1
TR30.3/01-03-015
Dear Mrs. Hoyler.
We ask you to put for discussion in TIA Engineering committee the
following
our offer or to inform us that it is necessary to undertake to us for
this
purpose
We offer in addition to standard interpretation results of the tests,
described in standard TSB-38 Test Procedure for Evaluation of 2-Wire
4-Kilohertz Voiceband Duplex Modems' to enter the following
additional
characteristics of work of modems.
1.
Statistical estimations of throughput mean value and
modem
function stability.
2.
An estimation of modem function stability in time and
revealing degradation, i.e. increasing of failure probability at each
subsequent test in comparison with preceding on long range of
connections.
Now our firm uses an own technique of a quantitative estimation of
throughput mean value and stability of work of modems. This
technique, using
methods of mathematical statistics, allows quantitatively to estimate
the
significance of a divergence between received in experiments
throughput mean
values, to estimate stability of functioning of compared modems and
the
significance of a divergence of these estimations. Application of
this
technique enables to compare compared modems more objectively.
Moreover in our firm new technique of estimation of modem functioning
stability in time and revealing of degradation (i.e. increasing of
failure
probability on each subsequent test in comparison with preceding one
, at
performance of a long series of connections) is elaborated.
It is represented to us expedient to add standard TSB38 at it next
revision
by the techniques developed by us that will provide uniform and more
objective approach to comparison of different modems and will allow
to
estimate quantitatively the significance of a divergence of
parameters of
functioning of compared modems
We count, that these techniques can serve as the prototype for these
additions.
If you will consider necessary, our firm is ready to participate in
works on
addition of standard TSB38 by offered techniques.
The mathematical description of offered techniques is applied.
Correspondence on the given question is charged to Alfred
Soloveychik. His
e-mail address is Alfred@Smlink.com.
Yours sincerely
2
MEAN VALUE THROUGHPUT ESTIMATION ON TEST
RESULT BASIS
Modem throughput is defined on a test result basis. Each test
consists of data file transfer under setting conditions (on lines in
which impairments are set by test equipment) and throughput
measurement under these conditions. In order to achieve a more
precise results each test may be repeated several times. The
throughput average is defined by weighting line throughputs in
accordance with the line probability at the Network
Communication Model, i.e.:
T = (p  tI ) / p I
Here:
tI – average throughput received as test results at the line
number i.
p – probability (score) of this line in Network
Communication Model,
Therefore throughput Mean Value is a Lineary Function of
throughputs received as test results on each lines. Then in
accordance with mathematical statistics laws his mathematical
expectation equals the same function of the mathematical
expectations of the tested lines throughputs:
M(T) = ( p  M(tI )) /pI
And variance of Mean Value equals
D(T) = ( p 2 D(tI )) / (p I)2
In this case throughput mathematical expectations are calculated
by known mathematical statistics formulas:
M (t)= ti / ni
and
D(t) = (ti – M(t))2 / (ni (ni – 1))
here
ni – test numbers at the line number i
Consequently the calculated accordance formula
T = (pI  tI ) / p I
I
I
I
I
3
Throughput Mean Value is distributed according to Student
(Gousset) distribution that draws nearer to the Normal
distribution of Gauss when test number is large enough. Such
test number usually completed at the modem testing.
Parameters of this distribution are Mean Value M(T) and
standard deviation that
 =  D(t)
Confidence interval of the throughput Mean Value is defined by
formula
M(T)  k
Here
.k – coefficient that dependent on accepted confidence
probability and number of degrees of freedom of computed
statistic. That is equal number of completed tests without one
and minus number of independent variables (in this case –
number of terms)
Since the calculated Throughput Mean Value are considered as
estimate of its mathematical expectation then it’s possible to
write the Confidence interval formula as
T  k
For usually accepted confidence probabilities P=0.95 value k =
1.96 , for P = 0.99 k = 2.58.
COMPARISON OF THROUGHPUT MEAN VALUES ON
TEST RESULT BASIS
.
Received results let us possibility to define significant of
difference between Throughput Mean Values obtained at
different tests. A Mathematical expectation of Throughput Mean
Value difference equals the difference of their mathematical
expectations:
M() =M(T1 – T2) = = M(T1 ) – M(T2 )
If throughputs of both modems are equal, then their
mathematical expectations are equal too
4
M(T1 ) = M(T2 )
And therefore mathematical expectation of their difference
equals zero
M() = 0
However because of influence of diverse factors that affect at
throughput its values T1 и T2 obtained by different tests will be
different. Their difference as well as difference of two normal
distributed quantities has normal distribution with mathematical
expectation M()
And variance equal a variance sum of the compared Mean
Values
D() = D(T1 ) + D(T2 )
Accordingly its standard deviation equal
() =  D()
The obtained throughput difference ought to be identified as
significant if counted difference
 = T1 – T2
statistic significantly distinguishes from zero (i.e zero places out
of the confidence interval). In order to check of significant of
obtained difference from zero it’s necessary to compute statistic
t =  / ()
that ought to be compared with table values corresponding to
acceptation values of confidence probability taking in account
number of freedom degree, that is equal sum of numbers of
freedom degree of comparing Mean Value, for example
accepted above values 1.96 and 2.58 accordingly for confidence
probability 0.95 and 0.99.If a computed value of t criterion
exceeds an according table value then difference is identified as
statistical significant under accepted confidence probability. In
other case obtained Mean Values may be identified as coincide.
COMPARISON OF MEAN VALUE THROUGHPUT
DISPERSION ON TEST RESULT BASIS
A Mean Value Throughput dispersion is characterized by
variance D(T) that is computed according to appointed above.
5
Besides of this for visual representation it may be used variance
coefficient that equal
V =  / M(T)
usually expressed in percent. Naturally the less variance
coefficient the more function quality of the tested modem.
For variance comparison of obtained results and estimation of a
difference significant it is used Fisher Criterion
F = D(T1 ) / D(T2 )
In this formula D(T1 ) is larger from comprised variances and
D(T1 ) - less. Numbers of the freedom degree ought to be
computed for each variance apart as number of completed test
without one minus number of terms.
An obtained Value of F criterion ought to be compared with a
Table Value for accepted Values of confidence Probability and
if it is less than Table Value then difference may be identified as
insignificant and therefore the throughput results dispersion is
the same in both tests.
EXAMPLE OF THE MEAN VALUE THROUGHPUT
ESTIMATION ON TEST RESULT BASIS
Computation of the described above estimation of Throughput
Mean Value and its variance produce no difficulties and may be
automatic by computing program. Below there is consideration
of using of the throughput Mean Value Estimation for modem
comparison and their quality definition
1. Initial Data
On the picture below is showed graphs of the throughput
providing (graph of probability throughput that is not less than
present) between tested modem and router Ascend, those were
obtained as tests result.
6
6000
5000
Throughput (cps)
4000
Modem A2350 . Mean Value =5077+/-49 (P=0,95). Failures NMC = 0.0%.
3000
2000
A2709 Modem. Mean Value = 5009+/-55 (P=0.95). Failures NMC = 0.571%.
1000
A2291 Modem. Mean Value = 5092+/-22 (P=0.95). Failures NMC = 0.146%.
0
0
10
20
30
40
50
60
70
80
90
Network Model Coverage (%)
In the table below it is represented statistical parameters of test
results that are showed at the graph. Those parameters are
computed in accordance to formulas above.
THE THROUGHPUT TEST RESULTS
Names of tests
and modems
Test number
Mean Value
cps
A2291 Modem
A2350 Modem
A2709 Modem
320
320
320
5092
5077
5009
Standart
Deviation of
Mean Value
cps
11.409
24.986
28.278
2. Computation of confidential interval
At first confidential intervals of the throughput Mean Values are
computed. Standard deviation of throughput Mean Value of the
A2291 Modem equal 11.409. In order to define of upper limit
deviation from Mean value it’s necessary to multiply the
standard deviation magnitude at coefficient, that is according to
100
7
accepted confidence Probability taking in account number of
freedom degree. In this case number of freedom degree equal
 = 320 –1 –160 = 159
When confidential probability is 0.95 and number of freedom
degree is 159 this coefficient equals 1.975. Therefore a upper
limit deviation from Mean Value is 11.4091.975 = 22.533 or
approximately 22. Confidential interval for obtained Mean
Value may be written as 509222 when confidential probability
is 0.95. In other words it may be said, when the same test series
are repeated with tested modem Throughput Mean Value will be
in the interval 509222 or between 5070 and 5114 in the 95% of
series. At the similar computations with confidence probability
0.95 it‘s obtained confidence interval for the A2350 modem is
507749 and for A2709 modem - 500955.
3. Estimation of Mean Value difference.
Now we shall study significant of the Throughput Mean Value
difference obtained at those tests
Difference between Throughput Mean Values of the A2291 and
A2350 Modems is
 = 5092 – 5077 = 15 cps
and its variance is
D = 11.4092 + 24.9862 = 754.466
And correspondingly standard deviation is
 = 754.466 = 27.468
At last in order to define significant of the obtained difference
criterion t is computed:
t = 15 / 27.468 = 0.546
Now by using Statistical Tables of the Student function or the
Excel function TDIST it is computed probability, that is
corresponding to the obtained magnitude of the t criteria and the
number of freedom degree. In studied case number of freedom
degree is accepted 318, i.e. sum of number freedom degree of
each Mean Value. On two tail significant degree this probability
equal
8
P(0.546 ) = 0.585
So when the same test series are repeated with compared
modems probability of events, that differences of Throughput
Mean Values that will be obtained as test result will be more
than difference obtained in studied tests, is 0.585, i.e. more than
half events. Off course this probability is very significant, and
because of it the obtained difference between Throughput Mean
Values of comparing series ought to be defined as statistical
insignificant. Or in other words Throughput Mean Value of
compared modems coincide.
Difference of throughput Mean Values of the A2291 and A2709
equal
 = 5092 – 5009 = 83 cps
and variance of this value is
D = 11.4092 + 28.2782 = 929.811
Accordingly standard deviation equal
 = 929.811 = 30.493
The t criteria magnitude is founded
t = 83 / 30.493= 2.722
and probability to surpass this magnitude taking in account
number of freedom degree and a 2 tail significant criteria is
P(2.722 ) = 0.0068
So probability to surpass occasionally the obtained difference
between Throughput Mean Values equal approximately 0.7%.
Events of this probability are rare events. Because of it obtained
difference is not result of some accidental factors but is
provocative by essential characterizes of those modems. In other
words obtained Throughput Mean Value difference is statistical
significant.
Difference between Throughput Mean Values of the A2350 and
A2709 modems is 68 cps, and its variance and standard
deviation equal 1423.945 and 37.735 accordingly. Under this
conditions the t criteria is 1.802 and the according probability
equal 0.072. Under usual significant degree of 5 % this
difference is not statistical significant and may be defined as
result of occasional fluctuations of test result. So difference
9
between Throughput Mean Values of the A2350 and A2709
modems is statistical insignificant.
4. Estimation of variance difference
The estimation of variance diversity is completed by using of
the Fisher criterion, that equal ratio of compared variances. For
the A2291 and A2350 Modems Fisher criterion equal
F = 24.9862 / 11.4092 =4.796
Probability that corresponds to this Fisher criterion value is
close to zero. I.e. an event to obtain by chance magnitude of
Fisher criterion more than it have obtained is an impossible
event practically. Therefore obtained value of this criterion is
very significant, and consequently obtained difference between
variances is very significantly too. Therefore stability of the
A2291 modem is significantly better than A2350 modem one.
Because of standard deviation of the A2709 modem Mean
Value is even more than of the A2350 modem one then variance
comparison of the A2291 and A2709 modems will obtain the
same result: the A2709 Modem Variance is significantly more
than the A2291 modem variance of Throughput Mean Value. Or
the A2709 modem stability is significantly worse than the
A2291 modem one.
Now we shall compare the A2350 and A2709 variances.
Magnitude of fisher Criterion for their variances equal
F = 28.2782 / 24.9862 = 1.281
When number of freedom degree for each variance is 159
probability that corresponds to this value equal
P(F > 1.281) = 0.0598
So an event probability that obtained Value of Fisher Criterion
will be surpassed is found 5.98%, i.e. when the same test series
are repeated with compared modems approximately at the 6%
Series a Fisher criterion will be more than obtained in studied
tests by chance. Under usual 5% grade of significant the
obtained value of the Fisher criterion is defined as insignificant.
Therefore obtained variances ought to be defined as not
distinguished one from other and stability of the compared
modems ought to be defined as equally.
10
5. Results of modem comparison
Results of modem characteristic comparison may be represented
in table to easy to read as below
Comparison of statistical characteristics of tested modems
reveals that the best characteristics of Throughput Mean Value
as well as functioning stability (the least variance Value) belong
to A2291 modem. The throughput Mean Value of the A2350
modem is close to the throughput Mean Value of the A2291
Modem, but its functioning stability is significantly worse than
the A2291 one.
Throughput of the A2709 modem is significantly less than
throughput of the A2291 modem as well as A2350 modem.
Throughput variance of the A2709 modem is significantly more
than throughput variance of the A2350, but it distinguished
statistical insignificantly from throughput variance of the A2350
modem.. Therefore functioning stability of this modem is
significantly lower than the A2291 modem one, but it is at the
same grade of functioning stability as A2350 modem
approximately.
11
Modem names
A2291 Modem
A2350 Modem
A2709 Modem
Test number
320
320
320
Throughpu
t Mean
Value
cps
Standard
deviation
of Mean
Value
cps
5092
5077
5009
11.409
24.986
28.278
Probability of surpass
of obtained Mean
Value difference
With
With
modem
modem
A2291
A2350
0.585
0.0068
0.072
Probability of surpass
of obtained variance
With
modem
A2291
With
modem
A2350
0.0
0.0
0.0598
12
RESUME
Using of the described statistical analyze methods for
throughput test processing gives a possibility to estimate an
achieved exactness of finite test results (to estimate confidence
interval of computed Mean Values) and to identify statistical
significant of different test result difference
13
ESTIMATION OF MODEM DEGRADATION AND
DEFINITION OF ITS PARAMETERS

DEGRADATION CRITERION
Notion “degradation” in this work means increasing of
probability of information transfer failures in long modem
work. When degradation is significant and modem was
functioning long time it is come moment, when each attempt
of information transfer comes to malfunction (failure) of
connection. The goal of this work are definition of
dependence of failure number from completed transfer
number and computing of parameters of this dependence
depending on completed test results.
This dependence is presented as equation of failure number
depending on completed failure number, that in this case is a
measure of duration of continues modem function. As
original assumption it is supposed that failure probability
increases after each transfer and at the end of long
sufficiently test range failure probability became close to
one. We used as mathematical model of such process
following formula for computing failure probability p in the
time of test number n:
p = 1 – exp(-n/k),
where k – degradation parameter.
This model matches to gradual accumulation of small
factors those together carry out transfer failure (“fatigue”).
Total failure number after n tests equals
f = 0np
for any number connections.
Suggesting that value n is continuous, that is possible when
values n is great enough (or connection number is many
sufficiently) it’s possible to write formula for computing of
total failure number as
f =0Npdn = 0 N (1 – exp(-n/k)dn),
14
where N is total number of completed connections.
After mathematical transformations this formula take up
form
f = N – k(1 – exp(-N/k))
When N = 0 value f equals zero too. This is according to
physical meaning of studying process: if there were not
connections, there were not failures. When N is sufficiently
great value exp(-N/k) draw nearer to zero, and, consequently,
failure number asymptotically draw nearer to straight line
f=N–k
This asymptote intersects axis X at the point (k,0) and has
incline 45 degree.
Thus value k describes modem degradation completely and
may be used as its criteria.
COMPUTATION OF DEGRADATION CRITERIA ON
TEST RESULTS BASIS
Initial data for detection of degradation criteria value are
results of test for throughput detection. As results of those
tests throughput values are detected under test conditions.
When there is failure throughput value is equal zero.
For detection of criteria degradation data of cumulative
failure number after fixed number tests is used.
Degradation criteria value must be defined so as function
computed according formula
f = N – k(1 – exp(-N/k))
is nearest to graph obtained as tests results. Criteria of
closeness (proximity) of theoretical and experimental
functions is sum of squares of their differences when their
arguments (in this case – number of executed tests) are the
same:
MIN(f - f )2 = MIN(f - (N – k(1 – exp(-N/k))))2
Using of usual method of minimum finding by computation
of first derivative of obtained function and equation decision
when this derivative equals zero in this case run to
r
c
r
15
transcendental equation that is not solved by elementary
methods. Therefore in order to define degradation criterion
it’s necessary to use digital methods. Research of
dependence between sum of squares of differences of
computed values of failure numbers and values obtained as
test results and criteria degradation value reveals following.
This dependence is continued function with one minimum.
Graph of this dependence (Fig 1) shows that at the left from
minimum point this function quickly decreases and at the
right of this point it slowly increases up to infinity.
Sum Squeris of diversits
Depend of Sum Squere of diversits between degradation curve and obtained integrates failures
curve from degradation coefficient Value
Degradation Coeffitient
Fig. 1
Due to small value of function gradient at the right side from
minimum point standard method of minimum finding –
method of steepest descent – is a weakly convergence and it’s
using isn’t profitable. Because of it for finding of point
minimum of studied function modified method of half
division ought to be used. When this method was used at the
beginning of calculation upper and lower values of
degradation coefficient are set. After this the studied
function value and its gradient are calculated at the middle
of degradation coefficient interval. In accordance with
16
gradient sign it is determined in which patch of studied
interval there is desired minimum. After this for patch that
contained minimum described computations repeated. This
computation repeats up to desired exactness of degradation
coefficient will be achieved.
COMPUTATION OF CUMULATIVE FAILURE
NUMBER WHEN DEGRADATION IS ABSENT
When degradation is absent failure probability p during of
each test is constant irrespective of preceding test number.
Because of it cumulative failure number after competing of
N test equals
f =1N p = p N
Consequently graph of cumulative failure number is straight
line that goes through origin of a coordinates system with
slope p.
For determination of p value data of cumulative failure
number after fixed number of throughput tests are used too.
Value p ought to be defined so as function computed
according formula
f==pN
is nearest to obtained test results. . Criteria of closeness
(proximity) of theoretical and experimental functions is sum
of squares of their differences when their arguments (in this
case – number of executed tests) are the same:
MIN(f - f )2 = MIN(f - p N)2
This problem is decided by standard mathematical methods
and its decision is
p = 1N ( i fi ) / 1N i2
where
i – ordinal test number;
fi – cumulative number of failures after executing of i
connections
r
c
r
DETECTION OF DEGRADATION PRESENCE BY
RESULTS OF EXPERIMENTS
17
For detection of modem degradation presence it’s necessary
to detect which from the two described equations is closer to
dependence between executed tests number and cumulative
failure number that obtained as test results. The degree of
affinity of the obtained test results and the equation that
describes this dependence is estimated by value of variance
of diversions cumulative failure number from rated values
calculated by using equations of degradation.
Variance Value is computed by dependence
s = 1N (fi – fic )2 / (N – 1)
where
fi - cumulative failure number after executing of i tests.;
fic – rated (computed according to studied formula)
cumulative failure number after executing of i tests;
N – total number of executed tests.
Variances computed for each of described above equations
are comprised among themselves by Fisher criterion that
equals ratio of the most dispersion to the least one. Fisher
criterion Value detects probability of event that obtained
ratio of variances would be surpassed by chance. If this
probability is small enough (usually it is used confidence
probability 5%), then difference of comprised variances is
admitted as significant. And if at the same time value of
variance calculated for degradation equation is smaller than
value variance calculated for linear equation (without
degradation) then it’s necessary to make conclusion that
there is degradation at tested modem function. If variance of
the linear equation is smaller than variance calculated for
degradation equation then it’s necessary to make conclusion
that there is no degradation at tested modem function. If the
probability of surpassing of obtained variances ratio is more
than confidence probability then it is impossible to admit
variance difference as significant and consequently it’s
impossible to make conclusion about degradation presence.
18
EXAMPLES OF ANALISEZ OF FUNCTION MODEM
DEGRADATION
1. Computation of modem function degradation
Calculations of coefficient of function modem degradation,
standard deviation of differences between rated failure
number and obtained in test failure number are executed in
the table. Sample of this table is shown below.
19
Calculation of rated failure number
Square of
Square of
Difference
Difference
Difference
Difference
between
between
between
between
Failure Number computed failure computed failure
computed failure computed failure
computed
number according number according Failure Number number according number according
Cumulative
according
degradation
degradation
computed
Linear formula
Linear formula
Test
Failure
degradation
formula and real formula and real according Linear and real failure
and real failure
Number Number
1-exp(-N/K) formula
failure number
failure number
formula
number
number
0
0
0
0
0
0
0
0
0
1
0 0.000474
0.000236986
-0.000236986
5.61621E-08
0.055851783
-0.055851783
0.003119422
2
0 0.000948
0.000947792
-0.000947792
8.9831E-07
0.111703566
-0.111703566
0.012477687
3
0 0.001421
0.002132196
-0.002132196
4.54626E-06
0.167555349
-0.167555349
0.028074795
4
0 0.001894
0.003789972
-0.003789972
1.43639E-05
0.223407133
-0.223407133
0.049910747
5
0 0.002367
0.005920896
-0.005920896
3.5057E-05
0.279258916
-0.279258916
0.077985542
6
0
0.00284
0.008524744
-0.008524744
7.26713E-05
0.335110699
-0.335110699
0.11229918
7
0 0.003313
0.011601291
-0.011601291
0.00013459
0.390962482
-0.390962482
0.152851662
8
0 0.003785
0.015150314
-0.015150314
0.000229532
0.446814265
-0.446814265
0.199642987
9
1 0.004257
0.019171588
0.980828412
0.962024374
0.502666048
0.497333952
0.24734106
10
1 0.004729
0.02366489
0.97633511
0.953230247
0.558517831
0.441482169
0.194906505
11
1 0.005201
0.028629997
0.971370003
0.943559684
0.614369615
0.385630385
0.148710794
12
2 0.005672
0.034066683
1.965933317
3.864893805
0.670221398
1.329778602
1.768311131
….
….
….
….
….
….
….
….
….
316
23 0.139119
22.52931401
0.47068599
0.221545301
17.64916347
5.350836529
28.63145156
317
24 0.139527
22.66863663
1.331363374
1.772528433
17.70501525
6.294984746
39.62683295
318
24 0.139934
22.80836715
1.191632854
1.419988858
17.76086704
6.239132963
38.92678012
319
25 0.140342
22.94850538
2.051494623
4.208630189
17.81671882
7.183281179
51.5995285
320
26 0.140749
23.08905113
2.910948875
8.473623353
17.8725706
8.127429396
66.05510859
Standard
Deviation
4.268215043
6.449112632
20
Initial data for calculation – test order number and
cumulative failure number after executing that test number
– are placed in the first two columns of this table. In the two
following columns rated failure number under accepted
degradation coefficient is calculated according to obtained
formula
f = N – k(1 – exp(-N/k)).
At first in the third column expression in parenthesis is
calculated and after this in fourth column rated failure
number is calculated. In fifth and sixth columns differences
of rated and obtained cumulative failure number and
squares of those differences are calculated. In last three
columns rated failure number for the case when degradation
is absent (according to linear dependence), differences of
rated and obtained cumulative failure number and squares
of those differences are calculated. In the footer of columns
where squares of differences of rated and obtained
cumulative failure number are calculated standard deviation
of this differences is computed.
Results of described calculations are used for detection of
degradation coefficient value and following analysis of
degradation presence.
2. Computation of degradation coefficient
Degradation coefficient should be chosen so that standard
deviation of diversions of rated cumulative failure number
from number obtained in test would be minimum. In order
to detect such coefficient value it’s necessary to execute the
described above calculation several times for computation of
standard deviation of diversions of rated cumulative failure
number from number obtained in test at different values of
degradation coefficient. By results of these calculations the
directed search of value of degradation coefficient with
which standard variation is minimum is carried out.
Process of search of degradation coefficient can be written
down in the below stated table. In the first two columns
upper and lower limits of interval where searching is
21
completed are written down. In following columns middle of
interval and standard
22
Detection of degradation coefficient
Standard
Standard
deviation for
deviation for
shifting
Interval
Interval
Middle of
middle
Shifting
value
Lower limit Upper limit interval
interval
Value
interval
Gradient
0
8192
4096
28.83381
4097
28.84681
0.013003
0
4096
2048
4.360214
2049
4.357191
-0.00302
2048
4096
3072
14.36769
3073
14.38231
0.014613
2048
3072
2560
7.413481
2561
7.425097
0.011615
2048
2560
2304
4.984035
2305
4.990833
0.006799
2048
2304
2176
4.36051
2177
4.363248
0.002738
2048
2176
2112
4.268289
2113
4.268398
0.000108
2048
2112
2080
4.289183
2081
4.287794
-0.00139
2080
2112
2096
4.272739
2097
4.272115
-0.00062
2096
2112
2104
4.269048
2105
4.268794
-0.00025
2104
2112
2108
4.268306
2109
4.268234
-7.2E-05
2108
2112
2110
4.268207
2111
4.268226
1.85E-05
2108
2110
2109
4.268234
2110
4.268207
-2.7E-05
2109
2110
2109.5
4.268215
deviation of diversions of rated cumulative failure number
from number obtained in test at degradation coefficient that
equals middle interval. After that the gradient of a standard
deviation is calculated. For this used value of degradation
coefficient is increased by one and for this value of
degradation coefficient standard deviation is computed. The
difference of these two values of standard deviation
determines its gradient. . Depending on a gradient sign new
interval borders for the further search of degradation
coefficient providing minimum of standard deviation are
defined. If the gradient positive, that is value of a standard
deviation is increased at increase of degradation coefficient,
then minimum of standard deviation will be placed in the
left half of studied interval at values of degradation
coefficient smaller middle of this interval. If the gradient is
negative search of degradation coefficient that provides
require of standard deviation minimum ought to be executed
at right half of studied interval. On the example given in the
table search of degradation coefficient was begun for an
23
interval from 0 up to 8192. For middle of this interval – 4096
- standard deviation of rated values from values obtained in
the test appeared equal 28.83381. To the value increased by
one 4097 there corresponds a standard deviation 28.84681.
Thus the gradient of standard deviation is equal 0.013003,
i.e. positive. Hence the minimum of a standard deviation
takes place when degradation coefficient is smaller than
middle of studied interval (smaller than 4096). Therefore on
the following step the interval from 0 up to 4096 should be
considered. At research of this interval the gradient appears
negative. Therefore on the third step the right half of this
interval - from 2048 up to 4096 - is examined. The course of
the further search is completely clear from the table.
The described calculations proceed until the length of an
interval will not correspond to required accuracy of
degradation coefficient. Taking into account practically
existing values of degradation coefficient its values needs to
be determined to within one, and, hence, the described
calculations should be stopped when length of studied
interval equal or smaller than one.
3. Analyses of degradation presence by results of
experiments.
Occurrence of failures in a test course and its representation
by settlement formulas are visualized by graph placed
below. (Fig 2). On an axis of abscissas of this diagram the
total of the executed tests is postponed, and on an axis of
ordinates – cumulative quantity of the failures which have
occurred at their performance. On the diagram three lines
are constructed: the diagram of the cumulative failure
number (step line) and diagrams of dependences
appropriate to the formulas degradation equation and linear
dependence. In dependence legend values of standard
deviation of diversions of rated values from values obtained
at the test are given. In this case the standard deviation for
the degradation equation equals 0.339394, and for linear
equation standard deviation is 0.40944. For detection of the
significance of a difference of these values it’s necessary to
24
use Fisher criteria that equal ratio of compared dispersions
(or squares of standard deviations). In examined case this
criterion is equal
F = 0.40944 2 / 0.339394 2 = 1.455366
10
9
Test 1. Grapf of cumulative number of failures
(Cumulative) Number Of Failures
8
Degradation function f = N - 8283(1 - exp(-N/8283)).
0.339394
7
6
Standard deviation from equation is
Linear function f = 0.014754*N. Stndard deviation from equation is 0.40944.
5
4
3
2
1
0
0
50
100
150
200
250
300
350
Number of executed tests
Fig 2
Probability casually to surpass the received value of Fisher
criterion (that is detected by statistical tables) taking in
account number of freedom degree (that equals number of
executed tests without one) in studied case equal 0.000421,
i.e. equals zero practically. Because of this obtained
dispersion diversion is considered as significant or, in other
words, diversions of rated values, computed according to
linear equation, from values, obtained in the test, is
significantly more than diversions of degradation equation.
Therefore it is necessary to admit that degradation equation
is closer to line of cumulative failure number linear
equation. It, in turn allows to draw a conclusion on presence
of degradation modem function in studied test
25
Let's consider the following example of the analysis of
degradation presence by test results. Diagram of cumulative
failure number and calculated diagrams of the linear
equation and the degradation equation are given on a fig 3.
In this case standard deviation of diversions from linear
equation is smaller than standard deviation of diversions
from degradation equation.
10
Test 2. Graph of cumulative number of failures
(Cumulative) Number Of Failures
8
Degradation function f = N - 7875(1 - exp(-N/7875)). Standard deviation from equation is
0.933493
6
Linear function f = 0.015806*N. Standard deviation from equation is 0.691425
4
2
0
0
50
100
150
200
250
300
350
Number of executed tests
Fig 3.
Significance of variance difference is checked by Fisher
criterion, that equal
F= 0.933493 2 / 0.691425 2 = 1.82277
Probability casually to surpass this value of variance ratio at
executed test number (320) equals 5*10-8, i.e. close to zero.
On this basis it is possible to draw a conclusion that obtained
difference is not accidental and linear equation describes test
results better than degradation equation. Because of this it’s
necessary to draw conclusion that in this case modem
function degradation is absent.
In studied examples differences of variances were significant
and probability of surpassing of obtained variances ratio
26
was close to zero. Now we shall consider once more example.
In this test according to results of242 connections values of
standard deviations were closed and equal 6.753664 for
degradation equation and 6.444786 for linear equation.
Fisher criterion for comprised variances equals 1.098151,
and probability of surpassing of this value equals 0.234.
50
Test 3. Graph of cumulative number of failures
45
(Cumulative) Number Of Failures
40
Degradation function f = N - 613(1 - exp(-N/613)). Standard
deviation from equation is 6.753664.
35
30
Linear function f = 0.136795*N. Standard deviation from
equation id 6.444786.
25
20
15
10
5
0
0
50
100
150
200
250
Number of executed tests
Fig 4
In statistics usually it takes into account of 5% level of
significance, i.e. difference is considered significant if
probability of surpassing of obtained Fisher criterion value
is lower than 5%. Obtained probability value is more than
5% and because of it variance difference impossible to draw
statistical valid conclusion about presence or absence of
modem function degradation. In this case it’s necessary to
execute additional tests in order to detect presence or
absence of degradation.
RESUME
As results of performed researches it is discovered
degradation criterion that detects modem function stability
300
27
during at the time of modem functioning. For its value
detection it is elaborated principles of computation of
degradation criterion value on test basis. Beside of this
statistical principles of proofing of presence or absence of
degradation are elaborated.