The failure rate in reliability; approximation and bounds

We consider models typical to the area of reliability, and a failure rate function for processes describing the dynamics of these models. Various approximations of the failure rate function are proposed and their accuracies are investigated. The basic case studied in the paper is a regenerative model. Some interesting particular cases (Markov, semi-Markov, etc.) are considered. All proposed estimates are stated in a tractable analytic form.


Introduction
Like many other great scientists, R.L. Dobrushin had a knack for getting to the root of a pro- blem due to his belief that only a simple and clear idea provides a foundation for studying any phenomena.His fantastic mathematical technique allowed him to convert such ideas into brill- iant solutions of intricate problems.This paper contains an example of how a general idea can be transformed into a solution lead- ing to numerical results.We took a particular problem from the area of reliability that consists of the comparison of two failure rate intensities.Such a problem can be solved directly given additional assumptions imposed on the underlying system (say, that the system is Markovian).
But such a solution is unsatisfactory, in general, because the additional assumptions lead to use tools which are not the best.
Our approach can be outlined in abstract terms as follows.First, we express the desired goal function in terms of auxiliary characteristics of the underlying process (which can be estimated 1This work was partly supported by Russian Foundation of Fundamental Research (grant 95- 01-00023), International Science Foundation and Russian Government (grant J76100), EEC (grant INTAS-93-893), and University of Marne-La-Vallfie.498 CHRISTIANE COCOZZA-THIVENT and VLADIMIR KALASHNIKOV separately) without imposing unnecessary restrictions.For example, the difference between failure rate intensities under comparison (a goal function) can be expressed in terms of passage times of various auxiliary processes (auxiliary characteristics).Second, we give estimates of auxiliary characteristics by any available method (analytical, numerical, or simulation).The choice of the method is dictated, in general, by the type of the underlying process.Note that the additional assumptions arise on the second stage only, while the expression of the goal function via auxiliary characteristics is obtained under general assumptions at the first stage.This enables us to enlarge the class of investigated systems.
The paper is concerned specifically with a so-called failure rate function that plays an impor- tant role in reliability and represents a conditional density to fail at time t provided that there was no failure until t.It seems clear to us and is reflected in the approach that before we can establish distributional properties of random processes, we must first study their structural features (such as regeneration or Markov property) and pathwise properties.
Throughout the paper, we deal with various random processes defined on [0, oo).It is conven- ient to denote by (r/s) a random process having state r/s at time s C [0, oo).Sometimes (when this cannot lead to a confusion) we will use a shorter notation r/= (%).The notation (r/s)0 < s <t refers to the restriction of (%) to the time segment [0, t] (so, (r/s) (r/s)0 <s < oo)" Let a random process (r/s) describe the dynamics of a system and its state space E be parti- tioned into two subsets: E Ag tO P, alg A q).We will view alg as a subset of "good" (or oper- ating) states and P as a subset of "bad" (or failed) states assuming that E is a complete separable metric space and all paths of (r/s) are cadlag (that is they are right-continuous and have limits from the left).Additional restrictions on process (r/s) will be imposed later on.Let er inf{s: r/, C D} (1.1) be the first break-down time (or lifetime) of the system.The following characteristic A(t) --A+01im P(r _< +AAIr > t)   (1.2) is called the failure rate function for process (r/s).Of course, it is nothing else but a hazard rate function for random lifetime r.We assume that A(t)is defined correctly by relation (1.2).Unfor- tunately, function A(t) defined this way cannot be calculated easily, even for simple systems that are of interest in engineering.Even if we approximate lifetime a by another random variable (r.v.) a' which is close to r, for example in a weak topology, this does not guarantee that the hazard rate function for er' is close to A(t) in a natural sense.
Because of this, the following characteristic, called the Vesely failure rate, is often used instead of A(t)" p(r/t + ( )lr/t ( ) v(t) lim  (1. 3) The paper is organized as follows.In Section 2 we consider a decomposition which is a key tool in obtaining basic results.Examples of the decomposition are given.The core of the decom- position is the representation of the dynamics of (r/s) (until its first entrance to P) as the dynamics of a process (r/s ) (obtained from (r/s) by tabooing transitions to the subset of failed states) subjected to failures.These failures form a Cox process (a Poisson process with an intensity a modulated by (r/s); that is, a -a((r/s)0 <s < t))" Such a decomposition is used in Section 3 to obtain the failure rate function A(t) (see CI.2)-) in terms of the intensity a t.In turn,   this enables us (in subsection 3.1) to write an equation determining A(oc)in the case where (r/s ) is a regenerative process with inter-regeneration times having an exponential moment.In subsection 3.2 we propose an upper bound 0 for A(oc) which is an expectation of c with respect to the stationary distribution of (r/s ) and we prove that Av(OC)in particular cases (the resulting inequality A(oc) _< Av(OC is very useful in practical studies).Subsection 3.3 deals with the Vesely failure rate (1.3) and its expression in terms of stationary characteristics of the process (r/s)" The remainder of the paper refers to the case where intensity c depends on the current state of (r/s): at-c(r/t) In this case, (ec) can be represented as an expectation of a(r/) with respect to a specific measure (.).Section 4 contains a construction of a regenerative process for which is a stationary distribution.Using the mentioned constructions, we give accuracy estimates of A and v in Section 5, under different restrictions imposed on process (r/s)" Various examples illuminating the estimates proposed will appear in [6] as the second part of this work.These examples are motivated by their applications in reliability area.In particular, they show how the derived estimates can be used in engineering.
2. Decomposition of he Initial Process

Definitions
Throughout the paper, we will assume that all random objects are defined on a common pro- bability space (2,,P) with elementary outcomes o E f.But the argument c0 will be skipped in most cases.
The following construction is crucial for us.Consider a random process (r/s ) with state space M1, and a random process (Ms) such that (1) for every t, the r.v.M is a function of 0 r/s)0 < s < t; (2) for every t, the r.v.M is non-negative; (3) the functional M is a nondecreasing function of t.
In other words, if t is the r-field generated by random variables r/s 0 < s < t, and 50-(5t)t > o is the natural filtration generated by the history of the process (r/s), then (Mr)is an increasirg process adapted to the filtration 5 (see Jacod and Shiryaev [13] (Chapter 1)).Let us call (Mr) an integral intensity function.As we will see, the integral intensity function will be typically represented as the integral M / cudu (2.1 0 with respect to the Lebesgue measure, and the intensity function a u -au((r/s)0 < s < u) is a non- negative and right-continuous process, adapted to the filtration 5. To avoid pgtho--logical cases, let us assume in this paper that representation (2.1) holds true.In fact, all results below can be reformulated in terms of M only, and therefore the requirement (2.1) can be relaxed.Definition 2.1: Say that ((r/s,r) can be decomposed into ((r/s), (as)) if P((r/s)O <s <t E ",r > t) EI((r/s) 0 <s <t e ", > t) E I(( o r/s)O_<s_<t C ") exp audu (2.2) 0 for any t>_0.
(.4) ()o((.Thus a can be viewed as a random conditional failure rate of r.v.T at time t given o Equation (2.2) is equivalent to )o ( ( ", > ).
(.) The following lemma is an obvious but useful consequence of Definition 2.1.
This lemma means that r is dominated stochastically by an r.v.having an exponential law with parameter e if the intensity function is bounded by e.The notion of stochastic domination can be found in Shaked and Shanthikumar [21] (Section 1.A) or Lindvall [18] (Chapter IV.l).

Examples
In this subsection we treat a variety of models partly motivated by their applications to relia- bility.They illustrate Definition 2.1 and contain corresponding decompositions which will be used in the sequel.
Example 2.3" Markov process on a discrete state space.Consider a time-homogeneous Markov process (r/s) on a discrete state space E with transition rates A(i,j) lira P(r/t + A J lr/t i) A-0 A j, i, j E E.
(2.6) Let (r/s ) be a Markov process with the same initial distribution as (r/s) and transition rates A(i, j) A(i, j), 7 j, i, j .Zig. (2.7) Define intensity a as where A o ., (,,.). (2.8) A(i, V) E A(i, j).
je Then decomposition (2.2) holds true; the proof will be given later for a more general case (see Examples 2.4 and 2.9).This decomposition is widely used in the theory of Markov processes, and in that case Lemma 2.2 is quite intuitive.
Example 2.4: Semi-Markov process on a discrete state space.Let (X, T)= (Xn, Tn) n > 0 be a Markov renewal process on a discrete state space E having a semi-Markov kernel with denfflty q, that is P(Xn+l-j, Tn+l-Tn<_tlXn-i )-/ q(i,j,x)dx.0 Denote by (r/s) the semi-Markov process associated with (X, T) and defined by the equality r/t Xn if T n <_ t < T n + 1" The conditional density of T, given X o i, has the form f(i, t) E q(i, j, t), jEE and its hazard rate function is equal to f(i, t) h(i, t) hm.,1--P(T1 _< t -F A IX0 -i, T 1 > t) OO f Let p(i j t) q(i, j, t) f(i, t) P(X1 j lX -i, T 1 t).
Then the density of the semi-Markov kernel can be represented as q(i,j,t)-p(i,j,t)Z(i,t)-p(i,j,t)h(i,t)exp h(i,s)ds 0 So the transition rate function, A(i, j, t) -ILmo-P(X1 j, T 1 _ t --/k IX0 -i, T 1 > t), can be expressed as A(i, j, t) = p(i, j, t)h(i, t).
(2.9) Since j e EP( i, J, t) 1, h(i, t) E A(i, j, t) and therefore, J E ( ) q(i,j,t)-A(i,j,t)exp E A(i,k,s)ds (2.10) kE It follows from relation (2.10) that a semi-Markov process can be defined either by its kernel or by transition rate function.
The decomposition of a semi-Markov process is given in the following proposition.
Proposition 2.5: Let (rls) be a semi-Markov process with a finite state space E and transition rate function A(i, j, x) and suppose ro E .For zp C E define A(i,,t)-E A(i,j,t)   and cr-inf{t:r/t E zp}.Let (r/s) be a semi-Markov process with state space Jig, the same initial distribution as (r/s) and transition rates Ao(i,j,t A(i,j,t), 5 j, i,j dig.
Define the intensity function a by the equality c A(Vt , 2, yt) (2.11) where Yt is the time spent by process (z/s) in its current state.Then ((z/t),r) can be decomposed into ((z/t) (st)).
H Example 2.6: Redundant system.Let a system consist of N elements.Each element can be in one of the two states 0 or 1.State 0 corresponds to the failure of an element while state 1 means that the element is operating.Assume that all elements are independent and let t-(t(1),...,t(N)) be the state vector of the system.Assume additionally that each marginal process t(i), 1  N, is an alternating renewal process.Then successive sojourn times of t(i) at states 0 and 1 are independent r.v.'s.Denote by $(i)(t) and (i)(t) failure and repair rates of element (1 N), respectively.In these terms, the density function of a sojourn time at state 1 (at state 0, resp.) for element can be represented in the form (2.16) be the vector of elapsed time of elements in their current states.For instance, yt(i) is the time elapsed from the last (before t)jump of (s(i)).In fact, Yt can be viewed as a functional of (rls)0 < s < t" Process (rls) can be regarded as a piecewise-constant process with jumps defined by the raesand A(i)(Yt(i)) -li_.m0-P(r]t+ A(J) ]t(J), Vj i; t + A(i) 0 (s)O <_ <_ t, t(i) 1) #(i)(Yt(i)) =Alim0-P('t + a(J) lt(J), Vj # i; 't + a(i) 1 ( 7)o <_ <_ t, ?t(i) 0).Now, let us assume that process (s) can reach set P from 1 only at times when one of the element fails.Let ]-((1),...,(N)) be a vector of binary components denoting a state of the system and let be the set of operating elements.For r/E , define another subset of elements (that may be empty) /(]) C l(r]), (2.17) such that k R()cv(k) 1, (V(1),..., r](k-1), 0, r(k + 1),...,(N)) @.
i R(,s)

Let
The proof of this result can be given by similar arguments as in the previous example.We will exhibit another approach in the following subsection and give an alternative proof for this result (see Example 2.11).
Observe that, in reliability theory, one often considers decomposed processes, where (rlt ) describes the system dynamics without failuring factors.The failure occurs within [t, t + A) with 0 probability chA and the intensity c to fail depends either on state rlt or even on prehistory 0 ()o < < -

A general method
We now present a general approach that can be applied for time-homogeneous Markov processes (s)" Some aspects that are intuitively reasonable are taken at face value, in order that the basic idea may be exposed without the burden of too much mathematical detail.The assump- tion that (rls) is a Markov process is not restrictive since a non-Markov process can be embedded into a Markov process by equipping it with supplementary coordinates (see Example 2.11 and CHRISTIANE COCOZZA-THIVENT and VLADIMIR KALASHNIKOV Kalashinikov [15] (Chapter 10)).Process (r/s) can be defined in terms of its transition probability Q(r/;t,.)P(s+t e Is ). (2.19) However, this characteristic is not convenient for practical purposes since it cannot be found in a closed form for many processes of interest.In practice, one prefers to define Markov processes with the help of so-called infinitesimal characteristics.One of the most useful infinitesimal characteristics is the generator A. A naive definition of the generator can be given as follows (see Oynkin [9] (Chapters 1 and 2), Feller [11] (Chapter X), and Ethier and Kurtz [10] (Chapter 4) for rigorous definitions).Let us view the transition probability (2.19) as an operator that maps the set of measurable bounded functions f: E-R 1 into itself by the formula gt() / f()Q(; t, d) E(f(t) 0 ), t 0, E where g0(): f()" The following characteristic is called the infinitesimal operator or generator of (): (2.20) provided the limit exists in the sup-normed topology l f ] -sups f() Af() is a derivative of the mean value of f(t) at t-0, along the path of process (s) given 0-; it is not defined on the set of all measurable bounded functions (in general) but only on a subset D A called the generator domain.This subset depends on the type of process under consideration.Actually, there are several definitions of the generator owing to the variety of meanings of the limit in (2.20) (see Oynkin [9]) and due to the expansion of the domain of A to unbounded functions (see Davis [7] and Kalashnikov [14]).We restrict ourselves with only Feller processes (see Dynkin [9] (Chapter 2) and Meyn and  Tweedie [19]).This class of processes can be characterized by the property that transition proba- bilities (2.19) map the set of bounded continuous functions into itself.It follows, in particular, that Feller processes have transition functions that are continuous with respect to the first argu- ment (initial state).
Operator A is linear and there is one-to-one correspondence between A and transition probability Q under wide conditions (see Dynkin [9] (Chapter 1)).Here, we will assume that such a correspondence takes place.Formula (2.20) is equivalent to the equation U(f(h) 0 ) f() + hAt(o) + o(h). (2.21) The following theorem gives a general way of decomposing a Markov process.
Theorem 2.7: Let (s) be a Markov process with state space E and generator A, and let be a subset of E, c and a inf{t:t }.Suppose that A can be decomposed into the 8urn A-A+A v, (2.22) such that Af() does not depend on values f((), , ff , and component A has the form Af() / f()A(; d) f()A(; ), E, (2.23) where A(;. is a finite positive measure with support for each , continuous with respect o and sup, e A(,, V) < .T ((,,),) ca dcood ito ((), ()), () a iaov o i* rator A (*ct o ) and , A(,; V).
(.4) Proof: Formula (2.24) can be regarded as a consequence of more general results contained in Dynkin [9] (Theorem 9.7 and Sections 10.1 to 10.3) if one treats (rs) as a Markov process that exits a time r.Our purpose here is to present an intuitive reasoning of the result rather than its rigorous version given in the cited book.
Let us estimate the probability (B)-P((,)0 < < B; > t) (2.) for fixed time t > 0 and subset B. Process (s) has right-continuous paths.Therefore, it is suffi- cient to find probability (2.25) for a cylindric set B that is to take B of the form {w:(s)0 < s < t(w) E B} {w:t .(w)B d C , 1 j n,O t < t 2 <... < t n t}, where n 1 and 0 t 1 < t 2 <... K t n t are arbitrary.
is not defined after stopping time This means that the state s (s)" In these terms, Pt(B) P('tj Bj, 1 <_ j <_ n, Let us take a function f from the domain of operator A that is equal to zero in (2.22) and (2.23), A/() f() A/() A(, )f(r),

(.)
Denote by A the generator of This means (see (2.21)) that, for any e , (rls;ZP)ds I-+ 0 (2.30) As there is one-to-one correspondence between transition functions and their generators, we have, for any CC, P(h G c IW0 w) E I(r e C)exp A(ou;) du O r + o(h). (2.31) 0 Using the Markov property of processes (s)and ()and equality (2.31), and letting h0, we have for any s-mh, m 1, P(s C)-i.E ( P((5 +a) 5)P(mh C 's-h) =iE P( e C l_h) To prove this, note that f(r/)-1, r/G tt, belongs to DA0.This function f definitely belongs to the domain O x (where A is defined in (2.29)) by (2.20), (2.21), and (2.22).In fact, A is the generator of the killed Markov process that exits at time a and D. DA0 by Theorem 9.3 from Dynkin [9].Let us extend f as follows: 1 Such an extended function belongs to D A just because of the definition of generators A,A, and equality (2.29).
Process (r/s ) may contain both jumpwise and continuous components (see Examples 2.10 and 2.11).But, due to form (2.23) of operator Av, process (r/s) may hit set V only by jumps, and the intensity of the jump at point x E E is equal to A(x;V).If such a jump leads to subset V, then the state of process (r/s) just after the jump is random with the probability distribution A(x;.)/A(x; V), while process (r/s ) stays in otto.
Example 2.9: Markov process on a discrete state space.
2.3.By definition (2.6), its generator has the form Let us decompose the operator A in the following way: A-A +A v, Af(i) E A(i, j)(f(j) f(i)), E, Then, all conditions of Theorem 2.7 are fulfilled and formulas (2.8) and (2.24) coincide.
Example 2.10: Semi-Markov process on a discrete state space.Consider a semi-Markov process (r/s) from Example 2.4 and denote by Ys the elapsed time of the process in its current state at time s.Then process r/s (r/s, Ys) s > 0 is Markov and its generator has the form Of(i, y__) + E A(i, j, y)(f(j, O) f(i, y)).
Af(i,y)-Oy jeE Let .Zig'= x [0, oc) and V'= V x [0, oc).This means that (i, y) tt' if and only if and (i,y) ' if and only if i .The term Of(i,y)/Oy in the above formula is responsible for deterministic behavior of Yt between successive jumps of (s), and, for (i,y) ', it does not depend on values of f(j,z), (j,z) '.We arrive at the decomposition A-A+A with A0f(i, y) Of(i, y + A(i j, y)(f(j, O) f(i y)) je APf( i' Y) E A(i, j, y)(f(j, O) f(i, y)).
(2.32) 3. The Asymptotic Failure Rate and its Approximations

The asymptotic failure rate
The failure rate A(t) and its limiting value A(oc) provide useful characteristics for doing a number of calculations in reliability.In this section we obtain explicit expressions for them and their approximations.To state our results, let us assume that the intensity function a at(( 0 r/s)O < s < t) satisfies the following conditions" Conditions C(a) 1.
For each >_ 0, function a s as(w is right-continuous at s t for almost all w E .

3.
There exists an absolute constant such that supa s < + oc The first condition is crucial to our approach.The second condition is quite natural because of the assumption that paths of (gs ) are cadlag.The third assumption is restrictive, of course.
For example, it may fail for semi-Markov models when holding time have IFR distributions with unbounded intensities.Moreover, some final formulas in this work hold true in the case when conditions C(a) are violated.This means that the third assumption can be relaxed.It seems that the most straightforward and general way to do this is to find comparison estimates which yield that the desired characteristics of the process with unbounded function a can be obtained as the limit of corresponding characteristics for the process with intensity min(at,) when -.
We will not touch this problem here but refer the reader to works of Kalashnikov [16] and [17] where such problems have been solved for specific characteristics of regenerative processes.Perhaps, comparison estimates wanted for the purposes of this paper may lead to different setups in comparison theorem.
We now express the failure rate function A(t) introduced in (1.2) in terms of the "decomposi- tion components" of ((gs, r).By condition C(a).2, for every t there exists a right derivative f au(w)du at(w (a.s.).0 As for any 0 < A <_ 1, the Lebesgue dominated convergence theorem yields 0 0 This completes the proof.
Let us find sufficient conditions ensuring the existence of the limit For this, we use a "regeneration technique" (see Kalashnikov [16]).Let us recall that a probability distribution u is spread out if there exists an n such that u *n has a non-zero absolutely continuous component (see Asmussen [1] (Chapter VI)).A random variable is said to be spread out if its probability distribution is spread out.
The Failure Iate in Reliability: Approximations and Bounds 509 Conditions CO 1.
Conditions C(a) hold true.

2.
Process (s) is regenerative (in the sense of Smith) with the sequence of regeneration times 0-So, S1,S2, Let S-S 1 be spread out, ES Assume that process (as) has the "lack-of-memory property" that is a u only depends on 0 N(u) <_ s <_ u" 3.
The following analog of Cramr's condition holds" there exists a real number n > 0 such that Eexp (a u-)du 1.
There exists a > t such that E(eas) < + c.
Let us discuss the above conditions.Assumption C0.2 that (s) is a regenerative process is quite natural from a practical point of view.In reliability, the regeneration property is often asso- ciated with returning of the process to the state where all components are operating.In accord- ance with this, the requirement of the lack-of-memory property for (as) seems to be natural.It follows that process (as) is also regenerative with the same regeneration times So,$1,$2, Hypothesis C0.4 requires the existence of an exponential moment for inter-regeneration time S.
This is necessary (but not sufficient, in general) for satisfying Cramr's condition (3.2) which is widely used in various applications since it enables one to simplify studying of underlying models.
As we will see, n is small for highly reliable systems.Therefore, assumption a > t in hypothesis C0.4 is not restrictive in many practical reliability problems.However, condition C0.4 does not seem quite satisfactory, even for Markov processes with a finite number of states.Because of this, we will relax it in some particular cases.
The following theorem contains the desired representation of the limiting value A(c).Now, take n > 0 to satisfy (3.2) and prove that there exist limits for both numerator N and deno- minator D t.Since (s) is a zero-delayed regenerative process and the same is true for (as) N satisfies the renewal equation As S is spread out, the distribution B has the same property.By C0.3 and C0.4,g(t) E atex p (au-n)du ,S > t 0 is bounded, Lebesgue integrable and limt__.g(t-O.By the key renewal theorem (see Asmussen  [1] (Chapter VI, Corollary 1.3)) the following limit exists: 1E / ctex p (c u-)du ) ) E / atex p (a u )du dt E exp (a u )du dt. (3.8) 0 0 0 0 Substituting (3.8)into (3.3) w arrive at (3.4).
3.2 An upper bound of (oo) Theorem 3.2 has the following useful corollaries giving upper bounds of A(c) in terms of a stationary expectation of regenerative process (r/s).
Proposition 3.3: Suppose that Conditions CO hold true.(3.10) Note that the right-hand side of (3.9) is nothing else but the stationary expectation of states of the regenerative process (as).
Remark 3.4: Inequality (3.10) holds under wider assumptions than those listed in Conditions CO.Indeed, it can be used whenever we know that Condition C0.3 holds true, A(x)< to, and ES < x.
Remark 3.5: Just to illustrate Remark 3.4, suppose that (/s) is an irreducible Markov or semi-Markov process on a finite state space E, with, for example, initial state 0 E E. Recall that (r/s) is supposed to be regenerative, therefore state 0 is recurrent for both processes (s) and (ris).
Then they are regenerative with the first regeneration time, respectively, S inf{s:s -0,s _ :0} and -inf{s:ris-0,s_ #0}" It is easy to verify (using the same method as in the proof of Proposition 2.5) that Eexp (()-g)du E (e 1( < )), 0 and the CramSr condition (3.2) is nothing else but the r-recurrence property (see Cocozza-Thivent  and Roussignol [3]) with v--a.Hence, the Cram6r condition (3.2) is true if distributions of sojourn time in the states have rational Laplace transforms.Moreover, Theorem 6.3 of Cocozza- Thivent and Roussignol [3] asserts that ()-a under these conditions.So all hypotheses in Proposition 3.3 are fulfilled (without the necessity to turn to Assumption C0.4).
The following remark gives a useful representation of 0 for Markov processes.It is an immediate consequence of definition (3.9) of 0.
Since the behavior of process () depends on the behavior of () until the first hitting time of the subset , the estimates contained in Proposition 3.3 and its Corollary 3.6 are attractive for practical use.Intensity 0 (see (3.9) to (3.11)) has a clear interpretation and can be calculated in terms of stationary characteristics of the auxiliary process ().We will investigate an accuracy of this approximation in Subsection 5.1.
The following lemma shows that stationary characteristics of () can be obtained in terms of stationary characteristics of process () provided that ()is time-reversible.Given a Markov process on a countable state space E with transition rates matrix A and subset C E, we say that is communicative if, for any and j in , there exists a path from to j, i.e., there exists n and il,... n in such that il-i,i -j, If E is communicative, the process is called irreducible.The process is time reversible if there exists a probability distribution such that for any and j, (i)A(i, j) (j)A(j, i).
(3.12)It can be easily seen that every distribution r satisfying (3.12) is stationary.Lemma 3.7: Let us consider a ime-reversible Markov process (s) on a discrete state space with stationary probability distribution (.), and let us suppose that set is communicative.
(3.13) Denote by A o the transition matrix of process (r/s).By (2.7), we have A(i, j) A(i, j) for all and j from .hl.Let us now define the distribution 7r on 1 by 7r(j)-(j)Tr(j0), j E 1.

The Vesely failure rate
We now derive some useful relations for the Vesely failure rate.Let us start with an ergodic Markov process (r/s) with a discrete state space E and generator A.Here "ergodic" means that the Markov process has a unique stationary probability distribution 7r such that lim E(f(r/t))-E for every bounded function f on E. Any irreducible Markov process on a finite state space is ergodic.It follows from (1.3) that the Vesely failure rate has the form and consequently, under Condition C(a).3, Av(OO) :tli_+mv(t 1 rr(r/) A" 2). ( rr(Ytl,) E a(r/)l (. j}rr(r/) E r--) (r/' o E E o E JI Proposition 3.8: Let (s) be a time-reversible Markov process on a finite state space with a stationary distribution of states (.) and a bounded generator A; let set be communicative.Then, v( o and Proof: The proof follows from Example 2.3, Remark 3.6, Lemma 3.7, and formula (3.16) by using the result indicated in Remark 3.5.El Remark 3.9: Engineers are often interested in the Vesely failure rate since it can easily be computed.Such calculations can be done with the help of a fault tree when the system consists of independent components (see Example 3.14).In such a case the system behavior can be described by a finite reversible Markov process, if the failure and repair rates of each component are constant and strictly positive.Engineers observed (from practical examples) that A(oc) < Ay(C) but, as far as we know, this result had not been proved.Now it is done and it is an important result for applications.
As we will see in Proposition 3.15, the equality A0_ Av(CX3 is still true for a general redun- dant system introduced in Example 2.6 (see also Example 3.14) in the case of constant failure rates and general repair rates.
We now come back to general Markov processes.The following proposition generalizes repre- sentation (3.16) of general Markov processes having the Feller property.
Proposition 3.10: Let (r/s) be an ergodic Feller process satisfying assumptions of Theorem 2.7 with stationary distribution of stales r.Then Y(CX) r(t)f a(r/; P)r(dr/). (3.17) Proof: By Remark 2.8, function f(r/)= ln p} belongs to the domain of the generator of (%).The generator's definition and (1.3) yield e Av(t) P(r h ) a--,olim P(r/t + a P r/t r/)P(r/t dr/) 1 lim / --z(E(f (r/A)/r/0 r/)-/(r/))P(r/t dr/) 1 P(r/t G dl As the process is ergodic, this yields the desired result. Example 3.11: Semi-Markov process on a discrete state space.We return to Examples 2.4 and 2.10 and preserve corresponding notations.Let the semi-Markov process by ergodic and let r be its stationary distribution of states.Example 2.10 and Proposition 3.10 yield 1 e.j f r o EA(i,j,y)r,(i, dy), (3.18) s where r' is the stationary distribution of Markov process (r/s, Ys)" for example, Disney and Kiessler [8] (Section 2.7))that r'(i, dy)= Ku(i) exp It is not difficult to prove (see, h(i, s)ds dy, 0 (3.19)where function h is defined in Example 2.4, u(. is the stationary distribution of the embedded Markov chain X with transition probabilities p(i, j)= f q(i, j, y)dy, and It" is a norming constant.
Let us denote by r the stationary distribution of the semi-Markov process.where the mean sojourn time in the state i, m(i)-E(Tllr]0-i), is finite.Thus, r can be viewed as the stationary distribution of states of a Markov process with the same embedded Markov chain X and the same mean sojourn times re(i).This Markov process has the generator P(i, J) a(i, j) n-(-5 Substituting (3.19)into (3.18) and applying (2.9) to (2.10), we get J0 0 This yields (.2) E E Ku(i) q(i, j,y)dy ,v(oo) () x--' .l:,( (3.20) The arguments above imply the following result.
Proposition 3.12: For an ergodic semi-Markov process on a discrete state space, the asymptotic Vesely failure rate is equal to the asymptotic Vesely failure rate of a Markov process having the same embedded Markov chain and the same mean sojourn times in all states.
Remark 3.13: It can be seen from (3.20) that, in the Markovian case, Iv(OC -1/MUT, where MUT means Mean Up Time of the system (see Pages and Gondran [20]).The last proposition shows that this is also true in the semi-Markovian case.
Example 3.14: Redundant system.Consider the redundant system described in Examples 2.6 and 2.11 and preserve the same notations.Let us embed (ris) into the Markov process (s, Ys)" Let r' be its stationary distribution of states.By Proposition 3.10 and equation (2.32 Since all components of the system are independent, the measure r' has the product form N "(', ) II '()((i), (i)), i--1 and the same is true for the stationary distribution r of states of process (rls) N ()-l] ()(()).i=1 For any i, function a (i) depends on r](i) and y(i) only.Hence, we can write a(i)((i),y(i))instead of a(i)(rl, y).Let 7i and 6 be the mean sojourn time of the ith component in "good" state 1 and in "bad" state O, respectively, and It can be easily seen from (3.19) that 1 cxp a(i)(l(i),s)ds + 0 and, consequently, m(J)(rl(J)) 1 (j 1 H 7 +bj +----H 7r )(/(j)).
o a(i)(' y)r'(r/, dy) 7i + 6i j: j # J 7i j: j Assume additionally that the system is coherent.This means that if the system is operating (resp., failed) and if one more component becomes operating (resp., failed) then the system remains operating (resp., failed).This assumption is typical and natural for reliability studies.
It can easily be computed with the help of fault trees and existing software.
The following result generalizes Remark 3.9.It uses PH-distributions (see Asmussen [1]  (Section II.6) and Kalashnikov [15] (Subsection 9.5.3)) which can be treated as distributions of absorption times for Markov processes with a finite state space.Proposition 3.15: Consider the redundant system described in Example 2.6.Let the system be coherent with constant failure rates and general strictly positive repair rates, and repair times having finite means.Then we have: ,X -Consequently, if the repair times have PH-distributions or if conditions C0.3 and C0.4 are satisfied, then () < .().
process r/1 is . (4.9) Quite similarly, one can prove that Moreover, it follows from (4.1), (4.5), and(4.7)that the initial distribution of We now construct a regene.rativeprocess that has as its stationary distribution.For this, let us introduce.processes*, >_ 1 and positive random variables T such that processes r/*, _> 0 are independent; process r/ is a probabilistic replica of the "original" process 0 in the sense that it has the same probability law (because of this, we did not change its notation); processes 7', >_ 1 are i.i.d, probabilistic replicas of the "original" process for any >_ O, Remark 4.3: If 0 is a Markov process, then the above construction shows that r] 1 is also a Markov process with the same transition probabilities as r] (and with the initial distribution ).
Remark 4.2: Equality (4.9) shows that all shifts Osl(ql,S 1) are distributed as (,S).By the terminology adopted in the theory of regenerative processes, this means that (rl,s1) is a version of (r,S), i.e., these two processes have identically distributed cycles but perhaps different delays.Regenerative process (I,sl) is delayed, in general; that is, P(S :/: O) > O.
where 1 A T O 1 / I(' E" )dt + E I( T < 1)/ I('t " )dr and S is the first regeneration time of the second type for the shifted process 0 o; evidently, T S > T o.Since S is measurable with respect to (s) it can be regarded as a constant when conditioning with respect to (s).Therefore, equality (4.10) yields (4.15) All shifts OTO +... + Tk" are identically distributed.Because of this we have, similarly to (4.14), Therefore, / o(rl)dt N 1. 0 E"f I(ql.)exp (f a(ql.)dv)dt(4.19) Pemark 4.5: The construction above is a generalization of the construction from Cocozza-Thivent and Roussignol [4] proposed there for Markov processes on a discrete state space.Let us notice that Conditions C1 are met for such processes (see Remark 3.5) except for Assumption C0.4 which is not needed.
ES1 o Now, let us give another expression for A(oo) in terms of processes r/ and r/1.Similarly to (4.The expectations E"exp (-f a(rl)dt and E"f exp (-f a(rllu)du)dt can be expressed in 0 0 0 terms of "ordinary" expectations E with the help of formulas (4.17) and (4.18) or Remark 4.3.
The estimates derived above can be used conveniently owing to the fact that, in many practical problems, one can estimate all terms involved in (4.21) and (4.22) and thus give bounds for )t() as we will see it in the following sections.
5. Accuracy Estimates of the Asymptotic Failure Rate In this section, we obtain accuracy estimates for the proposed approximations of the asymptotic failure rate.
PV <exp(E")(max(E1, E") + flmax(ES, ESg)).Note that in "typical" reliability problems, D is close to ES if the system is highly reliable.s Using the fact that fiN --< E (where f a(u)du is defined in (5.4)), we arrive at the bound  (5.18) e C)exp A(r/u;V)du 0 This and the Markov property imply that the probability (2.28) has the form pt(B) E I(r/tj E Bj, 1 <_ j <_ n