# On the Survival Time of a Duplex System: A Sokhotski-Plemelj Problem

Authors: Edmond J. Vanderperre
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2008)

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## Abstract

We analyze the survival time of a renewable duplex system characterized
by warm standby and subjected to a priority rule. In order
to obtain the Laplace transform of the survival function, we employ
a stochastic process endowed with time-dependent transition measures
satisfying coupled partial differential equations. The solution procedure
is based on the theory of sectionally holomorphic functions combined
with the notion of dual transforms. Finally, we introduce a security
interval related to a prescribed security level and a suitable risk criterion
based on the survival function of the system. As an example, we
consider the particular case of deterministic repair. A computer-plotted
graph displays the survival function together with the security interval
corresponding to a security level of 90%.


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## Body

## 1. Introduction

Standby provides a powerful tool to enhance the
reliability, availability, quality, and safety of operational plants (see,
e.g., [1–4]). Standby systems are often subjected to priority
rules. For instance, the external power supply station of a technical
plant has usually overall (break-in) priority in operation with regard to an internal (local) power generator kept in cold or warm standby; that is, the local
generator is only deployed if the external unit is down. The notion of
“cold” standby signifies that the local generator has a zero failure rate
in standby, whereas the notion of “warm” standby means that the
failure-free time of the local generator is stochastically larger [5] in
standby than in the operative state. Note that the warm standby mode of a unit
is often indispensable to perform an instantaneous switch from standby into the
operative state, allowing continuous operation of an operational system upon
failure of the online unit.Cold or warm standby systems, subjected to priority
rules, have received considerable attention in previous literature (see, e.g.,
[6–20]). As a variant, we introduce a duplex system consisting of a priority
unit (the p-unit) with a back-up nonpriority unit
(the n-unit) in warm standby and attended by a
repair facility. The p-unit has overall (break-in) priority in
operation with regard to the n-unit; that is, the n-unit is only deployed if and only if the p-unit is down. In order to avoid undesirable
delays in repairing failed units, we assume that the twin system is attended by
two heterogeneous repairmen. Each repairman has his own particular task.
Repairman 𝒩 is skilled at repairing the failed n-unit, whereas repairman 𝒫 is supposed to be an expert in repairing the failed p-unit. Both repairmen are jointly busy if both
units (the p-unit and n-unit) are down. Otherwise, at least one
repairman is idle. Any repair is assumed to be perfect. The entire system
(henceforth called the T-system) is up if at least one unit is up.
Otherwise, the T-system is down.In order to determine the survival function of theT-system,
we introduce a stochastic process endowed with time-dependent transition
measures satisfying coupled partial differential equations. The solution
procedure is based on a refined application of the theory of sectionally
holomorphic functions (see, e.g., [21]) combined with the notion of dual
transforms. Furthermore, we introduce a security interval [0,τ) related to a security level 0<δ<1 and a risk criterion based on the survival
function of the T-system. The security interval ensures a
survival of the T-system up to time τ with a probability larger than δ.
Finally, we consider the particular case of deterministic repair (replacement).
A computer-plotted graph displays the survival function together with the
security interval corresponding to a security level of 90%.


## 2. Formulation

Consider theT-system subjected to the
following conditions.(i) Thep-unit has a general failure-free
time distribution F(⋅) with finite mean and a general repair time
distribution R(⋅),R(0)=0.
The failure-free time and the repair time are denoted by f and r.
We assume that F(⋅) is Lebesgue absolutely continuous with a
density function (in the Radon-Nikodym sense) of bounded variation on [0,∞).(ii) Then-unit has a constant failure rate λ>0 in the operative state and a constant failure
rate 0<λs<λ in standby. Note that the inequality λs<λ is consistent with the notion of warm standby.
The failure-free time of the n-unit in warm standby (resp., in operation) is
denoted by fs (resp., fo). The common repair time of any n-failure is denoted by rs with (common) repair time distribution Rs(⋅),Rs(0)=0.
In addition, we assume that rs has finite mean and variance.(iii) The random variablesf,r,fs,fo, and rs are assumed to be statistically independent
and any repair is perfect.(iv) Characteristic functions are formulated in
terms of a complex transform variable. For instance,(2.1)Eeiωr=∫0∞eiωxdR(x),Imω≥0. Note that (2.2)Ee−iωr=∫−∞0eiωxd{1−R((−x)−)},Imω≤0. The corresponding Fourier-Stieltjes
transforms are called dual transforms. Without loss of generality (see Remark 7.4), we may assume that R and Rs have density functions of bounded variation on [0,∞).
Note that the bounded variation property implies that, for
instance, (2.3)|Eeiτf|=O(1|τ|),|τ|→∞.(v) In order to derive the survival function of
theT-system, we employ a stochastic process {Nt,t≥0} with discrete state space {A,B,C,D}⊂[0,∞) and absorbing state D characterized by the following exhaustive set
of mutually exclusive events.{Nt=A}:
the p-unit is operative and the n-unit is in warm standby at time t.{Nt=B}:
the n-unit is operative and the p-unit is under progressive repair at time t.{Nt=C}:
the p-unit is operative and the n-unit is under progressive repair at time t.{Nt=D}:
the T-system is down at time t.Note that the
absorbing stateD implies that a transition of the process {Nt} into state A is only possible via states B or C,
whereas a transition from state B or C into state D terminates the lifetime of the system.
Therefore, the inclusion of the absorbing state D into the state space of the process {Nt,t≥0} triggers the introduction of a so-called stopping
time. Consequently, we first define the non-Markovian process {Nt,t≥0} on a filtered probability space {Ω,𝒜,P,ℱ} where the historyℱ:={ℱt,t≥0} satisfies the Dellacherie conditions:(i)
ℱ0 contains the P-null sets of 𝒜;(ii)
for
allt≥0,ℱt=⋂u>tℱu;
that is, the family ℱ is right-continuous. Consider the ℱ-stopping time (2.4)θ:=inf{t:Nt=D∣N0=A,V0=0}, where Vt is the past failure-free time of the p-unit
being operative at time t.
We assume that the T-system starts functioning at some time origin t=0 in state A;
that is, let N0=A,V0=0, P-a.s. Thus, from t=0 onwards, θ is the survival time (lifetime) of the T-system.
The corresponding survival function is denoted by ℜ(⋅).
Clearly, ℜ(t)=P{θ>t},t≥0.
A (vector) Markov characterization of the non-Markovian process {Nt,t≥0} with absorbing state D is piecewise and conditionally defined by(1)
{(Nt,Ut)} if Nt=A (i.e., if the event {Nt=A} occurs), where Ut denotes the remaining failure-free time of the p-unit being up at time t;(2)
{(Nt,Xt)} if Nt=B,
where Xt denotes the remaining repair time of the p-unit
being under progressive repair at time t;(3)
{(Nt,Ut,Yt)} if Nt=C,
where Yt denotes the remaining repair time of the n-unit
being under progressive repair at time t;(4)
{Nt} if Nt=D (the absorbing state).The state space of the underlying Markov process, with
absorbing stateD,
is given by (2.5){(A,u)}⋃{(B,x)}⋃{(C,u,y)}⋃{D},u≥0,x≥0,y≥0. For K=A,B,C,D,
let pK(t):=P{Nt=K}, t≥0.(vi) Finally, we introduce the transition measures:(2.6)pA(t,u)du:=P{Nt=A,Ut∈du},pB(t,x)dx:=P{Nt=B,Xt∈dx},pC(t,u,y)dudy:=P{Nt=C,Ut∈du,Yt∈dy}. Note that, for instance, (2.7)pC(t)=∫0∞∫0∞dudyP{Nt=C,Ut≤u,Yt≤y}=∫0∞∫0∞pC(t,u,y)dudy.


## 3. Notations

(i) The indicator (function) of an event{Nt=K} is denoted by 1{Nt=K}.(ii) The complex plane and the real line are,
respectively, denoted byC and R with obvious superscript notations such as C+ and C−. For instance, C+:={ω∈C:Imω>0}.(iii) We frequently use the characteristic function:(3.1)γs+(τ):={Eeiτrs−1iτErsifτ≠0,1ifτ=0. Note that (3.2)γs+(ω)=1Ers∫0∞eiωx(1−Rs(x))dx,Imω≥0.Property 3.1  (see [22, Appendix]).
The function1+λsErsγs+(ω), Imω≥0, has no zeros in C+∪R.(iv) The Heaviside unit step function, with the
unit step att=t0,
is denoted by Ht0(⋅),
that is, (3.3)Ht0(t):={1ift≥t0>0,0ift<t0.(v) The greatest integer function is denoted by[⋅].(vi) The Laplace transform of any locally
integrable and bounded function on[0,∞) is denoted by the corresponding character
marked with an asterisk. For instance, (3.4)pA*(z):=∫0∞e−ztpA(t)dt,Rez>0. Observe that (3.5)ℜ*(z)=1−Ee−zθz,Rez>0. Moreover, by the product rule
for Lebesgue-Stieltjes integrals (see, e.g., [23, Appendix]) (3.6)zpD*(z)=∫0−∞e−ztdpD(t)=Ee−zθ,Rez>0.(vii) Letφ(τ),τ∈R,
be a bounded and continuous function. φ is called Γ-integrable if (3.7)limT→∞ϵ↓0∫ΓT,ϵφ(τ)dττ−u,u∈R, exists, where ΓT,ϵ:=(−T,u−ϵ]∪[u+ϵ,T).
The corresponding integral, denoted by (3.8)12πi∫Γφ(τ)dττ−u, is called a Cauchy principal
value in double sense.(viii) A functionφ(τ),τ∈R,
is called Hölder-continuous on R if for all τ1,τ2∈R,
there exists (β,A),0<β≤1,A>0 :(3.9)|φ(τ2)−φ(τ1)|≤A|τ2−τ1|β. The function φ(τ),τ∈R,
is called Hölder-continuous at infinity if there exists γ>0: (3.10)|φ(τ)|=O(1|τ|γ),|τ|→∞. Hölder-continuous functions with
exponent β=γ=1 are called Lipschitz-continuous.(ix) Note that the Hölder continuity ofφ(⋅) on R and at infinity is sufficient for the
existence of the Cauchy-type integral: (3.11)12πi∫Γφ(τ)dττ−ω,ω∈C.


## 4. Differential Equations

In order to derive a system of differential equations,
we observe the random behavior of theT-system in some time interval (t,t+Δ),Δ↓0.
Grouping terms of o(Δ) and taking the absorbing state D into account reveal that (4.1)pA(t+Δ,u−Δ)=pA(t,u)(1−λsΔ)+pB(t,0)dFdu(u)Δ+pC(t,u,0)Δ+o(Δ),pB(t+Δ,x−Δ)=pB(t,x)(1−λΔ)+pA(t,0)dRdx(x)Δ+o(Δ),pC(t+Δ,u−Δ,y−Δ)=pC(t,u,y)+λspA(t,u)dRsdy(y)Δ+o(Δ),pD(t+Δ)=pD(t)+λ∫0∞pB(t,x)dxΔ+∫0∞pC(t,0,y)dyΔ+o(Δ). Taking the
definition of directional derivative into account, for instance, (4.2)(∂∂t−∂∂u−∂∂y)pC(t,u,y):=limΔ↓0pC(t+Δ,u−Δ,y−Δ)−pC(t,u,y)Δ, entails that for t>0,u>0,x>0,
and y>0, (4.3)(λs+∂∂t−∂∂u)pA(t,u)=pB(t,0)dFdu(u)+pC(t,u,0),(4.4)(λ+∂∂t−∂∂x)pB(t,x)=pA(t,0)dRdx(x),(4.5)(∂∂t−∂∂u−∂∂y)pC(t,u,y)=λspA(t,u)dRsdy(y),(4.6)ddtpD(t)=λpB(t)+∫0∞pC(t,0,y)dy. Note that the initial condition N0=A,V0=0, P-a.s. implies that (4.7)pA(0,u)=dFdu(u),u>0. Moreover, P{θ≤t}=pD(t).
Finally, observe that (4.3)–(4.6) are consistent with the probability law ∑KpK(t)=1 and that pA(0)=1.


## 5. Functional Equation

First, we remark that our system of differential
equations is well adapted to a Laplace-Fourier transformation. As a matter of
fact, the transition functions are bounded on their appropriate regions and
locally integrable with respect tot.
Consequently, each Laplace transform exists for Rez>0.
Moreover, the integrability of the density functions and the transition
functions with regard to u,x,
and y also implies the integrability of the
corresponding partial derivatives.Applying a Laplace-Fourier transform technique to
(4.3)–(4.6) and taking the initial condition into account reveal that for Rez>0,Imω≥0,Imη≥0,
and Imζ≥0, (5.1)(λs+z+iζ)∫0∞e−ztE(eiζUt1{Nt=A})dt+pA∗(z,0)=pB∗(z,0)Eeiζf+∫0∞eiζupC∗(z,u,0)du+Eeiζf,(5.2)(λ+z+iω)∫0∞e−ztE(eiωXt1{Nt=B})dt+pB∗(z,0)=pA∗(z,0)Eeiωr,(5.3)(z+iζ+iη)∫0∞e−ztE(eiζUteiηYt1{Nt=C})dt+∫0∞eiζupC∗(z,u,0)du+∫0∞eiηypC∗(z,0,y)dy=λs∫0∞e−ztE(eiζUt1{Nt=A})dtEeiηrs,(5.4)zpD*(z)=λpB∗(z)+∫0∞eiηypC∗(z,0,y)dy|η=0. Adding (5.1) and (5.3) yields
the functional equation (5.5)(λs(1−Eeiηrs)+z+iζ)∫0∞e−ztE(eiζUt1{Nt=A})dt+pA∗(z,0)−pB∗(z,0)Eeiζf+∫0∞eiηypC∗(z,0,y)dy+(z+iζ+iη)∫0∞e−ztE(eiζUteiηYt1{Nt=C})dt=Eeiζf.


## 6. Survival Function

In order to obtain the Laplace transform of the
survival function, we first remark that by (5.4) and (3.6), (6.1)Ee−zθ=λpB∗(z)+∫0∞eiηypC∗(z,0,y)dy|η=0. Inserting ω=i(λ+z) (resp., ω=0) into (5.2) entails that (6.2)pB*(z,0)=pA*(z,0)Ee−(λ+z)r,(6.3)(z+λ)pB∗(z)+pB∗(z,0)=pA∗(z,0). Finally, inserting ζ=iz,η=0 into the functional equation (5.5) reveals
that (6.4)Ee−zf=pA*(z,0)−pB∗(z,0)Ee−zf+∫0∞eiηypC∗(z,0,y)dy|η=0. Invoking the relation (6.5)ℜ*(z)=1−Ee−zθz,Rez>0, yields by (6.1)–(6.4)
that (6.6)ℜ*(z)=1−Ee−zfz(1+pA*(z,0)Ee−(λ+z)r)+pA*(z,0)1−Ee−(z+λ)rz+λ. Hence, we only have to determine pA∗(z,0).


## 7. Methodology

In order to derive the unknownpA*(z,0),
we first eliminate the function (7.1)∫0∞e−ztE(eiζUteiηYt1{Nt=C})dt, by the substitution
ofη=τ,ζ=−τ+iz,τ∈R,Rez>0.
Noting that z+iζ+iη=0 reveals that by (5.5), (7.2)pA∗(z,0)(1−Ee−(z+λ)rEe−i(τ−iz)f)+∫0∞eiτypC∗(z,0,y))dy−(1+λsErsγs+(τ))iτ∫0∞e−ztE(e−i(τ−iz)Ut1{Nt=A})dt=Ee−i(τ−iz)f. Dividing  (7.2) by the factor 1+λsErsγs+(τ),
taking Property 3.1 into account, yields the boundary value
equation (7.3)ψ+(z,τ)−ψ−(z,τ)=(1+pA*(z,0)Ee−(λ+z)r)φ(z,τ), where (7.4)ψ+(z,ω):=ϕ+(z,ω)1+λsErsγs+(ω),ϕ+(z,ω):=∫0∞eiωypC∗(z,0,y)dy−λsErsγs+(ω)pA*(z,0),Imω≥0,ψ−(z,ω):=iω∫0∞e−ztE(e−i(ω−iz)Ut1{Nt=A})dt−pA*(z,0),Imω≤0,φ(z,τ):=Ee−i(τ−iz)f1+λsErsγs+(τ). Equation (7.3) constitutes a z-dependent Sokhotski-Plemelj problem on R,
solvable by the theory of sectionally holomorphic functions (see, e.g., [21]).
First, we need the following property.Lemma 7.1.
The functionφ(z,τ),Rez≥0, is Lipschitz-continuous onR and at infinity.Proof.
Note that Property3.1 implies that supτ∈R|1+λsErsγs+(τ)|−1<∞.
Hence, the existence of Ef,Ers,
and Ers2 entails that (7.5)supτ∈R|∂∂τφ(z,τ)|<∞. Consequently, by the mean value
theorem (see, e.g., [24]), there exists a constant K such that for all τ1,τ2∈R, (7.6)|φ(z,τ1)−φ(z,τ2)|≤K|τ1−τ2|. Hence, φ(z,τ) is Lipschitz-continuous on R.
Finally, note that the Lipschitz continuity ofφ(z,τ) at infinity follows from the boundedness of |1+λsErsγs+(τ)|−1 and (2.3).Corollary 7.2.
The function(7.7)12πi∫Γφ(z,τ)dττ−ω,ω∈C, is sectionally holomorphic
and regular.Moreover, by (7.3), (7.8)ψ−(z,ω)=(1+pA∗(z,0)Ee−(z+λ)r)12πi∫Γφ(z,τ)dττ−ω,ω∈C−. Note that (7.8) is only valid
for ω∈C−.
However, by the Sokhotski-Plemelj formula (see, e.g., [21, page 36]), (7.9)limω→0ω∈C−ψ−(z,ω)=(1+pA∗(z,0)Ee−(z+λ)r)α(z), where (7.10)α(z):=limω→0ω∈C−12πi∫Γφ(z,τ)dττ−ω=−12φ(z,0)+12πi∫Γφ(z,τ)dττ. On the other hand, we have by
continuity limω∈C−ω→0ψ−(z,ω)=ψ−(z,0)=−pA∗(z,0).
Hence, (7.11)pA∗(z,0)=−α(z)1+α(z)Ee−(z+λ)r. The function ℜ*(z) is now completely determined by (7.11) and
(6.6). We summarize the following result.Property 7.3.
The Laplace transform of the survival function is
given by(7.12)ℜ*(z)=11+α(z)Ee−(z+λ)r{1−Ee−zfz−1−Ee−(z+λ)rz+λα(z)}, where (7.13)α(z)=−12Ee−zf1+λsErs+12πi∫ΓEe−i(τ−iz)f1+λsErsγs+(τ)dττ,γs+(τ):={Eeiτrs−1iτErsifτ≠0,1ifτ=0.Remark 7.4.
It should be noted that Property3.1 also holds for an arbitraryRs with finite mean. Moreover, the existence of
moments does not depend on the canonical structure (Lebesgue decomposition) of
the underlying distribution. For instance, the inequality (7.14)|∂∂τγs+(τ)|≤1Ers∫0∞x(1−Rs(x))dx=12Ers2Ers<∞ also holds for an arbitrary Rs with finite mean and variance. Therefore,
Lemma 7.1 remains valid for arbitrary Rs.
The requirement of a finite variance σrs2 is extremely mild. In fact, the current
probability distributions of interest to statistical reliability engineering
even have moments of any order. Finally, the functional (7.15)Ee−(z+λ)r=∫0∞e−(z+λ)xdR(x) exists for an arbitrary R as a Lebesgue-Stieltjes integral on [0,∞) and has no impact on the existence of the
integral (7.16)12πi∫Γφ(z,τ)dττ−ω,ω∈C. Consequently, Property 7.3 holds
for arbitrary repair time distributions.


## 8. Risk Criterion

Along with the survival function of theT-system, we now introduce a security interval [0,τ),
where (8.1)τ:=sup{t≥0:ℜ(t−)>δ} for some 0<δ<1,
which is called the security level. In practice, δ is usually large. For instance, δ=0.9.
Therefore, we require that the T-system satisfy the risk criterion limt↑τℜ(t)>δ≫0.
Note that the security interval, corresponding to the security level δ,
ensures a continuous operation (survival) of the T-system up to time τ with probability larger than δ.
See the forthcoming example.


## 9. Deterministic Repair

As an example, we consider the particular case of
deterministic repair (replacement); that is, letR(⋅)=Rs(⋅)=Ht0(⋅),
where t0=1 is taken as time unit. Clearly, Ee−zr=Ee−zrs=e−z.
Furthermore, let F(u)=1−e−λu,u≥0.
Note that (9.1)Ee−i(τ−iz)f=−iλτ−i(λ+z). By Property 7.3, we
have (9.2)ℜ*(z)=1λ+z(1−α(z)1+α(z)e−(λ+z)). We recall that (9.3)α(z)=limω→0ω∈C−12πi∫Γφ(z,τ)dττ−ω=limω→0ω∈C−12πi∫Γ−iλ(τ−ω)−1(τ−i(λ+z))−11+λsErsγs+(τ)dτ. For ω∈C−,Rez≥0,
the integrand represents a meromorphic function in C+ with single pole i(λ+z).
Moreover, the function vanishes at infinity in C+∪R.
An application of the residue theorem entails that (9.4)α(z)=limω→0ω∈C−iλ(λ+z)(ω−i(λ+z))(λ+z+λs(1−e−(λ+z)))=−λλ+z+λs(1−e−(λ+z)). Hence, by (9.2), (9.5)ℜ*(z)=1z+λ(1+λz+λs+λ−(λs+λ)e−(z+λ)). Applying the inversion
technology presented in [25] yields the exact survival function (9.6)ℜ(t)=e−λt(1+λλs∑k=0[t](λλ+λs)k(1−e−λs(t−k)∑j=0k(λs(t−k))jj!)). Figure 1 displays the graph of ℜ(t),0≤t≤20,λ=0.5,
and λs=0.25 with the security interval [0,τ], τ=2.259.
The interval ensures a continuous operation of the T-system up to time τ=2.259 with a probability of at least 90%.Figure 1
Graph ofℜ(t),0≤t≤20,λ=0.5,
and λs=0.25 with security interval [0,τ], τ=2.259, and security level δ=0.9.



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