Reliability Analysis of a Repairable Parallel System with Repair Time Omission

Yu Pang, Hong-Zhong Huang*, Yu Liu, Qiang Miao, Zhonglai Wang School of Mechatronics Engineering

University of Electronic Science and Technology of China Chengdu, Sichuan, 610054, P.R. China

*Corresponding author. E-mail: hzhuang@uestc.edu.cn

AbstractBased on some practical problems in industrial engineering, a species of two-unit parallel repairable system model is introduced in this paper. The prominent feature of the model is described as follows: if a repair time is sufficiently short (less than some threshold value) that does not affect the system operation, i.e. the effect could be ignored, the system can be considered as being operating during this repair time. Two scenarios are discussed where the threshold value is regarded as a constant and non-negative random variable, respectively. Reliability indices such as instantaneous availability are formulated.

Keywords- availability; Markov model; parallel system; repair time omission

I. INTRODUCTION A great deal of research has been devoted to system

reliability for its wide applications in power plants, manufacturing systems, and military equipment [1][2]. Reliability analysis of repairable system is a momentous branch of reliability theory, maintaining a high level reliability is often an essential requisite for repairable system [3].

In the former literatures, two basic assumptions are involved in most of research on repairable system: 1) when the operating unit fails, repair begins immediately; 2) the system instantaneously falls into the failure state when it is out of work. However, in some practical situations, for example, if the demands on the system by customers are not too frequent, they more likely miss a small repair time, or are at most delayed by a negligibly short time in receiving service. In generally, if the system is temporarily under repair which has no effect on system operating, the system can be regarded as being operating during such a repair interval [4]. In other words, if a short repair time does not result in the system being out of work; such a repair interval can be omitted from the downtime period. Taking this into consideration, Zheng et al.[4] calculate system availability for a single-unit Markov repairable system.

In this paper, the reliability of a two-unit parallel system is analyzed with considering the repair time omission introduced in [4]. Reliability indices such as system availability are formulated. The remainder of this paper is organized as follows: In Section II, some basic assumptions of the original parallel system model are given. Difference between the original model and the new model are analyzed in Section III. A comprehensive reliability analysis is discussed in the next

section and the formulations of the indices are proofed in detail. A numerical example is presented in Section V, as well as the steady-state availability of the system. Conclusion is provided in Section VI.

II. BASIC ASSUMPTIONS OF THE ORIGINAL PARALLEL SYSTEM

Before formulating the new model, the original two-unit parallel repairable system is defined at first [5][6]:

The system is consisting of two different units as shown in Fig.1. The system fails if and only if the units are both in failure state, and there is only one repair facility available.

Assume that the failure time and the repair time of

each unit follow exponential distribution and with the parameter of

)(tFi )(tGii and i , respectively. All the

random variables involved are mutually independent. Thus, one has:

0;2,1 0,1)(

0,1)( =

>=>=

ti

etG

etF

it

i

it

i

i

i

The two units are new at the initial time ( t =0), and the repaired unit is restored into as good as new condition.

According to stochastic behavior of the entire system, there exist five system states:

State 1: unit 1 is up (operating), unit 2 is down (failed). In other words, unit 2 is under repair while unit 1 is working; the system is still operating;

Sponsored by the National Science Foundation of China (50775026) and the National Defense Key Foundation of China (9140A27020308JB34).

978-1-4244-4905-7/09/$25.002009 IEEE 41

State 2: unit 2 is up, unit 1 is down; the system is in the operating state;

State 3: unit 1 is under repair, unit 2 is waiting for repair; the system in its failure state;

State 4: unit 2 is under repair, unit 1 is waiting for repair; the system is also in failure state.

The state space is defined as , the working state space is while the failure state space is

{ 4,3,2,1,0=E }}{ 2,1,0=W { }4,3=F .

Let representing the stochastic process of the system state at time instant

)(tXt , thus one has:

=

; at timerepair for waitingis 1unit repair,under is 2unit ,4

; at timerepair for waitingis 2unit repair,under is 1unit ,3

; at timedown is 1unit up, is 2unit ,2 ; at timedown is 2unit up, is 1unit , 1

; at time operating are units both two ,0

)(

t

t

tt

t

tX

Apparently, forms a homogeneous continuous time Markov process in state space

{ 0),( ttX }E .

III. BASIC ASSUMPTIONS OF THE NEW MODEL The main difference between the original system model and

the new system model (parallel repairable system with considering repair time omission) is that: when the original system fails, the repair facility is still repairing the failed unit which fails earlier. Given a threshold value )0( , if any repair time involved is shorter than , the system can be regarded as in its operating state during the repair interval. That is to say the new system model is considered to be still in up state during this repair interval while the original system model is down state during the same period.

In contrast with the original system model, the new parallel repairable system model is described as follows:

The new system model consists of two units and only one repair facility is available. The two units are new at the initial time ( t =0); all the random variables are mutually independent; the failed units can be regarded as good as new after repair;

Whether the failed unit is waiting for or is receiving repair depends on the state of the repair facility. If the repair facility is available (no occupied), the failed unit will receive repair immediately after failure. The waiting period is called repair delay time. The failure time and repair time of these two units follow the exponential distribution and as in the original model. denotes the distribution of the repair delay time of the unit i which is formulated in the Section IV;

)(tFi )(tGi)(tQi

The new system model is operating if the original system model is operating;

The new system model is still operating if the original system model is failed but the repair time of the system is less than the threshold value ;

The new system model is failed only when the original system model is failed and repair time is greater than the threshold value ;

The threshold value can be either a constant or a non-negative random variable. The distribution is

)(H if is considered as a non-negative random variable.

In particular, when 0= , the new system model changes into the original system model; when = , the new system is never down. A possible sequence of system state changes of the original system model and the new system model is shown in Fig.2.

Let the stochastic process )(~ tX denoting the state of the new system model at any time instant t , and one has:

=

at timedown is system new the,1 at time up is system new the,0

)(~t

ttX

It is obvious that the Markov property is not held in the new

system model.

IV. RELIABILITY INDICES ASSESSMENT Considering the new system model, the instantaneous

system availability is discussed in this section. It is the derivation of the reliability indices that would be carried on in this section.

)(tA and denote the instantaneous availabilities of the original system model and the new system model, respectively. The instantaneous availability of a single repairable unit i (

)(~ tA

2,1=i ) is denoted by , and it is formulated as [5]: )(tAit

ii

i

ii

ii

iietA )()(

+

+++=

42

For a single repairable unit i ( 2,1=i ) with considering the repair delay, the Laplace form of the instantaneous availability

is well defined as [9]: )(' tAi

)()()(1))(1(/1

)(*sQsGsF

sFssA

iii

ii

= ,

where , and represent the Laplace-Stieltjes transforms of , and , respectively.

)( sFi )( sGi )( sQi)(tFi )(tGi )(tQi

Proposition 1: Assuming that the threshold value i ( 2,1=i ) are constants, thus the instantaneous availability of the new model is formulated as follows:

20

1222221'121

),min(

0

10

21112'21121

),min(

0

22

11

)()()(

)()()()()(~

dsdsYsPstAstA

dsdsYsPstAstAtAtA

st

st

( )

)()()()(

)()()()(

)()()()(

3)(,0)(~

110

21112'21121

0

110

21112'21121

00

110

21112'21121

),min(

00

1

11

11

dHdsdsYsPstAstA

dHdsdsYsPstAstA

tdHdsdsYsPstAstA

tXtXP

st

t

st

st

with time instant approaching infinity, the steady-state availability can be calculated as in (8).

ACKNOWLEDGMENT This research was partially supported by the National

Science Foundation of China under contract number 50775026 and the National Defense Key Foundation of China under contract number 9140A27020308JB34.

( )( ) ( )

[ ]( ) ( ) ( )

76.0

)46065.0*7(24175.0

43sin

31

43cos

32

43sin

31

43cos

32

31

1lim81

)(lim21

)()1()(lim21~

1

1143

43

1

21

211

0

10

22'

1

0

10

22'

1

1

01

1

11

111

1

+=

+

+

+

++=

++++=