INTRODUCTION
Reliability has been a major concern for the system designers. One of
the most important systems in reliability is the standby system. In this
system, the whole components are not employed at the moment, it means
that in each moment there is just one part or component that is employed
and as soon as the failure of the operating component, the system switch
on another well component.
Many systems consist of components having various failure modes. Several
studies have presented a KoutofN system subject to two failure modes.
Moustafa (1996) presented Markov models for analyzing the transient reliability
of KoutofN: G systems subject to two failure modes. Moustafa (1996)
proposed a procedure for obtaining closed form of the transient probabilities
and the reliability for nonrepairable systems. Another research effort
is the research of Pham and Pham (1991), which has considered [k, nk+1]out
ofn: F systems subject to two failure modes. Shao and Lamberson (1991)
presented a model for koutofn: G system with load sharing.
Zhang et al. (2000) presented circular consecutive 2outofn
repairable system with one repairman. They determined rate of occurrence
of failure, mean time between failures, reliability and mean time to first
failure. Li et al. (2006) presented a koutofn system with independent
exponential components. They assigned that some working components are
suspended as soon as the system is down, repair starts immediately when
a component fails and repair times are independent and exponentially distributed.
Also they determined mean time between failures, mean working time in
a failure repair cycle and mean down time in a failurerepair cycle.
Another attempt is the study conducted by Sarhan and Abouammoh (2001),
who applied the concept of shock model to derive the reliability function
of a koutofn nonrepairable system with nonindependent and nonidentical
components. Later ElGohary and Sarhan (2005) extended Sarhan and Abouammoh
(2001) study by proposing a Bayes estimator for of a three nonindependent
and nonidentical component series system under the condition of four
sources of fetal shock. Sarhan and Abouammoh (2001) supported their estimation
method by presenting a simulation study and showed how one can utilize
the theoretical results obtained in their study.
Azaron et al. (2006) introduced a new methodology, by using continuoustime
Markov processes and shortest path technique, for the reliability evaluation
of an Ldissimilarunit nonrepairable coldstandby redundant system.
Amiri and Ghassemi (2007a) introduced a method for transient analysis
of availability and survivability of a system with repairable components
using Markov models, eigen values and eigen vectors. The considered system
was supposed to consist of n identical components and k repairmen which
components are arranged in series or in koutofn or in parallel. they
proposed a methodology for obtaining availability, survivability, MTTF_{s}
(Mean time to system failure) of the system and calculating the duration
for the system to reach to its steady state. Amiri and Ghassemi (2007b)
introduced a method for analyzing the transient reliability of systems
with identical components and identical repairmen using Markov models,
eigen values and eigen vectors and assumed that the components of the
systems under consideration can have two distinct configurations, namely;
that can be arranged in series, or in parallel. they also considered third
case in which the system is up (good) if koutofn components are good.
For all three cases they proposed a procedure for calculating the transient
probability of the system availability and the duration of the system
to reach the steady state.
In this study, a methodology for transient analysis of availability and
survivability of a system with standby components is presented in two
cases: the identical components and the nonidentical components. In this
study, a methodology for obtaining availability, survivability, MTTF_{s}
(Mean time to system failure) of the system and calculating the duration
for the system to reach to its steady state is proposed.
NOMENCLATURE AND DEFINITIONS
N (t) 
= 
No. of components failed before time t 
N`(t) 
= 
No. of repaired components before time t 
X(t) 
= 
No. of failed components at time t 
p_{n}(t) 
= 
Probability of having n failed components at time t 
p_{n }(t) = P(X(t) = n) 
(2) 
A(t) 
= 
Probability of system to be up (good) at time t, regardless
of its historical components failure and/or repair 
A(∞) 
= 
Long time system availability or system reliability 
R_{s}(t) 
= 
Survivability function 
Determines the probability that a system does not leave the set B of
functioning states during the time interval (0 t);
MTTFs: Mean time to system failure;
Definition 1: If Q considered as the state transient rate matrix
and P(t) as the state transient probability in the exponential Markov
chain with the continuous time, then P`(t) and P_{n}(t) are defined
as follows:
where, Q and P(t) are square matrixes and P_{n}(t) and P_{n}(0)
are row vectors.
THE MODEL
In this study, our aim is the determining of availability, survivor function
and MTTF_{ }of a system with the following assumptions:
• 
The components in the system are standby 
• 
There are n independent components 
• 
The system components are repairable 
• 
There are k identical repairmen 
• 
The lifetime of each component is exponentially distributed with
the parameter λ 
• 
The service time of each component by each repairman is exponentially
distributed with the parameter μ 
• 
As soon as failure of an operating component, the system switch
on another well component 
THE PROPOSED METHODOLOGY
To describe the proposed methodology for analyzing the system`s transient
reliability, consider a system having n components and k repairmen. In
the case of standby system with identical components, it is considered
that the system fails when all n components fail. also In case of standby
system with nonidentical components, it is considered that system fails
when all n components fail. It is assumed that the time between two components
failure is a random variable having the exponential distribution with
the parameter λ. It is also assumed that there are k identical repairmen
providing services to the system. The service time of a component is also
an exponentially distributed random variable with the parameter μ.
Present goal is to provide a methodology for analyzing the transient availability
and survivability of the system and the time until the system is reached
to its steady state. Considering X(t) as the number of failed components
at time t, the following Markov models are considering (Fig.
1 for identical components and Fig. 2 for nonidentical
components):

Fig. 1: 
State transition diagram of the system with k repairmen
(identical components) 

Fig. 2: 
State transition diagram of the system with k repairmen
(nonidentical components) 
Figure 2 presents the standby system with two components,
but the system can also be evaluated with more components.
Letter O in the Fig. 2, is the abbreviation of operating
and letter F, is the abbreviation of failure and also letter S, is the
abbreviation of standby.
As an example if considering n = 5 and k = 1 the Markov model is represented
as follows:
The proposed methodology for obtaining the system availability and the
transient probabilities are based on several theorems. These theorems
are established to provide the underlying theory of our methodology. these
theorems are presented as the following:
Theorem 1: Considering a continuous time exponential Markov chain
in which P`(t) = e^{Q.t}, therefore:
Proof
where, I is an identity matrix. Since P(0) = I then: P(t) = e^{Q.t}
By Definition 1:
P_{n} (t) = P_{n} (0) · P(t) = P_{n} (0)
· e^{Q.t}.
Consider the following theorem [12].
Let consider Q as an nxn square matrix which has n nonrepeating
eigen values, then:
where, t represent time, V is a matrix of eigen vectors
of Q, V^{1} is the inverse of V and d is a diagonal eigen values
of Q defined as follows:
And the matrix e^{d.t} is as follows:
Theorem 2: Consider P(t) = e^{Q.t} in which Q is the transition
matrix. In matrix Q one of the eigen values is zero and the remaining
eigen values are the complex number with the negative real part.
Proof: Since in every row of transition matrix the summation of
row elements is zero, it can deduced that one its eigen value of matrix
Q is zero. By theorem 1 and relation (7):
where, λ_{k} is the kth eigen value, α_{ijk}`s
are constant values and π_{j} is the limiting probability.
Using the contradictory concept, if it is assumed that one of the eigen
values of Q is a complex number with positive real part then:
which contradicts
and therefore the eigen values of Q are complex numbers with the negative
real part.
Theorem 3: Consider P(t) = e^{Q.t} in which Q is the transition
matrix, the time elapse until system reaches to the steady state (P(t)
= II ) can be calculated by the following formula:
In which ε is a very small number (i.e., ε = 0.0001), S_{r
}is the largest real part of the eigen values excluding the zero
element of matrix Q and II is a square matrix representing
the limiting probabilities. The elements of matrix P(t) and II are
shown as follows:
Proof
By theorem 2 all S_{m }are negative and
(π_{j}, α_{kjm}, S_{m} and C_{m}
are constant numbers). Now suppose S_{r} is greater then S_{m},
then for large values of t:
Based on the proof of these theorems, an algorithmic procedure for calculating
the availability and survivability of the system is proposed.
Algorithm
• 
Let i = 0 
• 
Determine the transition matrix Q 
• 
Determine the eigen values and eigen vectors of the matrix Q and
Let i = i+1 
• 
Determine P(t) = V.e^{d.t}. V^{1} 
• 
Determine P(t) = P_{n}(0).P(t) and if i = 1 go to step 6
and if i = 2 go to step 7 
• 
Determine the availability of the system according to the type of
the system as follows: 
*For a system with standby components (identical components):
*For a system with standby components (nonidentical components):
where, n in the statement 11, is the state that all n components are
failed.
Then delete the nth row and nth column of the matrix Q and go to step
3.
• 
Determine the survivability and MTTF_{s} of the system with
parallel components as follows: 
First numerical example (identical components): Consider a system having
five identical components. There is one repairman for repairing this system.
It is assumed that time to failure of repaired component is a random variable
with exponential distribution function with the mean of 1/2 h. The repair time
is also considered to be a random variable distributed exponentially with the
mean of 1/10 h. The availability, survivability of the system at any given time
is calculated. It is assumed that the components of the system are identical
and as soon as the failure of the operating component, the system switch on
another well component.
Solution: For determining of the system availability, the graphical
Markov model can be presented as the Fig. 3.

Fig. 3: 
State transition diagram of the system with 1 repairmen
(identical components) 
According to the algorithm:
P_{n} (0) = (1 0 0 0 0 0)
P_{n }(t) = P_{n }(0). V . e^{d.t} . V^{1} = (p_{0}(t) p_{1}(t) p_{2}(t)
p_{3}(t) p_{4}(t) p_{5}(t))
p0(t)=0.8000+0.008440e^{19.7t}+0.0555e^{12t}+0.03917e^{4,25404t}+0.06642e^{7.527t}+0.03035e^{16.47t}
p1(t)=0.160010.01497e^{19.7t}0.05555e^{12t}0.008831e^{4.254t}0.036716e^{7.527t}0.043929e^{16.472t}
p2(t)=0.032002+0.009914e^{19.74t}0.01111e^{12t}0.01467e^{4.254t}0.02970e^{7.527t}+0.0135e^{16.472t}
p3(t)=0.00640040.004683e^{19.74t}+0.01111e^{12t}0.009601e^{4.254t}0.005940e^{7.52t}+0.002714e^{16.47t}
p4(t)=0.001280+0.001645e^{19.74t}+0.002222e^{12t}0.004502e^{4.254t}+0.003283e^{7.527t}0.003929e^{16.47t }
p5(t)=0.00025600.0003376e^{19.74t}0.002222e^{12t}0.001567e^{4.25t}+0.00265e^{7.52t}+0.001214e^{16.47t} 
(13) 
Now the system availability is calculated as follows:
Table 1 represents the probability of the system to
be up (good) at time t, for different values of t.
Table 1: 
Elapse until system reaches to the steady state 

By the following closed form formula, the time elapse until system reaches
to the steady state can be calculated as follow:
t = (LN(0.00001))/(4.254) = 2.706376 
The limiting probability can also calculated as follows:
π_{0} = 0.800057, π_{1} =
0.160012, π_{2} = 0.032002, π_{3} = 0.0064,
π_{4} = 0.00128, π_{5} = 0.000256 
The amounts of p_{n}(t) for different t values can be calculated.
The results are represented in Table 2.
For determining of the system survivability and MTTF_{s}, according
to the algorithm:
P_{0}(t)=0.79937e^{0.0020t}+0.069153e^{5.225t}+0.012153e^{19.42t}+0.042791e^{15.33t}+0.074421e^{10.01t}
P_{1}(t)=0.16093e^{0.00205t}0.022491e^{5.2256t}0.021048e^{19.420t}0.05700e^{15.338t}0.060236e^{10.013t}
P_{2}(t)=0.031587e^{0.00205t}0.028536e^{5.2256t}+0.01325e^{19.4207t}+
0.010399e^{15.3382t}0.026640e^{10.0133t}
P_{3}(t)=0.006186e^{0.00205t}0.01512e^{5.2256t}0.005609e^{19.4207t} +0.007919e^{15.338t}+0.0066195e^{10.013t}
P_{4}(t)=0.001026e^{0.00205t}0.004456e^{5.2256t}+0.001511e^{19.420t}0.004745e^{15.338t}+0.006663e^{10.013t}

(15) 
Now the system survivability and MTTF_{s}, can be calculated.
The results are represented in Table 3.
Second numerical example (nonidentical components): Consider
a system having two nonidentical components. There is one repairman for
repairing this system. It is assumed that time to failure of repaired
component is a random variable with exponential distribution function
with the mean of 1/2 h for the first component and 1/3 h for the second
component. The repair time is also considered to be a random variable
distributed exponentially with the mean of 1/10 h for the first component
and 1/15 h for the second component. The availability, survivability of
the system at any given time is calculated.
Table 2: 
Probability of having n failed components at time t 

Table 3: 
System survivability at time t 

It is assumed that the components
of the system are identical and as soon as the failure of the operating
component, the system switch on another well component.
Solution: For determining of the system availability, the graphical
Markov model can be presented as the Fig. 3 with λA
= 2; λB = 3; μA = 10; μB = 15;
According to the algorithm:
P_{n} (0) = (1 0 0 0 0 0) 
P_{n }(t) = P_{n }(0). V . e^{d.t}
. V^{1} = (p_{0}(t) p_{1}(t) p_{2}(t)
p_{3}(t) p_{4}(t) p_{5}(t))
P_{0}(t)=0.499072+0.00490144e^{29.6318t}0.0125423e^{15.0475t}0.386867e^{9.10368t}+0.895162e^{6.217t}
P_{1}(t)=0.09641940.0098725e^{29.6318t}0.00115655e^{15.0475t}0.40175e^{9.10368t}+0.31671e^{6.217t}
P_{2}(t)=0.06656830.00903142e^{29.6318t}+0.0109279e^{15.0475t}+
0.183268e^{9.10368t}0.251723e^{6.217t}
P_{3}(t)=0.0169387+0.0102971e^{29.6318t}+0.00181965e^{15.0475t}0.0527501e^{9.10368}+0.0236833e^{6.217t}
P_{4}(t)=0.321202+0.00370844e^{29.6318t}+0.000949792e^{15.0475t}+0.658218e^{0.10368t}0.984042e^{6.217t} 
Now the system availability can be calculated as follows:
A(t) = 1 − p_{3}(t) 
A(∞) = 0.983 
Table 4 represents the probability of the system to
be up (good) at time t, for different values of t.
Table 4: 
Elapse until system reaches to the steady state 

Table 5: 
Probability of having n failed components at time t 

By the following closed form formula, the time elapse until system reaches
to the steady state can be calculated.
t = (LN(0.00001))/(6.217) = 1.852 
The limiting probability can also calculated as follows:
π_{0} = 0.499072, π_{1} =
0.096419, π_{2} = 0.066568, π_{3} = 0.016939,
π_{4} = 0.321202 
The amounts of p_{n}(t) for different t, can be calculated. The
results are represented in Table 5.
For determining of the system survivability and MTTF_{s}, according
to the algorithm:
P_{0}(t)=0.541465e^{0.370399t}0.279643e^{10.9019t}+0.0398818e^{17.8014t}
+0.698258e^{5.9263t}
P_{1}(1)=0.0857736e^{0.370399t}0.266552e^{10.9019t}0.0166127e ^{17.8014t}+0.197411e^{5.9263t}
P_{2}(t)=0.182774e^{5.9263t}+0.0588278e^{0.370399t}0.0420119e^{17.8014t}+0.165959e^{10.9019t}
P_{4}(t)=0.326094e^{0.370399t}+0.337334e^{10.9019t}+
0.011221e^{17.8014t} 0.674644e^{5.9263t} 
The system survivability and MTTF_{s} can be calculated. The
results are represented in Table 6.
R_{s}(t) =
p_{0}(t) + p_{1}(t) + p_{2}(t) + p_{4}(t)

MTTF_{s }= 2.7346 
Table 6: 
System survivability at time t 

CONCLUSION
In this study, a methodology for analyzing the transient availability
and survivability of a system with the standby components was presented
in two cases: the identical components and the nonidentical components.
We employed the Markov models, eigen vectors and eigen values concepts
to develop the methodology for the transient reliability of such systems.
The proposed methodology can also be employed for determining MTTF_{s}
and the time elapse until system reaches to the steady state.