CC BY
13
Suppose that an initial state is 10. Hence the initial distribution is p(0) = [1 0] and the Laplace transform of the unconditional reliability function is R (s) = R0(s). Now the equation (20) takes the form of
P r
(s + ß o + 1o)
Pi
(s + ßj +1j)
a1
Ro( s)
R1 (s)
Pi
s + 1o (s + 1o)(s +P o +1o)a0 s + 11 (s + 11)(s + P1 + 11)a
For
ao = 2, a1 = 3, Po = o.2, ß1 = o.5,1o = o, 11 = o.2 : we have
1 o.o4 o.o4
-+-
Ro(s) =
s s (s+o.2)2 (s+o.2)2
1
o.125
s+o.2 (s+o.2)(s+o.7)3
1-
o.o4 o.125
(s+o.2)2 (s+o.7)3
Using the MATHEMATICA computer program we obtain the reliability function as the inverse Laplace transform.
R(t) = 1.33o23exp(-o.o6142931)
+ exp(-o.o21)(1.34oo7 • 1o-14 + 9.9198 • 1o-151)
- 2exp(-o.8439351)[o.o189459cos(o.1717891)
+ o.oo695828sin( o.1717891)]
- 2exp(-o.37535 t)[o.146168cos(o.2246991)
+ o. 128174 sin(o.2246991)] Figure 6 shows the reliability function.
20 4 0 60 80 100
Figure 6. The reliability function from example 2 The corresponding density function
f (t) = - R'(t) is shown in Figure 7
40 60 80 100
Figure 7. The density function from example 2
8. Conclusion
The semi-Markov processes theory is convenient for description of the reliability systems evolution through the time. The probabilistic characteristics of semi-Markov processes are interpreted as the reliability coefficients of the systems. If A represents the subset of failing states and i is an initial state, the random variable QiA designating the first passage
time from the state i to the states subset A, denotes the time to failure of the system. Theorems of semi-Markov processes theory allows us to find the reliability characteristic, like the distribution of the time to failure, the reliability function, the mean time to failure, the availability coefficient of the system and many others. We should remember that semi-Markov process might be applied as a model of the real system reliability evolution, only if the basic properties of the semi-Markov process definition are satisfied by the real system.
References
[1] Barlow, R. E. & Proshan, F. (1975). Statistical theory of reliability and life testing. Holt, Reinhart and Winston Inc. New York.
[2] Brodi, S. M. & Pogosian, I. A. (1973). Embedded stochastic processes in queuing theory. Kiev, Naukova Dumka.
a
o
1
a
1
o
1
20
[3] Cinlar, E. (1969). Markov renewal theory. Adv. Appl. Probab. 1,No 2, 123-187.
[4] Grabski, F. (2002). Semi-Markov models of reliability and operation. IBS PAN, v. 30. Warszawa.
[5] Grabski, F. & Kolowrocki, K. (1999). Asymptotic reliability of multistate systems with semi-Markov states of components. Safety and Reliability, A.A. Balkema, Roterdam, 317-322.
[6] Grabski, F. (2003). The reliability of the object with semi-Markov failure rate. Applied Mathematics and Computation, Elsevier.
[7] Grabski, F. & Jazwinski, J. (2003). Some problems of the transport system modelling. ITE.
[8] Grabski, F. (2005). Reliability model of the multistage operation. Advances in Safety and Reliability. Kolowrocki (ed.). Taylor & Fracis Group, London. ISBN 0 415 38340 4, 701-707
[9] Grabski, F. (2006). Random failure rate process. Submitted for publication in Applied Mathematics and Computation.
[10] Grabski, F. (2006). Semi-Markov Model of an Operation. International Journal of Materials & Structural Reliability. V. 4, 99-113.
[11] Kopocinska, I. & Kopocinski, B. (1980). On system reliability under random load of elements. Aplicationes Mathematicae, XVI, 5-15.
[12] Korolyuk, V. S. & Turbin, A.F. Semi-Markov processes end their applications, Naukova Dumka, Kiev.
[13] Limnios, N. & Oprisan, G. (2001). Semi-Markov Processes and Reliability. Boston, Birkhauser.
Grabski Franciszek
Navcd University, Gdynia, Poland
The random failure rate
Keywords
reliability, random failure rate, semi-Markov process Abstract
A failure rate of the object is assumed to be a stochastic process with nonnegative, right continuous trajectories. A reliability function is defined as an expectation of a function of a random failure rate process. The properties and examples of the reliability function with the random failure rate are presented in the paper. A semi-Markov process as the random failure rate is considered in this paper.
1. Introduction
Often, the environmental conditions are randomly changeable and they cause a random load of an object. Thus, the failure rate depending on the random load is a random process. The reliability function with semi-Markov failure rate was considered in the following papers Kopocinski & Kopocinska [5], [6], Grabski [3], [4].
2. Reliability function with random failure rate
Let { jt(/): t > 0} be a random failure rate of an object. We assume that the stochastic process has the nonnegative, right continuous trajectories. The reliability function is defined as
From Jensen's inequality we get very important result
R(t) = E
exp -Jji(x)dx
, f>0.
(1)
R(t) = E
exp - Jji(x)dx
> exp -\E[n{x)]dx = R(t), t > 0.
(4)
The above mentioned inequality means that the reliability function defined by the stochastic process {jt(/): i > 0} is greater than or equal to the reliability function with the deterministic failure rate, equal to the expectation A, (?) = A'| ji(7 ) |. It is obvious, that for the stationary stochastic process {jt(/): i > 0}. that has a constant mean value
A, (t) = A'|ji(7)| = X. the reliability function defined by (3) is
It means that the reliability function is an expectation of the process {o(0: t > 0}, where
o(t) = exp - Jji(x)dx , t > 0
(2)
Let
R(t) = exp - \E[n{xj\dx , t > 0.
(3)
R(t) = exp - X\dx = exp(-A, t), t > 0.
(5)
Hence, we come to conclusion: for each stationary random failure rate process, the according reliability function for each t > 0, has values greater than or equal to the exponential reliability function with parameter X.
Example 1.
Suppose that, the failure rate of an object is a stochastic process {ji(0 : t > 0}, given by ji(0 = C I. t > 0, where C is a nonnegative random variable. Trajectories of the process {o(0: t > 0}, are
£(0 = exp(-c—), f >0,
where c is a value of the random variable C. Assume that the random variable C has the exponential distribution with parameter P :
P(C<u) = \-e-V", u> 0.
Figure 2. Reliability function R(l)
Assume that the process {u(t): t > 0}has an ergodic mean, i.e.
lim i fu(x)dx = E[u(t)] = n .
Then, according to (1), we compute the reliability function
R{t) = E
exp -\Cxdx
= PJe
o
-"I y+P
w — U —
= je 2 pe^'du
2P
r +2P
Figure 1 shows that function.
Figure 1. Reliability function R(i) In that case the function (3) is
R(t) = exp - \E\Cx\dx = exp
( r^
2P
i>0.
Figure 2 shows that function.
Suppose that a failure rate process {jt(/):/>()} is a linear function of a random load process {u(t) :t> 0}:
Jl(i) = 8 ll(t) .
Then, [2], [3]
limii(-) = exp[-«i] .
e-»0 e
It means, that for small s R(x)« exp[-s Tix\.
3. Semi-Markov process as a random failure rate
The semi-Markov process as a failure rate and the reliability function with that failure rate was introduced by Kopocinski & Kopocinska [5]. Some extensions and developments of the results from [3] were obtained by Grabski [3], [4].
3.1. Semi-Markov processes with a discrete state space
The semi-Markov processes were introduced independently and almost simultaneously by P. Levy, W.L. Smith, and L.Takacs in 1954-55. The essential developments of semi-Markov processes theory were achieved by Cinlar [1], Koroluk & Turbin [8], Limnios & Oprisan [7], Silvestrov [9]. We will apply only semi-Markov processes with a finite or countable state space. The semi-Markov processes are connected to the Markov renewal processes. Let S be a discrete (finite or countable) state space and let R+ =[0,oo), N„={0,1,2,...}. Suppose, that
2,,,. . n = 0.1.2.... are the random variables defined
on a joint probabilistic space (fl , • , P ) with values
on S and R+ respectively. A two-dimensional random
sequence {(^„,S„), n = 0,1,2,...} is called a Markov renewal chain if for all
i0,....,i„_l,ieS,t0,...,tn eR+,neN0.
The equalities
1. = j, Jn+1 £11Xn = i, J = tn,...,X = i0, J0 = t0}
= p{ n+1 = j, Jn+1 £ 11X n = i}= Qj (t) (6)
2. P{X0 = i0, J0 = 0} = P{X0 = io} = p
(7)
hold.
It follows from the above definition that a Markov renewal chain is a homogeneous two-dimensional Markov chain such that the transition probabilities do not depend on the second component. It is easy to notice that a random sequence {Xn : n = 0,1,2,...} is a homogeneous one-dimensional Markov chain with the transition probabilities
Pj = P{X n+! = j IX n = i} = lim Qj (t). (8)
J t J
A matrix
Q(t) = [Qij (t): i, j e SJ
Is called a Markov renewal kernel. The Markov renewal kernel and the initial distribution p = [pi : i e S] define the Markov renewal chain.
That chain allows us to construct a semi-Markov
process.
Let
x 0 =J0 = 0,x n =J1 + ... +Jn , x» = suPi^ n : n e N 0}
A stochastic process {X(t): t > 0} given by the following relation
X(t) =Xn for t e [xn,xn+i)
(9)
is called a semi-Markov process on S generated by the Markov renewal chain related to the kernel Q(t), t > 0 and the initial distribution p. Since the trajectory of the semi-Markov process keeps the constant values on the half-intervals [x n, x n+1) and it is a right-continuous function, from equality X (x n) =X n, it follows that the sequence {X (x n): n = 0,1,2,...} is a Markov chain with the transition probabilities matrix
P = [pj : i, j e S].
(10)
The sequence {X(x n): n = 0,1,2,...} is called an embedded Markov chain in a semi-Markov process {X(t): t > 0}.
The function
Fj (t) = P{x n+1 -x n £ 11X(x n) = i,X(x n+1) = j}
Q« (t)
(11)
is a cumulative probability distribution of a holding time of a state i, if the next state will be j . From (11) we have
Qij (t) = pF (t).
The function
(12)
Gi (t) = P{xn+1 -xn £ 11 X(xn) = i}= EQj(t) (13)
jeS
is a cumulative probability distribution of an occupation time of the state i. A stochastic process {N(t): t > 0} defined by
N(t) = n for t e [x n, x n+1)
(14)
is called a counting process of the semi-Markov process {X(t): t > 0}.
The semi-Markov process {X(t): t > 0} is said to be regular if for all t > 0
P{N(t) < ¥} = 1.
(15)
It means that the process {X(t): t > 0} has the finite number of state changes on a finite period. Every Markov process {X(t): t > 0} with the discrete space S and the right-continuous trajectories keeping constant values on the half-intervals, with the generating matrix of the transition rates A = [a «j : i, j e S], 0 <-aii =ai < ¥ is the semi-
Markov process with the kernel
Q(t) = [Qj (t): i, j e S], where
Qj (t) = p« (1 - e -aii'), t > 0,
a,
p« = — for« * «
a
and
0
Pi = 0.
[I - q i (5)] = |5y. - q. (5 + 1i) : i, j e J\
3.2. Semi-Markov failure rate
Suppose that the random failure rate {1(t): t > 0} is the semi-Markov process with the discrete state space S = {1. : j e J}, J = {0,1,...,m} or J = {0,1,2,...},
0 £ 10 < 11 < ... with the kernel Q (t) = [Qj (t): i, j e J]
and the initial distribution p = [pt : i e J]. We define a conditional reliability function as
Ri (t) = E
expl - jn(u)du l|l(0) = l
t > 0, i e J. (16)
In [3] it is proved, that for the regular semi-Markov process {1(t): t > 0} the conditional reliability functions Ri (t), t > 0, i e J defined by (16), satisfy the system of equations
Ri (t) = ^[1-Gi (OJ+ZK^R, (t - x)JQ (x), (17)
j 0
i e J.
Using the Laplace transform we obtain the system of linear equations
R(s) = -V-G(s + 1) + (s)~j(s + 1), i e J (18)
s + 1- j j j
where
Ri (5) = Je-5tRi (t)dt,
H(5) =
5 + 1
- Gi (5 + 1i): i e J
The conditional mean times to failure we obtain from the formula
mi = lim Ri (p), p e (0, ¥), i e J
(20)
p ®0T
The unconditional mean time to failure has a form
m = I P(i (0) = 1) mi.
(21)
ieJ
3.3. 3-state random walk process as a failure rate
Assume that the failure rate is a semi-Markov process {^(t): t > 0} with the state space S = {10,11,12} and the kernel
Q(t) =
0 G0(t) 0 aG1 (t) 0 (1 - a)G1 (t) 0 G2(t) 0
where G0 (t), G1 (t),G2 (t) are the cumulative probability distribution functions with nonnegative support. Suppose that at least one of the functions is absolutely continuous with respect to the Lebesgue measure. Let p = [ p0 , p1 , p2 ] be an initial probability distribution of the process. That stochastic process is called the 3-state random walk process. In that case the matrices from the equation (19) are
[I - (s)] =
Gi (5) = Je-5tG(t)dt,
~j (5) = J e-5tdQj (t).
1
- aq10 + 11) 0
- q0(5 + 10) 1
- ~2(5 + 12)
0
- (1-a) ~1 (5 + I1) 1
(22)
In matrix notation we have
where
[I - q 1 (5)] R(5) = H(5),
where
R(5) = R (5): i e J[,
(19)
,(5) = Je-stdGt (t), i = 0,1,2.
R(5) =
R0 (5)
R~1(5)
R~2(5)
1
¥
¥
¥
¥
H(s) =
7+t - Go(s + Ao)
sir - ?i(s + 1>) ^ - G2(s +12)
(23)
The Laplace transform of unconditional reliability function is
R(s) = PoRo (s) + PR (s) + Pr R (s)
Example 2. Assume that
Po = 1 Pl = P2 = 0 and
G0(t) = 1 -(1 +a t)e-at,
Gr(t) = 1 - e t,
G2 (t) = 1 - (1 + g t)e-1, t > 0. The corresponding Laplace transforms are
Go(s) =
G1(s)=
a
s(s +a )2
b
s(s + b )
G2(s) =
,(s) =
(s) =
y
s(s + g )2
a2 (s + a)2
s + b
-(s) = -
r
(s + g )2 Let
P = [1, o,o], a = o.4 and
a = o.4, b = o.o4,g = o.o2, 1o = o, 11 = o.1, l = o.2 Since the matrices (22) and (23) are
[I - (s)] =
- o.4
o.o4
(o.o5+s)2 1
o.ooo4
— o.6
o.o4
2
H(s) =
(o.22+s)
1 o.oo25
" s (o.o5+sy 1 o.o4
s+o.1 (s+o.1)(o.14+s) 1 o.ooo4
s+l2
(s+o.2)(o.22+s )
From solution of equation (19), in this case, we obtain a(s )
R (s) = Ro( s) =
where
b (s)
a(s) = (o.o1623 + o.23349s + s2)
• (o.o5oo2 + o.44655s + s2)
b (s) = (o.o3o83 + s)(o.o7486 + s)(o. 13292 + s)
• (o.o4882 + o.44138s + s2)
Using the MATHEMATICA computer program we obtain the reliability function as the inverse Laplace transform
R(t) = o.51646e -a 132921 + o.23349e
-o.o74861
+ 2.28565e-
- 2 • o.o1539e
- 2 • o.o1343e"
-o.22o69t
cos(o.o1o75t) cos(o.o1o75t).
Figure 3 shows this reliability function.
o.oo25
1
o
o. 14+s
o. 14+s
o
1
o.132921
b
