# On Reliability of a Double Redundant Renewable System with a

## Transcript Of On Reliability of a Double Redundant Renewable System with a

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Article

On Reliability of a Double Redundant Renewable System with a Generally Distributed Life and Repair Times

Vladimir Rykov 1,2 , Dmitry Efrosinin 2,3,* , Natalia Stepanova 4 and Janos Sztrik 5 1 Department of Applied Mathematics and Computer Modelling, Faculty of Automation and Computer Science, Gubkin Russian State University of Oil and Gas, 119991 Moscow, Russia; [email protected] 2 Department of Information Technologies, Faculty of Mathematics and Natural Sciences, Peoples’ Friendship University of Russia (RUDN University), 117198 Moscow, Russia 3 Insitute for Stochastics, Johannes Kepler University Linz, 4030 Linz, Austria 4 Laboratory N17, Trapeznikov Institute of Control Sciences of RAS, 117997 Moscow, Russia; [email protected] 5 Department of Informatics and Networks, Faculty of Informatics, University of Debrecen, 4032 Debrecen, Hungary; [email protected] * Correspondence: [email protected]

Received: 3 February 2020; Accepted: 14 February 2020; Published: 19 February 2020

Abstract: The paper provides reliability analysis of a cold double redundant renewable system assuming that both life-time and repair time distributions are arbitrary. The proposed approach is based on the theory of decomposable semi-regenerative processes. We derive the Laplace–Stieltjes transform of two main reliability measures like the distribution of the time between failures and the time to the ﬁrst failure. The transforms are used to calculate corresponding mean times. It is further derived in closed form the time-dependent and time stationary state probabilities in terms of the Laplace transforms. Numerical results illustrate the effect of the type of distributions as well as their parameters on the derived reliability and probabilistic measures.

Keywords: redundant system; reliability; arbitrary distributions; time-dependent characteristics; decomposable semi-regenerative process

1. Introduction

The use of parallel or redundant units is an often recommended way of increasing the system reliability. In the last few decades, many authors have studied the redundant systems under different sets of assumptions about the system’s attributes and level of generality, the distribution of life, and repair times when one or more units are standby. In this paper, we formulate the model where both of distributions are assumed to be arbitrary. Such models are interesting both from theoretical and practical points of view. Theoretically, these studies are related to the creation and development of new mathematical methods such as regenerative methods, especially methods of the Decomposable Semi-Regenerative Processes (DSRP), and Markovization methods via introduction of the supplementary variables. In practical terms, the tasks under consideration form a basis for analyzing the reliability of different complex systems with generally distributed times.

Smith’s idea about regenerative stochastic processes [1] found many generalizations and applications. This approach still helps to solve many applied problems since it reduces the complexity of the task. Instead of a whole time period, the process can be analyzed within a separate regenerative period of time. However, if the process describing the system behavior is complicated enough in order to be exhaustively analyzed on the regenerative period, even in this case, different generalizations

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of the regenerative idea are possible. One possible generalization is to combine Smith’s idea with a semi-Markov approach [2], which seems quite appropriate for real models. The result of such combination is a semi-regenerative process. In Rykov and Yastrbrnrtsky [3], it appears under the name “Regenerative processes with several types of regeneration points”. Klimov [4] considered it as a “semi-Markov process with additional trajectories”. In accordance with Jakod [5] and Nummeline [6], in this paper, we will call such process a semi-regenerative process (SRP).

The next possible generalization is to construct within the basic regeneration cycle other regeneration points called embedded regeneration points. Sometimes, in a particular model, one can ﬁnd some embedded regeneration points and to represent the process distribution in a separate regeneration cycle in terms of its distribution in embedded regeneration cycles of the second level. Initially, such processes appeared in [7,8] under the name “Regenerative processes with embedded regeneration cycles” or “Decomposable Semi-Regenerative Processes” (DSRP). Regenerative processes with embedded regeneration cycles are widely used applications in operations research. In [7], regenerative processes have been applied to study priority queueing systems. In [9,10], the one-server queueing system with inﬁnite queue, recurrent input ﬂow, and general service time distribution (GI|GI|1, ∞—system in Kendall’s notation [11]) has been studied in terms of these processes (see also [12]). In paper [13], the authors investigated a model of a polling system by means of the DSRP. Another approach related to the study of complex systems (proposed in Belyaev [14]) contains the introduction of so-called supplementary variables. According to this method, the original model can be described by a continuous-time Markov chain (Markovization method). The implementation of this method to study reliability models is proposed by Rykov [15], Mazumdar [16], and Mine and Asakura [17].

The redundancy of some technical system means the duplication of critical components resulting in increasing reliability of the system. The double redundant systems are well known and were studied and applied under different assumptions like repair maintenance, preventive maintenance policy, exponential distributions for life and repair times, and so on; see, e.g., Gaver [18], Gnedenko et al. [19], Mine and Asakura [20], and Srinivasan [21]. In this paper, we study a homogeneous cold double redundant system with arbitrary life- and repair time distributions. In such a system, the cold standby redundancy means that the redundant component does not fail in a standby mode. The closed form solutions for the main reliability characteristics have been derived in terms of an integral transformation like a Laplace transform. The proposed analysis is based on the DSRP, the stochastic equations for the corresponding random variables (RVs), and the theory of regenerative processes [1].

The rest of the paper is organized as follows. In Section 2, the problem set, some assumptions, and notations are introduced. In Section 3, the Laplace–Stieltjes transforms (LSTs) of the regeneration period (the system life cycle), and the Laplace transforms (LTs) of the times to the ﬁrst failure and between system failures are calculated. In Sections 4 and 5, the time-dependent system state probabilities (TDSSPs) and stationary system state probabilities (SSSPs) are calculated. Finally, using some numerical examples, we demonstrate the effect of type of distributions and their parameter on the reliability and probabilistic measures of the system in Section 6. The paper ends with a conclusion.

2. The Problem Set: Assumptions and Notations

Consider a homogeneous cold double redundant renewable system with generally distributed life- and repair times, which, according to modiﬁed Kendall’s notations [11], will be denoted as. The system consists of two identical units which can be in two possible states: operational and failed. The system fails when both of units are in a failed state. For the renewable system, different strategies of its renovation are possible. In this paper, we consider a strategy when a new system cycle begins at the point in time when one unit starts working while the other unit failed and is being repaired.

Denote by Ai (i = 1, 2, . . . ), life times of the system units, by Bi (i = 1, 2, . . . ), its repair times, and suppose that all these random variables (RVs) are mutually independent and identically distributed.

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Thus, denote by A(t) = P[Ai ≤ t] and B(t) = P[Bi ≤ t] the corresponding cumulative distribution functions (CDFs). Denote also by A and B the RVs with the same CDFs as Ai and Bi, respectively. Suppose that the instantaneous failures and repairs are impossible and their mean times are ﬁnite:

A(0) = B(0) = 0,

∞

a = (1 − A(x))dx < ∞,

0

∞

b = (1 − B(x))dx < ∞.

0

Denote by E the set of system states. We assume that E = {i = 0, 1, 2}, where i stands for the number of failed units, and we introduce a random process J = {J(t), t ≥ 0}, where

J(t) = number of failed units at time t.

A simple state transition diagram is illustrated in Figure 1.

Figure 1. State transition diagram.

Denote by F the time between system failures, i.e., the period of time the system begins to operate again up to the next system failure. We remind that the system is again in an operational state when operation of one unit and repair of another one begin simultaneously. Denote by F1 the time to ﬁrst system failure (the time from the epoch when both units are in operational state up to the system failure) (see Figure 2). The functions F(t) = P[F ≤ t] and F1(t) = P[F1 ≤ t] are corresponding CDFs.

We are interesting in calculation of the reliability function

R(t) = P[F1 > t] = 1 − F1(t),

distribution of time to the ﬁrst failure F1(t), time-dependent system state probabilities (TDSSPs)

πj(t) = P[J(t) = j], j = 0, 1, 2,

and stationary system state probabilities (SSSPs)

πj = lim πj(t) ≡ lim P[J(t) = j], j = 0, 1, 2,

t→∞

t→∞

as well as the availability coefﬁcient

Kav. = π0 + π1 = 1 − π2.

The following notations will be used next.

• The Laplace–Stiltjes transforms (LSTs) of their CDFs are denoted as:

∞

∞

a˜(s) = E e−sA = e−sxdA(x), b˜(s) = E e−sB = e−sxdB(x),

0 ∞

f˜(s) = E e−sF = e−sxdF(x),

f˜1(s) = E e−sF(1)

0 ∞

= e−sxdF1(x).

0

0

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• In the case when the repair completion comes before the failure of other unit and when an operative unit fails before the repair completion of other unit, the distributions are equal, respectively,

t

t

aB(t) = B(x)dA(x) = P[B ≤ A ≤ t], bA(t) = A(x)dB(x) = P[A ≤ B ≤ t].

0

0

Applying the Laplace transform (LT) for aB(t) and bA(t), we get

∞

∞

a˜B(s) = e−stdaB(t) = e−sx B(x)dA(x),

(1)

0

0

∞

∞

b˜ A(s) = e−stdbA(t) = e−sx A(x)dB(x).

0

0

• Their derivatives with sign minus in point zero are denoted by:

d

∞

aB = − ds a˜B(s) s=0 = 0 xB(x)dA(x),

bA = − dds b˜ A(s) s=0 = 0∞ xA(x)dB(x). (2)

• The probabilities P[B ≤ A] and P[B ≥ A] are associated with transformations through the following ratios:

∞

a˜B(0) = B(x)dA(x) = P[B ≤ A] ≡ p,

0 ∞

b˜ A(0) = A(x)dB(x) = P[B > A] ≡ q = 1 − p.

0

• Note the property of transformations (1)

a˜1−B(s) = a˜(s) − a˜B(s), b˜1−A(s) = b˜(s) − b˜ A(s).

(3)

Figure 2. Trajectory of the process J. 3. Reliability Function

Process J is a regenerative one. A trajectory of this process is illustrated in Figure 2.

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Here, F stands for the time between system failures, i.e., the period of time the system begins to operate again (state 1) up to the next system failure (state 2). The variable G speciﬁes the length of a regenerative cycle according to a deﬁned renovation policy. The following lemma holds for the LSTs of the time F between failures and the time to the ﬁrst failure F1.

Lemma 1. The LST f˜(s) = E e−sF of the time F between failures and the LST f˜1(s) = E e−sF1 of the time to the ﬁrst failure F1 are of the form:

f˜(s) = a˜(s) − a˜B(s) , f˜1(s) = a˜(s) a˜(s) − a˜B(s) .

(4)

1 − a˜B(s)

1 − a˜B(s)

Proof. The life time of the system F is a time between two successive failures. After a failure in state 0, the system goes to state 1 where there can be two events: either the recovery of a unit in state 1 in time B and subsequent transition again to state 0 or a failure of a second unit in time A will happen. After a failure in state 1, the system comes back to state 1 in time B. Hence, as it can be seen from Figure 2, the system lifetime F satisﬁes the following stochastic equation:

F = A + F if B < A, (5)

A

if B > A.

The time to the ﬁrst system failure F1 satisﬁes

F1 = A + F.

(6)

Applying LT to the Equation (5) and taking into account notations (1), one can obtain

∞

f˜(s) = E e−sF = e−stdF(t) =

0 ∞

= e−sx B(x) f˜(s) + (1 − B(x)) dA(x) =

0

= f˜(s)a˜B(s) + a˜1−B(s).

(7)

Due to the last equation, one can ﬁnd that the LST of the system lifetime f˜(s) is given by the ﬁrst relation in of (4). The second equation of lemma directly follows from stochastic Equation (6).

Remark 1. Let ϕ˜i(s), i = 0, 1, 2, be the LST of the distribution of time to the ﬁrst failure of the system, starting from state i at t = 0. In previous statement f˜1(s) = ϕ˜0(s). The following equations hold:

ϕ˜0(s) = q˜0,1(s)ϕ˜1(s), ϕ˜1(s) = q˜1,1(s)ϕ˜1(s) + q˜1,2(s).

Here, q˜i,j(s) stands for the LST of the transition time distribution qi,j(t) from state i to state j. Obviously, we have q˜0,1(s) = a˜(s), q˜1,1(s) = a˜B(s) and q˜1,2(s) = a˜1−B(s). Expressing ϕ˜0(s), we get the relation

ϕ˜0(s) = q˜0,1(s)q˜1,2(s) , 1 − q˜1,1(s)

which coincide with the relation (4) for the ﬁrst time to failure.

The main result of this section is the following theorem:

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Theorem 1. The LT R˜ 1(s) of the system reliability function R1(t) = 1 − F1(t) is

R˜ 1(s) = (1 − a˜(s))(1 + a˜(s) − a˜B(s)) .

(8)

s(1 − a˜B(s))

Proof. Taking into account that the LT of any CDF connected with its LST through a relation

∞

∞

F˜1(s) = e−st F1(t)dt = 1 e−stdF1(t) = 1 f˜1(s),

s

s

0

0

the LT R˜ 1(s) of the reliability function R1(t) = 1 − F1(t) can be derived using (4) in form

R˜ 1(s) = 1 − F˜1(s) = 1 (1 − f˜1(s)) = 1 − a˜B(s) − a˜2(s) + a˜(s)a˜B(s)

s

s

s(1 − a˜B(s))

= (1 − a˜(s))(1 + a˜(s) − a˜B(s)) , s(1 − a˜B(s))

which ends the proof.

Corollary 1. The LT R˜ (s) of the survival probability R(t) = 1 − F(t) is of the form

R˜ (s) =

R˜ (s)

= 1 − a˜(s) ,

(9)

1 + a˜(s) − a˜B(s) s(1 − a˜B(s))

which follows directly from (4) and (8).

Corollary 2. The mean system life time between failures E[F] and mean time to the ﬁrst failure E[F1] are of

the form:

a

a

E[F] = q , E[F1] = a + q .

(10)

Proof. The ﬁrst relation follows from the equality

E[F] = R˜ (0) = lim 1 − a˜(s) = a = a , s→0 s(1 − a˜B(s)) 1 − a˜B(0) q

and the second one follows directly from the relation (6).

4. Time Dependent System State Probabilities

4.1. Deﬁnition

For calculation of the time dependent system state probabilities (TDSSP), we use a renewal theory. In our case, the process J is a regenerative process with a delay (see Figure 2) and its regeneration times are

S0 = 0, S1 = A1, S2 = S1 + G1, . . . , Sk+1 = Sk + Gk, . . . .

Here, regeneration cycles Gi (i = 1, 2, . . . ) are the time intervals between two successive returns of the process J into state 1 after a system failure, when one of the system units begins to operate and another one is being repaired. Thus, if the system functioning begins when both units are in operational state, the ﬁrst regeneration period differs from the others and has the duration A1. The state probabilities of the process J in any time t

πj(t) = P{J(t) = j}, j = 0, 1, 2

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can be represented in terms of its distribution over separate regeneration period

π(j 1)(t) = P[J(t) = j, t < G], j = 0, 1, 2,

(11)

and the renewal function H(t) can be deﬁned as follows:

t

πi(t) = πi(1,1)(t) +

d

H

(u)π

(1) j

(t

−

u).

(12)

0

The distribution of the process J on this period is of the form

π(j 1,1)(t) = P[J(t) = j, t < A1] = δj0(1 − A(t)).

(13)

In terms of the CDF G(t) = P[Gi ≤ t] of RVs Gi, a renewal function H(t) is determined as follows:

H(t) = ∑ P ∑ Gi ≤ t = ∑ G∗k(t).

(14)

k≥1

1≤i≤k

k≥1

On the basis of the foregoing, the next lemma can be proved. Consider, ﬁrst of all, the regeneration cycle distribution.

Lemma 2. The LST of the regeneration cycle is of the form:

g˜(s) = E[e−sG] = b˜ A(s) .

(15)

1 − a˜B(s)

Proof. The regeneration cycle is a time between two successive visits of state 1 where two events can occur: either the recovery of a unit in state 1 in time B and subsequent transition to state 0 or a failure of a second unit in time A happens. Figure 2 shows that RV G satisﬁes the following stochastic equation due to independence of RVs A and B:

G = A + G if A > B, (16)

B

if A ≤ B.

Applying LST to this stochastic equation, we obtain

∞

g˜(s) = E e−sG = e−stdG(t)

0

∞

= dA(x) B(x)e−sx g˜(s) + e−sydB(y) =

y>x 0

∞

∞

= g˜(s) e−sx B(x)dA(x) + e−sy A(y)dB(y) =

0 0

= g˜(s)a˜B(s) + b˜ A(s),

(17)

which implies the expression (15) for the LST of the regeneration period. As a corollary, we obtain the mean length of the regeneration period.

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Corollary 3. The mean length of the regeneration period can be deﬁned as:

E[G] = aB + bA .

(18)

q

Proof. The proof is obtained by differentiation

E[G] = − d g˜(s) = − b˜ A(0)(1 − a˜B(0)) + a˜Bb˜ A(0) .

ds

s=0

(1 − a˜B(0))2

This expression leads to expression (18) due to properties of a˜B(0) and b˜ A(0).

Lemma 3. The LST of the system renewal function is given by:

h˜ (s) =

b˜ A(s) .

(19)

1 − (a˜B(s) + b˜ A(s))

Proof. From the renewal theory, it is well known (and follows from (14)) that the LST of the renewal function H(t) is deﬁned as h˜ (s) = g˜(s)(1 − g˜(s))−1. Thus, substitution to this expression of (15) for

g˜(s) leads to (19).

Theorem 2. The LTs π˜ j(s) of the TDSSPs πj(t) (j = 0, 1, 2) are of the form:

1 − a˜(s)

b˜ a (s)

(1)

π˜ j(s) = δj0 s + 1 − (a˜B(s) + b˜ A(s)) π˜ j (s), j = 0, 1, 2,

(20)

where π˜ (j 1)(s) (j = 0, 1, 2) are the LTs of the TDSSPs π(j 1)(t) (j = 0, 1, 2) in a separate regeneration period. These probabilities will be calculated in the next subsection.

Proof. The form of Equations (12) and (14) prompts to calculate the process state probabilities in terms of their LTs and the renewal function LST

∞

π˜ j(s) = e−stπj(t)dt,

0

∞

h˜ (s) = e−stdH(t).

0

Applying LT to Equation (12) and taking into account Equation (13), one can obtain

π˜ j(s) = δj0 1 −sa˜(s) + h˜ (s)π˜ (j 1)(s). (21) Substitution into this equality of the expression for h˜ (s) (19) leads to (20) and completes the proof.

4.2. The State Probabilities in a Separate Regeneration Period

Now, we calculate the TDSSPs in a separate regeneration period. The probability π2(1)(t) can be calculated very easily in the main level regeneration period.

Lemma 4.

The

LT

of

the

second

TDSSP

(1)

π2

(t)

in

the

main

regeneration

period

is

given

by

(1)

a˜(s) − (a˜B(s) + b˜ A(s))

π˜ 2 (s) =

s(1 − a˜B(s)) .

(22)

Proof. Due to representation of the regeneration period by Formula (16), the event {J(t) = 2, t < G} occurs if and only if either the event {A1 ≤ t ≤ B1} occurs or the events {t > u = A1 > B1} and

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{J(t − u) = 2, t − u < G} occur. Thus, as it is shown in Figure 2, for the probability π2(1)(t), it holds that

t

π2(1)(t) = P[J(t) = 2, t < G] = P[A ≤ t < B] + dA(u)B(u)π2(1)(t − u).

0

From this equation for the LST, it follows

∞

π˜ 2(1)(s) = e−st A(t)(1 − B(t))dt + a˜B(s)π2(1)(s)

0

and ∞

π˜ (1)(s) = 1

e−st A(t)(1 − B(t))dt.

2

1 − a˜B(s)

0

Calculating LTs of the integral in these expressions using partial integration, we obtain

∞

π˜ (1)(s) =

1

e−st A(t)(1 − B(t))dt =

2

1 − a˜B(s)

0

=−

1

e−st A(t)(1 − B(t))dt ∞ +

s(1 − a˜B(s))

0

∞

+1

e−st[dA(t)(1 − B(t)) − A(t)dB(t)] =

s(1 − a˜B(s))

0

= a˜(s) − (a˜B(s) + b˜ A(s)) s(1 − a˜B(s))

that completes the proof.

Since

the

calculation

of

the

probabilities

(1)

πj

(j

=

0, 1)

is

not

a

trivial

task,

we

intend

to

apply

the

theory of DSRP [3,8,12]. According to this theory, the further analysis can be described as follows. As it

was shown in the Equation (12), the process distribution can be represented in terms of its distribution

in embedded regeneration cycles and embedded renewal function. This approach is useful if the

behavior of the process is too complex in order to derive a state distribution directly, but the embedded regeneration epochs are available. At known regeneration epochs Sk(1), the distribution in the basic regeneration period (regeneration period of the ﬁrst level) πi(1)(t) similar to the Equation (12) can be represented in terms of distributions in embedded regeneration cycles (regeneration cycles of the second level) π(j 2)(t) and embedded renewal function H(1)(t) in the following way:

∞

π(j 1)(t) = π(j 2)(t) +

d

H

(1)

(u)π

(2) j

(t

−

u),

j = 0, 1,

(23)

0

where

π(j 2)(t) = P{J(t) = j, t < G(2)}, j = 0, 1

is the process state distribution in a separate regeneration period of the second level. In this case

as embedded regeneration epochs, we choose the random number ν = min{n : An < Bn} of time

epochs

S1(1) = A11{A1>B1}, S2(1) = S1(1) + A21{A1>B1, A2>B2}, . . .

up to time, when the event {An ≤ Bn} occurs for the ﬁrst time. It means that the time interval G(1) between embedded regeneration epochs has a distribution G(1)(t) = A(t) and these epochs in the time

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interval F are determined by Equation (5). According to the theory of DSRP, the embedded renewal function H(1)(t) satisﬁes the equation

t

H(1)(t) = A(t) + dH(1)(u)A(t − u) − F(t),

(24)

0

where F(t) is a CDF of the time between system failures determined by its LST (4).

Similar to the basic case, the solution of Equations (23) and (24) can be represented in terms of

their LTs and LSTs:

∞

π˜ (j 2)(s) = e−stπ(j 2)(t)dt,

∞

h˜ (1)(s) = e−stdH(1)(t).

0

0

The next lemma speciﬁes connections between process distributions in the ﬁrst and in the second level regeneration cycles in terms of their LTs.

Lemma 5. The TDSSPs in the ﬁrst and in the second level regeneration cycles in terms of the LTs satisfy the

relation

π˜ (1)(s) = π˜ (2)(s) a˜B(s) j = 0, 1.

(25)

j

j 1 − a˜B(s)

Proof. Applying LT to Equation (23), we get

π˜ (j 1)(s) = (1 + h˜ (1)(s))π˜ (j 2)(s),

(26)

where, due to (24), the LST h˜ (1)(s) of the embedded renewal function H(1)(t) is of the form

h˜ (1)(s) = a˜(s) + h˜ (1)(s)a˜(s) − f˜(s),

which leads in turn to

h˜ (1)(s) = a˜(s) − f˜(s) . 1 − a˜(s)

Substitution of this relation to the Equation (26) and taking into account expressions for f˜(s) from (4) implies the result (22) that completes the proof.

We have to calculate now only π˜ (j 2)(s) (j = 0, 1).

Lemma 6. In notations (4), the LTs of the second level system state probabilities are:

π˜ 0(2)(s) = 1s (a˜(s) − (a˜B(s) + b˜ A(s))); π˜ 1(2)(s) = 1s [1 − (a˜(s) + b˜(s)) + (a˜B(s) + b˜ A(s))]. (27)

Proof. First, consider appropriate expressions for π˜ (j 2)(t) (j = 0, 1). From Figure 2, it follows that

• The event {J(t) = 0, t < G(1)} occurs if and only if {B ≤ t < A}; • The event {J(t) = 1, t < G(1)} occurs if and only if {t < B ≤ A} or {t < A ≤ B};

Article

On Reliability of a Double Redundant Renewable System with a Generally Distributed Life and Repair Times

Vladimir Rykov 1,2 , Dmitry Efrosinin 2,3,* , Natalia Stepanova 4 and Janos Sztrik 5 1 Department of Applied Mathematics and Computer Modelling, Faculty of Automation and Computer Science, Gubkin Russian State University of Oil and Gas, 119991 Moscow, Russia; [email protected] 2 Department of Information Technologies, Faculty of Mathematics and Natural Sciences, Peoples’ Friendship University of Russia (RUDN University), 117198 Moscow, Russia 3 Insitute for Stochastics, Johannes Kepler University Linz, 4030 Linz, Austria 4 Laboratory N17, Trapeznikov Institute of Control Sciences of RAS, 117997 Moscow, Russia; [email protected] 5 Department of Informatics and Networks, Faculty of Informatics, University of Debrecen, 4032 Debrecen, Hungary; [email protected] * Correspondence: [email protected]

Received: 3 February 2020; Accepted: 14 February 2020; Published: 19 February 2020

Abstract: The paper provides reliability analysis of a cold double redundant renewable system assuming that both life-time and repair time distributions are arbitrary. The proposed approach is based on the theory of decomposable semi-regenerative processes. We derive the Laplace–Stieltjes transform of two main reliability measures like the distribution of the time between failures and the time to the ﬁrst failure. The transforms are used to calculate corresponding mean times. It is further derived in closed form the time-dependent and time stationary state probabilities in terms of the Laplace transforms. Numerical results illustrate the effect of the type of distributions as well as their parameters on the derived reliability and probabilistic measures.

Keywords: redundant system; reliability; arbitrary distributions; time-dependent characteristics; decomposable semi-regenerative process

1. Introduction

The use of parallel or redundant units is an often recommended way of increasing the system reliability. In the last few decades, many authors have studied the redundant systems under different sets of assumptions about the system’s attributes and level of generality, the distribution of life, and repair times when one or more units are standby. In this paper, we formulate the model where both of distributions are assumed to be arbitrary. Such models are interesting both from theoretical and practical points of view. Theoretically, these studies are related to the creation and development of new mathematical methods such as regenerative methods, especially methods of the Decomposable Semi-Regenerative Processes (DSRP), and Markovization methods via introduction of the supplementary variables. In practical terms, the tasks under consideration form a basis for analyzing the reliability of different complex systems with generally distributed times.

Smith’s idea about regenerative stochastic processes [1] found many generalizations and applications. This approach still helps to solve many applied problems since it reduces the complexity of the task. Instead of a whole time period, the process can be analyzed within a separate regenerative period of time. However, if the process describing the system behavior is complicated enough in order to be exhaustively analyzed on the regenerative period, even in this case, different generalizations

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of the regenerative idea are possible. One possible generalization is to combine Smith’s idea with a semi-Markov approach [2], which seems quite appropriate for real models. The result of such combination is a semi-regenerative process. In Rykov and Yastrbrnrtsky [3], it appears under the name “Regenerative processes with several types of regeneration points”. Klimov [4] considered it as a “semi-Markov process with additional trajectories”. In accordance with Jakod [5] and Nummeline [6], in this paper, we will call such process a semi-regenerative process (SRP).

The next possible generalization is to construct within the basic regeneration cycle other regeneration points called embedded regeneration points. Sometimes, in a particular model, one can ﬁnd some embedded regeneration points and to represent the process distribution in a separate regeneration cycle in terms of its distribution in embedded regeneration cycles of the second level. Initially, such processes appeared in [7,8] under the name “Regenerative processes with embedded regeneration cycles” or “Decomposable Semi-Regenerative Processes” (DSRP). Regenerative processes with embedded regeneration cycles are widely used applications in operations research. In [7], regenerative processes have been applied to study priority queueing systems. In [9,10], the one-server queueing system with inﬁnite queue, recurrent input ﬂow, and general service time distribution (GI|GI|1, ∞—system in Kendall’s notation [11]) has been studied in terms of these processes (see also [12]). In paper [13], the authors investigated a model of a polling system by means of the DSRP. Another approach related to the study of complex systems (proposed in Belyaev [14]) contains the introduction of so-called supplementary variables. According to this method, the original model can be described by a continuous-time Markov chain (Markovization method). The implementation of this method to study reliability models is proposed by Rykov [15], Mazumdar [16], and Mine and Asakura [17].

The redundancy of some technical system means the duplication of critical components resulting in increasing reliability of the system. The double redundant systems are well known and were studied and applied under different assumptions like repair maintenance, preventive maintenance policy, exponential distributions for life and repair times, and so on; see, e.g., Gaver [18], Gnedenko et al. [19], Mine and Asakura [20], and Srinivasan [21]. In this paper, we study a homogeneous cold double redundant system with arbitrary life- and repair time distributions. In such a system, the cold standby redundancy means that the redundant component does not fail in a standby mode. The closed form solutions for the main reliability characteristics have been derived in terms of an integral transformation like a Laplace transform. The proposed analysis is based on the DSRP, the stochastic equations for the corresponding random variables (RVs), and the theory of regenerative processes [1].

The rest of the paper is organized as follows. In Section 2, the problem set, some assumptions, and notations are introduced. In Section 3, the Laplace–Stieltjes transforms (LSTs) of the regeneration period (the system life cycle), and the Laplace transforms (LTs) of the times to the ﬁrst failure and between system failures are calculated. In Sections 4 and 5, the time-dependent system state probabilities (TDSSPs) and stationary system state probabilities (SSSPs) are calculated. Finally, using some numerical examples, we demonstrate the effect of type of distributions and their parameter on the reliability and probabilistic measures of the system in Section 6. The paper ends with a conclusion.

2. The Problem Set: Assumptions and Notations

Consider a homogeneous cold double redundant renewable system with generally distributed life- and repair times, which, according to modiﬁed Kendall’s notations [11], will be denoted as

Denote by Ai (i = 1, 2, . . . ), life times of the system units, by Bi (i = 1, 2, . . . ), its repair times, and suppose that all these random variables (RVs) are mutually independent and identically distributed.

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Thus, denote by A(t) = P[Ai ≤ t] and B(t) = P[Bi ≤ t] the corresponding cumulative distribution functions (CDFs). Denote also by A and B the RVs with the same CDFs as Ai and Bi, respectively. Suppose that the instantaneous failures and repairs are impossible and their mean times are ﬁnite:

A(0) = B(0) = 0,

∞

a = (1 − A(x))dx < ∞,

0

∞

b = (1 − B(x))dx < ∞.

0

Denote by E the set of system states. We assume that E = {i = 0, 1, 2}, where i stands for the number of failed units, and we introduce a random process J = {J(t), t ≥ 0}, where

J(t) = number of failed units at time t.

A simple state transition diagram is illustrated in Figure 1.

Figure 1. State transition diagram.

Denote by F the time between system failures, i.e., the period of time the system begins to operate again up to the next system failure. We remind that the system is again in an operational state when operation of one unit and repair of another one begin simultaneously. Denote by F1 the time to ﬁrst system failure (the time from the epoch when both units are in operational state up to the system failure) (see Figure 2). The functions F(t) = P[F ≤ t] and F1(t) = P[F1 ≤ t] are corresponding CDFs.

We are interesting in calculation of the reliability function

R(t) = P[F1 > t] = 1 − F1(t),

distribution of time to the ﬁrst failure F1(t), time-dependent system state probabilities (TDSSPs)

πj(t) = P[J(t) = j], j = 0, 1, 2,

and stationary system state probabilities (SSSPs)

πj = lim πj(t) ≡ lim P[J(t) = j], j = 0, 1, 2,

t→∞

t→∞

as well as the availability coefﬁcient

Kav. = π0 + π1 = 1 − π2.

The following notations will be used next.

• The Laplace–Stiltjes transforms (LSTs) of their CDFs are denoted as:

∞

∞

a˜(s) = E e−sA = e−sxdA(x), b˜(s) = E e−sB = e−sxdB(x),

0 ∞

f˜(s) = E e−sF = e−sxdF(x),

f˜1(s) = E e−sF(1)

0 ∞

= e−sxdF1(x).

0

0

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• In the case when the repair completion comes before the failure of other unit and when an operative unit fails before the repair completion of other unit, the distributions are equal, respectively,

t

t

aB(t) = B(x)dA(x) = P[B ≤ A ≤ t], bA(t) = A(x)dB(x) = P[A ≤ B ≤ t].

0

0

Applying the Laplace transform (LT) for aB(t) and bA(t), we get

∞

∞

a˜B(s) = e−stdaB(t) = e−sx B(x)dA(x),

(1)

0

0

∞

∞

b˜ A(s) = e−stdbA(t) = e−sx A(x)dB(x).

0

0

• Their derivatives with sign minus in point zero are denoted by:

d

∞

aB = − ds a˜B(s) s=0 = 0 xB(x)dA(x),

bA = − dds b˜ A(s) s=0 = 0∞ xA(x)dB(x). (2)

• The probabilities P[B ≤ A] and P[B ≥ A] are associated with transformations through the following ratios:

∞

a˜B(0) = B(x)dA(x) = P[B ≤ A] ≡ p,

0 ∞

b˜ A(0) = A(x)dB(x) = P[B > A] ≡ q = 1 − p.

0

• Note the property of transformations (1)

a˜1−B(s) = a˜(s) − a˜B(s), b˜1−A(s) = b˜(s) − b˜ A(s).

(3)

Figure 2. Trajectory of the process J. 3. Reliability Function

Process J is a regenerative one. A trajectory of this process is illustrated in Figure 2.

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Here, F stands for the time between system failures, i.e., the period of time the system begins to operate again (state 1) up to the next system failure (state 2). The variable G speciﬁes the length of a regenerative cycle according to a deﬁned renovation policy. The following lemma holds for the LSTs of the time F between failures and the time to the ﬁrst failure F1.

Lemma 1. The LST f˜(s) = E e−sF of the time F between failures and the LST f˜1(s) = E e−sF1 of the time to the ﬁrst failure F1 are of the form:

f˜(s) = a˜(s) − a˜B(s) , f˜1(s) = a˜(s) a˜(s) − a˜B(s) .

(4)

1 − a˜B(s)

1 − a˜B(s)

Proof. The life time of the system F is a time between two successive failures. After a failure in state 0, the system goes to state 1 where there can be two events: either the recovery of a unit in state 1 in time B and subsequent transition again to state 0 or a failure of a second unit in time A will happen. After a failure in state 1, the system comes back to state 1 in time B. Hence, as it can be seen from Figure 2, the system lifetime F satisﬁes the following stochastic equation:

F = A + F if B < A, (5)

A

if B > A.

The time to the ﬁrst system failure F1 satisﬁes

F1 = A + F.

(6)

Applying LT to the Equation (5) and taking into account notations (1), one can obtain

∞

f˜(s) = E e−sF = e−stdF(t) =

0 ∞

= e−sx B(x) f˜(s) + (1 − B(x)) dA(x) =

0

= f˜(s)a˜B(s) + a˜1−B(s).

(7)

Due to the last equation, one can ﬁnd that the LST of the system lifetime f˜(s) is given by the ﬁrst relation in of (4). The second equation of lemma directly follows from stochastic Equation (6).

Remark 1. Let ϕ˜i(s), i = 0, 1, 2, be the LST of the distribution of time to the ﬁrst failure of the system, starting from state i at t = 0. In previous statement f˜1(s) = ϕ˜0(s). The following equations hold:

ϕ˜0(s) = q˜0,1(s)ϕ˜1(s), ϕ˜1(s) = q˜1,1(s)ϕ˜1(s) + q˜1,2(s).

Here, q˜i,j(s) stands for the LST of the transition time distribution qi,j(t) from state i to state j. Obviously, we have q˜0,1(s) = a˜(s), q˜1,1(s) = a˜B(s) and q˜1,2(s) = a˜1−B(s). Expressing ϕ˜0(s), we get the relation

ϕ˜0(s) = q˜0,1(s)q˜1,2(s) , 1 − q˜1,1(s)

which coincide with the relation (4) for the ﬁrst time to failure.

The main result of this section is the following theorem:

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Theorem 1. The LT R˜ 1(s) of the system reliability function R1(t) = 1 − F1(t) is

R˜ 1(s) = (1 − a˜(s))(1 + a˜(s) − a˜B(s)) .

(8)

s(1 − a˜B(s))

Proof. Taking into account that the LT of any CDF connected with its LST through a relation

∞

∞

F˜1(s) = e−st F1(t)dt = 1 e−stdF1(t) = 1 f˜1(s),

s

s

0

0

the LT R˜ 1(s) of the reliability function R1(t) = 1 − F1(t) can be derived using (4) in form

R˜ 1(s) = 1 − F˜1(s) = 1 (1 − f˜1(s)) = 1 − a˜B(s) − a˜2(s) + a˜(s)a˜B(s)

s

s

s(1 − a˜B(s))

= (1 − a˜(s))(1 + a˜(s) − a˜B(s)) , s(1 − a˜B(s))

which ends the proof.

Corollary 1. The LT R˜ (s) of the survival probability R(t) = 1 − F(t) is of the form

R˜ (s) =

R˜ (s)

= 1 − a˜(s) ,

(9)

1 + a˜(s) − a˜B(s) s(1 − a˜B(s))

which follows directly from (4) and (8).

Corollary 2. The mean system life time between failures E[F] and mean time to the ﬁrst failure E[F1] are of

the form:

a

a

E[F] = q , E[F1] = a + q .

(10)

Proof. The ﬁrst relation follows from the equality

E[F] = R˜ (0) = lim 1 − a˜(s) = a = a , s→0 s(1 − a˜B(s)) 1 − a˜B(0) q

and the second one follows directly from the relation (6).

4. Time Dependent System State Probabilities

4.1. Deﬁnition

For calculation of the time dependent system state probabilities (TDSSP), we use a renewal theory. In our case, the process J is a regenerative process with a delay (see Figure 2) and its regeneration times are

S0 = 0, S1 = A1, S2 = S1 + G1, . . . , Sk+1 = Sk + Gk, . . . .

Here, regeneration cycles Gi (i = 1, 2, . . . ) are the time intervals between two successive returns of the process J into state 1 after a system failure, when one of the system units begins to operate and another one is being repaired. Thus, if the system functioning begins when both units are in operational state, the ﬁrst regeneration period differs from the others and has the duration A1. The state probabilities of the process J in any time t

πj(t) = P{J(t) = j}, j = 0, 1, 2

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can be represented in terms of its distribution over separate regeneration period

π(j 1)(t) = P[J(t) = j, t < G], j = 0, 1, 2,

(11)

and the renewal function H(t) can be deﬁned as follows:

t

πi(t) = πi(1,1)(t) +

d

H

(u)π

(1) j

(t

−

u).

(12)

0

The distribution of the process J on this period is of the form

π(j 1,1)(t) = P[J(t) = j, t < A1] = δj0(1 − A(t)).

(13)

In terms of the CDF G(t) = P[Gi ≤ t] of RVs Gi, a renewal function H(t) is determined as follows:

H(t) = ∑ P ∑ Gi ≤ t = ∑ G∗k(t).

(14)

k≥1

1≤i≤k

k≥1

On the basis of the foregoing, the next lemma can be proved. Consider, ﬁrst of all, the regeneration cycle distribution.

Lemma 2. The LST of the regeneration cycle is of the form:

g˜(s) = E[e−sG] = b˜ A(s) .

(15)

1 − a˜B(s)

Proof. The regeneration cycle is a time between two successive visits of state 1 where two events can occur: either the recovery of a unit in state 1 in time B and subsequent transition to state 0 or a failure of a second unit in time A happens. Figure 2 shows that RV G satisﬁes the following stochastic equation due to independence of RVs A and B:

G = A + G if A > B, (16)

B

if A ≤ B.

Applying LST to this stochastic equation, we obtain

∞

g˜(s) = E e−sG = e−stdG(t)

0

∞

= dA(x) B(x)e−sx g˜(s) + e−sydB(y) =

y>x 0

∞

∞

= g˜(s) e−sx B(x)dA(x) + e−sy A(y)dB(y) =

0 0

= g˜(s)a˜B(s) + b˜ A(s),

(17)

which implies the expression (15) for the LST of the regeneration period. As a corollary, we obtain the mean length of the regeneration period.

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Corollary 3. The mean length of the regeneration period can be deﬁned as:

E[G] = aB + bA .

(18)

q

Proof. The proof is obtained by differentiation

E[G] = − d g˜(s) = − b˜ A(0)(1 − a˜B(0)) + a˜Bb˜ A(0) .

ds

s=0

(1 − a˜B(0))2

This expression leads to expression (18) due to properties of a˜B(0) and b˜ A(0).

Lemma 3. The LST of the system renewal function is given by:

h˜ (s) =

b˜ A(s) .

(19)

1 − (a˜B(s) + b˜ A(s))

Proof. From the renewal theory, it is well known (and follows from (14)) that the LST of the renewal function H(t) is deﬁned as h˜ (s) = g˜(s)(1 − g˜(s))−1. Thus, substitution to this expression of (15) for

g˜(s) leads to (19).

Theorem 2. The LTs π˜ j(s) of the TDSSPs πj(t) (j = 0, 1, 2) are of the form:

1 − a˜(s)

b˜ a (s)

(1)

π˜ j(s) = δj0 s + 1 − (a˜B(s) + b˜ A(s)) π˜ j (s), j = 0, 1, 2,

(20)

where π˜ (j 1)(s) (j = 0, 1, 2) are the LTs of the TDSSPs π(j 1)(t) (j = 0, 1, 2) in a separate regeneration period. These probabilities will be calculated in the next subsection.

Proof. The form of Equations (12) and (14) prompts to calculate the process state probabilities in terms of their LTs and the renewal function LST

∞

π˜ j(s) = e−stπj(t)dt,

0

∞

h˜ (s) = e−stdH(t).

0

Applying LT to Equation (12) and taking into account Equation (13), one can obtain

π˜ j(s) = δj0 1 −sa˜(s) + h˜ (s)π˜ (j 1)(s). (21) Substitution into this equality of the expression for h˜ (s) (19) leads to (20) and completes the proof.

4.2. The State Probabilities in a Separate Regeneration Period

Now, we calculate the TDSSPs in a separate regeneration period. The probability π2(1)(t) can be calculated very easily in the main level regeneration period.

Lemma 4.

The

LT

of

the

second

TDSSP

(1)

π2

(t)

in

the

main

regeneration

period

is

given

by

(1)

a˜(s) − (a˜B(s) + b˜ A(s))

π˜ 2 (s) =

s(1 − a˜B(s)) .

(22)

Proof. Due to representation of the regeneration period by Formula (16), the event {J(t) = 2, t < G} occurs if and only if either the event {A1 ≤ t ≤ B1} occurs or the events {t > u = A1 > B1} and

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{J(t − u) = 2, t − u < G} occur. Thus, as it is shown in Figure 2, for the probability π2(1)(t), it holds that

t

π2(1)(t) = P[J(t) = 2, t < G] = P[A ≤ t < B] + dA(u)B(u)π2(1)(t − u).

0

From this equation for the LST, it follows

∞

π˜ 2(1)(s) = e−st A(t)(1 − B(t))dt + a˜B(s)π2(1)(s)

0

and ∞

π˜ (1)(s) = 1

e−st A(t)(1 − B(t))dt.

2

1 − a˜B(s)

0

Calculating LTs of the integral in these expressions using partial integration, we obtain

∞

π˜ (1)(s) =

1

e−st A(t)(1 − B(t))dt =

2

1 − a˜B(s)

0

=−

1

e−st A(t)(1 − B(t))dt ∞ +

s(1 − a˜B(s))

0

∞

+1

e−st[dA(t)(1 − B(t)) − A(t)dB(t)] =

s(1 − a˜B(s))

0

= a˜(s) − (a˜B(s) + b˜ A(s)) s(1 − a˜B(s))

that completes the proof.

Since

the

calculation

of

the

probabilities

(1)

πj

(j

=

0, 1)

is

not

a

trivial

task,

we

intend

to

apply

the

theory of DSRP [3,8,12]. According to this theory, the further analysis can be described as follows. As it

was shown in the Equation (12), the process distribution can be represented in terms of its distribution

in embedded regeneration cycles and embedded renewal function. This approach is useful if the

behavior of the process is too complex in order to derive a state distribution directly, but the embedded regeneration epochs are available. At known regeneration epochs Sk(1), the distribution in the basic regeneration period (regeneration period of the ﬁrst level) πi(1)(t) similar to the Equation (12) can be represented in terms of distributions in embedded regeneration cycles (regeneration cycles of the second level) π(j 2)(t) and embedded renewal function H(1)(t) in the following way:

∞

π(j 1)(t) = π(j 2)(t) +

d

H

(1)

(u)π

(2) j

(t

−

u),

j = 0, 1,

(23)

0

where

π(j 2)(t) = P{J(t) = j, t < G(2)}, j = 0, 1

is the process state distribution in a separate regeneration period of the second level. In this case

as embedded regeneration epochs, we choose the random number ν = min{n : An < Bn} of time

epochs

S1(1) = A11{A1>B1}, S2(1) = S1(1) + A21{A1>B1, A2>B2}, . . .

up to time, when the event {An ≤ Bn} occurs for the ﬁrst time. It means that the time interval G(1) between embedded regeneration epochs has a distribution G(1)(t) = A(t) and these epochs in the time

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interval F are determined by Equation (5). According to the theory of DSRP, the embedded renewal function H(1)(t) satisﬁes the equation

t

H(1)(t) = A(t) + dH(1)(u)A(t − u) − F(t),

(24)

0

where F(t) is a CDF of the time between system failures determined by its LST (4).

Similar to the basic case, the solution of Equations (23) and (24) can be represented in terms of

their LTs and LSTs:

∞

π˜ (j 2)(s) = e−stπ(j 2)(t)dt,

∞

h˜ (1)(s) = e−stdH(1)(t).

0

0

The next lemma speciﬁes connections between process distributions in the ﬁrst and in the second level regeneration cycles in terms of their LTs.

Lemma 5. The TDSSPs in the ﬁrst and in the second level regeneration cycles in terms of the LTs satisfy the

relation

π˜ (1)(s) = π˜ (2)(s) a˜B(s) j = 0, 1.

(25)

j

j 1 − a˜B(s)

Proof. Applying LT to Equation (23), we get

π˜ (j 1)(s) = (1 + h˜ (1)(s))π˜ (j 2)(s),

(26)

where, due to (24), the LST h˜ (1)(s) of the embedded renewal function H(1)(t) is of the form

h˜ (1)(s) = a˜(s) + h˜ (1)(s)a˜(s) − f˜(s),

which leads in turn to

h˜ (1)(s) = a˜(s) − f˜(s) . 1 − a˜(s)

Substitution of this relation to the Equation (26) and taking into account expressions for f˜(s) from (4) implies the result (22) that completes the proof.

We have to calculate now only π˜ (j 2)(s) (j = 0, 1).

Lemma 6. In notations (4), the LTs of the second level system state probabilities are:

π˜ 0(2)(s) = 1s (a˜(s) − (a˜B(s) + b˜ A(s))); π˜ 1(2)(s) = 1s [1 − (a˜(s) + b˜(s)) + (a˜B(s) + b˜ A(s))]. (27)

Proof. First, consider appropriate expressions for π˜ (j 2)(t) (j = 0, 1). From Figure 2, it follows that

• The event {J(t) = 0, t < G(1)} occurs if and only if {B ≤ t < A}; • The event {J(t) = 1, t < G(1)} occurs if and only if {t < B ≤ A} or {t < A ≤ B};