Asymptotic behavior of the average recovery cost in models of recovery processes

To the problems of choosing the optimal strategy for the recovery process, one of the most significant optimality criteria is the cost of the recovery process implementation. According to this, we will consider the recovery processes taking into account the recovery cost.

Suppose X i , i = 1,2,... random developments of recovered elements from i -first up to the -nth failure, X 1 - element time to the first failure and F i (t~) - their distribution function.

Sequence of non-negative independent random variables X i with distribution functions F i (t~), i = 1,2, . is called the recovery process [1-6].

Suppose C i , i = 1,2,... - are e-recovery costs, c₀ is the cost of the element installed at the initial moment of time t = 0, and X₀ is a random variable that has got a distribution F 0 (t) < 0 whilst t < 0 and F 0 (t) = 1 whilst t > 0.

The sequence (X i , C i ), i = 0,1,... will be called the recovery process, taking into account the recovery cost [6–8]. This definition is natural for recovery processes in the theory of recovery of technical systems. There exist other definitions. Thus, [9] demonstrates this as a recovery process with incomes, [10] shows it as a generalized recovery process.

The recovery process together with the cost of recovery establishes a random value N (t) - the number of failures (restorations) and a random value C(t) which denotes a cost of recoveries for time from 0 to t :

c(t) = ЕЙ ) c_t,

P(N (t) = n) = F(n\t) - F(n+\t),

F ⁽ⁿ⁾ (t)-n-tiple resultant of distribution functions Fj_(t), i = 1,2,..., n,

F^t) = (F^ -V * F_n)(t) = /„ F ^(n-1\ t - x)dF_n(x),F⁽V(F) = F 1 (t).

We have to highlight, that -tiple resultant F ⁽ⁿ- ) (t) is a distribution function of the sum of considered independent random variables X j , i = 1,2, ...n.

In reliability theory, the mathematical expectation of the number of failures is called the recovery function H (t)

HV=E(N(t» = E Z ₌₁F ⁽ⁿ4 t).

Function S (t) = E(C(t)) will be called the cost function, S (t) - the average value of the recovery cost within the interval [0, t], and, following [6; 7],

S(t) = C o + E ” =1 C n F^(t).

Under the actual operating conditions, the distribution functions of random variables (the operating time of the recovered elements in case of failures), which determine the recovery process, are able not to match. Naturally, the recovery cost can also change. Assumptions concerning distribution functions lead to various mathematical models of the recovery processes.

The research considers the process of recovering the order (k 1 , k₂) with changing distribution functions [6; 9; 11–13], the order generalizes the simple and general recovery process well studied in probability theory and reliability theory [1–6].

In case of the process recovering the order (k₁,k₂), the distribution functions meet the requirements

F i (t) = F j (t)   for i = j (mod k₂), i,j > k 1 ).

Numbers i,j are congruent modulo a natural number k (i = j(mod k), if they give the same remainder when divided by k. In occasion (1,1), it is a simple process, occasion (2,1) provides the general recovery process.

If k 1 = 1 (order (1,k 2 ) ), we obtain a periodic process of the recovery order k₂,; if k₂ = 1 ( order (k 1 ,1) ), there is the recovery process of k 1 order.

For example, in case (1,3) (the periodic process of order 3), the sequence of distribution functions of a periodic process has got the form

Fi,F2,F3,Fi,F2,F3.....

and the sequence of distribution functions for the process of order (2,2) takes the form

F1,F2>F3,F2>F3,F2J:3.....

This case can be interpreted as a process when, after the first recovery, every two recoveries, the system returns to the state it was in after the first recovery.

We could denote Ц ; = Е(Х [ ) as an expectation function, O j = o(X j ) is a standard deviation of a random value X j .

Random variable distribution X is called lattice, if it can only take values of the form an, n = 0,1,..., and У ^да = ₀ P(X = na) = 1.

We could set HF(t~) as the recovery function of a simple process generated by the distribution function F (t~), HFG (t) is the recovery function of a general process generated by the first distribution function F (t~), the second and the further.

We could write the well-known theorems on the asymptotic behavior of the recovery function for a simple and general recovery process [4; 6].

The distribution F 2 (t) could be non-lattice. It is for any initial distribution F₁(t').

Theorem 1 (Elementary renewal theorem)

v HF1F2(f)1

lim= t—> да      tЦ

Theorem 2 (Fundamental renewal theorem). If g(t) as integrated per [0,да) is nonincreasing function, then lim L g(t - x)dHF1F2(x) = 1-L° g(x)dx.

t->да 0№

Theorem 3 (Blackwell theorem). For any h lim(H F^2(t + h}~ HF rF 2(t)) =-,Ц2< да. t^ да№

• Theorem 4.

iim(HFiF2(t)-^ = ^-;i + 1. ц2< да.(1)

t^да                 №2'    2^2   №2

For the processes of the -th order considered above, the periodic process of the k- th order, and the (k₁, k₂ ) order process, the theorems formulated above were proved in [6; 9; 11].

The further purpose is to prove an analog of the above theorems for the asymptotic behavior of the cost function S(t) of the (k₁, k₂) order recovery process, taking into account the cost of recoveries.

Theorems to the asymptotic behavior of the cost function S(t) of the (k₁,k₂) order recovery process with allowance for the recovery cost

Further, if F i (t~) = F j (t), then C j = C j , it is natural for the studied models of the recovery processes.

Following [6-8], we could write down an integral equation for the cost function S(t) of the recovery process under consideration:

S(t) = G(t) + ^ S(t - x)dФ ^{(k 2 )} (x),                        (2)

GV = C o (1- Ф ^{(к 2}^(к)) + У : '"   ¹ C n F^Ct) - Tig- C_n L F^(t - x)dФ ^{(k 2 )} (x), if k 1 > 1,

G(t) = C o (1 - Ф ^{(k 2 )} Ct)) + S^₁ C_nF ⁽ⁿ⁾ (t), when k 1 = 1,

Ф ^{(k 2)}(t) = ( ф ₁ * ф ₂ *.. .* Ф_k2)Ct) – is convolution of all distribution functions of random varia-bles/ = Xk_1-1+ t , defining the periodic part of the considered recovery process, Ф i_ (t) = F k ₁ —1 _+i(t),i = 1,2 k 2 .

The recovery function HFG (t) of the general process and the recovery function HG (t) of the simple process are bound by the relation [4; 6]

HFG(t) = F(t) + L HG(t - x')dF(x').                       (3)

In equation (2) we we make the substitution:

S(t) = V(t) + C o .                                      (4)

We obtain

V(t) + c o = c o (1 - Ф ^{(t 2 )} (t)) + C^? 1 c_nF^(t) - Xl^ ^cn Jo F ^k ( - x)dФ ^{(k 2 )} (x) + + J 0 ⁽ V ⁽ t - %) + Co)dФ ^{(k 2 ) (} x ⁾ .

After reduction

V(t) = Q(t) + J_o V(t - x)dФ ^{(k 2 )} (x),                       (5)

Q(t) = X^ ^{2 -1} c_nF^(l) - X k= ' c_n J F^(t - х^Ф^х) if k i >  1,

Q ⁽ t) = ^Xk= _i c k F ^{(k (} t' ⁾ , when k i = 1.

We consider the integral equation

U(t) = f(t) + f_QU(t-x)dg(x).                       (6)

If f(t) = f_i(t) - f₂(t), then the function U(t) = U_i(t) - U₂(t) is its solution, where functions U_i(t), U₂(t) are, respectively, solutions to the integral equations

U i (t) = f i (t) + J U i (t - x)dg(x), U 2 (t) = f 2 (t) + J U₂(t - x)dg(x).

Taking this into account, we seek the solution to the integral equation (6) in the form

V(t) = (X^^2-1 c_n)V i (t) - (X^ c_n)V 2 (t).                 (7)

The functions V^t), V₂(t) are, respectively, the solutions to the integral equations

• V i (t) = Q i (t) + £ V i (t - x)dФ ^{(k 2)}(x),V 2 (t) = Q 2 (t) + J V 2 (t - x)dФ ^{(k 2)}(x),   (8)

  Q i (t) =

  
  

  X k ¹ = ⁺ 1 ^{k 2 -1} c k F ^(k) (t) у ^k 1 + ^k 2 ^{-1 X}k=i    ^ck

  
  

  ,Q i (f) =

  
  

  ((X^k_n=i c_nF^* Ф ^(к2^ )(к)

  X^ c_n      '

  
  

The functions Q_i(t),Q₂(t) to construct are distribution functions since they are non-decreasing (F ⁽ⁿ⁾ (t) - distribution functions), Q i (0) = Q₂(0) = 0, lim_t ^ _m Q i (t) = lim_t4^(t) = 1.

At present, taking into account that the function Ф ^{(к 2 )} (t) is also a distribution function, in accordance with (4), we could conclude, that the solution of the integral equations (8) are the recovery functions of the general processes, given respectively by the first distribution functions Q_i(t), Q₂(t), the second and further Ф ^{(к 2 )} (t).

Therefore,

V i (t) = HQ 1 Ф ^{(к 2)}(t),V 2 (^) = HQ 2 Ф^(t)                    (9)

and considering (4), (7), (9)

S(t) = V(t) + c o = c o + (X k= ⁺ i ^{k 2 _i} cJHQ^²^ - (X^ c_n)HQ 2 Ф ^{(к 2)}(t).  (10)

Assuming in (10) c_o = 0, c j = 1, i > 1, we obtain a new formula for the recovery function of the (k_i, k₂) order process

H(t) = HG^^t) - HG^^F),

G^t) = X k ¹ = ⁺ ₁ ^{к 2 -1} F ^(k) (t),G 2 (t) = X¹^ (F ^(k * Ф ^{(к 2 )} )(t), supplementing the previously obtained formulae in [6; 12].

The resulting linear representation (10) of the cost function S(t) of the (k_i,k₂) order recovery process, taking into account the cost of recoveries through the recovery functions of two general recovery processes makes possible to extend the above theorems on the asymptotic behavior of the recovery function of the general recovery process to the cost function of the (k_i, k 2 ) order recovery process, taking into account the cost of recoveries.

We could denote к1-1       Vк1-1              C/'Vk2        Ук2

Ц х    ^ ^(Xi=i   ^ i ) ^Xi=i  ^F(xi⁾ , Ц у    F ^(Xj=i Y j )    ^Xj=i F^j i ),

/v1 — 1-1  9 7^7 X          \^^7 ЭГУГХ ox = ^21 o2 (Xi) oY = ^Lih 02 (Yi)

Theorem 1* (Elementary renewal theorem). For any initial distributions F₁(t),F₂(t), .. ,F-_r-1(t)

r S(t) lim---- t^да t

= li_m S ₊ (y^+ki-i tm t + ^(T n= ¹

HQi Ф^(k2) (t)

Cn)\im----;---- t^ ^    г

—

ry ^k 1 ^{-1 (T}n=1

Cn)lim“Ql^ = t^ да      t

^k i ^+k i ^-1 _c ¹ L ^k i ^-1 C —

у ^k1^+k2^-1

^T n=k₁    ^C n

V y

Here and below, the above mentioned corresponding theorems for the recovery function of the general process are taken into account, and that n-tiple convolution F ⁽ⁿ⁾ (t) is the distribution function of the sum of the considered independent random variables X i , i = 1,2, ...n, as well as mathematical expectation E(Y~) of the random value Y together with the distribution function Ф ^{(k l )} (t) is defined by the formula

E(Y) = E k 1 E(Yi) = R y -

Theorem 2* (Fundamental renewal theorem). If the distribution functions defining the periodic part of the considered (k1,k2) order restoration process are non-lattice, and g(t) integrable per [0,да) is a non-increasing function, then lim ^ g(t — x)dS(x) = lim ^ g(t — x)d(co + (T.-**2 1 Cn)HQ1Ф(k2) (t) — t^да 0                      t^да 0                        n

(L^k n 1 =-1 С пЖгФ^Чх» =

= T ^ i+i^{2 1} C n lim fi g(t — x)HQ 1 Ф ^{(k 2)}(x)dx — X ^{- 1}^ C n lim fi g(t — x)dHQ 2 Ф ^{(k 2 )} (x)d(x) = ^{n x}         t^ да 0                                      ⁿ ±     t^ да 0

k1+k2-1

—~ f ^ g ^(x)dx -

, k₁ +k ₂ -1    ( Q g ⁽ x}dx  k - -!¹   ^EXEx

' ^{n 1      nn}     v _Y       ⁿ=  ^{} nn}     v _Y

Theorem 3* (Blackwell theorem) . If the distribution function determining a periodic part of the studied (k 1 , k₂) order recovery process is non-lattice distributions, then for any h>0

lim(S(t + h) — S(t)) = t^ да

= lim((c o + (L - 1 ⁺ ₁ ^{k 2 -1} cJHQ^tt + h) — (T— c_n)HQ₂Ф ^(k^ (t + h)) — t^ да           ^{n                                       n}

—(C o + (Е П 1 + ^{- 2 -1} CnXHQ^(t) — (L^ C n )HQ 1 Ф ^(k2^ (^))) =

= (Е - ¹- ! ^{- 2 -1} C n )lim(HQ 1 Ф ^{(k 2 )} (t + h) — HQ^ ^{(k 2}\t) — ⁿ             t^ да

— CT— C n )imV^^k\t + h)— HQ₂ф V ^kг\t» = ^{n 1}       t^ да

,y k-t+k2-1          /-vki-1           ky ⁺ l~2^—2 ¹_r

_ ^(T n=1 ²   C nW _ ^(T n=1 ^C n )^ _ (E n=k1^^№

Vy            Vy            Vy for any initial distributions F1(t), F2(t),... F--1(t').

Theorem 4*. In case the operating times Yi have finite variances and the distribution functions Фi(t) specifying the periodic part of the considered (k1,k2) order recovery process , are non-lattice distributions. Then lim(S(t) — (Z^2- Cn)-t) = t^ да             n 1         Vy

₂                            ■y^k1^+k2^-1    nn

— _r I ¹ l-^ Y , -l\y^k 1 ^{+ k} 2 ^{-1       n}==^k_r     ⁿ^j = ^1V J    y^k 1 ^-1

- ^C°⁺2⁽V_Y^)Tn=k 1    ^Cn         V Y        ^+Tn=1 ^Cn .

Evidence. According to (1), we could write down

„                    ,      о    v k'+k2-1ш lim ^Ф^) -±Л=^- ^^f!—]^2^ + 1, t- A ^i     о vy) 2vY   (Sk^2 'cn)vY lim (HQ2

t^ co\                VY   2vY       (Zn^' cn)VY

Here we have taken into consideration, that for E(Z 1 ) and E(Z 2 ) of random values Z 1 and Z₂ with distribution functions respectively Q 1 (t), Q 2 (t)

B(ZJ = S^A

^—

¹ C n S n=1 P j , E^^AZ

.¹ C n ( ^Sn=1 P j +PY^.

Further, lim ta t^ да

k 1 +* 2 -

• n=1

¹ c n )V 1 - сАА

¹ C n )2-) = ^VY '

S k¹=j^k2~¹ c n ^ Y

2v Y

-

2^:4^=^+2^1

V y

^c n

,

lim ta t^ /

* ' —'

1 n=1

. ^Cn ^)V 2 ^{(t)   (2}n=1 ^Cn) ^ =

⁽² П=1^{1 c} n )° Y 2v Y

-

² n=1 ^c n ^{2 n}=1 ^v J

V y

-

S*1-1     I 2n=1 cn n=1 Cn +   2

Therefore, lim (c0 + (2*=+1*2-

¹ C n )V 1 - (S^

tOW) - (gZA

¹ c_n)^+(S^k_n¹₌ \ ¹ c_n)j-) =

V y      ^{n 1}       V y

S' = C o +-

^          ^•

2v Y

-

S^2-1^^ , S^2-1^

--1---

V y

-

(Z n^i¹ c n^Y

2v Y

, ²n=1 ^c n ² J=1 ^v J   yfc ' -1

⁺      v_Y     ^{+ 2}n=1 ^{C n}

-

У ^k1^-1c ² n=1 ^c n

_ - I V * 1 ⁺ * 2 = ^C 0 ^{+ 2}n=k ₁

-1

C ^Y-

^{C n} 2v Y

-

у ^k1 + ^k2^-1 yn

^S n=k₁    ^c n ^S j=1 ^v j I ^

V y

I * 1 -1 ' n=1

уk1+k2-1

, ²n=k'     ^c n

^Cn ⁺    2

— _r  -L ¹ л£х , -14 y* 1 ⁺ *2^-1

= ^C 0 ⁺ 2 ⁽ V 2 ⁺ 1) ²n=k 1

Cn

-

у ^k1+^k2^-1 yn ^S n=k₁    ^c n ² j=1 ^v j   „

+ S

V y

* 1 ^-1 _r ' n=1 ^Cn^.

Regarding (10), we obtain formula (11) for the asymptotic behavior of the cost function S(t) of the (k 1 , k 2 ) order recovery process, taking into account the cost of recoveries.

If formula (11) specifies c₀ = 0, C j = 1, i = 1,2, function of the (k 1 , k₂) order recovery process [6; 12]

..

., we obtain the asymptotic behavior of H(t)

l ^{im(H(t) -}^O = ^k1

t^ да           H y

-

* 2

-

2+*^

2hy

-

k2? + ;rS*=₁ iW)

H y    H y  ^{j 1}

We consider the recovery process, taking into account the cost of recovery, when full recovery occurs in the recovery process (F [ (t) = F₁(t), if an element fails, it is substituted by the element with the distribution function F 1 (t) similar to the function of the failed element), but the cost of the substituted elements changes in case of their failures, C j = C j when i = j (mod k₂), i,j > k 1 . We could note that this case is typical during operation.

This case from (11) results in

S ^k1+^k2^-1_r                   2

г    f c f n=ki    ⁿ          . ¹ ^± ।     pki+k2-1

l ^im(S(t)-- -¹---- ^t) = Co ⁺ - H ⁺ 1) ² _n ¹²

t^                * 2 V 1                   2 H '          ^{n k1}

Cn

-

у ^k1+^k2^-1

² n=k₁    ^nc ”

* 2

I v *1-1 ^{+ 2} j=1 ^Cj .

Alternating process of (k 1 , k₂) recovery order

In the theory of reliability, when determining the recovery process, it is assumed that the recovery is performed in a negligible time compared to the time of the element to the next failure, that is instantly. In practice, this is often not done. Therefore, along with the uptime, downtime, the time of finding out the reasons for the failure, the time of recovery could be equally important. Here, along with other characteristics, the cost of recovery is also significant during operation.

The sequences т (X_n), (Y_n~) form two simple recovery processes with distribution functions F(t), G(F), respectively. The sequence (X_n, Y_n) is called a simple alternating recovery process [3; 4; 6].

Y_n could be the recovery time after the n-th failure, and X_n could be the element operating time after the (n — 1)-th recovery. The intervals between successive failures (taking into account the recovery time) form a general recovery process with the first distribution function F(t), the second (P * G)(t). The intervals between successive recoveries form a simple recovery process with a distribution function (F * G)(t) [4; 6].

According to the models of recovery processes discussed above, we consider an alternating recovery process (X_n, Y_n) of (k 1 , k₂) order, where the sequences (X_n), (Y_n) form the (k 1 , k₂) order recovery processes with distribution functions F_n(t~) and G_n(t) [6]. In case k 1 = k₂ = 1, we have got a simple alternative recovery process.

If we consider the introduced alternating process as a sequence X 1 , Y 1 , X₂, Y 2 , ., X_n, Y_n, ., then we arrive at the recover process of the (2k 1 — 1,2k 2 ) order.

The intervals among successive failures (taking into account the recovery time) form the recover process of (k 1 + 1, k₂) order with distribution functions

Fi(t),(Gi*F2)(t).....(Gn-i*Fn)(t),., and the intervals among the successive recoveries form the recovery process of (k1,k2) order with distribution functions [6]

^(Fn * ^(t)

c_n could be the cost of the n-th recovery. This, in addition to the cost of the recovery, may include losses, penalties for failure, downtime. The sequence

(Xo,Co),(X1,y1.,c1),.(XnJn,Cn)^.

is called an alternating recovery process, taking into account the cost of recoveries.

If sequence (X_n,Y_n) is an alternating recovery process of (k 1 ,k 2 ) order and C j = C j , if P [ (t) = P j (t), then we obtain the alternating recovery process of (k 1 , k₂) order regarding the cost of recoveries [6]. Further, we suppose, that the cost of each recovery is fixed at the time the recovery is completed. Other approaches can be considered, for example, when the cost of recoveries is fixed at the moments of failures.

We could denote the random time of recovery completion by Z_n after the n — 1-th failure. Then Z_n = X_n + Y_n and T_n(t) = (F_n * G_n)(t) is the random variable distribution function Z_n .

We note again that sequence Z determines the recovery process of (     ) order, and, therefore, to k1, k2

calculate the cost function S ( t ) (the average cost of recoveries) of the considered alternating process, we move onto the recovery process ( ₇   ) of (k 1 , k₂) order with the cost of recoveries taken into ac-

Zn, cn count. After that, in accordance with theorems 1*–4*, we can write formulae for the asymptotic behavior of the cost function of the introduced alternating recovery process of the (k1, k2) order.

We write these formulae:

да

S (t )= c0 +Z Cn ^(n'( t), k1+ k2 -1

' n = k1    ^Cn

,

S (t)S'

lim

H p

t ^»

where

t lim Jg (t — x) dS (x) = t ^w 0

^? k i+ k 2 —1 ^^ n = k 1

lim ( S ( t + h ) - S ( t ) ) = t ^w

1   °

= c 0 +7 -

² [h,

2 ^p

lim S (t) — t ^w   v 7

A ki + k 2 - 1

H p

( ki + k 2 —1

² c n

I    ⁿ = ^k 1

n n = ki

—

C w n

— J g (x) dx

k1 +k2—1 c„}h n = k1     n)

H p        ’

t

H p

Ek1 + k2 —1 V'n n = k1  cn 2 j=1H j , k^

+ 2 c n , H p             n = 1

k 1 + k 2 — 1                j k 1 + k 2 — 1

H n = E ( X n ) + E ( Y n ) , H p

2 Hn Op =   2  (°2 (Xn ) + a; (Yn )) ■ n = k1                 у  n = k1

Many of the most important performance indicators of technical, information computing and many other systems are of a random nature. So, along with the random value of the failure number, an important characteristic in such systems when carrying out recovery processes is the cost of recovery from the start of operation to an arbitrary point in time t (especially in optimization problems for choosing a recovery strategy).

In this regard, the paper considers the extension of the recovery process to the recovery process taking into account the cost of recoveries.

For models of the recovery process with changing recovery costs and operating time distribution functions, a formula is obtained that linearly connects the average recovery cost (cost function) with the recovery functions of the two general recovery processes well-studied within probability theory and mathematical reliability theory of general recovery processes.

This results in possibility, within the framework of the mathematical theory of reliability, to simply transfer the well-known theorems on asymptotic behavior for the t ^ от recovery function (average number of failures) to the cost function in the models under consideration (with changing recovery costs and operating time distribution functions) of the recovery processes, taking into account the cost of recoveries.

The obtained theorems are generalized to the alternating recovery process, regarding the cost of recoveries, when the random time of recoveries is also taken into account.

We could note that the obtained asymptotic formulae will find application in the mathematical and operational reliability of rocket and space technology, electronic computing systems, power supply systems, heat supply systems, transport systems, and many other technical systems [14].

We also outline that along with the obtained formulae for the asymptotic behavior of the average cost of recoveries, limit theorems for the cost of recoveries (as a random variable), similar to those obtained in [6; 9; 13; 15], as well as finding the dispersion of the cost of recoveries in the considered models [16].

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@ Вайнштейн В. И., Вайнштейн И. И., Сафонов К. В., 2022