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

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%.


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 breakin 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 the T-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%.

Formulation
Consider the T-system subjected to the following conditions.
i The p-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 The n-unit has a constant failure rate λ > 0 in the operative state and a constant failure rate 0 < λ s < λ in standby.Note that the inequality λ s < 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 f s resp., f o .The common repair time of any n-failure is denoted by r s with common repair time distribution R s • , R s 0 0. In addition, we assume that r s has finite mean and variance.
iii The random variables f, r, f s , f o , and r s 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, The terminates the lifetime of the system.Therefore, the inclusion of the absorbing state D into the state space of the process {N t , t ≥ 0} triggers the introduction of a so-called stopping time.Consequently, we first define the non-Markovian process {N t , t ≥ 0} on a filtered probability space {Ω, A, P, F} where the history F : {F t , t ≥ 0} satisfies the Dellacherie conditions: i F 0 contains the P-null sets of A; ii for all t ≥ 0, F t u>t F u ; that is, the family F is right-continuous.
Consider the F-stopping time where V t 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 N 0 A, V 0 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 {N t , t ≥ 0} with absorbing state D is piecewise and conditionally defined by

2.6
Note that, for instance, 2.7

Notations
i The indicator function of an event {N t K} is denoted by 1{N t K}.
ii The complex plane and the real line are, respectively, denoted by C and R with obvious superscript notations such as C and C − .For instance, C : {ω ∈ C : Im ω > 0}.
iii We frequently use the characteristic function: iv The Heaviside unit step function, with the unit step at t t 0 , is denoted by 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 Observe that Moreover, by the product rule for Lebesgue-Stieltjes integrals see, e.g., 23, Appendix vii Let ϕ τ , τ ∈ R, be a bounded and continuous function.
exists, where Γ T, : −T, u − ∪ u , T .The corresponding integral, denoted by is called a Cauchy principal value in double sense.viii A function ϕ τ , τ ∈ R, is called H ölder-continuous on R if for all τ 1 , τ 2 ∈ R, there exists β, A , 0 < β ≤ 1, A > 0 : 3.9 The function ϕ τ , τ ∈ R, is called H ölder-continuous at infinity if there exists γ > 0: 3.10 H ölder-continuous functions with exponent β γ 1 are called Lipschitz-continuous. ix Note that the H ölder continuity of ϕ • on R and at infinity is sufficient for the existence of the Cauchy-type integral:

Differential equations
In order to derive a system of differential equations, we observe the random behavior of the T-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 Note that the initial condition N 0 A, V 0 0, P-a.s.implies that Moreover, P{θ ≤ t} p D t .Finally, observe that 4.3 -4.6 are consistent with the probability law K p K t 1 and that p A 0 1.

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 to t.Consequently, each Laplace transform exists for Re z > 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 Re z > 0, Im ω ≥ 0, Im η ≥ 0, and Im ζ ≥ 0, .

5.4
Adding 5.1 and 5.3 yields the functional equation 5.5

Survival function
In order to obtain the Laplace transform of the survival function, we first remark that by 5.4 and 3.6 , ∞ 0 e iηy p * C z, 0, y dy η 0 .

6.1
Inserting ω i λ z resp., ω 0 into 5.2 entails that Finally, inserting ζ iz, η 0 into the functional equation 5.5 reveals that Invoking the relation yields by 6.1 -6.4 that Hence, we only have to determine p * A z, 0 .

Methodology
In order to derive the unknown p * A z, 0 , we first eliminate the function by the substitution of η τ, ζ −τ iz, τ ∈ R, Re z > 0. Noting that z iζ iη 0 reveals that by 5.5 ,

7.2
Dividing 7.2 by the factor 1 λ s Er s γ s τ , taking Property 3.1 into account, yields the boundary value equation where On the other hand, we have by continuity lim ω→0 ω∈C − ψ − z, ω ψ − z, 0 −p * A z, 0 .Hence, 7.11 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 the current probability distributions of interest to statistical reliability engineering even have moments of any order.Finally, the functional exists for an arbitrary R as a Lebesgue-Stieltjes integral on 0, ∞ and has no impact on the existence of the integral Consequently, Property 7.3 holds for arbitrary repair time distributions.

Risk criterion
Along with the survival function of the T-system, we now introduce a security interval 0, τ , where τ : sup t ≥ 0 : R t− > δ 8.1 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 lim t↑τ R 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.

Deterministic repair
As an example, we consider the particular case of deterministic repair replacement ; that is, let R

1 1 3 ForFigure 1
Figure1displays the graph of R t , 0 ≤ t ≤ 20, λ 0.5, and λ s 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%.
corresponding Fourier-Stieltjes transforms are called dual transforms.Without loss of generality see Remark 7.4 , we may assume that R and R s have density functions of bounded variation on 0, ∞ .Note that the bounded variation property implies that, for instance, Note that the absorbing state D implies that a transition of the process {N t } into state A is only possible via states B or C, whereas a transition from state B or C into state D t A}: the p-unit is operative and the n-unit is in warm standby at time t.{N t B}: the n-unit is operative and the p-unit is under progressive repair at time t.{N t C}: the p-unit is operative and the n-unit is under progressive repair at time t.{N t D}: the T-system is down at time t.
if the event {N t A} occurs , where U t denotes the remaining failure-free time of the p-unit being up at time t; 2 { N t , X t } if N t B, where X t denotes the remaining repair time of the p-unit being under progressive repair at time t; 3 { N t , U t , Y t } if N t C, where Y t denotes the remaining repair time of the n-unit being under progressive repair at time t; 4 {N t } if N t D the absorbing state .For K A, B, C, D, let p K t : P{N t K}, t ≥ 0. vi Finally, we introduce the transition measures: p A t, u du : P N t A, U t ∈ du , p B t, x dx : P N t B, X t ∈ dx , p C t, u, y du dy : P N t C, U t ∈ du, Y t ∈ dy .
f 1 λ s Er s γ s 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.Finally, note that the Lipschitz continuity of ϕ z, τ at infinity follows from the boundedness of |1 λ s Er s γ s τ | −1 and 2.3 .
Lemma 7.1.The function ϕ z, τ , Re z ≥ 0, is Lipschitz-continuous on R and at infinity.Proof.Note that Property 3.1 implies that sup τ∈R |1 λ s Er s γ s τ | −1 < ∞.Hence, the existence of Ef, Er s , and Er 2 It should be noted that Property 3.1 also holds for an arbitrary R s 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 Graph of R t , 0 ≤ t ≤ 20, λ 0.5, and λ s 0.25 with security interval 0, τ , τ 2.259, and security level δ 0.9.
s with finite mean and variance.Therefore, Lemma 7.1 remains valid for arbitrary R s .The requirement of a finite variance σ 2 r s is extremely mild.In fact, • R s • H t 0 • , where t 0 1 is taken as time unit.Clearly, Ee −zr Ee −zr s e −z .Journal of Applied Mathematics and Stochastic Analysis Furthermore, let F u 1 − e −λu , u ≥ 0. Note that