Analysis for a two-dissimilar-component cold standby repairable system with repair priority

doi:10.1016/j.ress.2011.06.004

Reliability Engineering & System Safety

Volume 96, Issue 11, November 2011, Pages 1542-1551

https://doi.org/10.1016/j.ress.2011.06.004 Get rights and content

Abstract

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. Assume that working time distributions and repair time distributions of the two components are both exponential, and Component 1 has repair priority when both components are broken down. After repair, Component 1 follows a geometric process repair while Component 2 obeys a perfect repair. Under these assumptions, using the perfect repair model, the geometric process repair model and the supplementary variable technique, we not only study some important reliability indices, but also consider a replacement policy T, under which the system is replaced when the working age of Component 1 reaches T. Our problem is to determine an optimal policy T^⁎ such that the long-run average loss per unit time (i.e. average loss rate) of the system is minimized. The explicit expression for the average loss rate of the system is derived, and the corresponding optimal replacement policy T^⁎ can be found numerically. Finally, a numerical example for replacement policy T is given to illustrate some theoretical results and the model's applicability.

Highlights

► A two-dissimilar-component cold standby system with repair priority is formulated. ► The successive up/repair times of Component 1 form a decreasing/increasing geometric process. ► Not only some reliability indices but also a replacement policy are studied.

Introduction

Standby redundancy techniques are often used to improve reliability, enhance availability, or to reduce the cost of a system, in many practical applications. A two-component cold standby system with one repairman has been one of the classical models in the reliability theory. Under the conditions that each component after repair is “as good as new”, and the repair rule is “first in first out”, some reliability indices and repair-replacement policies for the system have been derived in the previous literature (see, e.g. [1], [2], for further details). Later, a priority rule in repair, or use of a component, has been considered. Nakagawa and Osaki [3] assumed that both working time and repair time of the priority component follow general distributions while both working time and repair time of the non-priority component follow exponential distributions, and the repairs are perfect. Under these assumptions, they developed some interesting reliability indices for the system, using the Markov renewal theory. Other similar studies include those by Osaki [4] and Buzacott [5], for generalizations. All these studies assumed that the repairs were perfect. This is indeed a perfect repair model. However, many repairable systems consisting of one or more components and one repairman are deteriorative in nature because of the effects of aging and damage accumulation. Barlow and Hunter [6] first presented a minimal repair model in which the minimal repair does not change the age of the system. Thereafter, Brown and Proschan [7] introduced an imperfect repair model in which the repair is perfect with probability p or minimal with probability $1 - p$ . For a deteriorating repairable system, it seems more reasonable to assume that successive working times after repair are stochastically decreasing, while consecutive repair times after failure are stochastically increasing. Lam [8], [9] introduced a geometric process repair model to represent the deteriorative behaviors of one-component or multi-component systems. Under this model, he studied two kinds of replacement policies for a simple repairable system (i.e. a one-component repairable system with one repairman), one based on working age T of the system, and the other based on number of failures N of the system. Explicit expressions of the average loss rate under these two kinds of policies are determined separately. Finkelstein [10] presented a general repair model based on scale transformation, after each repair, to generalize Lam's work. Zhang [11] generalized Lam's work using a bivariate replacement policy (T,N) under which the system is replaced at working age T or at the time of Nth failure, whichever occurs first. Leung [12] introduced the arithmetico-geometric process repair model to generalize Lam's work, and in this model, he also studied two kinds of replacement policies for T and N. The problem is to determine an optimal replacement policy T^⁎ or N^⁎ such that the long-run expected loss per unit time or per unit operation time is minimized. Various replacement policies have been examined by Stadje and Zuckerman [13], Lam [14], Stanley [15], Leung and Lee [16], Zhang et al. [17], [18], [19]. Zhang [20], [21], Castro and Pérez-Ocón [22], Chen and Li [23] and others.

All the above research works relate to a one-component repairable system. For multi-component systems, under the geometric process repair model, Lam and Zhang [24] reported a two-component series repairable system. When the working and repair times of both components follow exponential distributions, they derived some reliability indices for the system. Lam and Zhang [25] investigated a two-component parallel repairable system assuming that all working and repair times follow exponential distributions and that, after repair, one component is as good as new, while the other is not. Some important reliability indices of the system are obtained using the Laplace-transform technique. Further, Zhang and Wang [26] considered a replacement policy $M = (N_{1}, N_{2}, \dots, N_{k})$ for a k-dissimilar-component series repairable system, where $N_{1}, N_{2}, \dots$ , and N_k are the numbers of failures of Component 1, Component $2, \dots$ , and Component k, respectively. An optimal replacement policy, $M^{⁎} = (N_{1}^{⁎}, N_{2}^{⁎}, \dots, N_{k}^{⁎}),$ can be determined by minimizing the average cost rate of the system. Many researchers have worked on multi-component repairable systems along this direction, including Zhang and Lam [27], Zhang et al. [28], Lam and Zhang [29], Zhang and Wu [30] and others.

Using standby redundancy techniques, Lam [31] reported a maintenance model for a two-unit redundant system with one repairman. Under this model, he studied two kinds of replacement policies, based on the number of failures and the working age, for two units. The long-run average loss per unit time for each kind of replacement policies is derived. Zhang [32] applied the geometric process repair model to a two-component cold standby repairable system with one repairman, and assumed that, after repair, each component is not as good as new. Under this assumption, using a geometric process, he studied a replacement policy N based on the number of repairs of Component 1. The problem is to determine an optimal replacement policy N^⁎ such that the long-run expected profit per unit time is maximized. For further reference, see Zhang et al. [33] and Zhang and Wang [34]. A great deal of research work, based on the geometric process repair model, can be found in Lam [35].

In practical applications, a two-component cold standby repairable system with one repairman and priority in use or repair is often used. For example, in the operating theater of a hospital, an operation must be discontinued as soon as the power source is cut (i.e. power station failures). Usually, there is a standby power station (e.g. a storage battery) in the operating theater. Thus, the power station (e.g. Component 1) and the storage battery (e.g. Component 2) form a cold standby repairable lighting system. Obviously, it is reasonable to assume that the power station has repair priority as the area served by the power station is much wider than that served by the storage battery (only in the operating theater). Further, assume that the storage battery after repair is as good as new since its used time is shorter than the power station, and the repair of the storage battery is also convenient, while the power station's used time is longer and it takes longer to repair it, due to the complexity of the equipment. Other similar examples can be found in Lam [31]. Based on Zhang [32], Zhang and Wang [36] applied the geometric process repair model to a two-dissimilar-component cold standby repairable system with one repairman and priority in use and repair. Assume that either component after repair is not “as good as new” and follows a geometric process repair, and Component 1 has priority in use and repair. Under these assumptions, they considered a replacement policy N based on the number of repairs of Component 1 under which the system is replaced when the repair number of Component 1 reaches N. An optimal replacement policy N^⁎ by minimizing the average cost rate is determined. Now we may assume that Component 2 obeys a perfect repair, while Component 1 follows a geometric process repair, and is given repair priority. Furthermore, we assume that the working time and the repair time of both components are exponentially distributed. Under these assumptions, by using the perfect repair model, the geometric process repair model, and the supplementary variable technique, we not only develop some important reliability indices, such as the system availability, idle probability of the repairman, ROCOF, reliability and MTTFF, but also determine the replacement policy based on working age T of Component 1. The objective is to choose an optimal replacement policy T^⁎ such that the average loss rate is minimized. The explicit expression for the average loss rate of the system is derived, and the corresponding optimal replacement policy T^⁎ can be found numerically. Finally, a numerical example for replacement policy T is given to illustrate some theoretical results and the model's applicability.

For ease of reference, we provide a notation list, and state the definitions of stochastic order and geometric process as follows:

Section snippets

Notation

$X_{n}^{(i)}$	working time of Component $i (i = 1, 2)$ in the nth cycle, rv
$Y_{n}^{(i)}$	repair time of Component $i (i = 1, 2)$ in the nth cycle, rv
a, b	$a_{i} \geq 1, 0 < b < 1$ is called the ratio of geometric process
$F_{n}^{(i)} (t)$	cumulative distribution function (cdf) of $X_{n}^{(i)} (i = 1, 2)$
$G_{n}^{(i)} (t)$	cumulative distribution function (cdf) of $Y_{n}^{(i)} (i = 1, 2)$
$λ_{i}$	expectation of $X_{1}^{(i)}$ , i.e. ${EX}_{1}^{(i)} = λ_{i} (i = 1, 2)$
$μ_{i}$	expectation of $Y_{1}^{(i)}$ , i.e. ${EY}_{1}^{(i)} = μ_{i} (i = 1, 2)$
$N (t)$	state of the system at time t
$S (t)$	the number of cycles of Component 1 at time t
$p_{jk} (t)$	state probabilities of

Model assumptions

We study a two-dissimilar-component cold standby repairable system with one repairman and repair priority by making the following assumptions.

Assumption 1

At the beginning, both components are new, and Component 1 is in a working state while Component 2 is in a cold standby state.

Assumption 2

When both components are good, one is in a working state and the other is in a standby state. When the working component fails, the repairman will repair it immediately. At the same time, the standby component is switched into

System analysis

Based on the model's assumptions, the state of the system at time t is given by $N (t) = \{\begin{matrix} 0 & if Component 1 is working, \\ Component 2 is in cold standby at time t, \\ 1 & if Component 2 is working, \\ Component 1 is in cold standby at time t, \\ 2 & if Component 1 is working, \\ Component 2 is under repair at time t, \\ 3 & if Component 2 is working, \\ Component 1 is under repair at time t, \\ 4 & if Component 1 is under repair, \\ Component 2 is waiting for repair at time t . \end{matrix}$ Then ${N (t), t \geq 0}$ is a stochastic process with state space $Ω = {0, 1, 2, 3, 4}$ . The set of working states is $W = {0, 1, 2, 3}$ and the set of

System availability

By the definition of A(t), we have $A (t) = P {N (t) \in W} = \sum_{k = 1}^{\infty} [p_{0 k} (t) + p_{1 k} (t) + p_{2 k} (t) + p_{3 k} (t)] .$ The Laplace transform of A(t) is given by $A^{⁎} (s) = \sum_{k = 1}^{\infty} [p_{0 k}^{⁎} (s) + p_{1 k}^{⁎} (s) + p_{2 k}^{⁎} (s) + p_{3 k}^{⁎} (s)] = [p_{01}^{⁎} (s) + p_{11}^{⁎} (s) + p_{21}^{⁎} (s) + p_{31}^{⁎} (s)] + \sum_{k = 2}^{\infty} [p_{0 k}^{⁎} (s) + p_{1 k}^{⁎} (s) + p_{2 k}^{⁎} (s) + p_{3 k}^{⁎} (s)] = \frac{s + λ_{1} + λ_{2} + μ_{1}}{(s + λ_{1}) (s + λ_{2} + μ_{1})} + \frac{1}{s + λ_{1}} \sum_{k = 2}^{\infty} [1 + \frac{s + a^{k - 1} λ_{1}}{μ_{2}} + \frac{a^{k - 1} λ_{1}}{s + λ_{2} + b^{k - 1} μ_{1}} + g (s, λ_{1}, λ_{2}, μ_{1}, μ_{2}, a, b; k)] A_{k} .$ Then, according to the Tauberian theorem, the steady-state (or limiting) availability of the system is given by $A = \lim_{t \to + \infty} A (t) = \lim_{s \to 0} {sA}^{⁎} (s) = 0 .$ This is consistent

Replacement model under policy T

In this section, we consider a repair-replacement policy T based on the working age of Component 1. Component 1 (or 2) will (or may) be replaced by a new and identical one when the working age of Component 1 reaches T. Our objective is to determine an optimal replacement policy T^⁎ such that the average loss rate of the system is minimized. To do this, besides the assumptions in Section 2, we add the following:

Assumption 6

A replacement policy T based on the working age of Component 1 is used. Under policy T

A numerical example

In this section, we provide an example to illustrate the theoretical results and the model's applicability.

Now, we denote the density functions of $X_{j}^{(1)}, X_{j}^{(2)}, Y_{j}^{(1)}$ and $Y_{j}^{(2)}$ by $f_{j} (t), f (t)$ , g_j(t) and g(t) for $j = 1, 2, \dots$ , respectively. According to Assumption 3, the density functions of $Y_{j - 1}^{(2)} - X_{j}^{(1)}$ and $X_{j}^{(2)} - Y_{j}^{(1)}$ are, respectively, $ϕ_{j} (u) = g (u) ⋆ f_{j} (- u) = \{\begin{matrix} \frac{a^{j - 1} λ_{1} μ_{2}}{a^{j - 1} λ_{1} + μ_{2}} e^{- μ_{2} u}, & u \geq 0, \\ \frac{a^{j - 1} λ_{1} μ_{2}}{a^{j - 1} λ_{1} + μ_{2}} e^{a^{j - 1} λ_{1} u}, & u < 0, \end{matrix}$ $ψ_{j} (v) = f (v) ⋆ g_{j} (- v) = \{\begin{matrix} \frac{b^{j - 1} λ_{2} μ_{1}}{λ_{2} + b^{j - 1} μ_{1}} e^{- λ_{2} v}, & v \geq 0, \\ \frac{b^{j - 1} λ_{2} μ_{1}}{λ_{2} + b^{j - 1} μ_{1}} e^{b^{j - 1} μ_{1} v}, & v < 0, \end{matrix}$ respectively.

Conclusions

In this paper, we introduce a model for a two-dissimilar-component cold standby repairable system with one repairman and repair priority. We assume that Component 1, after repair, is not “as good as new”, such that the successive working times of Component 1 form a decreasing geometric process, and the consecutive repair times of Component 1 form an increasing geometric process. Obviously, the model in this paper is more realistic than the model in which both of the two components, after

Acknowledgments

The work described in this paper was supported by a Grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project no. CityU 1414/05H) and the National Natural Science Foundation of China (Project no. 10801032).

The authors are grateful for the editors and the referees valuable comments and suggestions, with which the presentation of the paper has been considerably improved.

References (38)

T. Nakagawa et al.
Stochastic behavior of a two-unit priority standby redundant system with repair
Microelectron Reliab
(1975)
M.S. Finkelstein
A scale model of general repair
Microelectron Reliab
(1993)
Y.L. Zhang et al.
A bivariate optimal replacement policy for a multistate repairable system
Reliab Eng Syst Saf
(2007)
Y.L. Zhang
A geometrical process repair model for a repairable system with delayed repair
Comput Math Appl
(2008)
I.T. Castro et al.
Reward optimization of a repairable system
Reliab Eng Syst Saf
(2006)
J. Chen et al.
An extended extreme shock maintenance model for a deteriorating system
Reliab Eng Syst Saf
(2008)
Y.L. Zhang et al.
A geometric process repair model for a series repairable system with k dissimilar components
Appl Math Model
(2007)
Y.L. Zhang et al.
Reliability analysis for a circular consecutive-2-out-of-n: F repairable system with priority in repair
Reliab Eng Syst Saf
(2000)
Y.L. Zhang et al.
Reliability analysis for a k/n(F) system with repairable repair-equipment
Appl Math Model
(2009)
Y. Lam
A maintenance model for two-unit redundant system
Microelectron Reliab
(1997)

Y.L. Zhang et al.

A geometric process repair model for a repairable cold standby system with priority in use and repair

Reliab Eng Syst Saf

(2009)

R.E. Barlow et al.

Mathematical theory of reliability

(1965)

H. Ascher et al.

Repairable systems reliability

(1984)

S. Osaki

Reliability analysis of a two-unit standby redundant system with priority

Can J Oper Res Soc

(1970)

J.A. Buzacott

Availability of priority standby redundant systems

IEEE Trans Reliab

(1971)

R.E. Barlow et al.

Optimum preventive maintenance policy

Oper Res

(1960)

M. Brown et al.

Imperfect repair

J Appl Prob

(1983)

Y. Lam

Geometric processes and replacement problem

Acta Math Appl Sin

(1988)

Y. Lam

A note on the optimal replacement problem

Adv Appl Prob

(1988)

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Citation Excerpt :
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Analysis for a two-dissimilar-component cold standby repairable system with repair priority

Abstract

Highlights

Introduction

Section snippets

Notation

Model assumptions

System analysis

System availability

Replacement model under policy T

A numerical example

Conclusions

Acknowledgments

Microelectron Reliab

Microelectron Reliab

Reliab Eng Syst Saf

Comput Math Appl

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Appl Math Model

Reliab Eng Syst Saf

Appl Math Model

Microelectron Reliab

Reliab Eng Syst Saf

Mathematical theory of reliability

Repairable systems reliability

Reliability analysis of a two-unit standby redundant system with priority

Can J Oper Res Soc

Availability of priority standby redundant systems

IEEE Trans Reliab

Optimum preventive maintenance policy

Oper Res

Imperfect repair

J Appl Prob

Geometric processes and replacement problem

Acta Math Appl Sin

A note on the optimal replacement problem

Adv Appl Prob