# Reliability evaluation of multi-component cold-standby redundant systems

**ABSTRACT** A new methodology for the reliability evaluation of an l-dissimilar-unit non-repairable cold-standby redundant system is introduced in this paper. Each unit is composed of a number of independent components with generalized Erlang distributions of lifetimes, arranged in any general configuration. We also extend the proposed model to the general types of non-constant hazard functions. To evaluate the system reliability, we construct a directed stochastic network with exponentially distributed arc lengths, in which each path of this network corresponds with a particular minimal cut of the reliability graph of system. Then, we present an analytical method to solve the resulting system of differential equations and to obtain the reliability function of the standby system. The time complexity of the proposed algorithm is O(2n), which is much less than the standard state-space method with the complexity of O(3n2). Finally, we generalize the proposed methodology, in which the failure mechanisms of the components are different.

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Reliability evaluation of multi-component

cold-standby redundant systems

Amir Azaron*, Hideki Katagiri, Kosuke Kato,

Masatoshi Sakawa

Department of Artificial Complex Systems Engineering, Graduate School of Engineering,

Hiroshima University, Kagamiyama 1-4-1, Higashi-Hiroshima, Hiroshima 739-8527, Japan

Abstract

A new methodology for the reliability evaluation of an l-dissimilar-unit non-repair-

able cold-standby redundant system is introduced in this paper. Each unit is composed

of a number of independent components with generalized Erlang distributions of life-

times, arranged in any general configuration. We also extend the proposed model to

the general types of non-constant hazard functions. To evaluate the system reliability,

we construct a directed stochastic network with exponentially distributed arc lengths,

in which each path of this network corresponds with a particular minimal cut of the reli-

ability graph of system. Then, we present an analytical method to solve the resulting sys-

tem of differential equations and to obtain the reliability function of the standby system.

The time complexity of the proposed algorithm is O(2n), which is much less than the stan-

dard state-space method with the complexity of O(3n2). Finally, we generalize the pro-

posed methodology, in which the failure mechanisms of the components are different.

? 2005 Elsevier Inc. All rights reserved.

Keywords: Reliability; Markov processes; Graph theory; Complexity

0096-3003/$ - see front matter ? 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2005.02.051

*Corresponding author.

E-mail address: azaron@msl.sys.hiroshima-u.ac.jp (A. Azaron).

Applied Mathematics and Computation 173 (2006) 137–149

www.elsevier.com/locate/amc

Page 2

1. Introduction

Many fielded systems use cold-standby redundancy as an effective system

design strategy. Cold-standby means that the redundant units cannot fail while

they are waiting. Space exploration and satellite systems achieve high reliability

by using cold-standby redundancy for non-repairable systems, see Sinaki [13].

Space inertial reference units are required to accurately monitor critical infor-

mation for extended mission times without opportunities for repair. Many

other systems use cold-standby redundancy as an effective strategy to achieve

high reliability including textile manufacturing systems, see Pandey et al.

[11], and carbon recovery systems used in fertilizer plants, see Kumar et al. [9].

Inthispaper,wepresentanewmethodologyforthereliabilityevaluationofan

l-dissimilar-unit multi-component non-repairable cold-standby redundant sys-

tem. Each unit is composed of a number of independent components, arranged

in any general configuration. The lifetimes of the components are assumed to

be independent random variables with generalized Erlang distributions. There-

fore, this methodology allows non-constant hazard functions.

To evaluate the system reliability, we construct a directed stochastic net-

work with exponentially distributed arc lengths, in which each path corre-

sponds with a particular minimal cut of the reliability graph of system.

Then, we prove that the system failure function is equal to the density function

of the shortest path, from the source node to the sink node, in the directed net-

work. Finally, we obtain a closed form for the system reliability.

Extensive research has been carried out on the reliability of redundant sys-

tems with similar/dissimilar units. Several methods and methodologies have

been discussed by Birolini [2] and Srinivasan and Subramanian [14]. Azaron

et al. [1] developed a new approach to evaluate the reliability function of a class

of dissimilar unit redundant systems with exponentially distributed lifetimes.

Multi-component systems have been analyzed by several authors including

Goel et al. [3] and Yamashiro [15]. Most of such studies deal with the analysis

Nomenclature

Ti

T

Cj

Xj

lifetime of the ith component of the system, i = 1,2,...,n

system lifetime

jth minimal cut of the reliability graph, j = 1,2,...,m

failure time of the jth minimal cut of the reliability graph,

j = 1,2,...,m

reliability function of the multi-component cold-standby system

distribution function of shortest path, from the source to the sink

node, in directed network

R(t)

F(t)

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A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149

Page 3

of a single unit system. Gupta et al. [6] investigated a single server two-unit

multi-component cold-standby system under the assumption that the cold-

standby unit becomes operative instantaneously upon the failure of operative

unit. Gupta et al. [7] analyzed a two dissimilar unit multi-component cold-

standby system with correlated failures and repairs. There has been little

research toward the study of l-unit multi-component systems because of the

complexity in the equations and not getting the results in closed form. An l-unit

multi-component system is analyzed by Goel and Gupta [4] assuming the failed

unit is replaced by the leading standby unit with a constant replacement rate.

The major limitations in the reliability evaluation approaches for l-dissimi-

lar-unit cold-standby systems thus far are:

1. Most available algorithms assume that each unit is composed of a single

component, but they also cannot get the results in closed form, see Goel

and Gupta [5].

2. Available algorithms that do address dissimilar units multi-component cold-

standby systems assume that each unit is composed of a number of compo-

nents arranged in a series configuration. Although this is a start, there are

many more complicated system configurations that should be examined.

The problem lies in the difficulty of presenting more complicated structures.

3. Most of these studies have been carried out using the standard state-space

method. In this method, the reliability analysis becomes complex and time

consuming as the number of components increases, because of the large

number of states involved.

In this paper, we extend the work of Azaron et al. [1] to evaluate the reliabil-

ity function of an l-dissimilar-unit multi-component cold-standby redundant

system. Our methodology not only gets the reliability function in closed form

for the complex structures (l-dissimilar-unit multi-component cold-standby

systems with non-constant hazard functions), but also the size of the state-

space and the corresponding computational time, accordingly, are much less

than the standard state-space method. Finally, we generalize the proposed

methodology, in which the failure mechanisms of the components are different

or the distribution parameters of the components are considered as the combi-

natorial design variables.

2. Reliability evaluation of multi-component cold-standby systems

A very efficient method to compute the reliability of a system is to express it

as a reliability graph, see Shooman [12] for the details. A reliability graph con-

sists of a set of arcs. Each arc represents a component of the system, while the

nodes of the graph tie the arcs together and form the structure. Corresponding

A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149

139

Page 4

with the ith arc (component) of the reliability graph, i = 1,2,...,n, there is a

random variable Tias the lifetime of this component with generalized Erlang

distribution of order niand the infinitesimal generator matrix Gias

2

Gi¼

?ki1

0

ki1

0

???

???

???

???

???

00

?ki2

?

0

ki2

00

?

0

?

0

??

?kini

0

kini

0000

66666664

3

77777775

.

In this case, Tiwould be the time until absorption in the absorbing state. An

Erlang distribution of order ni is a generalized Erlang distribution with

ki1¼ ki2¼ ??? ¼ kini¼ ki. When ni= 1, the underlying distribution becomes

exponential with the parameter ki. Ti, i = 1,2,...,n, are independent random

variables, due to the fact that the components work independently.

By definition, a cut of the reliability graph is a set of components, which

interrupts all connections between input and output when removed from the

graph. A minimal cut is a cut with the minimum number of terms. Each system

failure can be represented by the removal of at least one minimal cut from the

graph.

As mentioned, we consider a cold-standby system, i.e., not all of its compo-

nents are set to function at time zero. Initially, only the components of the first

path of the reliability graph work. Upon failing one component of this path,

the system is switched to the next path and the connection between the input

and the output is established through this second path. This process continues

until no other connection between the input and the output of the reliability

graph exists. In that case, the system fails.

Lemma 1. For j = 1,2,...,m, the following relation holds:

X

Xj¼

i2Cj

Ti.

ð1Þ

Proof. Refer to Azaron et al. [1] for the details of proof.

h

To evaluate the reliability function, we construct a directed stochastic net-

work with exponentially distributed arc lengths. There are m paths in this net-

work, in which the jth path of this directed network corresponds with the jth

minimal cut of the reliability graph of the system, j = 1,2,...,m. Clearly, by

Lemma 1, the length of each path in this directed network is equal to the failure

time of the corresponding cut. For constructing this network, we use the idea

that if the lifetime of the ith component of the system is distributed according

to a generalized Erlang distribution of order niand the infinitesimal generator

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A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149

Page 5

matrix Gi, it can be decomposed to niexponential serial arcs with parameters

ki1;ki2;...;kini. The following rule describes how to construct the proper direc-

ted network.

Rule 1. Arc i belongs to the jth path of the directed network, if and only if

i 2 Cj. If ni= 1, then the length of this arc is exponentially distributed with

parameter ki. Otherwise, if ni> 1, then this arc is replaced by niexponential

serial arcs with parameters ki1;ki2;...;kini.

Theorem 1. The system lifetime is given by

T ¼

min

j¼1;2;...;mfXjg.

ð2Þ

Proof. Upon the failure of the first minimal cut of the reliability graph of sys-

tem, all connections between the input and the output are interrupted, and con-

sequently the multi-component cold-standby system fails. Therefore, the

lifetime of the multi-component cold-standby system would be equal to the

failure time of the first minimal cut, which results in (2).

h

Corollary 1. The reliability function of the multi-component cold-standby system

is given by

RðtÞ ¼ 1 ? FðtÞ.

ð3Þ

Proof. Relation (3) follows from the definitions of R(t) and F(t).

h

3. Shortest path distribution in directed networks

Kulkarni?s method [8] is applied to obtain the distribution function of short-

est path, from the source to the sink node, in the directed network, and accord-

ingly the reliability function of the cold-standby system.

Let G = (V,A) be a directed network, in which V and A represent the sets of

nodes and arcs of the network, respectively. Let s and t represent the source

and the sink nodes of this network, respectively. The length of arc (u,v) 2 A

is indicated by T(u,v), which is an exponential random variable with parameter

k(u,v).

Definition 1. To describe the evolution of the stochastic process {X(t),t P 0},

for each X ? V, where s 2 X and t 2 X ¼ V ? X, we define the following

sets:

A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149

141

Page 6

1. X1? X, set of nodes not included in X with the property that each path

which connects any node of this set to the sink node t, contains at least

one member of X.

2. SðXÞ ¼ X [ X1.

Example 1. In the network depicted in Fig. 1, if we consider X = (1,2), then

X1¼ /, and S(X) = (1,2). However, if we consider X = (1,3,4), then the only

path that connects node ð2Þ 2 X to node (5) passes through node (4), which

belongs to X. Therefore, X1¼ ð2Þ, and S(X) = (1,2,3,4).

Definition 2

X ¼ fX ? V =s 2 X;t 2 X;X ¼ SðXÞg;

In Example 1, X* = {(1),(1,2),(1,3),(1,2,3),(1,2,4),(1,2,3,4),(1,2,3,4,5)}.

X?¼ X [ V .

ð4Þ

Definition 3. If X ? V such that s 2 X and t 2 X, then a cut is defined as:

CðX;XÞ ¼ fðu;vÞ 2 A=u 2 X;v 2 Xg.

There is a unique minimal cut contained in CðX;XÞ, denoted by C(X). If X 2 X,

then,

ð5Þ

CðX;XÞ ¼ CðXÞ.

It is shown that {X(t),t P 0} is a continuous-time Markov process with state

space X* and the infinitesimal generator matrix Q = [q(X,Y)](X,Y 2 X*), see

Kulkarni [8] for the details, where

P

?

ðu;vÞ2ðXÞ

0 otherwise.

qðX;YÞ ¼

ðu;vÞ2ðXÞ

kðu;vÞ

if Y ¼ SðX [ fvgÞ;

P

kðu;vÞ

if Y ¼ X;

8

>

>

>

>

>

>

>

>

:

<

ð6Þ

2

5 4

3

1

Fig. 1. Graph of Example 1.

142

A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149

Page 7

We assume that the states in X* are numbered 1,2,...,N = jX*j so that Q

matrix is upper triangular. State 1 is the initial state, and state N is the final

(absorbing) state. In Example 1, state 1 is (1), and state 7 is (1,2,3,4,5).

Let T represent the length of the shortest path in the directed network.

Clearly,

T ¼ minft > 0 : XðtÞ ¼ N=Xð0Þ ¼ 1g.

Therefore, the length of the shortest path in the directed network would be

equal to the time until {X(t),t P 0} gets absorbed in the final state N, starting

from state 1.

Chapman–Kolmogorov backward equations can be applied to compute

F(t) = P{T 6 t}. If we define

PiðtÞ ¼ PfXðtÞ ¼ N=Xð0Þ ¼ ig

then, F(t) = P1(t).

The system of linear differential equations is given by

ð7Þ

i ¼ 1;2;...;N;

ð8Þ

_PðtÞ ¼ Q ? PðtÞ;

Pð0Þ ¼ ½0;0;...;1?T;

where P(t) = [P1(t),P2(t),...,PN(t)]Trepresents the state vector of the system

and Q is the infinitesimal generator matrix.

Now, we present an efficient method to solve the system of differential equa-

tions with constant coefficients (9). Let M be the modal matrix of Q. That is, M

is the N · N matrix whose N columns are the eigenvectors of Q. Let

a1,a2,...,aNbe the eigenvalues of Q, which are the diagonal elements of Q

owning to its upper triangular nature. P(t) can be computed as follows:

ð9Þ

PðtÞ ¼ MeAtM?1Pð0Þ;

where eAtis the diagonal matrix as follows:

ea1t

0

ea2t

ð10Þ

eAt¼

?

?

?

?

0

0

?

??

0

?

?

eaNt

2

66664

3

77775

.

ð11Þ

As mentioned, F(t) = P1(t), and the reliability function of the multi-compo-

nent cold-standby system is computed from (3).

A system with multiple eigenvalues can be perturbed, by introducing a slight

change in some coefficients, to produce a system with distinct eigenvalues. In-

deed, the original system can be regarded as the limit of systems with distinct

eigenvalues, see Luenberger [10] for the details.

The proposed methodology is easily generalized, in which the distribution

parameters (kij,ni) are considered as the combinatorial design variables. In

A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149

143

Page 8

real-world problems, the designers sometimes use fundamentally different de-

signs or technologies with different distribution parameters, because the failure

mechanisms would be different. In this case, we first compute the reliability

function of the standby system, for all combinations of (kij,ni) for

j = 1,2,...,ni,

i = 1,2,...,n. Then,

i = 1,2,...,n, would be related to that combination, which results the maxi-

mum of the mean time to failure of the multi-component cold-standby redun-

dant system, considering the following equation:

Z1

It should also be noted that the infinitesimal generator matrix for each com-

bination of (kij,ni), for j = 1,2,...,ni, i = 1,2,...,n, would be different from the

other combinations, and this matter clearly increases the complexity of the

problem.

theoptimal

ðk?

ij;n?

iÞ,

j = 1,2,...,ni,

MTTF ¼

0

RðtÞdt.

ð12Þ

4. Numerical example

For controlling a spacecraft, there are 3 non-repairable dissimilar units in a

cold-standby redundancy scheme, depicted in Fig. 2.

At the beginning, the operating unit is unit 1, which is composed of a laptop

computer (component 1) and a power supply (component 2) arranged in a ser-

ies configuration. When this unit fails, the redundant unit 2, which is composed

of PC I (component 3), CD Drive I (component 4) and CD Drive II (compo-

nent 5), as the cold-standby redundant components, and also a monitor (com-

ponent 6), arranged in a general configuration, is put into operation. If unit 2

fails, then the redundant unit 3, which is composed of PC II (component 7) and

1 2

3

4

5

6

7

8

9

10

2

3

1

Fig. 2. Reliability graph of the multi-component cold-standby redundant system.

144

A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149

Page 9

Hard Drive I (component 8), Hard Drive II (component 9) and Hard Drive III

(component 10), as the cold standby redundant components, arranged in a gen-

eral configuration, goes into operation. The lifetimes of the components are

independent random variables with generalized Erlang distributions. There

are two combinations for the distribution parameters of this spacecraft control-

ler. Table 1 shows the parameters of the lifetime distributions, according to the

first combination. The time unit is in year. We are interested to find the reliabil-

ity function and the mean time to failure of this 3-dissimilar-unit multi-compo-

nent non-repairable cold-standbyredundant

combinations.

Fig. 3 shows the directed network, where each path is corresponding with a

particular minimal cut of the reliability graph. For example, the path 1–4–5–7

of this network is corresponding with the minimal cut (1,4,5,7) of the reliabil-

ity graph (2), which interrupts all connections between input and output of the

cold-standby system upon removing from the graph. Then, according to Rule

1, if ni> 1, we replace the particular arc by niexponential serial arcs with

parameters ki1;ki2;...;kini. The proper directed network of the first combina-

tion is shown in Fig. 4. The numbers above the arcs show the exponential

parameters of the corresponding arc lengths. For example, arc 1 in Fig. 3

system, considering both

Table 1

Parameters of the lifetime distributions (first combination)

ini

ðki1;ki2;...;kiniÞ

(2,4,3)

(1)

(2)

(5)

(3)

(1,6)

(3,2)

(4)

(5)

(6)

1

2

3

4

5

6

7

8

9

10

3

1

1

1

1

2

2

1

1

1

3

1 4 5 7

2 1

8 10

2 6 9

3

5

6

7 4

Fig. 3. Directed network.

A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149

145

Page 10

was replaced by 3 exponential serial arcs with parameters (2,4,3) in Fig. 4, be-

cause n1= 3, and (k11,k12,k13) = (2,4,3).

The stochastic process {X(t),t P 0} corresponding with the shortest path

analysis of the directed network, depicted in Fig. 4, has 14 states in the order

of

X* = {(1),(1,2),(1,2,3),(1,2,3,4),

(1,2,3,4,5,6,7),(1,2,3,4,5,6,7,8),

(1,2,3,4,5,6,7,8,10),(1,2,3,4,5,6,7,8,9,10),

Table 2 shows the corresponding infinitesimal generator matrix.

Then,weapplythe methoddescribed in Section 3,to solve the system of linear

differential equations_PðtÞ ¼ Q ? PðtÞ, P(0) = [0,0,...,1]T. Finally, the reliability

function and the mean time to failure of this multi-component cold-standby

redundant system, considering the first combination, are obtained as follows:

(1,2,3,4,5),(1,2,3,4,6),(1,2,3,4,5,6),

(1,2,3,4,5,6,7,9),(1,2,3,4,5,6,7,8,9),

(1,2,3,4,5,6,7,8,9,10,11)}.

RðtÞ ¼ 1 ? P1ðtÞ ¼ 106.66e?3t? 686.4e?4tþ 2911.7e?5tþ 707.15e?6t

? 4050te?6t? 4505e?7tþ 1419.96e?8tþ 673.92te?8tþ 46.37e?9t

þ 0.58e?11t? 0.02e?13t;

MTTF ffi 1.31.

Table 2

Matrix Q of the first combination

State12345678910 1112 1314

1

2

3

4

5

6

7

8

9

10

11

12

13

14

?320

4

1

1

4

0

0

0

5

0

0

0

1

0

0

0

0

0

1

5

0

0

0

2

5

8

0

0

0

0

0

0

0

4

0

0

0

0

0

0

0

3

0

0

0

0

0

0

0

0

0

3

4

0

0

0

0

0

0

0

0

5

0

0

0

0

0

0

0

0

0

0

0

0

5

3

0

0

0

0

0

0

0

0

0

2

2

6

8

0

0

0

0

0

0

0

0

0

0

0

0

0

0

?5

0

0

0

0

0

0

0

0

0

0

0

0

?4

0

0

0

0

0

0

0

0

0

0

0

?8

0

0

0

0

0

0

0

0

0

0

?6

0

0

0

0

0

0

0

0

0

?13

0

0

0

0

0

0

0

0

?11 11

?70

0

0

0

0

0

0

0

0

0

0

0

0

?8

0

0

0

0

0

?6

0

0

0

0

?7

0

0

0

?9

0

0

?8

0

2

2 4 3 5 3 3 2

2

3 4

1

1 6 4 6

5

8

1

5 9 7

6

10

11

Fig. 4. The proper directed network of the first combination, following Rule 1.

146

A. Azaron et al. / Appl. Math. Comput. 173 (2006) 137–149

Page 11

Table 3 shows the parameters of the lifetime distributions, according to the

second combination.

The size of the state space, considering the second combination, is equal to

8. Finally, the reliability function and the mean time to failure of this multi-

component cold-standby redundant system, considering the second combina-

tion, are obtained as follows:

RðtÞ ¼ ?13e?3tþ 14e?2t? 2te?3t? 9te?2tþ 6.5t2e?2t? 0.17t3e?2tþ 0.42t4e?2t;

MTTF ffi 2.07.

Therefore, comparing the two mentioned combinations for the spacecraft

controller results that the best would be the second combination, which has

the maximum mean time to failure.

5. Conclusion

In this paper, we introduced a new methodology, by using continuous-time

Markov processes and shortest path technique, for the reliability evaluation of

an l-dissimilar-unit non-repairable cold-standby redundant system, where each

unit is composed of a number of independent components arranged in any gen-

eral configuration.The lifetime of each component was assumed to be a ran-

domvariablewith generalizedErlang

methodology allows non-constant hazard functions. Finally, we generalized

the proposed methodology, in which the failure mechanisms of the components

are different or the distribution parameters of the components are considered

as the combinatorial design variables.

Our approach is a powerful approach for the reliability evaluation of

these complex systems. Computing the reliability of these systems, if is not

distribution. Therefore, this

Table 3

Parameters of the lifetime distributions (second combination)

ini

ðki1;ki2;...;kiniÞ

(1)

(1,1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

1

2

3

4

5

6

7

8

9

10

1

2

1

1

1

1

1

1

1

1

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impossible, is at least so complicated for most real case problems, because

either the convolution integrals are intractable or the size of the state space

would be enormous. For example, this problem could be solved by using clever

complete enumeration of network states, see Shooman [12] for the details.

According to our methodology, for a complete constructed network with k

nodes and k(k?1) arcs representing the components of the system (the worst

case example), the size of the state space is 2k?2+ 1, refer to Kulkarni [8] for

details, but the size of the state space using clever complete enumeration of net-

work states would be equal to 3k(k?1), because each component can be in one of

these three states: work, fail and standby. Therefore, although the proposed

algorithm is still exponential, but its time complexity is decreased from

O(3n2), corresponding with the standard state space method, to O(2n). In the

numerical example, corresponding with the first combination, the state space

has only 14 states, but according to the state space method, mentioned above,

the size of the state space would be 310.

In the case of general distribution of lifetime, it could be possible to approx-

imate the lifetime distribution by an appropriate generalized Erlang distribu-

tion, by matching the first three moments, because the class of generalized

Erlang distributions is a special case of Coxian distributions, and each general

distribution can be easily approximated by a Coxian distribution. After

approximating the general lifetime distributions by the appropriate generalized

Erlang distributions, our methodology can be applied to obtain the reliability

function of the multi-component cold-standby system.

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