rfc956.txt

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Mills                                                           [Page 5]



RFC 956                                                   September 1985
Algorithms for Synchronizing Network Clocks


3.  Clustering Algorithms

   Another method for developing a maximum-likelihood estimator is
   through the use of clustering algorithms.  These algorithms operate
   to associate points in a sample set with clusters on the basis of
   stochastic properties and are most useful when large numbers of
   samples are available.  One such algorithm operates on a sample set
   to selectively discard points presumed outside the cluster as
   follows:

      1.  Start with a sample set of n observations {x(1),x(2),...,x(n)

      2.  Compute the mean of the n observations in the sample set.
          Discard the single sample x(i) with value furthest from the
          mean, leaving n-1 observations in the set.

      3.  Continue with step 2 until only a single observation is left,
          at which point declare its value the maximum-likelihood
          estimator.

   This algorithm will usually (but not necessarily) converge to the
   desired result if the majority of observations are the result of
   "good" clocks, which by hypothesis are clustered about zero offset
   relative to the reference clock, with the remainder scattered
   randomly over the observation interval.

   The following Table 3 summarizes the results of this algorithm
   applied to the UDP data shown in Appendix A, which represents the
   measured clock offsets of 163 host clocks in the Internet system.
   These data were assembled using the UDP Time protocol [7], in which
   time is represented to a precision of one second.  Each line of the
   table represents the result of step 2 above along with the size of
   the sample set and its (unweighted) mean and variance.  The "Discard"
   column shows the value of the observation discarded at that step.















Mills                                                           [Page 6]



RFC 956                                                   September 1985
Algorithms for Synchronizing Network Clocks


                    Size    Mean    Var     Discard
                    -------------------------------
                    163     -210    9.1E+6  -38486 
                    162     26      172289  3728   
                    161     3       87727   3658   
                    160     -20     4280    -566   
                    150     -17     1272    88     
                    100     -18     247     -44    
                    50      -4      35      8      
                    20      -1      0       -2     
                    19      -1      0       -2     
                    18      -1      0       -2     
                    17      -1      0       1      
                    16      -1      0       -1     
                    15      -1      0       -1     
                    14      -1      0       -1     
                    13      0       0       0      
                    1       0       0       0      

       Table 3. Clustering Algorithm using UDP Time Protocol Data

   In Table 3 only a few of the 163 steps are shown, including those
   near the beginning which illustrate a rapid convergence as the
   relatively few outliers are discarded.  The large outlier discarded
   in the first step is almost certainly due to equipment or operator
   failure. The two outliers close to one hour discarded in the next two
   steps are probably simple operator mistakes like entering summer time
   instead of standard time.  By the time only 50 samples are left, the
   error has shrunk to about 4 sec and the largest outlier is within 12
   sec of the estimate.  By the time only 20 samples are left, the error
   has shrunk to about a second and the variance has vanished for
   practical purposes.

   The following Table 4 summarizes the results of the clustering
   algorithm applied to the ICMP data shown in Appendix A, which
   represents the measured clock offsets of 504 host clocks in the
   Internet system. These data were assembled using ICMP Timestamp
   messages [6], in which time is represented to a precision of one
   millisecond.  The columns in Table 4 should be interpreted in the
   same way as in Table 3, except that the data in Table 4 are in
   milliseconds, while the data in Table 3 are in seconds.








Mills                                                           [Page 7]



RFC 956                                                   September 1985
Algorithms for Synchronizing Network Clocks


                    Size    Mean    Var     Discard
                    -------------------------------
                    504     -3.0E+6 3.2E+14 8.6E+7 
                    500     -3.3E+6 2.9E+14 8.6E+7 
                    450     -1.6E+6 3.0E+13 -2.5E+7
                    400     29450   2.2E+11 3.6E+6 
                    350     -3291   4.1E+9  -185934
                    300     3611    1.6E+9  -95445 
                    250     2967    6.8E+8  66743  
                    200     4047    2.3E+8  39288  
                    150     1717    8.6E+7  21346  
                    100     803     1.9E+7  10518  
                    80      1123    8.4E+6  -4863  
                    60      1119    3.1E+6  4677   
                    50      502     1.5E+6  -2222  
                    40      432     728856  2152   
                    30      84      204651  -987   
                    20      30      12810   338    
                    15      28      2446    122    
                    10      7       454     49     
                    8       -2      196     24     
                    6       -9      23      0      
                    4       -10     5       -13    
                    2       -8      0       -8     

        Table 4. Clustering Algorithm using ICMP Timestamp Data

   As in Table 3 above, only some of the 504 steps are shown in Table 4.
   The distinguishing feature of the data in Table 4 is that the raw
   data are much more noisy - only some 30 host clocks are closer than
   one second from the reference clock, while half were further than one
   minute and over 100 further than one hour from it.  The fact that the
   algorithm converged to within 8 msec of the reference time under
   these conditions should be considered fairly remarkable in view of
   the probability that many of the outliers discarded are almost
   certainly due to defective protocol implementations.  Additional
   information on these experiments is presented in Appendix A.












Mills                                                           [Page 8]



RFC 956                                                   September 1985
Algorithms for Synchronizing Network Clocks


4.  Application to Time-Synchronization Data

   A variation of the above algorithms can be used to improve the offset
   estimates from a single clock by discarding noise samples produced by
   occasional retransmissions in the network, for example.  A set of n
   independent samples is obtained by polling the clock.  Then, a
   majority-subset table is used to compute the m(j) and s(j) statistics
   and the maximum-likelihood estimate determined as above.  For this
   purpose the majority-subset table could include larger subsets as
   well. In this manner the maximum-likelihood estimates from each of
   several clocks can be determined and used in the algorithm above.

   In order to test the effectiveness of this algorithm, a set of
   experiments was performed using two WIDEBAND/EISN gateways equipped
   with WWVB radio clocks and connected to the ARPANET.  These
   experiments were designed to determine the limits of accuracy when
   comparing these clocks via ARPANET paths.  One of the gateways
   (ISI-MCON-GW) is located at the Information Sciences Institute near
   Los Angeles, while the other (LL-GW) is located at Lincoln
   Laboratories near Boston.  Both gateways consist of PDP11/44
   computers running the EPOS operating system and clock-interface
   boards with oscillators phase-locked to the WWVB clock.

   The clock indications of the WIDEBAND/EISN gateways were compared
   with the DCNet WWVB reference clock using ICMP Timestamp messages,
   which record the individual timestamps with a precision of a
   millisecond. However, the path over the ARPANET between these
   gateways and the measurement host can introduce occasional
   measurement errors as large as several seconds.  In principle the
   effect of these errors can be minimized by using a large sample
   population;  however, use of the above algorithms allows higher
   accuracies to be obtained with far fewer samples.

   Measurements were made separately with each of the two gateways by
   sending an ICMP Timestamp Request message from the ARPANET address of
   DCN1 to the ARPANET address of the gateway and computing the
   round-trip delay and clock offset from the ICMP Timestamp Reply
   message.  This process was continued for 1000 message exchanges,
   which took from seven minutes to several hours, depending on the
   sample interval selected.

   The tables below summarize the results of both the majority-subset
   and clustering algorithms applied to the data from three experiments,
   one with ISI-MCON-GW and two with LL-GW.  The ISI-MCON-GW and LL-GW
   (a) experiments were designed to determine the limits of accuracy
   when using a continuous sequence of request/reply volleys, which



Mills                                                           [Page 9]



RFC 956                                                   September 1985
Algorithms for Synchronizing Network Clocks


   resulted in over two samples per second.  The remaining LL-GW (b)
   experiment was designed to determine the limits of accuracy using a
   much lower rate of about one sample every ten seconds.

   For each of the three experiments two tables are shown, one using the
   majority-subset algorithm and the other the clustering algorithm. The
   two rows of the majority-subset tables show the statistics derived
   both from the raw data and from the filtered data processed by a
   C(5,3) majority-subset algorithm.  In all cases the extrema and
   variance are dramatically less for the filtered data than the raw
   data, lending credence to the conjecture that the mean statistic for
   the filtered data is probably a good maximum-likelihood estimator of
   the true offset.

                              Mean    Var     Max     Min
             -------------------------------------------- 
             Raw data        637     2.1E+7  32751   -1096
             C(5,3)          -15     389     53      -103 

             Table 5. ISI-MCON-GW Majority-Subset Algorithm

                    Size    Mean    Var     Discard
                    -------------------------------
                    1000    637     2.1E+7  32751  
                    990     313     1.0E+7  32732  
                    981     15      1.0E+6  32649  
                    980     -18     2713    -1096  
                    970     -15     521     -122   
                    960     -15     433     -97    
                    940     -15     332     -75    
                    900     -15     239     26     
                    800     -15     141     12     
                    700     -16     87      5      
                    600     -17     54      -31    
                    500     -16     33      -5     
                    400     -18     18      -9     
                    300     -19     7       -12    
                    200     -19     2       -21    
                    100     -18     0       -19    
                    1       -17     0       -17    

               Table 6. ISI-MCON-GW Clustering Algorithm