Generalized Birth & death processes as degradation models Текст научной статьи по специальности «Математика»



GENERALIZED BIRTH & DEATH PROCESSES ASDEGRADATIONMODELS_

Vladimir Rykov, (Moscow, Russia)

Abstract

To model degradation processes in technical and biological objects generalized birth and death processes arc introduced and studied.

1 Introduction and Motivation

Traditional studies of technical and biological objects reliability mainly deals with their reliability function and steady state probabilities for renewable systems. Nevertheless, because there arc no infinitely long leaving objects and any repair is possible only from the state of partial failure, the modelling of degradation process during a life period of an object is a mostly interesting topic. From the mathematical point of view the degradation during object's life period can be described by the Birth & Death (B & D) type process with absorbing state, For this process the conditional state probability distribution given object's life period is a mostly interesting characteristic.

During last years ail intensive attention to the aging and degradation models for technical and biological objects has been attracted. The organization of special scientific conferences devoted to this topics testifies it. The aging and degradation models suppose the study of the systems with gradual failures for which multistate reliability models were elaborated (for tin: history and bibliography see, for example, [1]), In [2] - [5] the model of complex hierarchical system was proposed and the methods for its steady state and time dependent characteristics investigation was done. Controllable fault tolerance reliability systems were considered in [(:>] -[S], In the present paper a generalized B k, D process as a model for degradation and aging process for technical and biological objects is proposed. Conditional state probabilities given object's life period and their limiting values when t — x arc calculated. The variation oi the model parameters allows to consider various problems of aging and degradation control. Some simple example illustrate our approach.

2 A General Degradation Model

Most oi up-to-date complex technical systems also as biological objects with sufficiently high organization during their life period pass over different states of evolution and existence. From reliability point of view thc.sc states can be divided into three groups: the states of normal functioning, the dangerous (degradation) states and the failure states, sec fig,I, at which the states arc denoted by the letter s, normal, degradation and failure states are joined into the sets A", D. and F respectively, and possible transitions are shown with arrows.

^-

Fig, f. The structure of the degradation process.

In tin: simplest ease if the nature of the degradation process allows to completely order the states to admit the transition possibilities only to neighboring states it can be modelled by the process of B & D type.

1Thc paper was partially supported by tin: RFFI Grant No. 04-07-9011-5

3 Generalized Birth & Death Process

3.1 Definition. Basic Equalities

Suppose that the states of the object arc completely ordered, its transitions only into neighboring states are possible, and their intensities depend on the time spend in the present state. Consider firstly the general case of the process with denumerable set of states E = {1,2,...}. To describe the object behavior by a Markov process let us introduce an enlarged states space £ = Ex [0, oo) and consider two dimensional process Z(t) — {S'(t), X(t)}, where the first component S(t) £ E shows the object's state, and the second one X(t) £ [0, x ) denotes the time spent in the state Mince the last entrance into it. Denote by a;(i) and ii'jfi) (i t E) the transition intensities from the state i to the statcy i + 1 and i — 1 accordingly under the condition that the time spent at the state i equals to x.

Remark. If the stay time at. the state i is considered ay a minimum of two independent random variables (r.v): time .4; till to transition into the "next" state i + 1 and time B-. till to transition into the "previous" state i — 1 with cumulative distribution functions (c.d.f.) ¿¿(a:), Bi(x), probability density functions (p.d.f.) «¿(a;), b,(x), and mean values a, = — Ai(xdx)]1 b; = J1 — (.!•)] (fa, then the introduced process can be considered ay a special ease of semi-Markov process (SMP) [9], with conditional transition p.d.f.'s Oj(ar)

ai (,!')

ft(*) =

bj(x)

1 -Bi(x)

1 -A,ir)~

The c.d.f. of the stay time at the state i for this process equals

Qi{x) = 1 — (1 — ^(.rViU - B,\x)). and the elements of semi-Markov matrix are

X X

^^ Qii+10) = I Oi(ii)(l - BAu))du. (x) = I bi(«) (1 - Mu))du.

■o . . ". . ....b

Nevertheless, the given formalization open the new possibilities for the investigations and moreover in the degradation models we arc studying the conditional probability state distribution on given life period, that did not investigated previously.

T^f^ Denote by a.it.x) the p.d.f. of the process Z(t) at time t,

~i(t.x)dx - P{5(i) - i, x < X(t) < x + dx\. These functions satisfy to the Kolmogorov's system of differential equations

dn ¿(t, t) 0~,{t.x)

Ut Ox

with the initial and boundary conditions

- -(«¿(:ï) + ßi{x))-Ki{t,x),

Q < x <t < ao, i fc E

X! (t, 0) = S(t) + / JT3(i, x)ßz(x)dx, 0

I t

0) = J-ni_1(t,x)ai-i(x)dx + J Tvi+1(t,x)ßi+1(x)dx, 0 11

i e E.

In tin: following we will suppose the process to be lion reducible, non degenerated, i.e. it- is defined over all time axis. Sufficient condition for non reducibility it;

a,hi > 0

for all i ti E.

To avoid degeneration of the process it is needed, first of all, to eliminate instant states. For this we suppose that the process possesses the regularity property,

¿1(0) <1, 5,(0X1

for all i fc E.

Nevertheless, the absence ot instant states does not yet guarantee non-degeneration ot the process because of the possibility of jumps accumulating during the finite time interval, so that the process "can go to infinity" for the finite time interval. One of possible conditions for the non-reducible SMP without instant states to be □on-degenerated iy the recurrence of its embedded Markov chain [9]. But tliiy conditions arc hardly checked

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for general random walk in terms of its parameters. Therefore, we will use more strong, nevertheless very applicable sufficient condition for the process non-degenerateness. It is the condition of strong regularity, there exist constants c > 0 and e > 0 such, that

P{Ai > c} = 1 - Ai(c) > and P{B: > c} = 1 - Bi(c) > t for all i £ E.

The proof of sufficiency of the strong regularity of SMP for its non-degenerateness can be found, for example, in [10], [11].

For the non reducible, lion degenerated generalized B fc D process the Kolmogorov's system of equations (1) with initial and boundary conditions (2) lias a unique solution over all time axis. To find it we use the method of characteristics. Accordingly to [f 4] the characteristics of this system satisfied to the system of equations

_n-.(f.r)

= Ca.

dt = dx = — -—■—--———, i t E.

Thus, two first integrals for the system (1) arc

t~x = Ci> (1 — j4(x))(1 - Bix))

This means that at the line t — x = u the functions rr¿ it, x) (i £ E) have the form

ni(t, x) = g,,(t - ,1-)(1 - -4.,:(a'))(l - B;(x)), 0 <x<t< x , i £ E. (3)

where the functions g¡(t) in accordance with the initial and boundary conditions (2) satisfy to the system of equations

i

9i(í) = S(t) +/92(í - i)(l - A2{x)b-2{x))dx, o

i

9i{t) = f g-i-l(t - aj)ai_i(T))(l - Bi_i(a;))da;+ (4)

0

1

+ f 9i+i(t - a;)(l - j4i+i(a;)6¿+i(a;))¿c, ? = 2.3.....

6

The form of these equations shows that their solution should find in terms of its Laplace transforms (LT's). Therefore by passing to the LT's with respect to both variables into relations (3) after the change of the integration order one can get

f

(■5)

(6)

o o

where g, (s) arc the LT of the functions g,(t). and the functions 7;(s) are

7¡00 = / e_si(l — j4j(i))(l — Bi(t))dt.

o

From the other side by passing to the LT's with respect to variable t in the system (4) one get

gi(s) = f e~stgi (t)ctt = 1 + g2(s)v>2(s),

D

g,(s) = f e-stg,(t)dt = gi_1(s)^i-1(s) + gi+1(s)$i+1(s), i = 2,3..., o

where the functions and ii';(s) arc given by the relations

4>i(s) = j e_3''ttj(,i,)(l - B:(x))dx, jpi{s) - fa - Ai(x))bi(x)}dx, i £ E. 0 0 The relations (6) can be presented as a system of equations with respect to unknown functions gi(s)

f _ gi(s) - g2(s)Ms) = 1, (?)

\ + 9i(s) - ¿¿+i(s)V'i+i(s) = 0, i = 2,3 —

The closed form solution of this system in general ease even in the simplest case of usual B & D process does not possible.

In the ease of finite number n + 1 of states in the above system one should put ¿i„ + i(s) = 0. In spite of the above equations does not- possible to solve in closed form they provide calculation different characteristics of the process. Consider some of them.

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3.2 Stationary probability distribution

For calculation of the process Z(t) macro-states stationary probabilities

t t

= .V] t— t—

0 II

! !

lim 7Tj(t) = lim I tt(î,x)dx = lim f giif — x){l — Ai(x))(l — Bj(x))dx

t— " I— J " ' t— r-c J

we use the connection between asymptotic behavior of functions at infinity and their LT's at zero. Letting

7(0) =j and taking into account that accordingly to (5) S;(s) = 7Tj(s. 0). we find

lim 7T;(i) = lim s7fj(s) = lim sgi(s).

t—:■: s—Il s—0

Thus, for the problem solution it is necessary to calculate the values

g, = lim sg;(s),

a—kU

for what we use recursive relations (6). By multiplying equalities in these relations by s, and by passing to limit when s — 0 from these recursive relations we get

9l = 92>- •■>-

Si = 0Î-1&-1 + pt+ilfc+i. ? = 2.3...

(8)

where the following notations <pt = (.'>¿(0). ipi = ^¡(0) were used. Taking into account that c>, +ipi = 1. rewrite the last expressions in the form

9ii4>i + V-i) = Si-lPi-1 + + + i = 2,3----

that can be transformed to the following ones

gi4>, - gi+iVv+i = gt-\4>*-\ - giV>i, i = 2,3—

Now put i/'i = 0 (because this value does noe defined) and remark that from (8) it follows gid>i — gubl = 0, Then from the last recursive relation we find

9i<h ~ 9i+l^'i+i = 9101 - 92>i'2 =0. i = 2, 3.....

The last relation allows to calculate recursively coefficients g, in the form

4>i-1

9, = -Qi-i = ■ ■ ■ =

n

<p,-\

~ \ l<t<i Therefore, the stationary probabilities equal

Tj =7iGjgi, where G i = 1, G, = ]~[

With the help of normalizing condition X^-eE wc

91-

i<j<i

Vj

i = 2.3,

(9)

Thus, the convergence of the series

gi = I ^

V Ki<oo

(10)

(11)

is the condition for the stationary regime existence. The same result can be obtained by methods of the SMP theory [9]. We formulate these results as a Theorem.

Theorem 1. For the generalized B&D process stationary regime existence it is sufficient the convergence of the series (11). In this case the stationary probabilities are given by the formulas (9. 10). 'T!

Moreover, from the form of stationary probabilities it follows the next important corollary

Corollary, The maem-states stationary probabilities of generalized B&D process are insensitive to the shape of distributions Ai(x), Bi(x) and depend on r.v. A.¡, B~ and their distributions only by means of probabilities of jumps embedded random, walk up and down and mean time of the process stay in the given state.

& = P{A, < B, \, & = P{A, > B,}, and 7i = E[minA^B,]. (12)

For the process with finite number oi states n + 1 in the Kolmogorov's system of equations (1) one should put aiJ + i(;i) =0, In this case the stationary probabilities have the same form (9), but the normalizing constant (10) should lie changed by

91 = f E < •') ■

yi<t<n+l J

In the case of exponential distributions w4t(x) = 1 — e~'Ht and Bf(ar) = 1 — e~,3i> one get

& =

Cti+Pi'

V'i =

a, + ft

and the substitution of these expressions into the formulas (9) reduce their to the stationary probabilities of the usual B&D process.

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3.3 Distribution of the process states oil life period

For many phenomenons especially for degradation processes more appropriate is absorbing process model. For the generalized B&D process with absorbing state n + f in the Kolmogorov's system of equations (f) one should put a-,i + 1(:r) = ,3n+i(i) = 0. In this case the equation for the function 7r,l+i(i) takes the form

d~ll + iit,x) <jir... + i(f. .>;) = 0

&t

dx

with the initial and boundary condition

Tt.+i(î,0) = J nn(t,x)an(x)dx.

(13)

(14)

Thus, all functions -¡(t.x) (i = 1.n) have the same solution (3) as before. But the function ir„+i(i.x) is a constant over the characteristics, which arc determined by the equations

dnn+1

= dt = dx.

Kn+l

This means that

7rI1+i(i,œ) = g„+i(t - x), and from the boundary conditions (14) it follows that

f

9n+l(t) =

iW = j 9n(t~ sH(i)(l - Bn(x))dx. ii

The solution tt, (f) of the system of equations (1, 13, 14) gives the probability of the object stay in some state jointly with its life period T.

-i(t) = P{S(t) =i, t < T|, i = 1,2

For the degradation problems investigation more useful and adequate characteristic is the conditional state probability distribution on given object's liic period

ifi(i) = P{S(t) — i |i<T|, i = 1,2.....n.

From the above it follows that for ?r, it) the following representation is true

n{t)

ft,(t) = P{S(t) = i \t<T} =

R(t) •

where R(t) is the reliability (survival) function of the object, for which the following representation takes place

R(t) = l- irn+i(f). For the LT of the function ~„+1(i) one can find

i„+i(s) = i„+i(s,0) = ig,1+i(s) = -¿„(s)^„(s). (15)

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Therefore, for the LT R(s) of the reliability function R(t) = 1 - jrn+i(t) one has

R(s) = - - = -(f -

s s s

We will calculate the probability distribution of the object states conditionally given life period in terms of their LTV It follows from the relation (5) that the LT's tt.; (s) = jrf(s,0) of the macro-states probabilities jTf(i) have the form

Hi(s) = 9i(s) Ti(-s)- (16)

From the initial and boundary conditions it follows that the functions 9i(s) satisfy to the system of equations (7) for i = 1,2,... ,n and

9n-n(s) = 9n(s)^n(s)-

The solution of these equations can be done in terms of Kramer's rule, and it- can help to find the limits of the conditional probability states given life period.

To calculate their we should evaluate the asymptotic behavior of the functions 7Tj(t) (i — 1,2.....n) and

R(t) when t — x. We will do that with the help of their LT. Denote by A(s) the determinant of the matrix of coefficients of n first equations of the system (7) and by A, (s) the determinant of the same matrix in which i-th column is changed by the vector-column of the equation right side (vector e„). Then taking into account the expression (16) and the solution of the system (7) in terms of the Kramer's rule we got.

M3) = li(3)Ms) =li(s)lX7T

- , f.l — SsAll'n t-\ ¿n(j.)A„(j) TTn+llS) - ^Sj - ,

(t =

Theorem 2. Asymptotical behavior of the functions jr;(t) and R(t) when t — oo coincide and is determined by the maximal non-zero root si of the characteristic equation A(s) = 0. This provide the existence of the limit

V - M I i\, = hm jr; (i) = -s-.

' R(s i)

Proof. Since the functions are analytical in right half-plain, then accordingly to the inverse LT

formula the behavior of their originals arc determined by their singularities in the left half-plane [16]. Denote by si the maximal non-zero root oi the characteristic equation A(s) — 0, and suppose that in it neighboring the function A(s) lias a pole of the first order. Then the asymptotic behavior of the functions r-,it) when t — x are determined by the value si oi maximal root of the characteristic equation, and the coefficients in it are determined by the residuals of the functions ii(s) at this point. Therefore to find the limit of the functions jfj(f) it is necessary to compare the asymptotic behavior of the functions (t) and R(t) and to show that their asymptotic behavior when t — x are coincide.

From the relation (15) for the function if„+i(s) it follows that it can be represented in the form

1 ~ .. ¿ni>)A,.,(s) B Anis nn+1(s) = -9n{s)M*) = = 7 + —

with some coefficient B and some function Ar(s). Let us show that the coefficient B equals to one, B = 1. Really, from the definition it follows that

B = lim si,J+i(s) = lim 6n(s)g„(s) = 4>„g„(0) = 4>nA' ■

s— il B—0 ¿A(U i

Consider the matrix i''(s) of coefficients of the n first equations oi the system (7), denote by i'J, (s) the matrix, obtained from this one by changing its last column by the right hand side equation vector (vector e„), put 'I'' - i''(0). 'I'', — ^(0) and consider the matrix

Since elements of matrix ty are the transition probabilities of absorbing random walk, that is <p: + = 1. then the sum oi rows of the matrix i'' without its last- elements are equal to the zero row. Because the matrix 'I'', differs from 'I'' only with the last column, thus the sum of rows of the matrix — 4>n^'n without elements ol its la,st. column are also zero row. Consider now the elements of the last column. At the first place in it is an element — ij>n. at the before last one is ail element —>■>,,. and at the last one is ail element f = on + rpn■ Thus, the rows of the matrix — ^„i'J, arc linearly dependent, and, therefore, it.s determinant equals to zero. It follows from tills fact that = A(0) and consequently B = "'¿i,',"' = f. At least it follows from that

that LT .R(i) of the reliability function R(t) is presented in the form

R[») =

¿j?(a) A(>) '

Thus, the asymptotic behavior oi the reliability function R(t) when t — oc also as all others functions jrt(i) is determined by the maximal root sj of the characteristic equation A(s) — 0. In this case the coefficients in asymptotic representation of the functions ~;(f) and i?(t) when t — oc arc the residuals of the functions jr,(s) and R(s) at the point s = si,

.4;i = lim (s - si)ij(s) = lira (s - si)7;(s)

Aa£)

A(s)

7z(s'l jA;l>i i

M*i)

Am = lim (s — si)R(s) = lim (s — si

Ar(s)

Therefore, when t — oo there exists the limit

A (a) A(ai)

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= lim —-

f-x R(t.)

= lim

■—■"i R(s

7Tj(s) ^(¿i) 7i(si)Aj(si)

R(s i)

Aj?(si)

4 An Example

ill

Am'

9

To illustrate the above results let us consider a system with only three states, which can be considered as an example of the aggregated .states model (see [5], [12]), where all states of each group: normal functioning jY, degradation D, and failure F are joined into one. Suppose for the simplicity that the failures arise in accordance with the Poisson flow, but the repair times are generally distributed with c.d.f. B(x) and the hazard rate 3(x). Moreover suppose that the direct transition from normal state into the failure state are also possible with intensity 7. The marked transition graph for the process at the figure 2 is presented.

m

N

D

Fig, 2, The marked transition graph of the process with aggregated states.

In accordance with given transition graph the Kolmogorov's system of differential equations for system states probabilities lias the form

<l7TN(t) dt

—m--1--m—

/UF(t) dt

-(A + 7)7Tjv-(t) + J ¡3(x)iTD{t,x)dx, 11

-(v + i3(x))KD(t,x), 7TA'(i) +VKD{t,x)

with the initial and the boundary conditions

xD(tO) TJV (0)

A^v(i),

1, jrD(0,0) = jrjr(0)=0.

i! 18)

The reliability function of the system is

R(t) = 1 - nF(t) = 1 - f [7tt.v (w) + v-D («)] du. (19)

Jo

where the functions ir.vft) and ttxj (f- .r) are the solutions ot the two first equations of the system (17) and

7rD(i)= [ nD(t..v)dx. (20)

Jo

The solution of the second equation from the system (17) accordingly to (3) can be given in the form

7rd(m) = 9d(t-x)e~":r(l - B(x)), where the function go i t) is determined from the boundary condition (18). It gives

-D(t,x) = AnN(t - ;i-)e""'"(l - B(x)). (21)

Substitution of this solution into the first equation of the system (17) gives the following equation

i

d7V^t] - -(A + 7)7rff(i) + A I 0{x)nNit - x)e-"-' (1 - B(x))dx.

(22)

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The best method for its solution is a LT approach. In the terms oi LT with the initial condition (IS) an equation (22) after t-lic usual order of integration changing takes the form

s7ipf(s) — 1 = — (A + 7)i_v(fi) + \b(s + l.')ïï_\: (s), where bis) = J e~stb(x)dx is a LT oi the p.d.f.6(a;). ft follows from here that the solution of tin: last equation

M

lias a form

(23)

iiv (s) = [s + 7 + A(1 - b(s + v) jj

Next, the calculation of the LT io(s) of tin: function xu(i), given by the equality (20) after substitution into it oi the expression (23) gives

r r^ r

7rrj(s) = / e~st / nD(t,x)dxdt= / e~3t / XnN (t - x)e~,':r(l - B{x))dxdt = J 0 Jo Jo Jo

AirjV(s)(l - + v)) A(1 - b(s + v))

» + v (a + !/)(s + 7 + A(1 - b(s + u)) '

At least for the LT tf(s) of the function ~y(t) from the last of equations (17) one can find

.... _ , , T(s + ¡A + Ai/(1 — 6(s + v))

(s + i/)(s + 7 + A(1 - b(s + ¡A .1 )

Therefore, for the LT of the reliability function (19) we get-

(24)

(25)

= i-= ! s s

1 -

7(s + iA + Aî.v(1 - b(s + v))

(s + r;)(s +7 + A(1 - b(s +i/))) s + v + A(1 - b(s + ¡./))

(s + i/)(s + 7 + A(1 - b(s + i/))) From the last expression one can find the mean life time oi the object

mF = R( 0) =

K7 + A(l-b(fv)))'

(26)

(27

For calculation oi the limiting values of t-lic conditional state probabilities on life period we use the above procedure, which is based on the connection between asymptotic behavior of the functions Tr.vff), ^dM- -R(i)

at infinity and their LT at- the neighboring of the maximal non-zeros root ot the system characteristic equation A (s) = 0, In the considered ease the characteristic equation has a form

AO) = is + i/)(s + 7 + A(1 - b(s + i/))) = 0. One of its roots is a = — v. The second root is determined by the equation

(s + 7 + A(1 - Us + iv))) = 0

(28)

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his + ,.,)) = I +

A

(29)

Since the function bis + i')) is a quite monotone one [15], i.e. monotonically decreases, concave upward, takes the value 1 at the point s = v and i|(i-r)) < 1 then the equation (29) has a unique negative root, and his value depend on the sign of the difference 7 — v. If 7 > v the root of this equation, which we denote by sj is less than —v. si < — v. On the other hand if 7 < v the root of this equation si is grater than — v, s > —i>.

Therefore, when 7 > v the maximal root of the equation (29) is —v. and therefore,

, , ,. jr\'(i) ,. jfjv(s) Ji: = lim 7Tv(t) = lim - = lim —- =

I-« " f-x: Rit) s--v

= lim

v -

f

: + v + A(1 - b(s +!/))

1 + AtTi-s

Kpjs) _

Ttn = lim Tin (i) = lim L' ' = lim f-x i—foc Rit) s —,,

= lim

A(1 - his +

Aim

— -<■' s + v + A( 1 - b(s + //)) I + Amj

(30)

i.e. if the death intensity from the normal state is grater than the death intensity resulting by degradation, then the limiting distribution of the conditional state probabilities is determined by the parameter p = A m-£.

From another side, under condition 7 < v the greatest root of the characteristic equation (28) is the root of the equation (29). and consequently

,■ - y (0 ,. ¿¿V 0)

nx' = hm irjv(i) = urn -= lim —- =

i— = " ' t—oc Rit) 1—31

= lim

si

~d =

s—s 1 s + ii + A(1 — 6(s 4- v ) ) lim 7T r> ( t ) = lim ' jJ 'f ' = lim

t— 30 t— X Rit) 3—3! ll(s

S! +v + A(1 - b(si + v)) '

= lim

Ail -Jj(s + t/)

A(1 - b(Sl +1/})

—■"i s + u 4. A(1 - b(s 4- i>)) si + v + A(1 - b(si -I-v ))

(31)

Therefore, if the death intensity from the normal state is less than the same as a degradation result, then the limiting distribution of the conditional probabilities is strongly depend 011 the value si of the equation ([?]) root. Note that in the case if direct transitions from the normal .state to the failure state are impossible, i.e. when "r = 0 the second ease takes place.

5 Conclusion

Generalized Birth fe Death Processes, which arc special class of Semi-Markov Processes are introduced for modelling t-lic degradation processes. The special parametrization of the processes allows to give more convenient presentation of the results. The spccial attention is focused to the conditional state probabilities given life cycle, which are the mostly interesting for the degradation processes.

References

[1] A. Lisniansky. G. Levitin (2003) Multi-State System Reliability. Assessment. Optimization and Application. World Scientific. New Jersey, London, Singapore, Hong Kong, 358p.

[2] B, Dimitrov, V. Rykov. P, Stanchev (2002) On Multi-State Reliability Systems. In: Proceedings MMR-3002. Trondhcini (Norway) June 17-21, 2002.

[3] V. Rykov, B. Dimitrov (2002) On Multi-State Reliability Systems. In: Applied Stochastic Models and Information Processes. Proceedings of the International Seminar, Petrozavodsk, Sept. 8-13. Petrozavodsk, 2002. pp. 12S-135. See also http://ww.jip.ru/2002-2-2-2002.htm

[4] V. Rykov, B, Dimitrov, D. Green Jr.. P. Snanchev (2004) Reliability of complex hierarchical systems with fault tolerance units. In: Proceedings MMR-2004. Santa Fe (U.S.A.) June, 2004. (Printed in CD).

[5] B.Dimitrov, D.Green, V. Rykov and P.Stanchev. Reliability Model for Biological Objects. In: Longevity. Aging arid Degradation Models. Transactions of the First Russian-French Conference (LAD-2004), Saint Petersburg , June 7-9, 2004, Ed. by V. Antonov, C. Huber, M, Nikulin, V. Polischook, Saint Petersburg State Pol ¡technical University, SPB. 2004. Vol. 2. pp. 230-240,

[G] V. Rykov, D, Efrosinin. Reliability Control of Fault Tolerance Units. In: Abstracts of The 4~th International Conference on Mathematical Methods in Reliability (MMR-2004), Santa Fe, (USA), 21-25 July. 2004 (Published in CD),

[7] V. Rykov, D. Efrosinin (2004). Reliability Control of oi Biological Systems with failures. In: Longevity. Aging and Degradation Models. Transactions of the First Russian-French Con ference (LAD-2004), Saint Petersburg , June 7-9. 2004, Ed, V.Antonov, C.Huber, M.Nikulin, V.Polischook, Saint Petersburg State Pol ¡technical University, SPB, 2004. Vol. 2, pp. 241-255.

[S] V. Rykov. E, Buldaeva. On reliability controlof fault tolerance units: regenerative approach. In: Transactions of XXIV International Siminar on Stability Problems for Stochastic Modes. September 10-17. 2004, Jurmala, Latvia. Transport and Telecommunication Institute, Riga. Latvia, 2004.

[0] V.S, Korolyuk, A.F, Turbin. Semi-Markov processes and their applications. Kiev: "Naukova dumka". 1976. lS4p, (in Russian)

[10] D, McDonald. On semi-Markov and semi-regenerative processes. I. II.// Z. fur Wahrch. verw.Gcb.. 42 (1978), No. 2, pp. 261-377: Ann. of Prob.. 6 (1978), No.6. pp. 995-1014. '

[11] J.Jacod. Theorems dc renouvellement ct classification pour les chaines semi-Markoviennes.// Ann. inst. Hanri Pomcare, sect. B, 7 (1971). No.2, pp.83-129.

[12] V.S, Korolyuk, A.F. Turbin. Phase aggregation of complex systems. Kiev: "Vish shkola". 1978. 108p. (in Russian).

[13] V.S, Korolyuk, V.V. Korolyuk (1999) Stochastic Models of Systems. Kluver Academic Publishers.

[14] I.G. Penrovsky. Ledums on the theory of usual differential equations . M.-L.: GITTL, 1952, 232p. (in Russian).

[15] W. Feller, Introduction to Probability Theory and its Applications V. II John Wiley Sons, Inc. N.Y.-Lnd.-Sidney, 1966.

[16] M, A. Lavrent'ev, B.V, Shabat. Methods of the theory of complex variable functions. M.: Phismatgiz. 1958, 678p.