The H-Density Ratio for Address Assignment Efficiency An Update on the H ratio

Network Working Group                                          A. Durand
Request for Comments: 3194                              SUN Microsystems
Updates: 1715                                                 C. Huitema
Category: Informational                                        Microsoft
                                                           November 2001


       The Host-Density Ratio for Address Assignment Efficiency:
                        An update on the H ratio

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 (2001).  All Rights Reserved.

Abstract

   This document provides an update on the "H ratio" defined in RFC
   1715.  It defines a new ratio which the authors claim is easier to
   understand.

1. Evaluating the efficiency of address allocation

   A naive observer might assume that the number of addressable objects
   in an addressing plan is a linear function of the size of the
   address.  If this were true, a telephone numbering plan based on 10
   digits would be able to number 10 billion telephones, and the IPv4 32
   bit addresses would be adequate for numbering 4 billion computers
   (using the American English definition of a billion, i.e. one
   thousand millions.) We all know that this is not correct: the 10
   digit plan is stressed today, and it handles only a few hundred
   million telephones in North America; the Internet registries have
   started to implement increasingly restrictive allocation policies
   when there were only a few tens of million computers on the Internet.

   Addressing plans are typically organized as a hierarchy: in
   telephony, the first digits will designate a region, the next digits
   will designate an exchange, and the last digits will designate a
   subscriber within this exchange; in computer networks, the most
   significant bits will designate an address range allocated to a
   network provider, the next bits will designate the network of an
   organization served by that provider, and then the subnet to which
   the individual computers are connected.  At each level of the
   hierarchy, one has to provide some margins:  one has to allocate more
   digits to the region code than the current number of regions would
   necessitate, and more bits in a subnet than strictly required by the
   number of computers.  The number of elements in any given level of
   the   hierarchy will change over time, due to growth and mobility.
   If the current allocation is exceeded, one has to engage in
   renumbering, which is painful and expensive.  In short, trying to
   squeeze too many objects into a hierarchical address space increases
   the level of pain endured by operators and subscribers.

   Back in 1993, when we were debating the revision of the Internet
   Protocol, we wondered what the acceptable ratio of utilization was of
   a given addressing plan.  Coming out with such a ratio was useful to
   assess how many computers could be connected to the Internet with the
   current 32-bit addresses, as well as to decide the size of the next
   generation addresses.  The second point is now decided, with 128-bits
   addresses for IPv6, but the first question is still relevant:
   knowing the capacity of the current address plan will help us predict
   the date at which this capacity will be exceeded.

   Participants in the IPNG debates initially measured the efficiency of
   address allocation by simply dividing the number of allocated
   addresses by the size of the address space.  This is a simple
   measure, but it is largely dependent on the size of the address
   space.  Loss of efficiency at each level of a hierarchical plan has a
   multiplicative effect; for example, 50% efficiency at each stage of a
   three level hierarchy results in a overall efficiency of 12.5%.  If
   we want a "pain level indicator", we have to use a ratio that takes
   into account these multiplicative effects.

   The "H-Ratio" defined in RFC 1715 proposed to measure the efficiency
   of address allocation as the ratio of the base 10 logarithm of the
   number of allocated addresses to the size of the address in bits.
   This provides an address size independent ratio, but the definition
   of the H ratio results in values in the range of 0.0 to 0.30103, with
   typical values ranging from 0.20 to 0.28.  Experience has shown that
   these numbers are difficult to explain to others; it would be easier
   to say that "your address bits are used to 83% of their H-Density",
   and then explain what the H-Density is, than to say "you are hitting
   a H ratio of 0.25" and then explain what exactly the range is.

   This memo introduces the Host Density ratio or "HD-Ratio", a proposed
   replacement for the H-Ratio defined in RFC 1715.  The HD values range
   from 0 to 1, and are generally expressed as percentage points; the
   authors believe that this new formulation is easier to understand and
   more expressive than the H-Ratio.

2. Definition of the HD-ratio

   When considering an addressing plan to allocate objects, the host
   density ratio HD is defined as follow:

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

   This ratio is defined for any number of allocatable objects greater
   than 1 and any number of allocated objects greater or equal than 1
   and less than or equal the maximum number of allocatable objects.
   The ratio is usually presented as a percentage, e.g. 70%.  It varies
   between 0 (0%), when there is just one allocation, and 1 (100%), when
   there is one object allocated to each available address.  Note that
   for the calculation of the HD-ratio, one can use any base for the
   logarithm as long as it is the same for both the numerator and the
   denominator.

   The HD-ratio can, in most cases, be derived from the H ratio by the
   formula:

           H
   HD = --------
        log10(2)

3. Using the HD-ratio as an indicator of the pain level

   In order to assess whether the H-Ratio was a good predictor of the
   "pain level" caused by a specific efficiency, RFC1715 used several
   examples of networks that had reached their capacity limit.  These
   could be for example telephone networks at the point when they
   decided to add digits to their numbering plans, or computer networks
   at the point when their addressing capabilities were perceived as
   stretched beyond practical limits.  The idea behind these examples is
   that network managers would delay renumbering or changing the network
   protocol until it became just too painful; the ratio just before the
   change is thus a good predictor of what can be achieved in practice.
   The examples were the following:

   * Adding one digit to all French telephone numbers, moving from 8
   digits to 9, when the number of phones reached a threshold of 1.0
   E+7.

                                  log(1.0E+7)
      HD(FrenchTelephone8digit) = ----------- = 0.8750 = 87.5%
                                  log(1.0E+8)


                                  log(1.0E+7)
      HD(FrenchTelephone9digit) = ----------- = 0.7778 = 77.8%
                                  log(1.0E+9)

   * Expanding the number of areas in the US telephone system, making
   the phone number effectively 10 digits long instead of "9.2" (the
   second digit of area codes used to be limited to 0 or 1) for about
   1.0 E+8 subscribers.

                                log(1.0E+8)
      HD(USTelephone9.2digit) = ------------ = 0.8696 = 87.0 %
                                log(9.5E+9)


                                log(1.0E+8)
      HD(USTelephone10digit)  = ------------ = 0.8000 = 80.0 %
                                log(1E+10)

   * The globally-connected physics/space science DECnet (Phase IV)
   stopped growing at about 15K nodes (i.e. new nodes were hidden) in a
   16 bit address space.

                      log(15000)
      HD(DecNET IV) = ---------- = 0.8670 = 86.7 %
                      log(2^16)

   From those examples, we can note that these addressing systems
   reached their limits for very close values of the HD-ratio.  We can
   use the same examples to confirm that the definition of the HD-ratio
   as a quotient of logarithms results in better prediction than the
   direct quotient of allocated objects over size of the address space.
   In our three examples, the direct quotients were 10%, 3.2% and 22.8%,
   three very different numbers that don't lead to any obvious
   generalization.  The examples suggest an HD-ratio value on the order
   of 85% and above correspond to a high pain level, at which operators
   are ready to make drastic decisions.

   We can also examine our examples and hypothesize that the operators
   who renumbered their networks tried to reach, after the renumbering,
   a pain level that was easily supported.  The HD-ratio of the French
   or US network immediately after renumbering was 78% and 80%,
   respectively.  This suggests that values of 80% or less corresponds
   to comfortable trade-offs between pain and efficiency.

4. Using the HD-ratio to evaluate the capacity of addressing plans

   Directly using the HD-ratio makes it easy to evaluate the density of
   allocated objects.  Evaluating how well an addressing plan will scale
   requires the reverse calculation.  We have seen in section 3.1 that
   an HD-ratio lower than 80% is manageable, and that HD-ratios higher
   than 87% are hard to sustain.  This should enable us to compute the
   acceptable and "practical maximum" number of objects that can be
   allocated given a specific address size, using the formula:

   number allocatable of objects
               = exp( HD x log(maximum number allocatable of objects))
               = (maximum number allocatable of objects)^HD

   The following table provides example values for a 9-digit telephone
   plan, a 10-digit telephone plan, and the 32-bit IPv4 Internet:

                                             Very  Practical
                     Reasonable  Painful  Painful    Maximum
                         HD=80%   HD=85%   HD=86%     HD=87%
   ---------------------------------------------------------
   9-digits plan           16 M     45 M     55 M       68 M
   10-digits plan         100 M    316 M    400 M      500 M
   32-bits addresses       51 M    154 M    192 M      240 M

   Note: 1M = 1,000,000

   Indeed, the practical maximum depends on the level of pain that the
   users and providers are willing to accept.  We may very well end up
   with more than 154M allocated IPv4 addresses in the next years, if we
   are willing to accept the pain.

5. Security considerations

   This document has no security implications.

6. IANA Considerations

   This memo does not request any IANA action.

7. Author addresses

   Alain Durand
   SUN Microsystems, Inc
   901 San Antonio Road MPK17-202
   Palo Alto, CA 94303-4900
   USA

   EMail: Alain.Durand@sun.com


   Christian Huitema
   Microsoft Corporation
   One Microsoft Way Redmond, WA 98052-6399
   USA

   EMail: huitema@microsoft.com

8. Acknowledgment

   The authors would like to thank Jean Daniau for his kind support
   during the elaboration of the HD formula.

9. References

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

   [IANAV4]  INTERNET PROTOCOL V4 ADDRESS SPACE, maintained by the IANA,
             http://www.iana.org/assignments/ipv4-address-space

   [DMNSRV]  Internet Domain Survey, Internet Software Consortium,
             http://www.isc.org/ds/

   [NETSZR]  Netsizer, Telcordia Technologies, http://www.netsizer.com/

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