path: root/doc/rfc/rfc4692.txt

Network Working Group                                          G. Huston
Request for Comments: 4692                                         APNIC
Category: Informational                                     October 2006


             Considerations on the IPv6 Host Density Metric

Status of This Memo

   This memo provides information for the Internet community.  It does
   not specify an Internet standard of any kind.  Distribution of this
   memo is unlimited.

Copyright Notice

   Copyright (C) The Internet Society (2006).

Abstract

   This memo provides an analysis of the Host Density metric as it is
   currently used to guide registry allocations of IPv6 unicast address
   blocks.  This document contrasts the address efficiency as currently
   adopted in the allocation of IPv4 network addresses and that used by
   the IPv6 protocol.  Note that for large allocations there are very
   significant variations in the target efficiency metric between the
   two approaches.

Table of Contents

   1. Introduction ....................................................2
   2. IPv6 Address Structure ..........................................2
   3. The Host Density Ratio ..........................................3
   4. The Role of an Address Efficiency Metric ........................4
   5. Network Structure and Address Efficiency Metric .................6
   6. Varying the HD-Ratio ............................................7
      6.1. Simulation Results .........................................8
   7. Considerations .................................................10
   8. Security Considerations ........................................11
   9. Acknowledgements ...............................................11
   10. References ....................................................12
      10.1. Normative References .....................................12
      10.2. Informative References ...................................12
   Appendix A.  Comparison Tables ....................................13








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1.  Introduction

   Metrics of address assignment efficiency are used in the context of
   the Regional Internet Registries' (RIRs') address allocation
   function.  Through the use of a common address assignment efficiency
   metric, individual networks can be compared to a threshold value in
   an objective fashion.  The common use of this metric is to form part
   of the supporting material for an address allocation request,
   demonstrating that the network has met or exceeded the threshold
   address efficiency value, and it forms part of the supportive
   material relating to the justification of the allocation of a further
   address block.

   Public and private IP networks have significant differences in
   purpose, structure, size, and technology.  Attempting to impose a
   single efficiency metric across this very diverse environment is a
   challenging task.  Any address assignment efficiency threshold value
   has to represent a balance between stating an achievable outcome for
   any competently designed and operated service platform while without
   setting a level of consumption of address resources that imperils the
   protocol's longer term viability through consequent address scarcity.
   There are a number of views relating to address assignment
   efficiency, both in terms of theoretic analyses of assignment
   efficiency and in terms of practical targets that are part of current
   address assignment practices in today's Internet.

   This document contrasts the address efficiency metric and threshold
   value as currently adopted in the allocation of IPv4 network
   addresses and the framework used by the address allocation process
   for the IPv6 protocol.

2.  IPv6 Address Structure

   Before looking at address allocation efficiency metrics, it is
   appropriate to summarize the address structure for IPv6 global
   unicast addresses.

   The general format for IPv6 global unicast addresses is defined in
   [RFC4291] as follows (Figure 1).

    |         64 - m bits    |   m bits  |       64 bits              |
    +------------------------+-----------+----------------------------+
    | global routing prefix  | subnet ID |       interface ID         |
    +------------------------+-----------+----------------------------+

                          IPv6 Address Structure

                                 Figure 1



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   Within the current policy framework for allocation of IPv6 addresses
   in the context of the public Internet, the value for 'm' in the
   figure above, referring to the subnet ID, is commonly a 16-bit field.
   Therefore, the end-site global routing prefix is 48 bits in length,
   the per-customer subnet ID is 16 bits in length, and the interface ID
   is 64 bits in length [RFC3177].

   In relating this address structure to the address allocation
   function, the efficiency metric is not intended to refer to the use
   of individual 128-bit IPv6 addresses nor that of the use of the 64-
   bit subnet prefix.  Instead, it is limited to a measure of efficiency
   of use of the end-site global routing prefix.  This allocation model
   assumes that each customer is allocated a minimum of a single /48
   address block.  Given that this block allows 2^16 possible subnets,
   it is also assumed that a /48 allocation will be used in the overall
   majority of cases of end-customer address assignment.

   The following discussion makes the assumption that the address
   allocation unit in IPv6 is an address prefix of 48 bits in length,
   and that the address assignment efficiency in this context is the
   efficiency of assignment of /48 address allocation units.  However,
   the analysis presented here refers more generally to end-site address
   allocation practices rather than /48 address prefixes in particular,
   and is applicable in the context of any size of end-site global
   routing prefix.

3.  The Host Density Ratio

   The "Host Density Ratio" was first described in [RFC1715] and
   subsequently updated in [RFC3194].

   The "H Ratio", as defined in RFC 1715, is:

                            log (number of objects)
                        H = -----------------------
                                  available bits

                                 Figure 2

   The argument presented in [RFC1715] draws on a number of examples to
   support the assertion that this metric reflects a useful generic
   measure of address assignment efficiency in a range of end-site
   addressed networks, and furthermore that the optimal point for such a
   utilization efficiency metric lies in an H Ratio value between 0.14
   and 0.26.  Lower H Ratio values represent inefficient address use,
   and higher H Ratio values tend to be associated with various forms of
   additional network overhead related to forced re-addressing
   operations.



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   This particular metric has a maximal value of log base 10 of 2, or
   0.30103.

   The metric was 'normalized' in RFC 3194, and a new metric, the "HD-
   Ratio" was introduced, with the following definition:

                        log(number of allocated objects)
              HD = ------------------------------------------
                   log(maximum number of allocatable objects)

                                 Figure 3

   HD-Ratio values are proportional to the H ratio, and the values of
   the HD-Ratio range from 0 to 1.  The analysis described in [RFC3194]
   applied this HD-Ratio metric to the examples given in [RFC1715] and,
   on the basis of these examples, postulated that HD-Ratio values of
   0.85 or higher force the network into some form of renumbering.  HD-
   Ratio values of 0.80 or lower were considered an acceptable network
   efficiency metric.

   The HD-Ratio is referenced within the IPv6 address allocation
   policies used by the Regional Internet Registries, and their IPv6
   address allocation policy documents specify that an HD-Ratio metric
   of 0.8 is an acceptable objective in terms of address assignment
   efficiency for an IPv6 network.

   By contrast, the generally used address efficiency metric for IPv4 is
   the simple ratio of the number of allocated (or addressed) objects to
   the maximum number of allocatable objects.  For IPv4, the commonly
   applied value for this ratio is 0.8 (or 80%).

   A comparison of these two metrics is given in Table 1 of Attachment
   A.

4.  The Role of an Address Efficiency Metric

   The role of the address efficiency metric is to provide objective
   metrics relating to a network's use of address space that can be used
   by both the allocation entity and the applicant to determine whether
   an address allocation is warranted, and provide some indication of
   the size of the address allocation that should be undertaken.  The
   metric provides a target address utilization level that indicates at
   what point a network's address resource may be considered "fully
   utilized".

   The objective here is to allow the network service provider to deploy
   addresses across both network infrastructure and the network's
   customers in a manner that does not entail periodic renumbering, and



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   in a manner that allows both the internal routing system and inter-
   domain routing system to operate without excessive fragmentation of
   the address space and consequent expansion of the number of route
   objects carried within the routing systems.  This entails use of an
   addressing plan where at each level of structure within the network
   there is a pool of address blocks that allows expansion of the
   network at that structure level without requiring renumbering of the
   remainder of the network.

   It is recognized that an address utilization efficiency metric of
   100% is unrealistic in any scenario.  Within a typical network
   address plan, the network's address space is exhausted not when all
   address resources have been used, but at the point when one element
   within the structure has exhausted its pool, and when augmentation of
   this pool by drawing from the pools of other elements would entail
   extensive renumbering.  While it is not possible to provide a
   definitive threshold of what overall efficiency level is obtainable
   in all IP networks, experience with IPv4 network deployments suggests
   that it is reasonable to observe that at any particular level within
   a hierarchically structured address deployment plan an efficiency
   level of between 60% to 80% is an achievable metric in the general
   case.

   This IPv4 efficiency threshold is significantly greater than that
   observed in the examples provided in conjunction with the HD-Ratio
   description in [RFC1715].  Note that the examples used in the HD-
   Ratio are drawn from, among other sources, the Public Switched
   Telephone Network (PSTN).  This comparison with the PSTN warrants
   some additional examination.  There are a number of differences
   between public IP network deployments and PSTN deployments that may
   account for this difference.  IP addresses are deployed on a per-
   provider basis with an alignment to network topology.  PSTN addresses
   are, on the whole, deployed using a geographical distribution system
   of "call areas" that share a common number prefix.  Within each call
   area, a sufficient number blocks from the number prefix must be
   available to allow each operator to draw their own number block from
   the area pool.  Within the IP environment, service providers do not
   draw address blocks from a common geographic number pool but receive
   address blocks from the Regional Internet Registry on a 'whole of
   network' basis.  This difference in the address structure allows an
   IP environment to achieve an overall higher level of address
   utilization efficiency.

   In terms of considering the number of levels of internal hierarchy in
   IP networks, the interior routing protocol, if uniformly deployed,
   admits a hierarchical network structure that is only two levels deep,
   with a fully connected backbone "core" and a number of satellite
   areas that are directly attached to this "core".  Additional levels



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   of routing hierarchy may be obtained using various forms of routing
   confederations, but this is not an extremely common deployment
   technique.  The most common form of network structure used in large
   IP networks is a three-level structure using regions, individual
   Points of Presence (POPs), and end-customers.

   Also, note that large-scale IP deployments typically use a relatively
   flat routing structure, as compared to a deeply hierarchical
   structure.  In order to improve the dynamic performance of the
   interior routing protocol the number of routes carried in the
   interior routing protocol, is commonly restricted to the routes
   corresponding to next-hop destinations for iBGP routes, and customer
   routes are carried in the iBGP domain and aggregated at the point
   where the routes are announced in eBGP sessions.  This implies that
   per-POP or per-region address aggregations according to some fixed
   address hierarchy is not a necessary feature of large IP networks, so
   strict hierarchical address structure within all parts of the network
   is not a necessity in such routing environments.

5.  Network Structure and Address Efficiency Metric

   An address efficiency metric can be expressed using the number of
   levels of structure (n) and the efficiency achieved at each level
   (e).  If the same efficiency threshold is applied at each level of
   structure, the resultant efficiency threshold is e^n.  This then
   allows us to make some additional observations about the HD-Ratio
   values.  Table 2 of Appendix A (Figure 8) indicates the number of
   levels of structure that are implied by a given HD-Ratio value of 0.8
   for each address allocation block size, assuming a fixed efficiency
   level at all levels of the structure.  The implication is that for
   large address blocks, the HD-Ratio assumes a large number of elements
   in the hierarchical structure, or a very low level of address
   efficiency at the lower levels.  In the case of IP network
   deployments, this latter situation is not commonly the case.

   The most common form of interior routing structure used in IP
   networks is a two-level routing structure.  It is consistent with
   this constrained routing architecture that network address plans
   appear to be commonly devised using up to a three-level hierarchical
   structure, while for larger networks a four-level structure may
   generally be used.










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   Table 3 of Attachment A (Figure 9) shows an example of address
   efficiency outcomes using a per-level efficiency metric of 0.75 (75%)
   and a progressively deeper network structure as the address block
   expands.  This model (termed here "limited levels") limits the
   maximal number of levels of internal hierarchy to 6 and uses a model
   where the number of levels of network hierarchy increases by 1 when
   the network increases in size by a factor of a little over one order
   of magnitude.

   It is illustrative to compare these metrics for a larger network
   deployment.  If, for example, the network is designed to encompass 8
   million end customers, each of which is assigned a 16-bit subnet ID
   for their end site, then the following table Figure 4 indicates the
   associated allocation size as determined by the address efficiency
   metric.

         Allocation:  8M Customers

                                   Allocation    Relative Ratio

         100% Allocation Efficiency   /25               1
         80%  Efficiency (IPv4)       /24               2
         0.8  HD-Ratio                /19              64
         75%  with Limited Level      /23               4
         0.94 HD-Ratio                /23               4

                                    Figure 4

   Note that the 0.8 HD-Ratio produces a significantly lower efficiency
   level than the other metrics.  The limited-level model appears to
   point to a more realistic value for an efficiency value for networks
   of this scale (corresponding to a network with 4 levels of internal
   hierarchy, each with a target utilization efficiency of 75%).  This
   limited-level model corresponds to an HD-Ratio with a threshold value
   of 0.945.

6.  Varying the HD-Ratio

   One way to model the range of outcomes of taking a more limited
   approach to the number of levels of aggregateable hierarchy is to
   look at a comparison of various values for the HD-Ratio with the
   model of a fixed efficiency and the "Limited Levels" model.  This is
   indicated in Figure 5.








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          Prefix Length (bits)
          |
          |
          | Limited    HD-Ratio
          |  Levels    0.98    0.94    0.90    0.86    0.82    0.80
          |       |       |       |       |       |       |       |
          1   0.750   0.986   0.959   0.933   0.908   0.883   0.871
          4   0.750   0.946   0.847   0.758   0.678   0.607   0.574
          8   0.750   0.895   0.717   0.574   0.460   0.369   0.330
         12   0.563   0.847   0.607   0.435   0.312   0.224   0.189
         16   0.563   0.801   0.514   0.330   0.212   0.136   0.109
         20   0.422   0.758   0.435   0.250   0.144   0.082   0.062
         24   0.422   0.717   0.369   0.189   0.097   0.050   0.036
         28   0.316   0.678   0.312   0.144   0.066   0.030   0.021
         32   0.316   0.642   0.264   0.109   0.045   0.018   0.012
         36   0.237   0.607   0.224   0.082   0.030   0.011   0.007
         40   0.237   0.574   0.189   0.062   0.021   0.007   0.004
         44   0.178   0.543   0.160   0.047   0.014   0.004   0.002
         48   0.178   0.514   0.136   0.036   0.009   0.003   0.001

                                 Figure 5

   As shown in this figure, it is possible to select an HD-Ratio value
   that models IP level structures in a fashion that behaves more
   consistently for very large deployments.  In this case, the choice of
   an HD-Ratio of 0.94 is consistent with a limited-level model of up to
   6 levels of hierarchy with a metric of 75% density at each level.
   This correlation is indicated in Table 3 of Attachment A.

6.1.  Simulation Results

   In attempting to assess the impact of potentially changing the HD-
   Ratio to a lower value, it is useful to assess this using actual
   address consumption data.  The results described here use the IPv4
   allocation data as published by the Regional Internet Registries
   [RIR-Data].  The simulation work assumes that the IPv4 delegation
   data uses an IPv4 /32 for each end customer, and that assignments
   have been made based on an 80% density metric in terms of assumed
   customer count.  The customer count is then used as the basis of an
   IPv6 address allocation, using the HD-Ratio to map from a customer
   count to the size of an address allocation.










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   The result presented here is that of a simulation of an IPv6 address
   allocation registry, using IPv4 allocation data as published by the
   RIRs spanning the period from January 1, 1999 until August 31, 2004.
   The aim is to identify the relative level of IPv6 address consumption
   using a IPv6 request size profile based on the application of various
   HD-Ratio values to the derived customer numbers.

   The profile of total address consumption for selected HD-Ratio values
   is indicated in Figure 6.  The simulation results indicate that the
   choice of an HD-Ratio of 0.8 consumes a total of 7 times the address
   space of that consumed when using an HD-Ratio of 0.94.

                 HD-Ratio       Total Address Consumption
                 |        Prefix Length   Count of
                 |        Notation        /32 prefixes
                 0.80    /14.45          191,901
                 0.81    /14.71          160,254
                 0.82    /15.04          127,488
                 0.83    /15.27          108,701
                 0.84    /15.46           95,288
                 0.85    /15.73           79,024
                 0.86    /15.88           71,220
                 0.87    /16.10           61,447
                 0.88    /16.29           53,602
                 0.89    /16.52           45,703
                 0.90    /16.70           40,302
                 0.91    /16.77           38,431
                 0.92    /16.81           37,381
                 0.93    /16.96           33,689
                 0.94    /17.26           27,364
                 0.95    /17.32           26,249
                 0.96    /17.33           26,068
                 0.97    /17.33           26,068
                 0.98    /17.40           24,834
                 0.99    /17.67           20,595

                                 Figure 6

   The implication of these results imply that an IPv6 address registry
   will probably see sufficient distribution of allocation request sizes
   such that the choice of a threshold HD-Ratio will impact the total
   address consumption rates, and the variance between an HD-Ratio of
   0.8 and an HD-Ratio of 0.99 is a factor of one order of magnitude in
   relative address consumption over an extended period of time.  The
   simulation also indicates that the overall majority of allocations
   fall within a /32 minimum allocation size (between 74% to 95% of all
   address allocations), and that the selection of a particular HD-Ratio
   value has a significant impact in terms of allocation sizes for a



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   small proportion of allocation transactions (the remainder of
   allocations range between a /19 to a /31 for an HD-Ratio of 0.8 and
   between a /26 and a /31 for an HD-Ratio of 0.99).

   The conclusion here is that the choice of the HD-Ratio will have some
   impact on one quarter of all allocations, while the remainder are
   serviced using the minimum allocation unit of a /32 address prefix.
   Of these 'impacted' allocations that are larger than the minimum
   allocation, approximately one tenth of these allocations are 'large'
   allocations.  These large allocations have a significant impact on
   total address consumption, and varying the HD-Ratio for these
   allocations between 0.8 to 0.99 results in a net difference in total
   address consumption of approximately one order of magnitude.  This is
   a heavy-tail distribution, where a small proportion of large address
   allocations significantly impact the total address consumption rate.
   Altering the HD-Ratio will have little impact on more than 95% of the
   IPv6 allocations but will generate significant variance within the
   largest 2% of these allocations, which, in turn, will have a
   significant impact on total address consumption rates.

7.  Considerations

   The HD-Ratio with a value of 0.8 as a model of network address
   utilization efficiency produces extremely low efficiency outcomes for
   networks spanning of the order of 10**6 end customers and larger.

   The HD-Ratio with a 0.8 value makes the assumption that as the
   address allocation block increases in size, the network within which
   the addresses will be deployed adds additional levels of hierarchical
   structure.  This increasing depth of hierarchical structure to
   arbitrarily deep hierarchies is not a commonly observed feature of
   public IP network deployments.

   The fixed efficiency model, as used in the IPv4 address allocation
   policy, uses the assumption that as the allocation block becomes
   larger, the network structure remains at a fixed level of levels; if
   the number of levels is increased, then efficiency achieved at each
   level increases significantly.  There is little evidence to suggest
   that increasing a number of levels in a network hierarchy increases
   the efficiency at each level.

   It is evident that neither of these models accurately encompass IP
   network infrastructure models and the associated requirements of
   address deployment.  The fixed efficiency model places an excessive
   burden on the network operator to achieve very high levels of
   utilization at each level in the network hierarchy, leading to either
   customer renumbering or deployment of technologies such as Network
   Address Translation (NAT) to meet the target efficiency value in a



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   hierarchically structured network.  The HD-Ratio model using a value
   of 0.8 specifies an extremely low address efficiency target for
   larger networks, and while this places no particular stress on
   network architects in terms of forced renumbering, there is the
   concern that this represents an extremely inefficient use of address
   resources.  If the objective of IPv6 is to encompass a number of
   decades of deployment, and to span a public network that ultimately
   encompasses many billions of end customers and a very high range and
   number of end use devices and components, then there is legitimate
   cause for concern that the HD-Ratio value of 0.8 may be setting too
   conservative a target for address efficiency, in that the total
   address consumption targets may be achieved too early.

   This study concludes that consideration should be given to the
   viability of specifying a higher HD-Ratio value as representing a
   more relevant model of internal network structure, internal routing,
   and internal address aggregation structures in the context of IPv6
   network deployment.

8.  Security Considerations

   Considerations of various forms of host density metrics create no new
   threats to the security of the Internet.

9.  Acknowledgements

   The document was reviewed by Kurt Lindqvist, Thomas Narten, Paul
   Wilson, David Kessens, Bob Hinden, Brian Haberman, and Marcelo
   Bagnulo.






















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10. References

10.1.  Normative References

   [RFC1715]   Huitema, C., "The H Ratio for Address Assignment
               Efficiency", RFC 1715, November 1994.

   [RFC3177]   IAB and IESG, "IAB/IESG Recommendations on IPv6 Address
               Allocations to Sites", RFC 3177, September 2001.

   [RFC3194]   Durand, A. and C. Huitema, "The H-Density Ratio for
               Address Assignment Efficiency An Update on the H ratio",
               RFC 3194, November 2001.

   [RFC4291]   Hinden, R. and S. Deering, "IP Version 6 Addressing
               Architecture", RFC 4291, February 2006.

10.2.  Informative References

   [RIR-Data]  RIRs, "RIR Delegation Records", February 2005,
               <ftp://ftp.apnic.net/pub/stats/>.






























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Appendix A.  Comparison Tables

   The first table compares the threshold number of /48 end user
   allocations that would be performed for a given assigned address
   block in order to consider that the utilization has achieved its
   threshold utilization level.

   Fixed Efficiency Value  0.8
   HD-Ratio Value          0.8

                               Number of /48 allocations to fill the
                                address block to the threshold level

   Prefix          Size              Fixed Efficiency      HD-Ratio
                                       0.8                     0.8

   /48                 1                 1 100%              1  100%
   /47                 2                 2 100%              2   87%
   /46                 4                 4 100%              3   76%
   /45                 8                 7  88%              5   66%
   /44                16                13  81%              9   57%
   /43                32                26  81%             16   50%
   /42                64                52  81%             28   44%
   /41               128               103  80%             49   38%
   /40               256               205  80%             84   33%
   /39               512               410  80%            147   29%
   /38             1,024               820  80%            256   25%
   /37             2,048             1,639  80%            446   22%
   /36             4,096             3,277  80%            776   19%
   /35             8,192             6,554  80%          1,351   16%
   /34            16,384            13,108  80%          2,353   14%
   /33            32,768            26,215  80%          4,096   13%
   /32            65,536            52,429  80%          7,132   11%
   /31           131,072           104,858  80%         12,417    9%
   /30           262,144           209,716  80%         21,619    8%
   /29           524,288           419,431  80%         37,641    7%
   /28         1,048,576           838,861  80%         65,536    6%
   /27         2,097,152         1,677,722  80%        114,105    5%
   /26         4,194,304         3,355,444  80%        198,668    5%
   /25         8,388,608         6,710,887  80%        345,901    4%
   /24        16,777,216        13,421,773  80%        602,249    4%
   /23        33,554,432        26,843,546  80%      1,048,576    3%
   /22        67,108,864        53,687,092  80%      1,825,677    3%
   /21       134,217,728       107,374,180  80%      3,178,688    2%
   /20       268,435,456       214,748,365  80%      5,534,417    2%
   /19       536,870,912       429,496,730  80%      9,635,980    2%
   /18     1,073,741,824       858,993,460  80%     16,777,216    2%
   /17     2,147,483,648     1,717,986,919  80%     29,210,830    1%



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   /16     4,294,967,296     3,435,973,837  80%     50,859,008    1%
   /15     8,589,934,592     6,871,947,674  80%     88,550,677    1%
   /14    17,179,869,184    13,743,895,348  80%    154,175,683    1%
   /13    34,359,738,368    27,487,790,695  80%    268,435,456    1%
   /12    68,719,476,736    54,975,581,389  80%    467,373,275    1%
   /11   137,438,953,472   109,951,162,778  80%    813,744,135    1%
   /10   274,877,906,944   219,902,325,556  80%  1,416,810,831    1%
   /9    549,755,813,888   439,804,651,111  80%  2,466,810,934    0%
   /8  1,099,511,627,776   879,609,302,221  80%  4,294,967,296    0%
   /7  2,199,023,255,552 1,759,218,604,442  80%  7,477,972,398    0%
   /6  4,398,046,511,104 3,518,437,208,884  80% 13,019,906,166    0%
   /5  8,796,093,022,208 7,036,874,417,767  80% 22,668,973,294    0%

           Table 1.  Comparison of Fixed Efficiency Threshold vs
                     HD-Ratio Threshold

                                 Figure 7

   One possible assumption behind the HD-Ratio is that the
   inefficiencies that are a consequence of large-scale deployments are
   an outcome of an increased number of levels of hierarchical structure
   within the network.  The following table calculates the depth of the
   hierarchy in order to achieve a 0.8 HD-Ratio, assuming a 0.8
   utilization efficiency at each level in the hierarchy.

   Prefix          Size              0.8 Structure
                                HD-Ratio    Levels
   /48                 1               1         1
   /47                 2               2         1
   /46                 4               3         2
   /45                 8               5         2
   /44                16               9         3
   /43                32              16         4
   /42                64              28         4
   /41               128              49         5
   /40               256              84         5
   /39               512             147         6
   /38             1,024             256         7
   /37             2,048             446         7
   /36             4,096             776         8
   /35             8,192           1,351         9
   /34            16,384           2,353         9
   /33            32,768           4,096        10
   /32            65,536           7,132        10
   /31           131,072          12,417        11
   /30           262,144          21,619        12
   /29           524,288          37,641        12
   /28         1,048,576          65,536        13



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   /27         2,097,152         114,105        14
   /26         4,194,304         198,668        14
   /25         8,388,608         345,901        15
   /24        16,777,216         602,249        15
   /23        33,554,432       1,048,576        16
   /22        67,108,864       1,825,677        17
   /21       134,217,728       3,178,688        17
   /20       268,435,456       5,534,417        18
   /19       536,870,912       9,635,980        19
   /18     1,073,741,824      16,777,216        19
   /17     2,147,483,648      29,210,830        20
   /16     4,294,967,296      50,859,008        20
   /15     8,589,934,592      88,550,677        21
   /14    17,179,869,184     154,175,683        22
   /13    34,359,738,368     268,435,456        22
   /12    68,719,476,736     467,373,275        23
   /11   137,438,953,472     813,744,135        23
   /10   274,877,906,944   1,416,810,831        24
   /9    549,755,813,888   2,466,810,934        25
   /8  1,099,511,627,776   4,294,967,296        25

          Table 2: Number of Structure Levels Assumed by HD-Ratio

                                 Figure 8

   An alternative approach is to use a model of network deployment where
   the number of levels of hierarchy increases at a lower rate than that
   indicated in a 0.8 HD-Ratio model.  One such model is indicated in
   the following table.  This is compared to using an HD-Ratio value of
   0.94.

   Per-Level Target Efficiency: 0.75

   Prefix           Size Stepped      Stepped Efficiency      HD-Ratio
                         Levels          0.75                   0.94

   /48                 1  1                1 100%                 1 100%
   /47                 2  1                2 100%                 2 100%
   /46                 4  1                3  75%                 4 100%
   /45                 8  1                6  75%                 7  88%
   /44                16  1               12  75%                13  81%
   /43                32  1               24  75%                25  78%
   /42                64  1               48  75%                48  75%
   /41               128  1               96  75%                92  72%
   /40               256  1              192  75%               177  69%
   /39               512  2              384  75%               338  66%
   /38             1,024  2              576  56%               649  63%
   /37             2,048  2            1,152  56%             1,244  61%



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   /36             4,096  2            2,304  56%             2,386  58%
   /35             8,192  2            4,608  56%             4,577  56%
   /34            16,384  2            9,216  56%             8,780  54%
   /33            32,768  2           18,432  56%            16,845  51%
   /32            65,536  2           36,864  56%            32,317  49%
   /31           131,072  3           73,728  56%            62,001  47%
   /30           262,144  3          110,592  42%           118,951  45%
   /29           524,288  3          221,184  42%           228,210  44%
   /28         1,048,576  3          442,368  42%           437,827  42%
   /27         2,097,152  3          884,736  42%           839,983  40%
   /26         4,194,304  3        1,769,472  42%         1,611,531  38%
   /25         8,388,608  3        3,538,944  42%         3,091,767  37%
   /24        16,777,216  3        7,077,888  42%         5,931,642  35%
   /23        33,554,432  4       14,155,776  42%        11,380,022  34%
   /22        67,108,864  4       21,233,664  32%        21,832,894  33%
   /21       134,217,728  4       42,467,328  32%        41,887,023  31%
   /20       268,435,456  4       84,934,656  32%        80,361,436  30%
   /19       536,870,912  4      169,869,312  32%       154,175,684  29%
   /18     1,073,741,824  4      339,738,624  32%       295,790,403  28%
   /17     2,147,483,648  4      679,477,248  32%       567,482,240  26%
   /16     4,294,967,296  4    1,358,954,496  32%     1,088,730,702  25%
   /15     8,589,934,592  5    2,717,908,992  32%     2,088,760,595  24%
   /14    17,179,869,184  5    4,076,863,488  24%     4,007,346,185  23%
   /13    34,359,738,368  5    8,153,726,976  24%     7,688,206,818  22%
   /12    68,719,476,736  5   16,307,453,952  24%    14,750,041,884  21%
   /11   137,438,953,472  5   32,614,907,904  24%    28,298,371,876  21%
   /10   274,877,906,944  5   65,229,815,808  24%    54,291,225,552  20%
   /9    549,755,813,888  5  130,459,631,616  24%   104,159,249,331  19%
   /8  1,099,511,627,776  5  260,919,263,232  24%   199,832,461,158  18%

                   Table 3: Limited Levels of Structure

                                 Figure 9

Author's Address

   Geoff Huston
   APNIC

   EMail: gih@apnic.net











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