Our company has a service contract for network availabiity. The
service level is for a ?minimum average of 99.9%?. The calculated
measurement last month(using very precise and unrounded information)
was 99.8614%.

The vendor maintains that the 99.8614% should be "rounded-up" to conform
with the significant digits in the contract (nn.n%) and therfore the
service level was achieved. Unfortunately, the contract did not
address rounding, calculation precision, or significant digits.

Our position is that the word ?minimum? implies a floor threshold  -
where the measurement is either below it or above(equal). In this
situation, 99.8614% is below the  99.9% floor.

Two questions, from a math perspective:

1. In math theory, Is there a implied traling zero which makes 99.9%
become 99.90%? Are these numbers in fact the same?

2. Are there other rounding principles that might apply in this
situation? How about rounding down (99.8614% to 99.8%)? 

3. Is the whole issue of rounding in this situation not applicable
given that the raw data can be measured with incredible precision?

There are potential monetary credits so I would like as much
justification for our position as possible.

---

Request for Question Clarification by pafalafa-ga on 30 Jan 2005 05:51 PST

Charles,

This strikes me as much more a matter of contracts than it does of mathematics.

Does the contract specify a time period for the ?minimum average of
99.9%?...that is, does the contract make clear that this is the
MONTHLY minimum?

"Average" implies the summing (and subsequent division) of numerous
values, some of which may well be BELOW the stated 99.9% level, as
long as others are above it, and can "cancel out" the effects of the
lower numbers.

So...does one average over the course of a week...a month...a year? 
What does your contract say to this.

As for the rounding, seems to me both you and your contractor can make
a reasonable case.  I wouldn't rely on this particular argument as a
slam dunk.

As for the monetary credits involved, you may want to seriously
consider whether the potential gains are worth souring your relations
with your service provider, and the potential headaches that can
cause.

Let me know your thoughts?

pafalafa-ga

---

Clarification of Question by charles46-ga on 30 Jan 2005 08:21 PST

Thanks for your quick response..MY CLARIFICATONS IN UPPER CASE

Charles,

This strikes me as much more a matter of contracts than it does of mathematics.

IT IS BOTH MATH AND CONTRACTS FOR SURE...

I WANT TO FOCUS ON THE MATH ASPECTS AS PER MY ORIGINAL QUESTIONS.. 

Does the contract specify a time period for the ?minimum average of
99.9%?...that is, does the contract make clear that this is the
MONTHLY minimum.

YES IT DOES. THE CONTRACT IS ON A MONTHLY BASIS AND IT SPECIFIES A
"MINIMUM AVERAGE OF 99.9%". THE  EXACT FORMULA IS:

"(TOTAL TIME - DOWN TIME)/TOTAL TIME =99.9%"

AS I MENTIONED IN MY QUESTION, THE MEASUREMENTS ARE VERY PRECISE AND
WE AGREE ON THE NUMBERS FOR THE FORMULA (BELOW).  FOR EXAMPLE, IN OUR
CASE THE NUMBERS WOULD BE:

"(TOTAL TIME (1,000,000 hrs) - DOWN TIME (1386 hrs)/TOTAL TIME
(1,000,000 hrs) =99.8614%"

THE FUNDAMENTAL QUESTION IS HOW TO ROUND THIS NUMBER? OBVIOUSLY, THE
VENDOR WOULD LIKE TO ROUND IT UP. I STILL SAY THAT 99.8614% IS LESS
THAN 99.9% REGARDLESS.

"Average" implies the summing (and subsequent division) of numerous
values, some of which may well be BELOW the stated 99.9% level, as
long as others are above it, and can "cancel out" the effects of the
lower numbers.

SEE ABOVE...THE MEASUREMENTS ARE VERY PRECISE. 

So...does one average over the course of a week...a month...a year? 
What does your contract say to this.

MONTH

As for the rounding, seems to me both you and your contractor can make
a reasonable case.  I wouldn't rely on this particular argument as a
slam dunk.

As for the monetary credits involved, you may want to seriously
consider whether the potential gains are worth souring your relations
with your service provider, and the potential headaches that can
cause.

FOR SURE..I WANT THE MATH ANGLE..

Let me know your thoughts?

pafalafa-ga

---

Request for Question Clarification by hedgie-ga on 01 Feb 2005 04:56 PST

I was going to give you an answer - but noticed that

comment 
 From: parker_79-ga on 31 Jan 2005 03:36 PST 

 provided that same answer . Math really is silent on this
- but natural sciences have clear convention as described,
one favoring the vendor.

You may withdraw the question, or I can find you couple sites
which describe this fact -as an  answer.

So far, the comments and this RFC were free.

So - do you want an answer?

No matter how the numbers are measured, 99.8614% is less than 99.9%.
There are statistical rules about what are "significant figures" and
there are many various ways of rounding -- you could equally
legitimately argue that the ...614 part should be dropped altogether,
leaving 99.8% if you wish. However, 99.9% really only means "999 out
of 1000" and 99.8614% means "998614 out of 1000000". The 99.8614%
measurement is only significant if 1000000 or more measurements were
made.

First, it shouldn't be necessary to say 99.90, because then what about
specifying 99.900?  If adding zeroes to specify an exact number was
necessary, you could never specify an exact number because you can
always add another zero.  So I would say "99.86..." does not meet the
requirements.

Also, there is more than one method of "rounding" a number:
http://www.roguewave.com/support/docs/sourcepro/currencyug/6-6.html
Even though one method is more common, it is not the only thing that
could be used to limit decimal places in a final number.  So even if
the number is rounded to one decimal place, it does not mean the
vendor wins.

BUT, you should probably ask a lawyer familiar with the specific
business before making trouble.  The most important thing would be the
common language used for whatever type of contract you are talking
about.  If the industry standard for network availablity is to round
to the specified number, that's probably the final answer.

In scientific measurement, a 99.9% has a significant figure only up to
the last digit specified. This means that the last sig fig has an
uncertainty of a +/- 5 meaning that it's 99.9% +/-.05 making the range
99.85% to 99.95%. Had the percentage been specified as 99.90%, the
range would have been 99.895% to 99.905%. That's just my input though.
good luck.

At my work, the last 0 is important.

99.9% means you can round to the tenth.
99.90% means that rounding to the tenth will not be accepted, however
rounding to the hundreth will.
99.900% means rounding to the hundreth is ok, but rounding to the
thousanth is not and so on...

If you had no reason to know that these rounding practices were in
place then the precident needs to be set before you a penalized for
it.  However if this knowledge should have been commonly accepted in
your field or if you have done work in the company before with these
limits then you should be completely responsible for knowing this.

I think the math is more on the vendor's side here.  99.9 means closer
to 99.9 than 99.8.  I can see how an argument could be made for the
other side though.

Considering that the total time is 1,000,000 hours, I think some
consessions should be made for the fact that most networks were a
little unreliable in the 1890's.

Who wrote the contract?  Is it a standard contract that they provide
to all their customers, or was it negotiated?

For standard one-size-fits-all contracts, courts traditionally will
construe ambiguities against the drafter of the contract.  The
question would be whether a reasonable person would only be able to
interpret it the way the drafter intended.  If a reasonable person
could interpret it the other way (i.e. the way you expect it), then
the drafter would likely lose the battle.

If, on the other hand, you have personally negotiated this contract,
there would be a lot of debate in court re: intent at signing.  That's
harder to predict.

--Joey

Charles46,

This is a very interesting question. I hope at the end of the piece
you?ll let us know ?who won.?

I think your problem is more of a contractual one, not a numerical
one. My reasoning is as follows:

 99.8614% is clearly less than 99.9% and does not meet the percentage
requirement.  Let?s look at the actual values. You said ?last month?
so I assume you mean December 2004. December has 31 day, each with 24
hours and each hour with 3600 seconds. So the number of seconds in
December is 31x24x3600 = 2,687,400. Multiplying that number by
0.998614 and subtracting the result from 2,687,400 yields 3712.263
seconds of actual network unavailability. This is just over one hour;
1:01:52 more or less. Doing the same with .999 yields 2678.4 seconds
that the network could have been down in December and still met the
contractual requirement. The difference in the allowable network
unavailability and the actual network unavailability is 3712.263 ?
2678.4 = 1033.863 or 17 minutes 13 seconds more or less.

Look at a more striking analogy. If you sold an item for an agreed
upon price of $2,675,721, but the buyer only paid you $2,674,687, a
difference of $1034, you would cry foul.

I think any jury in the country would side with you in that you did
not receive the network availability that you contracted for.

The caveat to the above is, is there any language in the contract that
specifies or implies that rounding of the value is to be done? You?ve
stated that it doesn?t. A second concern in this area is case
precedence.

You?ve told us that the contract states a ?minimum average of 99.9%?
and that this is over a month. As you only have one data set, the
month of December as a whole, and you?re averaging it over itself, you
don?t end up with much of an average. This really seems to be a case
of poor choice of contractual language. A better choice might have
been ?a minimum of 99.9%?. Notwithstanding that point, you?ve given us
the exact formula: "(TOTAL TIME - DOWN TIME)/TOTAL TIME =99.9%".
Clearly this is on your side. This is also the way I made my
calculations.

Several comments have supported arguments for rounding using various
methodologies. Parker_79?s make the most sense to me. However, some
case precedence, some standard methodology of rounding or maybe
something in the UCC should be applied to make the determination if
99.98614% meets the 99.9% criteria. Unfortunately, this is exactly
your question, and I don?t think I?ve helped you in that regard.

One last point that may be on your side; You?ve told us that the
measurement are made with ?with incredible precision.? Why are they
made with such precision? If the measurements are made with such
precision so that they can be applied to the contractual formula, then
I think that?s a point for your side. It would certainly have been
possible to take a measurement (is the network available or not) every
one minute, one hour or at whatever frequency you desired. If your
contract specifies that the measurements are to be taken at a high
frequency (every second or less) then, again, I think that is a point
for your side.

Good luck.

Request for Question Clarification by  hedgie-ga 

Sure..please take a crack at the question

To the best of my knowledge in mathematics a trailing zero is not a
necessity. The two numbers (99.9% and 99.90%) are of the same value
however 99.90% conveys a higher degree of accuracy. In scientific
measurements however in scientific calculations the number of
significant figures is determined by the operations through which the
numbers are put. In your particular case given that you can account
for network availability every hour and that it was recorded and
tested every hour the rules for rounding to significant figures deem
that your calculations are accurate to the ten-thousandths place as
your calculation reveal. Other rounding principles such as truncation
also further your position in this argument. And truncating to the
tenths place as in your contract the percent would then be 99.8%.
Given that the number of hours total and those hours of down time are
measured precisely and accurately the issue of rounding truly does not
apply based on any scientific principle of rounding or by truncation.

1) Yes there is, in <i>number theory</i> 99.9 = 99.90, or any other
quantity of zeroes you feel like appending.  In number theory, these
are <b>not</b> significant digits.  It is in <i>measurement</i> that
99.90 conveys a higher precision than 99.9 alone.  Your problem is in
measurement, and in that context, there is an argument (a weak one)
that 99.9 = 99.8614.

2) Although in number theory, you are free to "round down" and state
that 99.8614 = 99.8, however, in measurement, this is contrary to
established scientific practices and reasonable logic.  And no, it
does not apply in this situation that other than the commonly-accepted
method of rounding (taught to every elementary-school student) should
be reasonably expected of you or them.

3) Rounding is not applicable in this situation, but not for the
reason you state.  For your provider to advertise a minimum of 99.9%
uptime, but practice a minimum of 99.85%, and to base this discrepency
on "rounding errors" is both false and misleading.  If they meant a
minimum of 99.85% uptime, they should have (and easily could have)
said so.  No, in the language of contracts, minimum means minimum when
no language to the contrary is included (such as if the contract
literally states a rounding mechanism in use).

If this is a reasonably large provider, you should have no problem
finding a reputable firm to take this on as a <i>class action</i>,
entitling you as class representative to substantially more in damages
than you actually suffered.

If their argument was enforceable, and they bill you $100 a month, you
would then be entitled to pay only $99.50 by the same argument.  It is
an absurd assertion, and any court will agree.

The math part has a very simple answer: 99.8614 is always less than 99.9 .

--

Even for sufficiently large values of 98.8614  :)

I haven't addressed the contractual and human issues, and I agree with
the several things the commenters have already said.

ah..typos on numbers are bad.  I am sorry about the 98.8614 on the
line with the smily (should be 99.8614).

Charles46

I've been watching this question to see if you were going to let us
know the outcome. I have also had the following thought.

If the rounding arguement is allowed to stand and is applied to it's
fullest extent, then your service provider could have only 99.85%
uptime as this would "round" to 99.9%. Their allowable downtime would
therefore be 4017.6 seconds. This is fully 50% more downtime than is
allowed at 99.9%.