allocated to a single network provider, "RA". A "supernetted" route
+ to this block of network numbers would be described as 192.24.0.0
+ with mask of 255.248.0.0 (0xFFF80000).
+
+
+
+
+
+Fuller, Li, Yu & Varadhan [Page 15]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ Assume this service provider connects six clients in the following
+ order (significant because it demonstrates how temporary "holes" may
+ form in the service provider's address space):
+
+ "C1" requiring fewer than 2048 addresses (8 class C networks)
+
+ "C2" requiring fewer than 4096 addresses (16 class C networks)
+
+ "C3" requiring fewer than 1024 addresses (4 class C networks)
+
+ "C4" requiring fewer than 1024 addresses (4 class C networks)
+
+ "C5" requiring fewer than 512 addresses (2 class C networks)
+
+ "C6" requiring fewer than 512 addresses (2 class C networks)
+
+ In all cases, the number of IP addresses "required" by each client is
+ assumed to allow for significant growth. The service provider
+ allocates its address space as follows:
+
+ C1: allocate 192.24.0 through 192.24.7. This block of networks is
+ described by the "supernet" route 192.24.0.0 and mask
+ 255.255.248.0
+
+ C2: allocate 192.24.16 through 192.24.31. This block is described
+ by the route 192.24.16.0, mask 255.255.240.0
+
+ C3: allocate 192.24.8 through 192.24.11. This block is described
+ by the route 192.24.8.0, mask 255.255.252.0
+
+ C4: allocate 192.24.12 through 192.24.15. This block is described
+ by the route 192.24.12.0, mask 255.255.252.0
+
+ C5: allocate 192.24.32 and 192.24.33. This block is described by
+ the route 192.24.32.0, mask 255.255.254.0
+
+ C6: allocate 192.24.34 and 192.24.35. This block is described by
+ the route 192.24.34.0, mask 255.255.254.0
+
+ Note that if the network provider uses an IGP which can support
+ classless networks, he can (but doesn't have to) perform
+ "supernetting" at the point where he connects to his clients and
+ therefore only maintain six distinct routes for the 36 class C
+ network numbers. If not, explicit routes to all 36 class C networks
+ will have to be carried by the IGP.
+
+ To make this example more realistic, assume that C4 and C5 are
+ multi-homed through some other service provider, "RB". Further assume
+
+
+
+Fuller, Li, Yu & Varadhan [Page 16]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ the existence of a client "C7" which was originally connected to "RB"
+ but has moved to "RA". For this reason, it has a block of network
+ numbers which are allocated out "RB"'s block of (the next) 2048 class
+ C network numbers:
+
+ C7: allocate 192.32.0 through 192.32.15. This block is described
+ by the route 192.32.0, mask 255.255.240.0
+
+ For the multi-homed clients, we will assume that C4 is advertised as
+ primary via "RA" and secondary via "RB"; C5 is primary via "RB" and
+ secondary via "RA". To connect this mess together, we will assume
+ that "RA" and "RB" are connected via some common "backbone" provider
+ "BB".
+
+ Graphically, this simple topology looks something like this:
+
+
+ C1
+192.24.0.0 -- 192.24.7.0 \ _ 192.32.0.0 - 192.32.15.0
+192.24.0.0/255.255.248.0 \ / 192.32.0.0/255.255.240.0
+ \ / C7
+ C2 +----+ +----+
+192.24.16.0 - 192.24.31.0 \| | | |
+192.24.16.0/255.255.240.0 | | _ 192.24.12.0 - 192.24.15.0 _ | |
+ | | / 192.24.12.0/255.255.252.0 \ | |
+ C3 -| |/ C4 \| |
+192.24.8.0 - 192.24.11.0 | RA | | RB |
+192.24.8.0/255.255.252.0 | |___ 192.24.32.0 - 192.24.33.0 ___| |
+ /| | 192.24.32.0/255.255.254.0 | |
+ C6 | | C5 | |
+192.24.34.0 - 192.24.35.0 | | | |
+192.24.34.0/255.255.254.0 | | | |
+ +----+ +----+
+ \\ \\
+192.24.12.0/255.255.252.0 (C4) || 192.24.12.0/255.255.252.0 (C4) ||
+192.32.0.0/255.255.240.0 (C7) || 192.24.32.0/255.255.254.0 (C5) ||
+192.24.0.0/255.248.0.0 (RA) || 192.32.0.0/255.248.0.0 (RB) ||
+ || ||
+ VV VV
+ +--------------- BACKBONE PEER BB ---------------+
+
+ 5.2 Routing advertisements
+
+ To follow rule #1, RA will need to advertise the block of addresses
+ that it was given and C7. Since C4 is multi-homed and primary
+ through RA, it must also be advertised. C5 is multi-homed and
+ primary through RB. It need not be advertised since longest match by
+ BB will automatically select RB as primary and the advertisement of
+
+
+
+Fuller, Li, Yu & Varadhan [Page 17]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ RA's aggregate will be used as a secondary.
+
+ Advertisements from "RA" to "BB" will be:
+
+ 192.24.12.0/255.255.252.0 primary (advertises C4)
+ 192.32.0.0/255.255.240.0 primary (advertises C7)
+ 192.24.0.0/255.248.0.0 primary (advertises remainder of RA)
+
+ For RB, the advertisements must also include C4 and C5 as well as
+ it's block of addresses. Further, RB may advertise that C7 is
+ unreachable.
+
+ Advertisements from "RB" to "BB" will be:
+
+ 192.24.12.0/255.255.252.0 secondary (advertises C4)
+ 192.24.32.0/255.255.254.0 primary (advertises C5)
+ 192.32.0.0/255.248.0.0 primary (advertises remainder of RB)
+
+ To illustrate the problem alluded to by the "note" in section 4.2,
+ consider what happens if RA loses connectivity to C7 (the client
+ which is allocated out of RB's space). In a stateful protocol, RA
+ will announce to BB that 192.32.0.0/255.255.240.0 has become
+ unreachable. Now, when BB flushes this information out of its routing
+ table, any future traffic sent through it for this destination will
+ be forwarded to RB (where it will be dropped according to Rule #2) by
+ virtue of RB's less specific match 192.32.0.0/255.248.0.0. While
+ this does not cause an operational problem (C7 is unreachable in any
+ case), it does create some extra traffic across "BB" (and may also
+ prove confusing to a network manager debugging the outage with
+ "traceroute"). A mechanism to cache such unreachability information
+ would help here, but is beyond the scope of this document (such a
+ mechanism is also not implementable in the near-term).
+
+6. Extending CIDR to class A addresses
+
+ At some point, it is expected that this plan will eventually consume
+ all of the remaining class C address space. As of this writing, the
+ upper half of the class A address space has already been reserved for
+ future expansion. This section describes how the CIDR plan can be
+ used to utilize this portion of the class A space efficiently. It is
+ expected that this contingency would only be used if no long term
+ solution has become apparent by the time that the class C address
+ space is consumed.
+
+ Fundamentally, there are two differences between using a class A
+ address and a block of class C's. First, the configuration of DNS
+ becomes somewhat more complicated than it is without the aggregation
+ of class A subnets. The second difference is that the routers within
+
+
+
+Fuller, Li, Yu & Varadhan [Page 18]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ the class A address would need to support and use a classless IGP.
+
+ Maintenance of DNS with a subnetted class A is somewhat painful. As
+ part of the mechanism for providing reverse address lookups, DNS
+ maintains a "IN-ADDR.ARPA" reverse domain. This is configured by
+ reversing the dotted decimal network number, appending "IN-ADDR.ARPA"
+ and using this as a type of pseudo-domain. Individual hosts then end
+ up pointing back to a host name. Thus, for example, 131.108.1.111
+ has a DNS record "111.1.108.131.IN-ADDR.ARPA." Since the pseudo-
+ domains can only be delegated on a byte boundary, this becomes
+ painful if a stub domain receives a block of address space that does
+ not fall on a byte boundary. The solution in this case is to
+ enumerate all of the possible byte combinations involved. This is
+ painful, but workable. This is discussed further below.
+
+ Routing within a class A used for CIDR is also an interesting
+ challenge. The usual case will be that a domain will be assigned a
+ portion of the class A address space. The domain can either use an
+ IGP which allows variable length subnets or it can pick a single
+ subnet mask to be used throughout the domain. In the latter case,
+ difficulties arise because other domains have been allocated other
+ parts of the class A address space and may be using a different
+ subnet mask. If the domain is itself a transit, it may also need to
+ allocate some portion of its space to a client, which might also use
+ a different subnet mask. The client would then need routing
+ information about the remainder of the class A.
+
+ If the client's IGP does not support variable length subnet masks,
+ this could be done by advertising the remainder of the class A's
+ address space in appropriately sized subnets. However, unless the
+ client has a very large portion of the class A space, this is likely
+ to result in a large number of subnets (for example, a mask of
+ 255.255.255.0 would require a total of 65535 subnets, including those
+ allocated to the client). For this reason, it may be preferable to
+ simply use an IGP that supports variable length subnet masks within
+ the client's domain.
+
+ Similarly, if a transit has been assigned address space from a class
+ A network number, it is likely that it was not assigned the entire
+ class A, and that other transit domains will get address space from
+ this class A. In this case, the transit would also have to inject
+ routing information about the remainder of the class A into it's IGP.
+ This is analogous to the situation above, with the same
+ complications. For this reason, we recommend that the use of a class
+ A for CIDR only be attempted if IGP's with variable length subnet
+ mask support be used throughout the class A. Note that the IGP's
+ need not support supernetting, as discussed above.
+
+
+
+
+Fuller, Li, Yu & Varadhan [Page 19]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ Note that the technique here could also apply to class B addresses.
+ However, the limited number of available class B addresses and their
+ usage for multihomed networks suggests that this address space should
+ only be reserved for those large single organizations that warrant
+ this type of address. [2]
+
+7. Domain Service considerations
+
+ One aspect of Internet services which will be notably affected by a
+ move to either "supernetted" class-C network numbers or subdivided
+ class-A's will be the mechanism used for address-to-name translation:
+ the IN-ADDR.ARPA zone of the domain system. Because this zone is
+ delegated on octet boundaries only, any address allocation plan which
+ uses bitmask-oriented addressing will cause some degree of difficulty
+ for those which maintain parts of the IN-ADDR.ARPA zone.
+
+ 7.1 Procedural changes for class-C "supernets"
+
+ At the present time, parts of the IN-ADDR.ARPA zone are delegated
+ only on network boundaries which happen to fall on octet boundaries.
+ To aid in the use of blocks of class-C networks, it is recommended
+ that this policy be relaxed and allow the delegation of arbitrary,
+ octet-oriented pieces of the IN-ADDR.ARPA zone.
+
+ As an example of this policy change, consider a hypothetical large
+ network provider named "BigNet" which has been allocated the 1024
+ class-C networks 199.0.0 through 199.3.255. Under current policies,
+ the root domain servers would need to have 1024 entries of the form:
+
+ 0.0.199.IN-ADDR.ARPA. IN NS NS1.BIG.NET.
+
+ 1.0.199.IN-ADDR.ARPA. IN NS NS1.BIG.NET.
+
+ ....
+
+ 255.3.199.IN-ADDR.ARPA. IN NS NS1.BIG.NET.
+
+ By revising the policy as described above, this is reduced only four
+ delegation records:
+
+ 0.199.IN-ADDR.ARPA. IN NS NS1.BIG.NET.
+
+ 1.199.IN-ADDR.ARPA. IN NS NS1.BIG.NET.
+
+ 2.199.IN-ADDR.ARPA. IN NS NS1.BIG.NET.
+
+ 3.199.IN-ADDR.ARPA. IN NS NS1.BIG.NET.
+
+
+
+
+Fuller, Li, Yu & Varadhan [Page 20]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ The provider would then maintain further delegations of naming
+ authority for each individual class-C network which it assigns,
+ rather than having each registered separately. Note that due to the
+ way the DNS is designed, it is still possible for the root
+ nameservers to maintain the delegation information for individual
+ networks for which the provider is unwilling or unable to do so. This
+ should greatly reduce the load on the domain servers for the "top"
+ levels of the IN-ADDR.ARPA domain. The example above illustrates
+ only the records for a single nameserver. In the normal case, there
+ are usually several nameservers for each domain, thus the size of the
+ examples will double or triple in the common cases.
+
+ 7.2 Procedural changes for class-A subnetting
+
+ Should it be the case the class-A network numbers are subdivided into
+ blocks allocated to transit network providers, it will be similarly
+ necessary to relax the restriction on how IN-ADDR.ARPA naming works
+ for them. As an example, take a provider is allocated the 19-bit
+ portion of address space which matches 10.8.0.0 with mask
+ 255.248.0.0. This represents all addresses which begin with the
+ prefixes 10.8, 10.9, 10.10, 10.11, 10.12, 10.13, 10.14, an 10.15 and
+ requires the following IN-ADDR.ARPA delegations:
+
+ 8.10.IN-ADDR.ARPA. IN NS NS1.MOBY.NET.
+
+ 9.10.IN-ADDR.ARPA. IN NS NS1.MOBY.NET.
+
+ ....
+
+ 15.10.IN-ADDR.ARPA. IN NS NS1.MOBY.NET.
+
+ To further illustrate how IN-ADDR.ARPA sub-delegation will work,
+ consider a company named "FOO" connected to this provider which has
+ been allocated the 14-bit piece of address space which matches
+ 10.10.64.0 with mask 255.255.192.0. This represents all addresses in
+ the range 10.10.64.0 through 10.10.127.255 and will require that the
+ provider implement the following IN-ADDR.ARPA delegations:
+
+ 64.10.10.IN-ADDR.ARPA. IN NS NS1.FOO.COM.
+
+ 65.10.10.IN-ADDR.ARPA. IN NS NS1.FOO.COM.
+
+ ....
+
+ 127.10.10.IN-ADDR.ARPA. IN NS NS1.FOO.COM.
+
+ with the servers for "FOO.COM" containing the individual PTR records
+ for all of the addresses on each of these subnets.
+
+
+
+Fuller, Li, Yu & Varadhan [Page 21]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+8. Transitioning to a long term solution
+
+ This solution does not change the Internet routing and addressing
+ architectures. Hence, transitioning to a more long term solution is
+ not affected by the deployment of this plan.
+
+9. Conclusions
+
+ We are all aware of the growth in routing complexity, and the rapid
+ increase in allocation of network numbers. Given the rate at which
+ this growth is being observed, we expect to run out in a few short
+ years.
+
+ If the inter-domain routing protocol supports carrying network routes
+ with associated masks, all of the major concerns demonstrated in this
+ paper would be eliminated.
+
+ One of the influential factors which permits maximal exploitation of
+ the advantages of this plan is the number of people who agree to use
+ it.
+
+ If service providers start charging networks for advertising network
+ numbers, this would be a very great incentive to share the address
+ space, and hence the associated costs of advertising routes to
+ service providers.
+
+10. Recommendations
+
+ The NIC should begin to hand out large blocks of class C addresses to
+ network service providers. Each block must fall on bit boundaries
+ and should be large enough to serve the provider for two years.
+ Further, the NIC should distribute very large blocks to continental
+ and national network service organizations to allow additional levels
+ of aggregation to take place at the major backbone networks. In
+ addition, the NIC should modify its procedures for the IN-ADDR.ARPA
+ domain to permit delegation along arbitrary octet boundaries.
+
+ Service providers will further allocate power-of-two blocks of class
+ C addresses from their address space to their subscribers.
+
+ All organizations, including those which are multi-homed, should
+ obtain address space from their provider (or one of their providers,
+ in the case of the multi-homed). These blocks should also fall on
+ bit boundaries to permit easy route aggregation.
+
+ To allow effective use of this new addressing plan to reduce
+ propagated routing information, appropriate IETF WGs will specify the
+ modifications needed to Inter-Domain routing protocols.
+
+
+
+Fuller, Li, Yu & Varadhan [Page 22]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+ Implementation and deployment of these modifications should occur as
+ quickly as possible.
+
+11 References
+
+ [1] Moy, J, "The OSPF Specification Version 2", RFC 1247, Proteon,
+ Inc., January 1991.
+
+ [2] Rekhter, Y., and T. Li, "An Architecture for IP Address
+ Allocation with CIDR", RFC 1518, T.J. Watson Research Center, IBM
+ Corp., cisco Systems, September 1993.
+
+12. Security Considerations
+
+ Security issues are not discussed in this memo.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Fuller, Li, Yu & Varadhan [Page 23]
+
+RFC 1519 CIDR Address Strategy September 1993
+
+
+13. Authors' Addresses
+
+ Vince Fuller
+ BARRNet
+ Pine Hall 115
+ Stanford, CA, 94305-4122
+
+ EMail: vaf@Stanford.EDU
+
+
+ Tony Li
+ cisco Systems, Inc.
+ 1525 O'Brien Drive
+ Menlo Park, CA 94025
+
+ EMail: tli@cisco.com
+
+
+ Jessica (Jie Yun) Yu
+ Merit Network, Inc.
+ 1071 Beal Ave.
+ Ann Arbor, MI 48109
+
+ EMail: jyy@merit.edu
+
+
+ Kannan Varadhan
+ Internet Engineer, OARnet
+ 1224, Kinnear Road,
+ Columbus, OH 43212
+
+ EMail: kannan@oar.net
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Fuller, Li, Yu & Varadhan [Page 24]
+ \ No newline at end of file