International Journal of Recent Advances in Engineering & Technology (IJRAET)
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Profit-Function of Two- Identical Cold Standby Missile System
subject to Failure caused by power and failure due to Time Error
Ashok Kumar Saini
BLJS COLLEGE, TOSHAM (BHIWANI) HARYANA, INDIA
Email : [email protected]
Abstract : A power failure on October 23, 2010 meant that
one-ninth of America’s nuclear arsenal went offline for
almost an hour.
A U.S. Air Force spokesman said there had been a
'hardware issue' relating to an underground cable linking
the command centre with the missiles.
The Patriot Missile Failure
On February 25, 1991, during the Gulf War, an American
Patriot Missile battery in Dharan, Saudi Arabia, failed to
track and intercept an incoming Iraqi Scud missile. The
Scud struck an American Army barracks, killing 28
soldiers and injuring around 100 other people. A report of
the General Accounting office, GAO/IMTEC-92-26,
entitled Patriot Missile Defense: Software Problem Led to
System Failure at Dhahran, Saudi Arabia reported on the
cause of the failure. It turns out that the cause was an
inaccurate calculation of the time since boot due to
computer arithmetic errors. Specifically, the time in tenths
of second as measured by the system's internal clock was
multiplied by 1/10 to produce the time in seconds. This
calculation was performed using a 24 bit fixed point
register. In particular, the value 1/10, which has a nonterminating binary expansion, was chopped at 24 bits after
the radix point. The small chopping error, when multiplied
by the large number giving the time in tenths of a second,
led to a significant error. Indeed, the Patriot battery had
been up around 100 hours, and an easy calculation shows
that the resulting time error due to the magnified chopping
error was about 0.34 seconds. (The number 1/10 equals
1/24+1/25+1/28+1/29+1/212+1/213+.... In other words, the
binary
expansion
of
1/10
is
0.0001100110011001100110011001100.... Now the 24 bit
register
in
the
Patriot
stored
instead
0.00011001100110011001100 introducing an error of
0.0000000000000000000000011001100... binary, or about
0.000000095 decimal. Multiplying by the number of tenths
of
a
second
in
100
hours
gives
0.000000095×100×60×60×10=0.34.)
In this paper we have taken Failure caused by power and
failure due to Time Error. When the main unit fails due to
power then cold standby system becomes operative. Failure
caused by power cannot occur simultaneously in both the
units and after failure the unit undergoes Type-I or TypeII or Type-III repair facility immediately. Applying the
regenerative point technique with renewal process theory
the various reliability parameters MTSF, Availability,
Busy period,
evaluated.
Benefit-Function
analysis
have
been
Keywords: Cold Standby, failure caused by power and
failure due to Time Error. first come first serve, MTSF,
Availability, Busy period, Benefit -Function.
INTRODUCTION
The Patriot Missile Failure
On February 25, 1991, during the Gulf War, an
American Patriot Missile battery in Dharan, Saudi
Arabia, failed to track and intercept an incoming Iraqi
Scud missile. The Scud struck an American Army
barracks, killing
28 soldiers and
injuring around
100
other
people. Failure
at
Dhahran,
Saudi
Arabia reported
on the cause of
the failure. It
turns out that
the cause was
an inaccurate calculation of the time since boot due to
computer arithmetic errors. Specifically, the time in
tenths of second as measured by the system's internal
clock was multiplied by 1/10.
A Scud travels at about 1,676 meters per second, and so
travels more than half a kilometer in this time. This was
far enough that the incoming Scud was outside the
"range gate" that the Patriot tracked. Ironically, the fact
that the bad time calculation had been improved in some
parts of the code, but not all, contributed to the problem,
since it meant that the inaccuracies did not cancel.
The following paragraph is excerpted from the GAO
report.
The range gate's prediction of where the Scud will next
appear is a function of the Scud's known velocity and
the time of the last radar detection. Velocity is a real
number that can be expressed as a whole number and a
decimal (e.g., 3750.2563...miles per hour). Time is kept
continuously by the system's internal clock in tenths of
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seconds but is expressed as an integer or whole number
(e.g., 32, 33, 34...). The longer the system has been
running, the larger the number representing time. To
predict where the Scud will next appear, both time and
velocity must be expressed as real numbers. Because of
the way the Patriot computer performs its calculations
and the fact that its registers are only 24 bits long, the
conversion of time from an integer to a real number
cannot be any more precise than 24 bits. This conversion
results in a loss of precision causing a less accurate time
calculation. The effect of this inaccuracy on the range
gate's calculation is directly proportional to the target's
velocity and the length of the the system has been
running. Consequently, performing the conversion after
the Patriot has been running continuously for extended
periods causes the range gate to shift away from the
center of the target, making it less likely that the target,
in this case a Scud, will be successfully intercepted.
Did UFO cause power failure at nuclear missile base?
Missile technicians claim sightings coincided with
October outage
could only have done so from an airborne command and
control centre, he said.
Another official said there had been similar breakdowns
on other bases in the past.
But Robert Hastings says more was involved.
According to Hastings, three missile maintenance
technicians have agreed to speak to him on the condition
of anonymity, revealing the military has kept UFO
sightings that occurred during the power outage under
wraps.
The witnesses, he said, reported sightings of 'a large
cigar-shaped object high above the missile field'.
Minuteman missile: Computer breakdown meant the
U.S. Air Force lost control for 59 minutes, officials say
Hastings told AOL: 'They said the object was seen in the
sky above the field, throughout the weekend, both
during the (missile) disruption and the following day.'
His witnesses claim the power outage lasted several
hours longer than officials reported.
When Warren Air Force Base in Wyoming lost control
of 50 nuclear, inter-continental missiles last October,
officials said a communication failure between the
control centre and the weapons was to blame.
'I have detailed information about the events. The Air
Force said this (missile) disruption lasted 59 minutes. It
actually lasted the better part of 26 hours,' he said.
However, three missile technicians stationed at the base
have raised fresh questions in the case, amid reports
UFO sightings coincided with the incident.
'It was intermittent and involved a very specific
sequence of these five missile alert facilities going on
and offline. I have all of that down to the most minute
detail.'
UFO
researcher
Robert
Hastings
says
eyewitnesses claim the interruption to the power supply
also lasted much longer than the Air Force admits.
Cover-up claims: Missile technicians stationed at
Warren Air Force Base in Wyoming have reported UFO
sightings coinciding with October's power outage
A power failure on October 23, 2010 meant that oneninth of America’s nuclear arsenal went offline for
almost an hour.
A U.S. Air Force spokesman said there had been a
'hardware issue' relating to an underground cable linking
the command centre with the missiles.
This disrupted ‘communication between the control
centre and the missiles, but during that time they were
still able to monitor the security of the affected
missiles’.
Defence officials insisted there was never any danger of
an accidental launch. But the incident was deemed
serious enough for Barack Obama to be briefed on it
later.
Speaking out: UFO researcher Robert Hastings says
three eyewitnesses have come forward to him on the
condition of anonymity
There was no evidence of foul play and the U.S. never
lost the capability to launch the missiles, although it
Blast: A Minuteman nuclear-tipped missile takes off
from its launch facility at Cape Canaveral Air Force
Station, Florida
The eyewitnesses agreed that what they saw 'was not a
commercial blimp.'
'It had no passenger gondola and no advertising on its
hull,' Hastings said.
'Further, its aspect ratio (length to width) was very
similar to a WWI Zeppelin: long and thin, and not at all
like the squat shape of a corporate blimp.'
The witnesses did not, however, claim the alleged UFO
was connected with the outage.
It is not the first time Hastings has reported UFO
sightings at nuclear missile sites.
Going public: Captain Robert Salas accused the U.S. Air
Force of lying in September
He organized a press conference last September, when
six former Air Force officers stepped forward to reveal
they had seen or had been involved with sightings at
missile sites.
They claim that since 1948, aliens have been hovering
over UK and U.S. nuclear missile sites and deactivating
the weapons - once even landing in a British base.
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The men said they were encouraged not to speak to the
media about their sightings.
4.
The repair facility does no damage to the units and
after repair units are as good as new.
Captain Robert Salas, who was among the six,
confirmed: ‘We’re talking about unidentified flying
objects, as simple as that.'
5.
The switches are perfect and instantaneous.
6.
All random variables are mutually independent.
7.
When both the units fail, we give priority to
operative unit for repair.
8.
Repairs are perfect and failure of a unit is detected
immediately and perfectly.
9.
The system is down when both the units are nonoperative.
However, Lieutenant Colonel John Thomas, director of
public affairs for Air Force Global Strike Command
headquarters at Barksdale Air Force Base in Louisiana,
denies there is a policy to silence eyewitnesses to
unexplained phenomena at Air Force bases.
'I have no reason to dispute anybody's claims of
anything they may seen historically, because those
occurrences and reports took place decades into the past
and probably will decades into the future,' he told AOL.
'This incident is separate from all of that. We took it
very seriously and we're very confident that we
understand fully what happened.'
'If people see things that are unusual, they are
encouraged to report them,' he said, adding: 'When
people join the military, they don't give up their First
Amendment rights.'
Stochastic behavior of systems operating under
changing environments has widely been studied.
Dhillon , B.S. and Natesan, J. (1983) studied an outdoor
power systems in fluctuating environment . Kan Cheng
(1985) has studied reliability analysis of a system in a
randomly changing environment. Jinhua Cao (1989) has
studied a man machine system operating under changing
environment subject to a Markov process with two
states. The change in operating conditions viz.
fluctuations of voltage, corrosive atmosphere, very low
gravity etc. may make a system completely inoperative.
Severe environmental conditions can make the actual
mission duration longer than the ideal mission duration.
In this paper we have taken failure caused by power
and failure due to Time Error. When the main
operative unit fails then cold standby system becomes
operative. Failure due to Time Error cannot occur
simultaneously in both the units and after failure the unit
undergoes repair facility of Type- II by ordinary
repairman or Type III by multispecialty repairman in
case of failure caused by power
immediately. The
repair is done on the basis of first fail first repaired.
Assumptions
1.
2.
3.
1, 2 are constant failure rates for failure caused
by power and failure due to Time Error
respectively. The CDF of repair time distribution
of Type I, Type II and multispecialty repairmen
Type-III are G1(t), G2(t) and G3(t).
The failure due to Time Error is non-instantaneous
and it cannot come simultaneously in both the
units.
The repair starts immediately after failure caused
by power and failure due to Time Error and works
on the principle of first fail first repaired basis.
Notations
1 , 2 - failure rates for failure caused by power , failure
due to Time Error respectively.
G1(t), G2(t), G3(t) – repair time distribution Type –I,
Type-II, Type III due to power failure, due to Error
Time, repair by the multispecialty repairman
respectively.
p, q - probability of failure caused by power
and
failure due to Time Error respectively such that p+ q=1
Mi(t) System having started from state i is up at time t
without visiting any other regenerative state
Ai (t) state is up state at instant t
Ri (t) System having started from state i is busy for
repair at time t without visiting any other regenerative
state.
Bi (t) the server is busy for repair at time t.
Hi(t) Expected number of visits by the server for
repairing given that the system initially starts from
regenerative state i
Symbols for states of the System
Superscripts
O, CS, PF, TEF,
Operative, Cold Standby, failure caused by power,
failure due to Time Error respectively
Subscripts npf, pf, tef, ur, wr, uR
No failure caused by power , failure caused by power ,
failure due to Time Error , under repair, waiting for
repair, under repair continued from previous state
respectively
Up states – 0, 1, 2, 3, 8,9 ; Down states – 4, 5, 6, 7
regeneration point – 0,1,2, 3, 8, 9
States of the System
0(Onpf, CSnpf)
One unit is operative and the other unit is cold standby
and there is no failure caused by power in both the
units.
1(PF pf, urI , Onpf)
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The operating unit fails due to failure caused by power
and is under repair immediately of Type- I and standby
unit starts operating with no failure caused by power.
p27 = q- qG2*( 2) = p28(7),
2(TEFtef, urII , Onpf)
We can easily verify that
The operative unit fails due to Time Error and
undergoes repair of type II and the standby unit becomes
operative with no failure caused by power.
p01 + p02 = 1,
3(TEFtef, urIII , Onpf)
p30 = p82 = p91 = 1
(1)
p10 + p14 (=p11(4)) + p15 (=p12(5) ) = 1,
p23 + p26 (=p29(6)) + p27 (=p28(7) ) = 1
And mean sojourn time is
The first unit fails due to Time Error and under Type-III
multispecialty repairman and the other unit is operative
with no failure caused by power .
µ0 = E(T) =
Mean Time To System Failure
4(PF pf,uR1 , PF pf,wrI)
The unit failed due to PF resulting from Failure caused
by power is under repair of Type- I continued from
state 1and the other unit failed due to PF resulting from
Failure caused by power is waiting for repair of Type-I.
Ø0(t) = Q01(t)[s] Ø1(t) + Q02(t)[s] Ø2(t)
Ø1(t) = Q10 (t)[s] Ø0(t) + Q14(t) + Q15(t)
Ø2(t) = Q23 (t)[s] Ø3(t) + Q26(t) + Q27(t)
Ø3(t) = Q30(t)[s] Ø0(t)
5(PF pf,uR1 , TEFtef, wrII)
The unit failed due to PF resulting from Failure caused
by power is under repair of Type- I continued from
state 1and the other unit fails also due to Failure due to
Time Error is waiting for repair of Type- II.
6(TEFtef, uRII , PF pf ,wrI)
(3-6)
We can regard the failed state as absorbing
Taking Laplace-Stiljes transform of eq. (3-6) and
solving for
ø0*(s)
= N1(s) / D1(s)
(6)
where
The operative unit fails due to Failure due to Time Error
and under repair continues from state 2 of Type –II and
the other unit is failed due to PF resulting from Failure
caused by power and waiting for repair of Type-I
7(TEFtef ,uRII , PFpf,wrII)
The one unit fails due to Failure due to Time Error is
continued to be under repair of Type II and the other
unit failed due to PF resulting from Failure caused by
power is waiting for repair of Type-II
*
N1(s) = Q01*[ Q14 * (s) + Q15 * (s) ] + Q02*[ Q26 * (s) + Q27
(s) ]
D1(s) = 1 - Q01* Q10* - Q02* Q23* Q30*
Making use of relations (1) & (2) it can be shown that
ø0*(0) =1, which implies that ø0 (t) is a proper
distribution.
(s)
MTSF = E[T] =
8(PFpf,urIII , TEFtef, wrII)
s=0
The one unit fails due to power is under multispecialty
repair of Type-III and the other unit is failed due to
Time Error is waiting for repair of Type-II.
9(PFpf,urIII, TEFtef, wrI)
(D1’(0)
=
=
(
’
- N1 (0)) / D1 (0)
+p01
+ p02
) / (1 - p01 p10 - p02 p23 )
where
The one unit fails due to power is under multispecialty
repair of Type-III and the other unit is failed due to
Time Error is waiting for repair of Type-I
μ0 = μ01+ μ02
Transition Probabilities
μ2 = μ23+μ28(7)+ μ29(6)
Simple probabilistic considerations yield the following
expressions:
Availability analysis
p01 = 1 / 1 + 2 ,
1)+q G2*( 2) ,
p02 = 2 / 1 + 2 ,
p10 = pG1*(
,
μ1 = μ10 + μ11 + μ12(5),
(4)
Let Mi(t) be the probability of the system having started
from state i is up at time t without making any other
regenerative state. By probabilistic arguments, we have
p14 = p- pG1*( 1) = p11(4) ,
M0(t) = e− 1 t e− 2 t ,
p15 = q- q G1*( 2) = p12(5),
M1(t) =p G1(t) e - 1 t
p23 = pG2*( 1)+q G2*( 2) ,
M2(t) =q G2(t) , M3(t) = G3(t)
p26 = p- pG2 ( 1) = p29
*
(2)
(6)
,
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The point wise availability Ai(t) have the following
recursive relations
A0(t) = M0(t) + q01(t)[c]A1(t) + q02(t)[c]A2(t)
A1(t) = M1(t) + q10(t)[c]A0(t) + q12(5)(t)[c]A2(t)+
q11(4)(t)[c]A1(t) ,
A2(t) = M2(t) + q23(t)[c]A3(t) + q28(7)(t)[c] A8(t) +
q29(6)(t)] [c]A9(t)
A3(t) = M3(t) + q30(t)[c]A0(t)
The expected busy period of the server when there is
failure due to Time Error and failure caused by
power in (0,t]
R0(t) = q01(t)[c]R1(t) + q02(t)[c]R 2(t)
q12(5)(t)[c] R2 (t) +
R1(t) = S1(t) + q10(t)[c]R0 (t) +
q11(4)(t)[c]R1(t)
R2(t) =
S2(t) + q23(t)[c]R3(t) + q28(7)(t) R8(t)
(6)
+q29 (t)][c]R9(t)
R3(t) = S3(t) + q30(t)[c]R0(t)
A8(t) = q82(t)[c]A2(t)
A9(t) = q91(t)[c]A1(t)
R8(t) = S8(t) + q82(t)[c]R2(t)
(7-11)
Taking Laplace Transform of eq. (7-11) and solving for
R9(t) = S9(t) + q91(t)[c]R1(t)
(16-21)
where
=
N2(s) / D2(s)
S1(t) =p G1(t) e - 1 t , S1(t) =q G2(t) e - 2 t
S8(t)= S9(t) = G3(t)
(22)
(12)
where
Taking Laplace Transform of eq. (16-21) and solving for
(4)
11 }{1-
N2(s) =
0
91 ]
01[
1{1
82}
(5)
+
(7)
28
[{1 -
3+
23
+
(5)
12
82 }-
(6)
29
–
= N3(s) / D2(s)
12
2}{1
(7
28
3]+
23
(4)
11 }+
–
02[{
N 3(s) =
28
1]
D2(s) = {1 (5)
12
(6)
29
(7)
28
(4)
11
(4)
11
}+
91
-
01[
(5)
12
82} +
(6)
29
(7
28
}{1-
91
23
{1 –
]-
01[
S1(1 –
82)
+
(5)
12 [
S2 +
23
S3+
23S3
+
(7)
28
S8 + S9
(6)
29
)(1-
and D 2(s) is already defined.
02[{
23
30
{1 –
(Omitting the arguments s for brevity)
10]
In the long run, R0 =
(Omitting the arguments s for brevity)
The steady state availability
A0 =
=
=
Using L’ Hospitals rule, we get
A0 =
=
(13)
The expected up time of the system in (0,t] is
(24)
The expected period of the system under failure due to
Failure due to Time Error and failure caused by power
is
(t) =
So that
The expected number of visits by the repairman
Type-I or Type-II for repairing the identical units in
(0,t]
H0(t) = Q01(t)[s][1+ H1(t)] + Q02(t)[s][1+ H2(t)]
H1(t) = Q10(t)[s]H0(t)] + Q12(5)(t)[s] H8(t) + Q11(4)(t)]
[s]H1(t) ,
(t) =
So that
H2(t) = Q23(t)[s]H3(t) + Q28(7)(t) [s] H8(t) +Q29(6)(t)]
[c]H9(t)
(14)
The expected down time of the system in (0,t] is
(t) = tSo that
(7)
28
( S8+
(6)
S9)]+ 02 [ ( S2+
29
(4)
(6)
11 )+ S1
29
91]
82 }-
10
(23)
where
(6)
29 )
(7)
91
S3(t) =
H3(t) = Q30(t)[s]H0(t)
(t)
H8(t) = Q82(t)[s]H2(t)
(15)
H9(t) = Q91(t)[s]H1(t)
(25-30)
Taking Laplace Transform of eq. (25-30) and solving for
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=
N4(s) / D3(s)
(31)
N4(s) = { Q01* + Q02*}[ { 1 – Q28(7)* Q82* } – Q12(5)*
Q29(6)* Q91* ]
- expected busy period of the system under failure due to
Failure due to Time Error and failure caused by power
for repairing the units in (0,t ]
And
- expected number of visits by the repairman Type- I
or Type- II for repairing of identical the units in (0,t]
D3(s) = {1 – Q11(4)*} { 1- Q28(7)* Q82*} – Q12(5)* Q29(6)*
Q91* - Q01*[ Q10*{ 1 – Q28(7)* Q82* }+ Q12(5)* Q23* Q30*] Q02* Q30*{1 – Q11(4)*}+ Q29(6)* Q91* Q10*]
expected number of visits by the multispecialty
repairman Type- III for repairing of identical the units
in (0,t]
(Omitting the arguments s for brevity)
The expected total cost per unit time in steady state is
In the long run,
C=
H0 = N4(0) / D3’(0)
(32)
=
= K1A0 - K 2R0 - K 3H0 - K 4W0
where
where
N4(0) = {1 – p 11(4)} { 1- p 28(7) } – p 12(5) p 29(6)
K1 - revenue per unit up-time,
The expected number of visits by the multispecialty
repairman Type-III for repairing the identical units
in (0,t]
K2 - cost per unit time for which the system is busy
under repairing,
W 0(t) = Q01(t)[s][1+ W 1(t)] + Q02(t)[s][1+ W 2(t)]
K3 - cost per visit by the repairman type- I or type- II
for units repair,
W 1(t) = Q10(t)[s]W 0(t)] + Q12(5)(t)[s] W 8(t) + Q11(4)(t)]
[s]W 1(t) ,
W 2(t) = Q23(t)[s]W 3(t) + Q28(7)(t) [s] W 8(t) +Q29(6)(t)]
[c]W 9(t)
W 3(t) = Q30(t)[s]W 0(t)
W 8(t) = Q82(t)[s]W 2(t)
W 9(t) = Q91(t)[s]W 1(t)
(33-38)
K4 cost per visit by the multispecialty repairman
Type- III for units repair
CONCLUSION
After studying the system, we have analyzed graphically
that when the failure rate failure due to Time Error and
failure caused by power increases, the MTSF, steady
state availability decreases and the Profit-function
decreased as the failure increases.
Taking Laplace Transform of eq. (33-38) and solving for
REFERENCES
[1]
Dhillon, B.S. and Natesen, J, Stochastic Anaysis
of outdoor Power Systems in fluctuating
environment, Microelectron. Reliab. ,1983; 23,
867-881.
[2]
Kan, Cheng, Reliability analysis of a system in a
randomly changing environment, Acta Math.
Appl. Sin. 1985, 2, pp.219-228.
[3]
Cao, Jinhua, Stochatic Behaviour of a Man
Machine System operating under changing
environment subject to a Markov Process with
two states, Microelectron. Reliab. ,1989; 28, pp.
373-378.
Benefit- Function Analysis
[4]
The Benefit-Function analysis of the system considering
mean up-time, expected busy period of the system under
failure due to Failure due to Time Error and failure
caused by power , expected number of visits by the
repairman for unit failure.
Barlow, R.E. and Proschan, F., Mathematical
theory of Reliability, 1965; John Wiley, New
York.
[5]
Gnedanke, B.V., Belyayar, Yu.K. and Soloyer ,
A.D. , Mathematical Methods of Relability
Theory, 1969 ; Academic Press, New York.
=
N5(s) / D3(s)
(39)
N5(s) = Q01* Q12(5)* [ Q23* Q30* + Q28(5)* Q82* + Q29(6)*
Q91* ] + Q02* [ Q23* Q30* + Q28(5)* Q82* + Q29(6)* Q91* ]{1 –
Q11(4)*}]
(Omitting the arguments s for brevity)
In the long run,
W 0 = N5(0) / D3’(0)
(40)
Where
N5(0) = p 01 p 12(5) + p 02{1 – p 11(4)}
The expected total Benefit-Function incurred in (0,t] is
C(t) = Expected total revenue in (0,t]
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ISSN (Online): 2347 - 2812, Volume-3, Issue -2, 2015
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International Journal of Recent Advances in Engineering & Technology (IJRAET)
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Fig. The State transition Diagram
Up-State
Down-State
 regeneration point

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ISSN (Online): 2347 - 2812, Volume-3, Issue -2, 2015
7