A model is proposed to describe DSS-14 outage times. Discrepancy Reporting System outage data for the period from January 1986 through September 1988 are used to estimate the parameters of the model. The model provides a probability distribution for the duration of outages, which agrees well with observed data. The model depends only on a small number of parameters, and has some heuristic justification. This shows that the Discrepancy Reporting System in the Deep Space Network (DSN) can be used to estimate the probability of extended outages in spite of the discrepancy reports ending when the pass ends. The probability of an outage extending beyond the end of a pass is estimated as around 5 percent

Posner, E. C.

Rumsey, H. C.

Stevens, R.

English

NASA Technical Reports Server

TDA Progress Report 42-98
,, .,_
,/
N90-12802
August 15, 1989
A Simple Model for DSS-14 Outage Times
H. C. Rumsey,1 R. Stevens,and E. C. Posner
Officeof Telecommunicationsand Data Acquisition
A model is proposed to describe DSS-14 outage times. Discrepancy Reporting
System outage data for the period from January 1986 through September 1988 are
used to estimate the parameters of the model. The model provides a probability
distribution for the duration of outages, which agrees well with observed data.
The model depends only on a small number of parameters, and has some heuristic
justification. This shows that the Discrepancy Reporting System in the DSN can
be used to estimate the probability of extended outages in spite of the discrepancy
reports ending when the pass ends. The probability of an outage extending beyond
the end of a pass is estimated as around 5 percent.
I. Introduction
A model is proposed to describe DSS-14 outage times.
Outage data for the period from January 1986 through
September 1988 are used to estimate the parameters of the
model. The model provides a probability distribution for
the duration of the outages. The model does not address
questions about the mean time between outages. However,
it does allow estimation of the probability of major outages
even though the Discrepancy Reporting (DR) system stops
recording them at the end of a pass. The philosophy has
been that the best model is the simplest one that fits well
enough.
The nature of the model, as above, is affected by the
way outages are reported. If an extended outage occurs,
1 Consultant to the TDA Office from Laguna Niguel,
California.
the time to restore service is not reported. Only the time
lost for that pass is recorded. This limitation makes it
difficult to determine the actual outage durations from the
DR data and must be accounted for in the model. In the
model, it is assumed that the actual time to restore service
is hidden by a "cutoff" process which corresponds to the
end of the pass. A typical pass runs for 9 hours. This
means that the actual time to restore service is masked
by a 9-hour cutoff window. (In developing the model, 8-,
9-, and 10-hour cutoffs were tried, with 9 fitting best and
being reasonable on other considerations.)
II. Outage Distribution
Let R(t) be the distribution function for reported out-
age durations, and let A(t) be the distribution function for
actual outage durations. Assuming the "cutoff" process
and outage durations are independent, then for t >_ 0,
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https://ntrs.nasa.gov/search.jsp?R=19900003486 2020-03-19T23:46:36+00:00Z
or
Pr(reported time > t)
= Pr(actual time > t)Pr(cutoff time > t)
1 - R(t) = (1 - A(t))(1 - t/U)+ (1)
Pr(reported time > t) = 1 - R(t)
= ((1 - a)(1 + t/a)e -`la
+ a(1 + rib)e-rib)(1 -- t/U)+
(2)
where U is the duration of the pass (nominally 9 hours)
and (1 - t/U)+ is 0 for t > U. Equation (1) assumes that
the start time of an outage is uniformly distributed over
the duration of the pass.
Equation (1) deals with the "cutoff" problem but says
nothing about the actual distribution of outages A(t). The
model for A(t) is based on the reported outage data and a
desire to minimize the number of parameters in the model.
Figure 1 shows this measured outage data from the DR
system, accurate to one minute. Thus, some minutes show
multiple outages. There are 498 outages presented in this
figure. The mean time to restore service is about 40 min-
utes.
The simplest model to fit such data would be an ex-
ponential distribution. The distribution function of an ex-
ponential random variable X of a mean a is given by
F(t) = Pr(X < t) = 1- e-'la
The fit is not accurate for short outages because Fig. 1
shows the density goes to 0 at 0-length outages, but an
exponential has maximum density at 0-length outages. An
exponential random variable with mean around 40 minutes
fits the first part of the outage data fairly well beyond a few
minutes up to about 100 minutes, but there are too many
extreme values (t > 100 minutes) in the outage data. This
suggests that there are two (or more) classes of failures, in
addition to the short failures. For the first class of long
failures, service can be restored quickly, in less than 40
minutes on the average. The second class of long failure
requires more time to overcome.
The form for this tail distribution has been taken to make
the density function (essentially) zero at t = 0. Observe
that if Ta is a random variable with distribution function
1- (1 +t/a)e -t/a, then the expected value of Ta is 2a while
the maximum of the density function of Ta occurs at t = a.
The parameters a and b occur symmetrically in
Eq. (2). If a is chosen to be the smaller value, then out-
ages of class "a" can be arranged to peak around t = 10
minutes. Outages of class "b" can be arranged to fit the
tails of the observed outages. More heuristics appear in
Section VI.
IV. Parameter Estimation
Equation (2) defines the model. To complete the
model, good values for the parameters U, a, a, and b must
be found. The maximum likelihood method will be used to
estimate the parameters. Let (Q,...,tn) be the reported
outage times. The ts are reported by the DR system to
the nearest minute. Let t be one of the outage times. The
models specify the probability p(t) that an outage has du-
ration t, where t is measured in minutes. Namely,
p(t) = R(t + 1/2) - R(t - 1/2)
where R is determined by Eq. (2). The probability that
the observed outages occurred is the product of the prob-
abilities of the separate outages,
1-Ip(t )
j=l
III. Model
The following model has been adopted, with three pa-
rameter a, a, and b to be estimated, since U is 9 hours and
not estimated:
This product is the likelihood function of the observations.
It is a function of the model parameters U, a, a, and b.
Maximum likelihood says to choose these parameters to
maximize this product. The maximization is easy in this
case because there are so few parameters.
203
There is a minor problem in determining the best
value for the parameter U. Recall that U is the cutoff time
for the pass. Finding U is like finding the end point of an
interval (0, U) in which a uniform random variable occurs.
A little thought shows that the likelihood function for such
a problem is maximized by taking U to be as small as pos-
sible. In the case of interest here, this would correspond
to taking U to be slightly less than 8 hours (the maximum
reported outage is 462 minutes). If extreme outages were
common, this "defect" in the maximum-likelihood method
would be no problem. Yet, the number of extended out-
ages is small. So rather than estimate U from the likeli-
hood function, U has been taken to be 9 hours throughout.
To see how this selection affects the results, U = 8 and 10
hours for the case Fb -- 1 were also tried. As expected,
U = 8 hours gives a larger value for the likelihood func-
tion, but the other parameters a and a are hardly changed.
The results for U = 10 hours were not as good. Only
U = 9 hours is considered below. This is consistent with
the known distribution of pass lengths.
V. Goodness of Fit
A grid of points was used to find the maximum-
likelihood values for the parameters. The maximum-
likelihood values found were
a = 0.186
a = 11.4 minutes
b = 77.5 minutes
The corresponding distribution function is then given by
played in Fig. 2. Qualitatively, the model fits the observed
outages very well, for short, medium, and long outages.
The mean time MTR to restore service for the model.
is given by
where
MTR = (1 - a)aT(Ula) + abT(U/b)
T(x) = 2 - 3Ix + (1 + 3/x)e -_
Substituting for a, a, b, and U = 9 hours (540 minutes) as
always gives
MTR = 40.6 minutes
This is in excellent agreement with the observed MTR of
40.4 minutes.
The probability that an outage exceeds 150 minutes
was computed. With 498 outages, the model predicts there
should be 28.4 outages of duration 150 minutes or longer.
In the actual data of 498 outages there were 24 outages
of this duration. Since the extended outage statistics are
expected to be Poisson with an estimated mean of 28.4 and
thus a sigma of _ = 5.33, the discrepancy of 4.4 is less
than one sigma. This fits as well as could be expected
with only 24 events. The probability of short and medium
outages is also seen to fit very well. Hence, the use of the
model seems indicated.
Pr(reported outage _ t) =
0.814(1 + t/ll.4)e -t/ll'4
+0.186(1 + t/77.5)e -t/77s] (1 - t/540)+
This equation represents the model. To see how well
this model fits the observed outage data, the outage data
was smoothed with a 5-point smoothing filter. The same
filter was applied to the model as well. The results are dis-
VI. Heuristics
It can be expected that the density of outages near
zero outage time is very nearly 0, because it takes some
minimum time to notice an outage and to respond to it,
even by switching in a hot standby automatically. The
(1 + t/a)e -'/a term in Eq. (2) does just this--it has den-
sity 0 at 0, for the corresponding density is (t/a2)e -fla.
This density also covers intermediate outages, but so does
the "b" class. One might think that yet another distribu-
tion should be mixed in to cover these intermediate outages
that cannot be recovered merely by switching something
204
in. But as has been seen, adding an extra one or two pa-
rameters is not necessary--the fit is very good with just
the single time parameter a, the location of the maximum
density of the short and medium outages, and the addi-
tional parameter a which gives the relative fraction of long
outages.
The long outages are described by a similar distribu-
tion (1 + t/b)e-Ub with b >> a. Why the tail of this and
the shorter-outage distribution should be exponential is
less clear, but the fit is good, and hard to tell from a dis-
tribution with a long low constant tail given the amount
of data. It can be observed that the form of the a and
b distribution arises as a difference of pure exponentials
with infinitesimally close memoryless repair rates, but this
does not seem to help the heuristics. The long tail can
arise from certain failures such as low-noise maser warm
up that takes a certain minimum time, e.g., 12 hours to
recover from, when hot standbys for switching in are not
provided.
Finally, as explained, the (1 - t/U)+ multiplier term
in Eq. (2) arises from the truncation of outage data at the
end of a pass, where it was assumed that failures occur uni-
formly over the duration (U = 9 hours) of a pass. It is this
truncation which makes it hard to distinguish a negative
exponential tail from a long fiat tail. The three-parameter
model has been adopted even though the heuristics are not
perfect.
VII. Summary
It has been shown that the Discrepancy Reporting
System in the DSN can be used to give good estimates of
the probability of extended outages in spite of the discrep-
ancy reports ending when the pass ends. The probability of
a major outage (one extending beyond the end of a pass)
is estimated by the best-fit model as around 5 percent.
The model also gives good estimates for the probability of
short, medium, and long outages. It is simple and yet fits
very well.
Acknowledgments
We thank D. W. Ginavan and G. C. Smith of the Bendix Field Engineering
Corp. for organizing and providing the outage data used in this study.
205
25
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0
0
-0
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-0 0
(3O
0
000 0
-
_OC>O
0_00_ 0 O0 0
000_0 () 0 0 0
i_>_°°_>_ _°° i° _>i o i ,
OUTAGE TIME, minutes
Fig. 1. DSS-14 outage times, January 1986-September 1988.
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18
16
14
(/3
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_ 6
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I I I I
SMOOTHED FREQUENCIES FOR 498 OUTAGES
I I I
f SMOOTHED FREQUENCIES FOR 498 OUTAGESFROM THE DISTRIBUTION:
1 ° (0.814F(t/11.4) + 0.186F(t/77.5)) (1 -t/540)+
WHERE F(t) = (1 + t) e -t
\
I
o __'v_,_-_.___,____,__ __ .m
0 60 120 180 240 300 360
OUTAGE TIME, minutes
Fig. 2. Smoothed DSS-]4 outage times, January 1986-September 1988
compared with model.
I
420 480
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A simple model for DSS-14 outage times

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