Hazard rate function in dynamic environment

Lu XiaoFei a, Liu Min b,n

a JiuQuan Satellite Launch Center, Gansu 732750, Chinab Department of Automation, TNList, Tsinghua University, Beijing 100084, China

a r t i c l e i n f o

Article history:Received 3 June 2013Received in revised form23 April 2014Accepted 27 April 2014Available online 14 May 2014

Keywords:MaintenanceHazard rate functionDynamic environmentMarkov additive processEffective age

a b s t r a c t

The hazard rate function is always applied to make maintenance policy, and the usual hazard ratefunction is computed by the data of failure times of systems working in constant environment, thus forsystems working in dynamic environment it cannot be directly applied. In this paper, hazard ratefunction of system in the dynamic environment is computed, and the effects of current environmentstatus and the environmental history on hazard rate function are explicitly presented. For system withthe known degradation process, hazard rate function is studied by the Markov additive process. Theenvironment evolution process is modeled as a stochastic process with two states, one state representsnormal environment, the other represents severe environment, and system degrades more quicklyunder severe environment than under normal environment. The relationship between hazard ratefunctions of system in time-invariant and dynamic environment is researched, from which threeimportant facts are revealed, firstly hazard rate function jumps as the environment jumps, secondly theform of hazard rate function is determined Wby current environment state, and thirdly the effective ageof system is determined by the environmental history. For system with the unknown degradationprocess, based on the above facts, this paper derives the hazard rate function in dynamic environment,and proposes a method to compute the effective age under given environmental history. Finally theoptimal maintenance policy for system in dynamic environment is studied.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Maintenance is applied to slow the degradation level of thesystem, when making optimal maintenance policy, hazard ratefunction (HRF) is always used [5,8,10,12]. Nakagawa [29] definedthe classic HRF mathematically and proposed several classicalmaintenance policies based on HRF for non-repairable system.Also based on the hazard rate function, many imperfect main-tenance models for repairable system have also been proposed[19,24,32]. Lie and Nakagawa extended HRF by introducing themultiply factor and the effective age adjustment factors [20,27,28].Furthermore, Lin et al. proposed a hybrid HRF model, in whichboth the multiply factor and effective age of the HRF are adjustedafter imperfect preventive maintenance [21,22,34].

However, there is a disadvantage in the above works, that is,both external environment and operational modes of the systemare time-invariant [11,13,39,40]. External environment meanshumidity, temperature, stress, wind speed, radiation of sunlight,shock and so on; operational mode refers to how the system

works, such as continuously works or not, works with full capacityor partial capacity and so on. For simplicity, both externalenvironmental and operational modes are called as the “environ-ment” in this paper. Besides the systems working in time-invariantenvironment, there are many systems working in dynamic envir-onment [25,30]. Lu et al. [23] proposed two examples whichshows the dynamic environment, the first example is the steelrolling machine under stochastic stress; the second example iscargo vehicles under stochastic road conditions. Many articlesconsidered the dynamic environment, but seldom studied theHRF of system in dynamic environment and applied it to makemaintenance policy [13]. Hence, it is necessary to study the HRF ofsystem in dynamic environment.

Proportional hazard rate model (PHM) is often applied todescribe the effects of environment on HRF [2,4,17,18,38]. Cox[7] firstly proposed PHM in the field of biomedicine, and then it isextensively used in field of reliability engineering to deal withdifferent stresses [33]. The classical format of PHM isrðtÞ ¼ r0ðtÞeβðtÞ, in which r0ðtÞ is the baseline HRF, eβðtÞ shows theeffect of external environment βðtÞ on the baseline HRF. Byintroducing healthy condition into HRF, the failure probability ismore accurately computed. Jardine et al. [15] applied a WeibullPHM to model the failure data of aircraft engine and marine gas

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Reliability Engineering and System Safety

http://dx.doi.org/10.1016/j.ress.2014.04.0200951-8320/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail address: lium@tsinghua.edu.cn (L. Min).

Reliability Engineering and System Safety 130 (2014) 50–60

turbine, also applied the PHM to optimize maintenance policy formine haul truck wheel motors [16]. Vlok et al. [36] applied PHM tomodel vibration monitoring data of pumps in coal wash plant.

Besides PHM, the Markov additive process (MAP) is alsoapplied to describe the effect of environment on degradationprocess of the system [35]. MAP was introduced by Cinlar [6],who applied it to model the life length of systems in dynamicenvironment. MAP includes two stochastic processes: XðtÞ and ZðtÞ.XðtÞ represents the degradation level of the system, and ZðtÞrepresents dynamic environment process. XðtÞ is influenced byZðtÞ. Lu et al. [23] proved the discontinuity of the HRF of thesystem in dynamic environment through MAP, where the dynamicenvironment was described by a two-state homogeneous Markovprocess. One state represents normal environment, and the otherrepresents severe environment. Under normal environment theHRF of the system is r0ðtÞ, and under severe environment it isKr0ðtÞ, K41, see Fig. 1. But the assumption that K is constantmeans that the current hazard rate is only determined by thecurrent external environment, and the history of environment hasno effect on it, that is not reasonable. Ghasemi et al. [14]also considered the dynamic environment when making main-tenance policy, and proposed that when the environment jumpedfrom z1 to z2 at time t, the hazard rate keeps the same, whereas

the form of HRF follows r2ðtÞ (see Fig. 2). According to theconclusion of [23], the hazard rate cannot keep the same whenenvironment jumps.

In this paper, the effect of both current environment andenvironmental history on HRF of system in dynamic environmentis researched, and three important facts are found out: when theenvironment jumps the hazard rate function of system also jumps,the form of hazard rate function is determined by currentenvironment status, the effective age of system is determined bythe environmental history.

This paper is organized as follows. In Section 2, the HRF ofsystem in dynamic environment is computed with the Markovadditive process, and the degradation process follows typicalPoisson stochastic processes. In Section 3, the relationshipbetween HRF of system in time-invariant environment and theone in dynamic environment is researched. In Section 4, forconstant HRF of system in time-invariant environment, HRF ofsystem in dynamic environment is computed. In Section 5, basedon the property of estimated HRF from failure time data, thecondition is given, under which HRF of system in dynamicenvironment can be analytical computed. In Section 6, mainte-nance policy of system in dynamic environment is optimizedbased on HRF.

Nomenclature

HRF hazard rate functionMAP Markov additive process

Notations

Xt degradation level of system at time tXthreshold degradation thresholdTf time of failurez1 normal environmentz2 severe environmentr1ðtÞ HRF of system in normal environmentr2ðtÞ HRF of system in severe environment

ðzi; t1Þ-ðzj; t2Þ-⋯ environment evolution process, the envir-onment is zi during ð0; t1�, then changes into zj duringðt1; t1þt2�, and so on

P environmental processH history of external environmenth1ðtjHÞ HRF of system at t, the system is in dynamic environ-

ment with history H, and environment is normal at th2ðtjHÞ HRF of system at t, the system is in dynamic environ-

ment with history H, and environment is severe at tλ1 parameter of Poisson process of system in normal

environmentλ2 parameter of Poisson process of system in severe

environment

Fig. 1. HRF of system in dynamic environment in [23].

L. XiaoFei, L. Min / Reliability Engineering and System Safety 130 (2014) 50–60 51

2. HRF of system in dynamic environment in view of MAP

In this section, HRF of system in dynamic environment isdiscussed in view of MAP. We assume that the system suffersexternal shock, and each shock makes the system suffer constantdamage [9]. Arrivals of these external shocks follow the time-homogeneous Poisson process, and the parameter of the Poissonprocess is determined by environment, in normal environment theparameter is λ1, in severe environment the parameter is λ2 [3].Then at time t the degradation level Xt ¼Nðλ1; t1ÞþNðλ2; t2Þ, wheret1 is the total time of system in normal environment, t2 is the totaltime of system in severe environment, t1þt2 ¼ t, Nðλ1; t1Þ is thenumber of jumps of the Poisson process during t1, Nðλ2; t2Þ isthe number of jumps of Poisson process during t2. If XtZXthreshold,the system fails. The environment process ZðtÞ is a two-statesstochastic process, where z1 represents normal environment andz2 represents severe environment.

Since the environment is dynamic, at time t the externalenvironment may be z1 or z2, which divides the hazard rate at tinto two cases. We firstly compute the hazard rate of system whenZðtÞ ¼ z1. Although ZðtÞ has infinite realization paths, because thereare only two states z1 and z2, these realization paths can besummarized and classified into five scenes.

Scene A: The environment is normal all the time, that isP1 ¼ ðz1;1Þ, for hazard rate at time t, the environment historyH1 ¼ ðz1; tÞ, see Fig. 3.

Scene B: The environment is severe during ð0; tÞ, and changesinto normal at t, that is P2 ¼ ðz2; tÞ-ðz1;1Þ, for hazard rate at time t,the environment history H2 ¼ ðz2; tÞ, see Fig. 4.

Scene C: The environment is normal during ð0; t1Þ, severeduring ½t1; t1þt2Þ, and change into normal at t1þt2, that isP3 ¼ ðz1; t1Þ-ðz2; t2Þ-ðz1;1Þ, for hazard rate at time t, the envir-onment history H3 ¼ ðz1; t1Þ-ðz2; t2Þ, see Fig. 5.

Scene D: The external environment history is P4 ¼ ðz1; t1Þ-ðz2; t2Þ-ðz1; t3Þ-ðz2; t4Þ-ðz1;1Þ, for hazard rate at time t, theenvironment history H4 ¼ ðz1; t1Þ-ðz2; t2Þ-ðz1; t3Þ-ðz2; t4Þ, seeFig. 6.

Scene E: The external environment history is the same withscene D, but we compute the hazard rate at t ¼ t1þt2þt3þt4þt5instead of t ¼ t1þt2þt3þt4, that is P5 ¼ P4, the environmenthistory H5 ¼ ðz1; t1Þ-ðz2; t2Þ-ðz1; t3Þ-ðz2; t4Þ-ðz1; t5Þ, see Fig. 7.

2.1. Compute r1ðtÞ and h1ðtjH2Þ

By the property of the Poisson process, the survival function ofsystem in normal environment is

PðXtoXthresholdÞ ¼ΓðX0 þ1; λ1tÞ

X0!ð1Þ

Fig. 2. HRF of system in dynamic environment in [14].

t

1( )r t

Fig. 3. Environmental process of Scene A.

t

1 2( | )h t H

Fig. 4. Environmental process of Scene B.

1 2t t+

1 3( | )h t H

1t

Fig. 5. Environmental process of Scene C.

2t

1 4( | )h t H

1t

3t 4t

t

Fig. 6. Environmental process of Scene D.

L. XiaoFei, L. Min / Reliability Engineering and System Safety 130 (2014) 50–6052

Here Xthreshold ¼ X0 þ1, ΓðX0 þ1; λ1tÞ ¼R1λ1t

uX0e�udu, and

ddΔt

ΓðX 0 þ1; λ1tþλ1ΔtÞ

¼ ddΔt

Z 1

λ1tþλ1ΔtuX 0

e�udu¼ �λ1ðλ1tþλ1ΔtÞX 0e�λ1t�λ1Δt ð2Þ

By the definition, the HRF at t of scene A is

r1ðtÞ ¼ limΔt-0

PðTf Aðt; tþΔt�jH1ÞPðXtrX0jH1ÞΔt

¼ limΔt-0

PðXtrX0jH1Þ�PðXtþΔtrX0jH1-ðz1;ΔtÞÞPðXtrX 0jH1ÞΔt

¼ limΔt-0

ΓðX0 þ1;λ1tÞ�ΓðX0 þ1; λ1tþλ1ΔtÞΓðX0 þ1; λ1tÞΔt

¼ limΔt-0

�ðd=dΔtÞΓðX0 þ1;λ1tþλ1ΔtÞΓðX0 þ1;λ1tÞ

¼ λ1ðλ1tÞX0e�λ1tR1

λ1tuX 0

e�uduð3Þ

The HRF at t of scene B is

h1ðtjH2Þ ¼ limΔt-0

PðTf A ðt; tþΔt�jH2ÞPðXtrX0jðz2; tÞÞΔt

¼ limΔt-0

PðXtrX0jðz2; tÞÞ�PðXtþΔtrX 0jðz2; tÞ-ðz1;ΔtÞÞPðXtrX0jðz2; tÞÞΔt

¼ limΔt-0

ΓðX0 þ1;λ2tÞ�ΓðX0 þ1; λ2tþλ1ΔtÞΓðX0 þ1; λ2tÞΔt

¼ limΔt-0

� ddΔtΓðX0 þ1; λ2tþλ1ΔtÞ

ΓðX0 þ1; λ2tÞ

¼ λ1ðλ2tÞX0e�λ2tR1

λ2tuX 0

e�uduð4Þ

The ratio of h1ðtjH2Þ and r1ðtÞ is

h1ðtjH2Þr1ðtÞ

¼ λX02 e�λ2t

λX01 e�λ1t

R1λ1t

uX 0e�uduR1

λ2tuX 0

e�uduð5Þ

Fig. 8 shows the ratio for tA ½0:1;5�, when λ1 ¼ 1, λ2 ¼ 2, andX0 ¼ 4. From this figure, although the environment is the same attime t, the HRF at t of two scenes is different in view of the Markovadditive process.

Besides the Poisson degradation process, we give anotherexample to show the effect of environmental history.

Example 2. The degradation process follows Brown Motion with-out drift [37], the survival function is

PðTf rtÞ ¼ 1�2 1�ΦX0

sffiffit

p� �� �

¼ 2ΦX0

sffiffit

p� �

�1

Here ΦðxÞ ¼ ð1=ffiffiffiffiffiffi2π

pÞ R x�1 eð�u2=2Þdu, s is the parameter of Brown

Motion. We assume that s¼s1 when system is in normalenvironment, and s¼s2 when system is in severe environment.

The HRF at t in scene A is

r1ðtÞ ¼ limΔt-0

PðXtrX0jH1Þ�PðXtþΔtrX0jH1-ðz1;ΔtÞÞPðXtrX0jH1ÞΔt

¼ limΔt-0

�2ðd=dΔtÞðPðXtþΔtrX0jH1-ðz1;ΔtÞÞÞPðXtrX0jH1Þ

¼ limΔt-0

�2ðd=dΔtÞðΦððX0=s1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðtþΔtÞ

pÞÞÞ

2ΦððX0=s1ffiffit

p ÞÞ�1

¼ ð1=ffiffiffiffiffiffi2π

pÞðX0=s1Þt�ð3=2Þe�ðX02=2s1

2tÞ=ð2ΦððX0=s1ffiffit

pÞÞ�1Þ ð6Þ

The HRF at t in scene B is

h1ðtjH2Þ ¼ limΔt-0

PðXtrX0jðz2; tÞÞ�PðXtþΔtrX0jðz2; tÞ-ðz1;ΔtÞÞPðXtrX0jðz2; tÞÞΔt

¼ limΔt-0

�2ðd=dΔtÞðΦððX0=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis22tþs2

1Δtq

ÞÞÞ2ΦððX0=s2

ffiffit

p ÞÞ�1

¼ X0ffiffiffiffiffiffi2π

p s21ðs2

2tÞ�3=2e�ðX 02=2s22tÞ=ð2ΦððX0=s2

ffiffit

pÞÞ�1Þ ð7Þ

The ratio of h1ðtjH2Þ and r1ðtÞ ish1ðtjH2Þr1ðtÞ

¼ s31

s32

2ΦððX0=s1ffiffit

p ÞÞ�12ΦððX0=s2

ffiffit

p ÞÞ�1eðX

02=2tÞ ð1=s21Þ� ð1=s2

2Þð Þ ð8Þ

Fig. 9 shows the ratio for tA ½0:1;5�, when s1 ¼ 1, s2 ¼ 2, andX0 ¼ 4.

2.2. Compute h1ðtjH3Þ

Let Φ¼ λ1t1þλ2t2, then

h1ðtjH3Þ ¼ limΔt-0

PðTf A ðt; tþΔt�jH3ÞPðXtrX0jH3ÞΔt

¼ limΔt-0

ΓðX0 þ1;ΦÞ�ΓðX0 þ1;Φþλ1ΔtÞΓðX0 þ1;ΦÞΔt

¼ limΔt-0

RΦþλ1ΔtΦ uX0

e�uduΔtR1Φ uX0

e�udu

¼ λ1ðλ1t1þλ2t2ÞX0e�λ1t1 �λ2t2R1

λ1t1 þλ2t2uX0

e�uduð9Þ

Substitute t ¼ t1þt2 into Eq. (3),

r1ðt1þt2Þ ¼λ1ðλ1t1þλ1t2ÞX

0e�λ1t1 �λ1t2R1

λ1t1 þλ1t2uX 0

e�uduð10Þ

Fig. 8. h1ðtjH2Þ=r1ðtÞ of Example 1.

2t

1 5( | )h t H

1t

3t 4t

t

5t

Fig. 7. Environmental process of Scene E.

L. XiaoFei, L. Min / Reliability Engineering and System Safety 130 (2014) 50–60 53

The ratio of h1ðtjH3Þ and r1ðtÞ at t ¼ t1þt2 is

h1ðtjH3Þr1ðtÞ

¼ ðλ1=λ2t1þt2ÞX0

ðt1þt2ÞX0

λ2X 0e�λ2t2

λ1X 0e�λ1t2

R1λ1ðt1 þ t2Þ u

X0e�uduR1

λ2ðλ1=λ2 U t1 þ t2Þ uX0e�udu

ð11Þ

Fig. 10 shows the ratio versus t2 for λ1 ¼ 1, λ2 ¼ 2, X0 ¼ 4 andt1 ¼ 0:5, t1 ¼ 1, t1 ¼ 2. The smaller t1, the larger h1ðtjH3Þ=r1ðtÞ forthe same t2.

2.3. Compute h1ðtjH4Þ

Let Φ¼ λ1t1þλ2t2þλ1t3þλ2t4,

h1ðtjH4Þ ¼ limΔt-0

PðTf A ðt; tþΔt�jH4ÞPðXtrX0jH4ÞΔt

¼ limΔt-0

RΦþλ1ΔtΦ uX0

e�uduΔtR1Φ uX0

e�udu

¼ λ1ðλ1t1þλ2t2þλ1t3þλ2t4ÞX0e�λ1t1 �λ2t2 �λ1t3 �λ2t4R1

λ1t1 þλ2t2 þλ1t3 þλ2t4uX0

e�uduð12Þ

2.4. Compute h1ðtjH5Þ

Let Φ¼ λ1t1þλ2t2þλ1t3þλ2t4þλ1t5,

h1ðtjH5Þ ¼ limΔt-0

PðTf Aðt; tþΔt�jH5ÞPðXtrX 0jH5ÞΔt

¼ limΔt-0

RΦþλ1ΔtΦ uX 0

e�uduΔtR1Φ uX0

e�udu

¼ λ1ðλ1t1þλ2t2þλ1t3þλ2t4þλ1t5ÞX0e�λ1t1 �λ2t2 �λ1t3 �λ2t4 �λ1t5R1

λ1t1 þλ2t2 þλ1t3 þλ2t4 þλ1t5uX 0

e�udu

ð13Þ

By Eqs. (4), (9), (12) and (13), HRF of system in dynamicenvironment can be summarized as for ZðtÞ ¼ z1

h1ðtjHÞ ¼λ1Φ

X0e�ΦR1

Φ uX0e�udu

ð14Þ

By Eq. (14), the HRF of system in dynamic environment whenZðtÞ ¼ z2

h2ðtjHÞ ¼λ2Φ

X0e�ΦR1

Φ uX0e�udu

ð15Þ

Here Φ¼ λ1τ1þλ2τ2, τ1 is the total time of normal environment,τ2 is the total time of severe environment, τ1þτ2 ¼ t.

2.5. HRF of system when external shocks follow non-homogeneousPoisson process

Let λ1ðtÞ and λ2ðtÞ be the intensity functions of non-homogeneous function of the external shock process, andmiðtÞ ¼

R t0 λiðuÞdu, miðt1; t2Þ ¼

R t2t1λiðuÞdu, i¼ 1;2. For system

always in normal or severe environment, the probability of nshocks arrive during ½s; sþt� is

PðNðsþtÞ�NðsÞ ¼ njziÞ ¼ðmiðsþtÞ�miðsÞÞn

n!exp �ðmiðsþtÞ�miðsÞÞ

� �ð16Þ

Since mið0Þ ¼ 0, Eq. (1) also keeps for the non-homogeneousPoisson process, for system always in normal or severe environ-ment, the survival function is

PðXtoXthresholdÞ ¼ΓðX0 þ1;miðtÞÞ

X0!ð17Þ

Here X0 ¼ Xthreshold�1, ΓðX0 þ1;miðtÞÞ ¼R1miðtÞ u

X0e�udu and

ddΔt

ΓðX0 þ1;miðtþΔtÞÞ ¼ ddΔt

Z 1

miðtþΔtÞuX0

e�udu

¼ �λ1ðtþΔtÞmiðtþΔtÞX0e�miðtþΔtÞ

Let τi ¼∑ij ¼ 1tj, then

r1ðtÞ ¼λ1ðtÞðm1ðtÞÞX

0e�m1ðtÞR1

m1ðtÞ uX0e�udu

ð18Þ

h1ðtjH2Þ ¼λ1ðtÞðm2ðtÞÞX

0e�m2ðtÞR1

m2ðtÞ uX 0e�udu

ð19Þ

h1ðtjH3Þ ¼λ1ðm1ðτ1Þþm2ðτ1; τ2ÞÞX

0e�m1ðτ1Þ�m2ðτ1 ;τ2ÞR1

m1ðτ1Þþm2ðτ1 ;τ2Þ uX0e�udu

ð20Þ

It is obvious that Eqs. (18)–(22) are similar to Eqs. (3), (4), (9),(12) and (13). HRF of system in dynamic environment can be

Fig. 10. External environment of scene C.

Fig. 9. h1ðtjH2Þ=r1ðtÞ of Example 2.

L. XiaoFei, L. Min / Reliability Engineering and System Safety 130 (2014) 50–6054

summarized as for ZðtÞ ¼ zi

hiðtjHÞ ¼λiΦ

X0e�ΦR1

Φ uX0e�udu

ð23Þ

Here Φ¼ R t0 λðujzðuÞÞdu, when zðtÞ ¼ z1, λðtjzðtÞÞ ¼ λ1ðtÞ; when

zðtÞ ¼ z2, λðtjzðtÞÞ ¼ λ2ðtÞ. When λ1ðtÞ ¼ λ1, λ2ðtÞ ¼ λ2, Eq. (23) isEq. (14) or (15), that is to say, HRF of system suffering externalshocks which follow the homogeneous Poisson process is thespecial case of HRF of system suffering external shocks whichfollow the non-homogeneous Poisson process. For simplicity, weassume the Poisson process is homogenous in the followingsections.

3. Relationship between r1ðtÞ and h1ðtjHÞ

By Eqs. (3) and (14), r1ðtÞ has the same format as h1ðtjHÞ,

r1ðtÞ ¼λ1ðλ1tÞX

0e�λ1tR1

λ1tuX0

e�udu

h1ðtjHÞ ¼λ1Φ

X 0e�ΦR1

Φ uX0e�udu

Since

Φ¼ λ1τ1þλ2τ2 ¼ λ1 τ1þλ2λ1τ2

� �ð24Þ

Let A1 ¼ τ1þλ2λ1τ2, h1ðtjHÞ can be rewritten as

h1ðtjHÞ ¼λ1Φ

X 0e�ΦR1

Φ uX0e�udu

¼ λ1ðλ1A1ÞX0e�λ1A1R1

λ1A1uX 0

e�udu¼ r1ðA1Þ ð25Þ

Then h1ðtjHÞ can be constructed based on r1ðtÞ by adjust thevirtual age, that is to say the HRF of system in dynamic environ-ment can be computed by HRF of system in constant environmentby adjusting the virtual age. Since λ24λ1, then A14τ1þτ2, whichmeans the virtual age of system in dynamic environment is largerthan the one in normal environment.

Such as the environmental process is P ¼ z1; t1ð Þ- z2; t2ð Þ-z1; t3ð Þ- z2; t4ð Þ- z1;1ð Þ, let τ1 ¼ t1þt3þu, τ2 ¼ t2þt4. Then forhazard rate of system at t ¼ τ1þτ2, the virtual age of system innormal environment is τ1þτ2, whereas the one in the dynamicenvironment process is τ1þðλ2=λ1Þτ2, see Fig. 11.

Let A2 ¼ ðλ1=λ2Þτ1þτ2, h2ðtjHÞ can be rewritten as

h2ðtjHÞ ¼λ2Φ

X 0e�ΦR1

Φ uX0e�udu

¼ λ2ðλ2A2ÞX0e�λ2A2R1

λ2A2uX 0

e�udu¼ r2ðA2Þ ð26Þ

Since λ1oλ2, then A2oτ1þτ2, which means the HRF of systemin dynamic environment has less virtual age than HRF in severeenvironment. Fig. 12 shows h2ðt1þt2þt3þt4þ⋯jHÞ.

With Eqs. (25) and (26), Fig. 13 shows the HRF of system indynamic environment.

4. Derivation of HRF in dynamic environment from constantHRF

Since the degradation process is difficult to obtain, the failuremechanism of system is always made by life test, with which theHRF of system in determined environment is estimated by largeamount of failure time data. Under environment s, M systems runto failure, these failure times are recorded as xi, 1r irM, then the

Fig. 11. h1ðtjHÞ and X1.

Fig. 12. h2ðtjHÞ and X2ðHÞ.

h1ðtjH4Þ ¼λ1ðm1ðτ1Þþm2ðτ1; τ2Þþm1ðτ2; τ3Þþm2ðτ3; τ4ÞÞX

0e�m1ðτ1Þ�m2ðτ1 ;τ2Þ�m1ðτ2 ;τ3Þ�m2ðτ3 ;τ4ÞR1

m1ðτ1Þþm2ðτ1 ;τ2Þþm1ðτ2 ;τ3Þþm2ðτ3 ;τ4Þ uX0e�udu

ð21Þ

h1ðtjH5Þ ¼λ1ðm1ðτ1Þþm2ðτ1; τ2Þþm1ðτ2; τ3Þþm2ðτ3; τ4Þþm1ðτ4; τ5ÞÞX

0e�m1ðτ1Þ�m2ðτ1 ;τ2Þ�m1ðτ2 ;τ3Þ�m2ðτ3 ;τ4Þ�m1ðτ4 ;τ5ÞR1

m1ðτ1Þþm2ðτ1 ;τ2Þþm1ðτ2 ;τ3Þþm2ðτ3 ;τ4Þþm1ðτ4 ;τ5Þ uX0e�udu

ð22Þ

L. XiaoFei, L. Min / Reliability Engineering and System Safety 130 (2014) 50–60 55

HRF under s is computed as

rsðtÞ ¼ limΔt-0

Nsðt; tþΔtÞðM�Nsð0; tÞÞΔt

ð27Þ

Here Nsðt1; t2Þ represents the number of systems failing duringðt1; t2Þ in environments.

In the following section, the HRF of system in dynamicenvironment is derived from the HRF of system in time-invariantenvironment. Let RiðtÞ be the survival function of system in time-invariant environment, RdðtjHÞ be the survival function of systemin dynamic environment.

First we consider one simple case, the HRF of system in time-invariant environment is constant, such as in normal environment,r1ðtÞ ¼ r1; in severe environment, r2ðtÞ ¼ r2, then survival functionsfollows exponential distribution,

R1ðtÞ ¼ e� r1t ; R2ðtÞ ¼ e� r2t ð28Þ

According to Eq. (28), the failure mechanism can be explainedby a special Markov additive process. Suppose there is a systemwhich suffers external catastrophic shock, that is to say, the systemfails when the first shock arrives, thus Xthreshold ¼ 1, X0 ¼ 0 and thearrive of the catastrophic shock follows the time-homogeneousPoisson process, since the pdf of first jump of Poisson Process attime t also follows exponential distribution, the survival functionof system is the same with Eq. (2). Thus the conclusion in Section 2can be applied to system with constant HRF in determinedenvironment.

For H¼ ðz1; t1Þ⋯ðz2; tnÞ-ðz1; τÞ, the hazard rate at t ¼ t1þ⋯þtnþτ is

h1 tjHð Þ ¼ λ1ðΦþλ1τÞX0e�Φ�λ1τR1

Φþλ1τuX0

e�udu¼ λ1e�Φ�λ1τR1

Φþλ1τe�udu

¼ λ1e�Φ�λ1τ

e�Φ�λ1τ¼ λ1:

For H¼ ðz1; t1Þ⋯ðz1; tnÞ-ðz2; τÞ, the hazard rate at t ¼ t1þ⋯þtnþτ is

h2ðtjHÞ ¼λ2ðΦþλ2τÞX

0e�Φ�λ2τR1

Φþλ2τuX0

e�udu¼ λ2e�Φ�λ2τR1

Φþλ2τe�udu

¼ λ2e�Φ�λ2τ

e�Φ�λ2τ¼ λ2

Fig. 14 shows the HRF of system in dynamic environment whenr1ðtÞ ¼ r1, r2ðtÞ ¼ r2.

5. Derivation of HRF in dynamic environment from HRF intime-invariant environment

In this section, we consider one general case that the HRFs ofsystem in time-invariant environment are not constant, such asthe Weibull HRF r1ðtÞ ¼ λ1tα , r2ðtÞ ¼ λ2tα [1].

Suppose there are M the same type of systems working indynamic environment, the HRF is also estimated by Eqn.(27),rdðtÞ ¼ lim

Δt-0ðNdðt; tþΔtÞ=ðM�Ndð0; tÞÞΔtÞ, whereas Ndðt1; t2Þ

represents the number of systems failing during ðt1; t2Þ in dynamicenvironment. At t, if ZðtÞ ¼ z1, then

h1ðtjHÞ ¼ limΔt-0

Ndðt; tþΔtÞðM�Ndð0; tÞÞΔt

¼ limΔt-0

Nz1 ðt; tþΔtÞðM�Ndð0; tÞÞΔt

ð29Þ

Since the system fails more quickly in severe environment thanin normal environment, Ndð0; tÞ4Nz1 ð0; tÞ, which means thereexists one virtual age A1 such that Ndð0; tÞ4Nz1 ð0;A1Þ, then

h1ðtjHÞ ¼ limΔt-0

Nz1 ðt; tþΔtÞðM�Ndð0; tÞÞΔt

¼ limΔt-0

Nz1 ðt; tþΔtÞðM�Nz1 ð0;A1ÞÞΔt

ð30Þ

Take the Weibull HRF as example, the environmental history isH¼ ðz1; t1Þ⋯ðz2; tnÞ-ðz1; τÞ, then at time t ¼ t1þt2þ⋯tn,

h1ðtjHÞ ¼ λ1ðA1ðHÞþτÞα

The same with ZðtÞ ¼ z1, for H ¼ ðz1; t1Þ⋯ðz1; tnÞ-ðz2; τÞ,h2ðtjHÞ ¼ λ2ðA2ðHÞþτÞα

Here A1ðHÞ and A2ðHÞ are determined by environmental history.If the survival function can be written as RiðtÞ ¼ϕðgiðzi; tÞÞ, and

giðzi; tÞhas the following properties:

(1) giðzi; tÞ ¼ giðzi;XþtÞ(2) giðzi; t1Þþgiðzi; t2Þ ¼ giðzi; t1þt2Þ

Then the survival function of system in dynamic environmentcan be computed by summing contribution of each constantenvironment section, that is

RdðtjHÞ ¼ϕð∑ig1ðz1; tiÞþ∑

jg2ðz2; tjÞÞ ¼ϕ g1ðz1; τ1Þþg2ðz1; τ2Þ

� �ð31Þ

For H¼ ðz1; t1Þ⋯ðz2; tnÞ-ðz1; τÞ, let YH ¼ g1ðz1; τ1Þþg2ðz1; τ2Þ,then the hazard rate at t ¼ t1þt2þ⋯þτ is

h1ðtjHÞ ¼ limΔt-0

RdðtjHÞ�RdðtþΔtjH-ðz1;ΔtÞÞRdðtjHÞΔt

¼ limΔt-0

ϕðYHÞ�ϕðYHþg1ðz1;ΔtÞÞϕðYHÞΔt

¼ dϕðuÞdu

u ¼ YH

dg1ðz1;uÞdu

u ¼ 0

1ϕðYHÞ

Fig. 13. HRF of system in dynamic environment.

Fig. 14. HRF of system in dynamic environment when r1ðtÞ ¼ r1, r2ðtÞ ¼ r2.

L. XiaoFei, L. Min / Reliability Engineering and System Safety 130 (2014) 50–6056

¼ d ln ϕðuÞdu

u ¼ YH

dg1ðz1;uÞdu

u ¼ 0

ð32Þ

For H ¼ ðz1; t1Þ⋯ðz1; tnÞ-ðz2; τÞ, the hazard rate at t ¼ t1þt2þ⋯þτ is

h2ðtjHÞ ¼ limΔt-0

RdðtjHÞ�RdðtþΔtjH-ðz2;ΔtÞÞRdðtjHÞΔt

¼ limΔt-0

ϕðYHÞ�ϕðYHþg2ðλ2;ΔtÞÞϕðYHÞΔt

¼ dϕðuÞdu u ¼ YH

dg2ðλ2;uÞdu u ¼ 0

1ϕðYHÞ

¼ d ln ϕðuÞdu u ¼ YH

dg2ðλ2;uÞdu

u ¼ 0

ð33Þ

The following example is given to show the HRF of system indynamic environment by HRF of system in determinedenvironment.

Example 2. r1ðtÞ ¼ 2λ1t, r2ðtÞ ¼ 2λ2t.The survival functions of system at normal and severe environ-

ment are

R1ðtÞ ¼ e�λ1t2 ¼ e�ðffiffiffiffiλ1

ptÞ2

R2ðtÞ ¼ e�λ2t2 ¼ e�ðffiffiffiffiλ2

ptÞ2

8<: ð34Þ

By Eq. (31), ϕðuÞ ¼ e�u2 ,ffiffiffiffiλi

pt can be considered as the

contribution of section of ðzi; tÞ to survival function. Thus thecontribution of ðzi; tÞ with arbitrary initial time to the survivalfunction of system is the same, for H ¼ ðz1; t1Þ⋯ðz2; tnÞ-ðz1; τÞ,

YH ¼ffiffiffiffiffiλ1

pt1þ

ffiffiffiffiffiλ2

pt2þ⋯

ffiffiffiffiffiλ2

ptnþ

ffiffiffiffiffiλ1

pτ

The survival function at t ¼∑itiþτ is

RdðtjHÞ ¼ e�ffiffiffiffiλ1

pt1 þ

ffiffiffiffiλ2

pt2 þ⋯

ffiffiffiffiλ2

ptn þ

ffiffiffiffiλ1

pτ

� �2ð35Þ

The HRF at t ¼ t1þt2þ⋯þτ is

h1ðtjHÞ ¼ limΔt-0

RdðtjHÞ�RdðtþΔtjH-ðz1;ΔtÞÞRdðtjHÞΔt

¼ limΔt-0

e�Y2H �e�ðYH þ

ffiffiffiffiλ1

pΔtÞ2

e�Y2HΔt

¼ limΔt-0

2ðYHþffiffiffiffiffiλ1

pΔtÞ

ffiffiffiffiffiλ1

pe�ðYH þ

ffiffiffiffiλ1

pΔtÞ2

e�YH2

¼ 2YH

ffiffiffiffiffiλ1

p¼ 2

ffiffiffiffiffiλ1

pðffiffiffiffiffiλ1

pt1þ

ffiffiffiffiffiλ2

pt2þ⋯

ffiffiffiffiffiλ2

ptnþ

ffiffiffiffiffiλ1

pτÞ

¼ 2λ1 t1þffiffiffiffiffiλ2λ1

st2þ⋯

ffiffiffiffiffiλ2λ1

stnþτ

0@

1A ð36Þ

Since r1ðtÞ ¼ 2λ1t, h1ðtjHÞ has the same form with r1ðtÞ, andh1ðtjHÞ ¼ r1ðX1Þ ð37Þ

X1 ¼ t1þffiffiffiffiffiλ2λ1

st2þ⋯

ffiffiffiffiffiλ2λ1

stnþτ ð38Þ

For H ¼ ðλ1; t1Þ⋯ðλ1; tnÞ-ðλ2; τÞ, the hazard rate atτ¼ t1þt2þ⋯þt is

h2ðtjHÞ ¼ limΔt-0

RdðtjHÞ�RdðtþΔtjH-ðz2;ΔtÞÞRdðtjHÞΔt

¼ limΔt-0

e�Y2H �e�ðYH þ

ffiffiffiffiλ2

pΔtÞ2

e�Y2HΔt

¼ 2YH

ffiffiffiffiffiλ2

p¼ 2

ffiffiffiffiffiλ2

pðffiffiffiffiffiλ1

pt1þ

ffiffiffiffiffiλ2

pt2þ⋯

ffiffiffiffiffiλ1

ptnþ

ffiffiffiffiffiλ2

pτÞ

¼ 2λ2

ffiffiffiffiffiλ1λ2

st1þt2þ⋯

ffiffiffiffiffiλ1λ2

stnþτ

0@

1A ð39Þ

Since r2ðtÞ ¼ 2λ2t, h2ðtjHÞ has the same form with r2ðtÞ, andh2ðtjHÞ ¼ r2ðX2Þ ð40Þ

X2 ¼ffiffiffiffiffiλ1λ2

st1þt2þ⋯

ffiffiffiffiffiλ1λ2

stnþτ ð41Þ

By Eqs. (37) and (40), the HRF of system in dynamic environ-ment has the similar form as Fig. 11 shows.

Example 3. r1ðtÞ ¼ 6λ1ðtþt2Þ; r2ðtÞ ¼ 6λ2ðtþt2Þ.The survival functions of system at normal and severe environ-

ment are

R1ðtÞ ¼ e�3λ1t2 �2λ1t3 ¼ e�3ðffiffiffiffiλ1

ptÞ2 �3ð

ffiffiffiffiλ1

3p

tÞ3

R2ðtÞ ¼ e�3λ2t2 �2λ2t3 ¼ e�2ðffiffiffiffiλ2

ptÞ2 �2ð

ffiffiffiffiλ2

3p

tÞ3:

(ð42Þ

By Eq. (40), ϕðu; vÞ ¼ e�3u2e�2v3 ,ffiffiffiffiffiλ1

pt and

ffiffiffiffiffiλ2

pt can be con-

sidered as the contribution of section of ðz1; tÞ and ðz2; tÞ to e�3u2 ;ffiffiffiffiffiλ1

3p

t andffiffiffiffiffiλ2

3p

t can be considered as the contribution of section ofðz1; tÞ and ðz2; tÞ to e�2v3 , thus the contribution of ðzi; tÞ witharbitrary initial time to the survival function of system is thesame, and the contribution of dynamic environment is dividedinto two parts, Y1H and Y2H .

For H ¼ ðz1; t1Þ⋯ðz2; tnÞ-ðz1; τÞ,Y1H ¼

ffiffiffiffiffiλ1

pt1þ

ffiffiffiffiffiλ2

pt2þ⋯

ffiffiffiffiffiλ2

ptnþ

ffiffiffiffiffiλ1

pτ

Y2H ¼ffiffiffiffiffiλ1

3p

t1þffiffiffiffiffiλ2

3p

t2þ⋯ffiffiffiffiffiλ2

3p

tnþffiffiffiffiffiλ1

3p

τ:

(ð43Þ

The survival function at t ¼∑itiþτ is

RðtjHÞ ¼ e�3ðY1H Þ2e�2ðY2H Þ3 ð44Þ

The HRF at t ¼ t1þt2þ⋯þτ is

h1ðtjHÞ ¼ limΔt-0

PðXtoX0jHÞ�PðXtþΔtoX0jH-ðz1;ΔtÞÞPðXtoX0jHÞΔt

¼ limΔt-0

e�3ðY1H Þ2e�2ðY2H Þ3 �e�3ðY1H þffiffiffiffiλ1

pΔtÞ2e�2ðY2H þ

ffiffiffiffiλ1

3p

ΔtÞ3

e�3ðY1H Þ2e�2ðY2H Þ3Δt

¼ limΔt-0

6ðffiffiffiffiffiλ1

pðY1Hþ

ffiffiffiffiffiλ1

pΔtÞþ

ffiffiffiffiffiλ13

pðY2Hþ

ffiffiffiffiffiλ13

pΔtÞ2Þe�3ðY1H þ

ffiffiffiffiλ1

pΔtÞ2e�2ðY2H þ

ffiffiffiffiλ1

3p

ΔtÞ3

e�3ðY1H Þ2e�2ðY2H Þ3

¼ 6ffiffiffiffiffiλ1

pY1Hþ6

ffiffiffiffiffiλ1

3p

ðY2HÞ2

¼ 6λ1 t1þffiffiffiffiffiλ2λ1

st2þ⋯þτ

0@

1Aþ6λ1 t1þ

ffiffiffiffiffiλ2λ1

3

st2þ⋯þτ

!20@ ð45Þ

Since r1ðtÞ ¼ 6λ1ðtþt2Þ, h1ðtjHÞ has the same form with r1ðtÞ.For H ¼ ðz1; t1Þ⋯ðz1; tnÞ-ðz2; τÞ, the HR at τ¼ t1þt2þ⋯þt is

h2ðtjHÞ ¼ limΔt-0

PðXtoX0jHÞ�PðXtþΔtoX0jH-ðz2;ΔtÞÞPðXtoX0jHÞΔt

¼ limΔt-0

e�3ðY1H Þ2e�2ðY2H Þ3 �e�3ðY1H þffiffiffiffiλ2

pΔtÞ2e�2ðY2H þ

ffiffiffiffiλ2

3p

ΔtÞ3

e�3ðY1H Þ2e�2ðY2H Þ3Δt

¼ limΔt-0

6ðffiffiffiffiffiλ2

pðY1Hþ

ffiffiffiffiffiλ2

pΔtÞþ

ffiffiffiffiffiλ23

pðY2Hþ

ffiffiffiffiffiλ23

pΔtÞ2Þe�3ðY1H þ

ffiffiffiffiλ2

pΔtÞ2e�2ðY2H þ

ffiffiffiffiλ2

3p

ΔtÞ3

e�3ðY1H Þ2e�2ðY2H Þ3

¼ 6ffiffiffiffiffiλ2

pY1Hþ6

ffiffiffiffiffiλ2

3p

Y2Hð Þ2

¼ 6λ2

ffiffiffiffiffiλ1λ2

st1þt2þ⋯þτ

0@

1Aþ6λ2

ffiffiffiffiffiλ1λ2

3

st1þt2þ⋯þτ

!20@ ð46Þ

Since r2ðtÞ ¼ 2λ2t, h2ðtjHÞhas the same form with r2ðtÞ.

L. XiaoFei, L. Min / Reliability Engineering and System Safety 130 (2014) 50–60 57

By Eqns. (43) and (44), the HRF of system in dynamic environ-ment has the similar form as Fig. 11 shows.

The HRF of system in constant environment and the one indynamic environment can be integrated into one function r t; τ; zð Þ,where t is the operation time, τ is the total time of severeenvironment, z is the condition of environment at t. Then

rðt;0;0Þ ¼ r1ðtÞrðt;0;1Þ ¼ r2ðtÞ

rðt; τ;0Þ ¼ h1ðtjHÞ;H ¼ ðz2; τÞ-ðz1; t�τÞrðt; τ;1Þ ¼ h2ðtjHÞ;H ¼ ðz1; t�τÞ-ðz2; τÞ

8>>>><>>>>:

ð47Þ

Remark 1. When the contributions of ðzi; tÞ with different initialtimes to survival function are not the same, according to thedefinition of HRF Eq. (27), the HRF of system in dynamic environ-ment also jumps between r1ðtÞ and r2ðtÞ when the externalenvironment changes, except it is difficult to determine the virtualage of system.

6. Optimal maintenance policy of system in dynamicenvironment

The system is under optimal preventive maintenance, at time Tthe system is preventively perfect maintained with cost cr , and thesystem is minimal repaired with cost cm if failed before T, theexternal environment follows a two-state homogeneous Markovprocess. The transition rate matrix of this continuous-time Markov

process is Q ¼�λ λμ �μ

" #, where ð1=λÞ; ð1=μÞare expected

sojourn time in normal and severe state respectively. Then,

s-expected cost during ½0; T � is crþcmR T0 hðtÞdt, where hðtÞ is the

hazard rate function of system in dynamic environment at time t.Then, the s-expected cost rate is

crþcmR T0 hðtÞdtT

ð48Þ

The derivative of Eq. (48) with respect to T is

ddT

crþcmR T0 hðtÞdtT

!¼ cmhðTÞT�cr�cm

R T0 hðtÞdt

T2 ð49Þ

Since the optimal preventive maintenance period T means theminimal s-expected cost rate, then the numerator of Eq. (48)should be zeros, that is to say

hðTÞT�Z T

0hðtÞdt ¼ cr

cmð50Þ

Since the external environment is stochastic, Zð0Þ may benormal or severe. For a two-state homogeneous Markov process,the stationary distribution ½p1; p2� exists, where p1 and p2 are theprobabilities of Zð0Þ ¼ z1 and Zð0Þ ¼ z2, respectively, p1 ¼ ðμ=μþλÞ,p2 ¼ ðλ=μþλÞ. At time t, toT , ZðtÞ may also be normal or severe.Let f ijðt; xÞ be the probability density function that Zð0Þ ¼ zi,ZðtÞ ¼ zj, and x is the total time of environment is normal during½0; t�, then

hðtÞ ¼ p1

Z t

0r1 xþλ2

λ1ðt�xÞ

� �f 11ðt; xÞdx

þp1

Z t

0r2

λ1λ2xþðt�xÞ

� �f 12ðt; xÞdx

þp2

Z t

0r1 xþλ2

λ1ðt�xÞ

� �f 21ðt; xÞdx

þp2

Z t

0r2

λ1λ2xþðt�xÞ

� �f 22ðt; xÞdx

¼Z t

0r1 xþλ2

λ1ðt�xÞ

� �ðp1f 11ðt; xÞþp2f 21ðt; xÞÞdx

þZ t

0r2

λ1λ2xþðt�xÞ

� �ðp1f 12ðt; xÞþp2f 22ðt; xÞÞdx ð51Þ

Mckinlay [26] studied the distribution of time spent in normalstate during time interval ½0; t�, for 0rxrt,

f ðt; xÞ ¼ f½p1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμx=ðt�xÞ

qþp2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμðt�xÞ=x

q�I1ð2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμxðt�xÞ

qÞ

þðp1λþp2μÞI0ð2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμxðt�xÞ

qÞ

þp1δðt�xÞþp2δðxÞge�λx�μðt� xÞ ð52ÞHere IrðzÞ ¼∑1

k ¼ 0ðz=2Þ2kþ r

k!ðkþ rÞ! is the modified Bessel function of order rand

f 1ðt; xÞ ¼ fδðt�xÞþλI0ð2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμxðt�xÞ

pÞþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμx=ðt�xÞ

pI1ð2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμxðt�xÞ

pÞge�λx�μðt�xÞ

f 2ðt; xÞ ¼ fδðxÞþμI0ð2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμxðt�xÞ

pÞþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμðt�xÞ=x

pI1ð2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμxðt�xÞ

pÞge�λx�μðt� xÞ

By analysis, we found

f 11ðt; xÞ ¼ δðt�xÞe�λtþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμx=ðt�xÞ

qI1ð2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμxðt�xÞ

qÞe�λx�μðt�xÞ

ð53Þ

f 21ðt; xÞ ¼ μI0ð2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμxðt�xÞ

qÞe�λx�μðt� xÞ ð54Þ

f 12ðt; xÞ ¼ λI0ð2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμxðt�xÞ

qÞe�λx�μðt� xÞ ð55Þ

f 22ðt; xÞ ¼ δðxÞe�μtþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμðt�xÞ=x

qI1ð2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμxðt�xÞ

qÞe�λx�μðt�xÞ ð56Þ

hðtÞ ¼ μμþλ

e�λtr1ðtÞþλ

μþλe�μtr2ðtÞ

þZ t

0r1 xþ

ffiffiffiffiffiλ2λ1

sðt�xÞ

0@

1A λμμþλ

I0ð2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμxðt�xÞ

qÞe�λx�μðt� xÞdx

þZ t

0r2

ffiffiffiffiffiλ1λ2

sxþðt�xÞ

0@

1A λμμþλ

I0ð2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμxðt�xÞ

qÞe�λx�μðt� xÞdx

þZ t

0r1 xþ

ffiffiffiffiffiλ2λ1

sðt�xÞ

0@

1Aμ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

λμx=ðt�xÞp

μþλI1

ð2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμxðt�xÞ

qÞe�λx�μðt�xÞdx

þZ t

0r2

ffiffiffiffiffiλ1λ2

sxþðt�xÞ

0@

1Aλ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

λμðt�xÞ=xp

μþλ

I1ð2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλμxðt�xÞ

qÞe�λx�μðt�xÞdx ð57Þ

For simplicity, Let λ¼ μ¼ 1, also substitute r1ðtÞ ¼ t, r2ðtÞ ¼ 4t,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðλ2=λ1Þ

p¼ 2,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðλ1=λ2Þ

p¼ 1=2 into hðtÞ, we obtain

hðtÞ ¼ 12e� t tþ1

2e� t4t

þZ t

0ðxþ2ðt�xÞÞ1

2I0ð2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðt�xÞ

pÞe� tdx

þZ t

04

12xþðt�xÞ

� �12I0ð2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðt�xÞ

pÞe� tdx

þZ t

0ðxþ2ðt�xÞÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix=ðt�xÞ

p2

I1ð2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðt�xÞ

pÞe� tdx

þZ t

04

12xþðt�xÞ

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt�xÞ=x

p2

I1ð2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðt�xÞ

pÞe� tdx

L. XiaoFei, L. Min / Reliability Engineering and System Safety 130 (2014) 50–6058

¼ e� t

25tþ

Z t

0ð6t�3xÞI0ð2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðt�xÞ

pÞdx

þZ t

0ðð2t�xÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix=ðt�xÞ

pþð4t�2xÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt�xÞ=x

pÞI1ð2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðt�xÞ

pÞdx�

ð58Þ

With Eq. (58), the s-expected cost versus T is computed andplotted in Fig. 15, the optimal preventive interval is T ¼ 2:1, whichcorresponds to minimal s-expected cost rate. We also computedthe s-expected cost rate by Monte Carlo Simulation (MCS), Fig. 16shows the flow chart, and the results is also plotted in Fig. 15. Thes-expected cost rate computed by Eq. (48) and Monte CarloSimulation are the same.

7. Conclusions

In this paper, the HRF of system in dynamic environment iscomputed. By analyzing the relationship of HRF of system in time-invariant and the one in dynamic environment, three facts arefound out, firstly the HRF of system jumps when the externalenvironment changes, secondly the HRF form is determined bycurrent environment condition, and thirdly the effective age isdetermined by environmental history. For system which hasconstant HRF in time-invariant environment, the HRF is computedby considering degradation process as a special Markov additiveprocess. For system with unknown degradation process, HRF ofsystem in dynamic environment has the same property with HRFresearched by Markov additive process, but the effective age isdifficult to be computed when external environment changes.Section 4 proposes the conditions under which the effective agecan be analytically computed. Finally maintenance policy forsystem in dynamic environment is optimized based on HRFstudied in this paper.

Acknowledgments

This work is partially supported by the National Natural ScienceFoundation of China (Nos. 61025018 and 60834004), the NationalKey Basic Research and Development Program of China (No.2009CB320602), the National Science and Technology MajorProject of China (No. 2011ZX02504-008).

References

[1] Almalki SJ, Nadarajah S. Modifications of the Weibull distribution: a review.Reliab Eng Syst Saf 2014;124:32–55.

[2] Barker K, Baroud H. Proportional hazards models of infrastructure systemrecovery. Reliab Eng Syst Saf 2014;124:201–6.

[3] Bjerga T, Aven T, Zio E. An illustration of the use of an approach for treatingmodel uncertainties in risk assessment. Reliab Eng Syst Saf 2014;125:46–53.

[4] Bris R, Medonos S, Wilkins C, Zdráhala A. Time-dependent risk modeling ofaccidental events and responses in process industries. Reliab Eng Syst Saf2014;125:54–66.

[5] Cha JH, Finkelstein M. The failure rate dynamics in heterogeneous populations.Reliab Eng Syst Saf 2013;112:120–8.

[6] Çinlar E. Markov additiveprocesses II. Z Wahrscheinlichkeitstheorie Verw Geb1972;24:95–121.

Fig. 15. s-Expected cost rate of system in dynamic environment computed byEq. (40) and Monte Carlo simulation.

Fig. 16. Flow chart of computing s-expected cost rate of system in dynamicenvironment.

L. XiaoFei, L. Min / Reliability Engineering and System Safety 130 (2014) 50–60 59

[7] Cox DR. Regression models and life tables. Journal of the Royal StatisticalSociety. B. 1972;34:187–202.

[8] Cui L, Kuo W, Loh HT, Xie M. Optimal allocation of minimal and perfect repairsunder resource constraints. IEEE Trans Reliab 2004;53:193–9.

[9] Delia MC, Rafael PO. Matrix stochastic analysis of the maintainability of amachine under shocks. Reliab Eng Syst Saf 2014;121:11–7.

[10] Doyen L, Gaudoin O. Classes of imperfect repair models based on reduction offailure intensity or virtual age. Reliab Eng Syst Saf 2004;84:45–56.

[11] Elferik S, Ben-Daya M. Age-based hybrid model for imperfect preventivemaintenance. IIE Trans 2006;38:365–75.

[12] Finkelstein M. Failure rate modeling for reliability and risk. London: Spinger-Verlag; 2008.

[13] García-Díaz JC, Gozalvez-Zafrilla JM. Uncertainty and sensitive analysis ofenvironmental model for risk assessments: an industrial case study. ReliabEng Syst Saf 2012;107:16–22.

[14] Ghasemi A, Yacout S, Ouali MS. Evaluating the reliability function and themean residual life for equipment with unobservable states. IEEE Trans Reliab2010;59:45–54.

[15] Jardine AKS, Anderson PM, Mann DS. Application of the weibull proportionalhazards model to aircraft and marine engine failure data. Qual Reliab Eng Int1987;3:77–82.

[16] Jardine AKS, Banjevic D, Wiseman M, Buck S, Joseph T. Optimizing a mine haultruck wheel motors' condition monitoring program. J Qual Maint Eng2001;7:286–301.

[17] Jiang XM, Yuan Y, Liu X. Bayesian inference method for stochastic damageaccumulation modeling. Reliab Eng Syst Saf 2013;111:126–38.

[18] Kumar D, Westberg U. Proportional hazards modeling of time-dependent covari-ates using linear regression: a case study. IEEE Trans Reliab 1996;45:386–92.

[19] Levitin G, Lisnianski A. Optimization of imperfect preventive maintenance formulti-state systems. Reliab Eng Syst Saf 2000;67:193–203.

[20] Lie CH, Chun YH. An algorithm for preventive maintenance policy. IEEE TransReliab 1986;35:71–5.

[21] Lin D, Zuo MJ, Yam RCM. General sequential imperfect preventive main-tenance models. Int J Reliab Qual Saf Eng 2000;7:253–66.

[22] Lin D, Zuo MJ, Yam RCM. Sequential imperfect preventive maintenance modelswith two categories of failure modes. Naval Res Logist 2001;48:172–83.

[23] Lu XF, Chen MY, Liu M, Zhou DH. Optimal imperfect periodic preventivemaintenance for systems in time-varying environments. IEEE Trans Reliab2012;61:426–39.

[24] Lugtigheid D, Jiang X, Jardine AKS. A finite horizon model for repairablesystems with repair restrictions. J Oper Res Soc 2008;59:1321–31.

[25] Mattrand C, Bourinet JM. The cross-entropy method for reliability assessmentof cracked structures subjected to random Markovian loads. Reliab Eng SystSaf 2014;123:171–82.

[26] McKinlay SA. Markov-modulated models for derivatives pricing [PostgraduateDiploma Thesis]. Australia: The University of Melbourne; 2009.

[27] Nakagawa T. A summary of imperfect preventive maintenance policies withminimal repair. RAIRO Oper Res 1980;14:249–55.

[28] Nakagawa T. Sequential imperfect preventive maintenance policies. IEEE TransReliab 1988;37:295–8.

[29] Nakagawa T. Maintenance theory of reliability. Springer-Verlag LondonLimited; 2005.

[30] Park JH, Chang W, Lie CH. Stress-reducing preventive maintenance model for aunit under stressful environment. Reliab Eng Syst Saf 2012;108:42–8.

[31] Peysson F, Ouladsine M, Outbib R. A generic prognostic methodology usingdamage trajectory models. IEEE Trans Reliab 2009;58:277–85.

[32] Pham H, Wang H. Imperfect maintenance. Eur J Oper Res 1996;94:425–38.[33] Samrout M, Chatelet E, Kouta R, Chebbo N. Optimization of maintenance

policy using the proportional hazard model. Reliab Eng Syst Saf2009;94:44–52.

[34] Sheu SH, Chang CC. A periodic replacement model based on cumulativerepair-cost limit for a system subjected to shocks. IEEE Trans Reliab2010;59:374–82.

[35] Singpurwalla ND. Survival in dynamic environments. Stat Sci 1995;10:86–103.[36] Vlok PJ, Goetzee JL, Jardine AKS, Makis V. Optimal component replacement

decisions using vibration monitoring and the proportional hazards model.J Oper Res Soc 2002;53:193–202.

[37] Wang XL, Balakrishnan N, Guo B. Residual life estimation based on a general-ized Wiener degradation process. Reliab Eng Syst Saf 2014;124:13–23.

[38] Wang ZQ, Wang PF. A new approach for reliability analysis with time-variantperformance characteristics. Reliab Eng Syst Saf 2013;115:70–81.

[39] Wu S, Clements-Croome D. Optimal maintenance policies under differentoperational schedules. IEEE Trans Reliab 2005;54:338–46.

[40] Wu SM, Zuo MJ. Linear and nonlinear preventive maintenance models. IEEETrans Reliab 2010;59:242–9.

L. XiaoFei, L. Min / Reliability Engineering and System Safety 130 (2014) 50–6060