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

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%.

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 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%.

2. 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, 𝐄𝑒^𝑖𝜔𝑟=^∞0𝑒^𝑖𝜔𝑥𝑑𝑅(𝑥),Im𝜔≥0.(2.1) Note that 𝐄𝑒^{−𝑖𝜔𝑟}=^0−∞𝑒^𝑖𝜔𝑥𝑑1−𝑅(−𝑥)−,Im𝜔≤0.(2.2) The 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, |||𝐄𝑒_𝑖𝜏𝑓|||=𝑂1||𝜏||,||𝜏||→∞.(2.3)

(v) In order to derive the survival function of the T-system, we employ a stochastic process {𝑁_𝑡,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.

{𝑁_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.

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 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)ℱ₀ contains the P-null sets of 𝒜;(ii)for all 𝑡≥0,ℱ_𝑡=⋂_𝑢>𝑡ℱ_𝑢; that is, the family ℱ is right-continuous. Consider the ℱ-stopping time 𝜃∶=inf𝑡∶𝑁_𝑡=𝐷∣𝑁₀=𝐴,𝑉₀=0,(2.4) 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 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 {𝑁_𝑡,𝑡≥0} with absorbing state 𝐷 is piecewise and conditionally defined by

(1){(𝑁_𝑡,𝑈_𝑡)} if 𝑁_𝑡=𝐴 (i.e., 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).

The state space of the underlying Markov process, with absorbing state 𝐷, is given by (𝐴,𝑢)(𝐵,𝑥)(𝐶,𝑢,𝑦)𝐷,𝑢≥0,𝑥≥0,𝑦≥0.(2.5) For 𝐾=𝐴,𝐵,𝐶,𝐷, let 𝑝_𝐾(𝑡)∶=𝐏{𝑁_𝑡=𝐾}, 𝑡≥0.

(vi) Finally, we introduce the transition measures: 𝑝𝑝_𝐴(𝑡,𝑢)𝑑𝑢∶=𝐏𝑁_𝑡=𝐴,𝑈_𝑡∈𝑑𝑢_𝐵,𝑝(𝑡,𝑥)𝑑𝑥∶=𝐏𝑁_𝑡=𝐵,𝑋_𝑡∈𝑑𝑥_𝐶,(𝑡,𝑢,𝑦)𝑑𝑢𝑑𝑦∶=𝐏𝑁_𝑡=𝐶,𝑈_𝑡∈𝑑𝑢,𝑌_𝑡∈𝑑𝑦.(2.6) Note that, for instance, 𝑝_𝐶(𝑡)=^∞0^∞0=𝑑_𝑢𝑑_𝑦𝐏𝑁_𝑡=𝐶,𝑈_𝑡≤𝑢,𝑌_𝑡≤𝑦_∞0^∞0𝑝_𝐶(𝑡,𝑢,𝑦)𝑑𝑢𝑑𝑦.(2.7)

3. 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∶Im𝜔>0}.

(iii) We frequently use the characteristic function: 𝛾^+𝑠(𝜏)∶=⎧⎪⎨⎪⎩𝐄𝑒^{𝑖𝜏𝑟𝑠}−1𝑖𝜏𝐄𝑟_𝑠1iiff𝜏𝜏=≠00.,(3.1) Note that 𝛾^+𝑠(𝜔)=1𝐄𝑟_𝑠^∞0𝑒^𝑖𝜔𝑥1−𝑅^𝑠(𝑥)𝑑𝑥,Im𝜔≥0.(3.2)

Property 3.1 (see [22, Appendix]). The function 1+𝜆_𝑠𝐄𝑟_𝑠𝛾^+𝑠(𝜔), Im𝜔≥0, has no zeros in 𝐂⁺∪𝐑.

(iv) The Heaviside unit step function, with the unit step at 𝑡=𝑡₀, is denoted by 𝐻_𝑡0(⋅), that is, 𝐻_𝑡0(𝑡)∶=1if𝑡≥𝑡⁰0if𝑡<𝑡₀>0,.(3.3)

(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, 𝑝^∗𝐴(𝑧)∶=^∞0𝑒^−𝑧𝑡𝑝_𝐴(𝑡)𝑑𝑡,Re𝑧>0.(3.4) Observe that ℜ^∗(𝑧)=1−𝐄𝑒^−𝑧𝜃𝑧,Re𝑧>0.(3.5) Moreover, by the product rule for Lebesgue-Stieltjes integrals (see, e.g., [23, Appendix]) 𝑧𝑝^∗𝐷(𝑧)=^∞0−𝑒^−𝑧𝑡𝑑𝑝_𝐷(𝑡)=𝐄𝑒^−𝑧𝜃,Re𝑧>0.(3.6)

(vii) Let 𝜑(𝜏),𝜏∈𝐑, be a bounded and continuous function. 𝜑 is called Γ-integrable if lim_{𝑇→∞𝜖↓0}_Γ𝑇,𝜖𝜑(𝜏)𝑑𝜏𝜏−𝑢,𝑢∈𝐑,(3.7) exists, where Γ_𝑇,𝜖∶=(−𝑇,𝑢−𝜖]∪[𝑢+𝜖,𝑇). The corresponding integral, denoted by 12𝜋𝑖_Γ𝜑(𝜏)𝑑𝜏𝜏−𝑢,(3.8) is called a Cauchy principal value in double sense.

(viii) A function 𝜑(𝜏),𝜏∈𝐑, is called Hölder-continuous on 𝐑 if for all 𝜏₁,𝜏₂∈𝐑, there exists (𝛽,𝐴),0<𝛽≤1,𝐴>0 :|||𝜑𝜏₂−𝜑𝜏¹|||≤𝐴|||𝜏²−𝜏₁|||^𝛽.(3.9) The function 𝜑(𝜏),𝜏∈𝐑, is called Hölder-continuous at infinity if there exists 𝛾>0: |||𝜑(𝜏)|||=𝑂(1||𝜏||^𝛾),||𝜏||→∞.(3.10) Hölder-continuous functions with exponent 𝛽=𝛾=1 are called Lipschitz-continuous.

(ix) Note that the Hölder continuity of 𝜑(⋅) on 𝐑 and at infinity is sufficient for the existence of the Cauchy-type integral: 12𝜋𝑖_Γ𝜑(𝜏)𝑑𝜏𝜏−𝜔,𝜔∈𝐂.(3.11)

4. 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 𝑝_𝐴(𝑡+Δ,𝑢−Δ)=𝑝_𝐴(𝑡,𝑢)1−𝜆_𝑠Δ+𝑝_𝐵(𝑡,0)d𝐹d𝑢(𝑢)Δ+𝑝^𝐶𝑝(𝑡,𝑢,0)Δ+𝑜(Δ),_𝐵(𝑡+Δ,𝑥−Δ)=𝑝_𝐵(𝑡,𝑥)(1−𝜆Δ)+𝑝_𝐴(𝑡,0)d𝑅𝑝d𝑥(𝑥)Δ+𝑜(Δ),_𝐶(𝑡+Δ,𝑢−Δ,𝑦−Δ)=𝑝_𝐶(𝑡,𝑢,𝑦)+𝜆_𝑠𝑝_𝐴(𝑡,𝑢)d𝑅^𝑠𝑝d𝑦(𝑦)Δ+𝑜(Δ),_𝐷(𝑡+Δ)=𝑝_𝐷(𝑡)+𝜆∫^∞0𝑝_𝐵(𝑡,𝑥)𝑑𝑥Δ+∫^∞0𝑝_𝐶(𝑡,0,𝑦)𝑑𝑦Δ+𝑜(Δ).(4.1) Taking the definition of directional derivative into account, for instance, 𝜕𝜕𝑡−𝜕𝜕𝑢−𝜕𝜕𝑦𝑝^𝐶(𝑡,𝑢,𝑦)∶=lim_Δ↓0𝑝_𝐶(𝑡+Δ,𝑢−Δ,𝑦−Δ)−𝑝_𝐶(𝑡,𝑢,𝑦)Δ,(4.2) entails that for 𝑡>0,𝑢>0,𝑥>0, and 𝑦>0, 𝜆^𝑠+𝜕𝜕𝜆𝑡−+𝜕𝜕𝜕𝑢𝑝^𝐴(𝑡,𝑢)=𝑝_𝐵(𝑡,0)dd𝐹𝑢(𝑢)+𝑝^𝐶(𝑡,𝑢,0),(4.3)𝜕𝑡−𝜕𝜕𝑥𝑝^𝐵(𝑡,𝑥)=𝑝_𝐴(𝑡,0)d𝑅𝜕d𝑥(𝑥),(4.4)𝜕𝑡−𝜕𝜕𝑢−𝜕𝜕𝑦𝑝^𝐶(𝑡,𝑢,𝑦)=𝜆_𝑠𝑝_𝐴(𝑡,𝑢)d𝑅^𝑠dd𝑦(𝑦),(4.5)d𝑡𝑝^𝐷(𝑡)=𝜆𝑝_𝐵(𝑡)+∫^∞0𝑝_𝐶(𝑡,0,𝑦)𝑑𝑦.(4.6) Note that the initial condition 𝑁₀=𝐴,𝑉₀=0, P-a.s. implies that 𝑝_𝐴(0,𝑢)=d𝐹d𝑢(𝑢),𝑢>0.(4.7) Moreover, 𝐏{𝜃≤𝑡}=𝑝_𝐷(𝑡). Finally, observe that (4.3)–(4.6) are consistent with the probability law ∑_𝐾𝑝_𝐾(𝑡)=1 and that 𝑝_𝐴(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 to 𝑡. 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, (𝜆_𝑠+=𝑧𝑝+_∗𝐵𝑖𝜁)^∞0𝑒^−𝑧𝑡𝐄𝑒^{𝑖𝜁𝑈𝑡}𝟏𝑁_𝑡=𝐴𝑑𝑡+𝑝^∗𝐴(𝑧,0)(𝜆+𝑧+𝑖(𝜔𝑧),0_∞0)𝐄𝑒^𝑖𝜁𝑓+^∞0𝑒^𝑖𝜁𝑢𝑝^∗𝐶(𝑧,𝑢,0)𝑑𝑢+𝐄𝑒^𝑖𝜁𝑓,(5.1)(𝑧+𝑖𝜁+𝑖𝜂)_∞0𝑒^−𝑧𝑡𝐄𝑒^{𝑖𝜔𝑋𝑡}𝟏𝑁_𝑡=𝐵𝑑𝑡+𝑝^∗𝐵(𝑧,0)=𝑝^∗𝐴(𝑧,0)𝐄𝑒^𝑖𝜔𝑟,(5.2)=𝜆_𝑠𝑒^−𝑧𝑡𝐄𝑒^{𝑖𝜁𝑈𝑡}𝑒^{𝑖𝜂𝑌𝑡}𝟏𝑁_𝑡=𝐶𝑑𝑡+^∞0𝑒^𝑖𝜁𝑢𝑝^∗𝐶(𝑧,𝑢,0)𝑑𝑢+^∞0𝑒^𝑖𝜂𝑦𝑝^∗𝐶(𝑧,0,𝑦)𝑑𝑦^∞0𝑧𝑝_∗𝐷𝑒^−𝑧𝑡𝐄𝑒^{𝑖𝜁𝑈𝑡}𝟏𝑁_𝑡=𝐴𝑑𝑡𝐄𝑒^{𝑖𝜂𝑟𝑠},(5.3)(𝑧)=𝜆𝑝^∗𝐵(𝑧)+^∞0𝑒^𝑖𝜂𝑦𝑝^∗𝐶(𝑧,0,𝑦)𝑑𝑦|||_𝜂=0.(5.4) Adding (5.1) and (5.3) yields the functional equation 𝜆^𝑠1+−𝑝𝐄_∗𝐴𝑒^{𝑖𝜂𝑟𝑠}+𝑧+𝑖𝜁^∞0𝑒^−𝑧𝑡𝐄𝑒^{𝑖𝜁𝑈𝑡}𝟏𝑁_𝑡=𝐴𝑑𝑡+(𝑧(+𝑧,𝑖𝜁0)+−𝑖𝜂𝑝)^∗𝐵(_∞0𝑧,0)𝐄𝑒^𝑖𝜁𝑓+^∞0𝑒^𝑖𝜂𝑦𝑝^∗𝐶(𝑧,0,𝑦)𝑑𝑦𝑒^−𝑧𝑡𝐄𝑒^{𝑖𝜁𝑈𝑡}𝑒^{𝑖𝜂𝑌𝑡}𝟏𝑁_𝑡=𝐶𝑑𝑡=𝐄𝑒^𝑖𝜁𝑓.(5.5)

6. Survival Function

In order to obtain the Laplace transform of the survival function, we first remark that by (5.4) and (3.6), 𝐄𝑒^−𝑧𝜃=𝜆𝑝^∗𝐵(𝑧)+^∞0𝑒^𝑖𝜂𝑦𝑝^∗𝐶(𝑧,0,𝑦)𝑑𝑦|||_𝜂=0.(6.1) Inserting 𝜔=𝑖(𝜆+𝑧) (resp., 𝜔=0) into (5.2) entails that (𝑧+𝑝^∗𝐵𝜆()𝑧𝑝,_∗𝐵0)=𝑝^∗𝐴(𝑧,0)𝐄𝑒^{−(𝜆+𝑧)𝑟},(6.2)(𝑧)+𝑝^∗𝐵(𝑧,0)=𝑝^∗𝐴(𝑧,0).(6.3) Finally, inserting 𝜁=𝑖𝑧,𝜂=0 into the functional equation (5.5) reveals that 𝐄𝑒^−𝑧𝑓=𝑝^∗𝐴(𝑧,0)−𝑝^∗𝐵(𝑧,0)𝐄𝑒^−𝑧𝑓+^∞0𝑒^𝑖𝜂𝑦𝑝^∗𝐶(𝑧,0,𝑦)𝑑𝑦|||_𝜂=0.(6.4) Invoking the relation ℜ^∗(𝑧)=1−𝐄𝑒^−𝑧𝜃𝑧,Re𝑧>0,(6.5) yields by (6.1)–(6.4) that ℜ^∗(𝑧)=1−𝐄𝑒^−𝑧𝑓𝑧(1+𝑝^∗𝐴(𝑧,0)𝐄𝑒^{−(𝜆+𝑧)𝑟})+𝑝^∗𝐴(𝑧,0)1−𝐄𝑒^{−(𝑧+𝜆)𝑟}𝑧+𝜆.(6.6) Hence, we only have to determine 𝑝^∗𝐴(𝑧,0).

7. Methodology

In order to derive the unknown 𝑝^∗𝐴(𝑧,0), we first eliminate the function ^∞0𝑒^−𝑧𝑡𝐄𝑒^{𝑖𝜁𝑈𝑡}𝑒^{𝑖𝜂𝑌𝑡}𝟏𝑁_𝑡=𝐶𝑑𝑡,(7.1) by the substitution of𝜂=𝜏,𝜁=−𝜏+𝑖𝑧,𝜏∈𝐑,Re𝑧>0. Noting that 𝑧+𝑖𝜁+𝑖𝜂=0 reveals that by (5.5), 𝑝^∗𝐴(𝑧,0−)11−+𝐄𝜆𝑒^{−(𝑧+𝜆)𝑟}_𝑠𝐄𝑒^{−𝑖(𝜏−𝑖𝑧)𝑓}+^∞0𝑒^𝑖𝜏𝑦𝑝^∗𝐶(𝑧,0,𝑦))𝑑𝑦𝐄𝑟_𝑠𝛾^+𝑠(𝜏)𝑖𝜏^∞0𝑒^−𝑧𝑡𝐄𝑒^{−𝑖(𝜏−𝑖𝑧)𝑈𝑡}𝟏𝑁_𝑡=𝐴𝑑𝑡=𝐄𝑒^{−𝑖(𝜏−𝑖𝑧)𝑓}.(7.2) Dividing (7.2) by the factor 1+𝜆_𝑠𝐄𝑟_𝑠𝛾^+𝑠(𝜏), taking Property 3.1 into account, yields the boundary value equation 𝜓⁺(𝑧,𝜏)−𝜓⁻(𝑧,𝜏)=(1+𝑝^∗𝐴(𝑧,0)𝐄𝑒^{−(𝜆+𝑧)𝑟})𝜑(𝑧,𝜏),(7.3) where 𝜓⁺(𝑧,𝜔)∶=𝜙⁺(𝑧,𝜔)𝜙1+𝜆_𝑠𝐄𝑟_𝑠𝛾^+𝑠(𝜔)₊,𝜓(𝑧,𝜔)∶=^∞0𝑒^𝑖𝜔𝑦𝑝^∗𝐶(𝑧,0,𝑦)𝑑𝑦−𝜆_𝑠𝐄𝑟_𝑠𝛾^+𝑠(𝜔)𝑝^∗𝐴(𝑧,0),Im𝜔≥0,₋𝜑((𝑧𝑧,,𝜔𝜏))∶∶==𝑖𝜔𝐄^∞0𝑒𝑒^−𝑧𝑡𝐄_{−𝑖(𝜏−𝑖𝑧)𝑓}𝑒^{−𝑖(𝜔−𝑖𝑧)𝑈𝑡}𝟏𝑁_𝑡=𝐴𝑑𝑡−𝑝^∗𝐴(𝑧,0),Im𝜔≤0,1+𝜆_𝑠𝐄𝑟_𝑠𝛾^+𝑠(𝜏).(7.4) 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 function 𝜑(𝑧,𝜏),𝑅𝑒𝑧≥0, is Lipschitz-continuous on 𝐑 and at infinity.

Proof. Note that Property 3.1 implies that sup_𝜏∈𝐑|1+𝜆_𝑠𝐄𝑟_𝑠𝛾^+𝑠(𝜏)|⁻¹<∞. Hence, the existence of 𝐄𝑓,𝐄𝑟_𝑠, and 𝐄𝑟^2𝑠 entails that sup_𝜏∈𝐑|||𝜕𝜕𝜏𝜑(𝑧,𝜏)|||<∞.(7.5) Consequently, by the mean value theorem (see, e.g., [24]), there exists a constant 𝐾 such that for all 𝜏₁,𝜏₂∈𝐑, ||𝜑(𝑧,𝜏₁)−𝜑(𝑧,𝜏₂)||≤𝐾||𝜏₁−𝜏₂||.(7.6) Hence, 𝜑(𝑧,𝜏) is Lipschitz-continuous on R.
Finally, note that the Lipschitz continuity of 𝜑(𝑧,𝜏) at infinity follows from the boundedness of |1+𝜆_𝑠𝐄𝑟_𝑠𝛾^+𝑠(𝜏)|⁻¹ and (2.3).

Corollary 7.2. The function 12𝜋𝑖_Γ𝜑(𝑧,𝜏)𝑑𝜏𝜏−𝜔,𝜔∈𝐂,(7.7) is sectionally holomorphic and regular.

Moreover, by (7.3), 𝜓⁻(𝑧,𝜔)=(1+𝑝^∗𝐴(𝑧,0)𝐄𝑒^{−(𝑧+𝜆)𝑟})12𝜋𝑖_Γ𝜑(𝑧,𝜏)𝑑𝜏𝜏−𝜔,𝜔∈𝐂⁻.(7.8) Note that (7.8) is only valid for 𝜔∈𝐂⁻. However, by the Sokhotski-Plemelj formula (see, e.g., [21, page 36]), lim_{𝜔→0𝜔∈𝐂−}𝜓⁻(𝑧,𝜔)=(1+𝑝^∗𝐴(𝑧,0)𝐄𝑒^{−(𝑧+𝜆)𝑟})𝛼(𝑧),(7.9) where 𝛼(𝑧)∶=lim_{𝜔→0𝜔∈𝐂−}12𝜋𝑖_Γ𝜑(𝑧,𝜏)𝑑𝜏𝜏−𝜔=−12𝜑(𝑧,0)+21𝜋𝑖_Γ𝜑(𝑧,𝜏)𝑑𝜏𝜏.(7.10) On the other hand, we have by continuity lim_{𝜔→0𝜔∈𝐂−}𝜓⁻(𝑧,𝜔)=𝜓⁻(𝑧,0)=−𝑝^∗𝐴(𝑧,0). Hence, 𝑝^∗𝐴(𝑧,0)=−𝛼(𝑧)1+𝛼(𝑧)𝐄𝑒^{−(𝑧+𝜆)𝑟}.(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 ℜ^∗(𝑧)=11+𝛼(𝑧)𝐄𝑒^{−(𝑧+𝜆)𝑟}{1−𝐄𝑒^−𝑧𝑓𝑧−1−𝑧𝐄+𝑒𝜆^{−(𝑧+𝜆)𝑟}𝛼(𝑧)},(7.12) where 𝛼(𝑧)=−121+𝐄𝑒𝜆_𝑠𝐄^−𝑧𝑓𝑟_𝑠+12𝜋𝑖_Γ𝐄𝑒^{−𝑖(𝜏−𝑖𝑧)𝑓}𝛾_+𝑠1+𝜆_𝑠𝐄𝑟_𝑠𝛾^+𝑠(𝜏)𝑑𝜏𝜏,(𝜏)∶=⎧⎪⎨⎪⎩𝐄𝑒^{𝑖𝜏𝑟𝑠}−1𝑖𝜏𝐄𝑟_𝑠1iiff𝜏𝜏=≠00.,(7.13)

Remark 7.4. 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 |||𝜕𝜕𝜏𝛾^+𝑠(𝜏)|||≤1𝐄𝑟_𝑠^∞0𝑥1−𝑅_𝑠(𝑥)𝑑𝑥=12𝐄𝐄𝑟𝑟^2𝑠_𝑠<∞(7.14) also holds for an arbitrary 𝑅_𝑠 with finite mean and variance. Therefore, Lemma 7.1 remains valid for arbitrary 𝑅_𝑠. The requirement of a finite variance 𝜎^2𝑟_𝑠 is extremely mild. In fact, the current probability distributions of interest to statistical reliability engineering even have moments of any order. Finally, the functional 𝐄𝑒^{−(𝑧+𝜆)𝑟}=^∞0𝑒^{−(𝑧+𝜆)𝑥}d𝑅(𝑥)(7.15) exists for an arbitrary 𝑅 as a Lebesgue-Stieltjes integral on [0,∞) and has no impact on the existence of the integral 12𝜋𝑖_Γ𝜑(𝑧,𝜏)𝑑𝜏𝜏−𝜔,𝜔∈𝐶.(7.16) Consequently, Property 7.3 holds for arbitrary repair time distributions.

8. 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↑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, let 𝑅(⋅)=𝑅_𝑠(⋅)=𝐻_𝑡0(⋅), 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 𝐄𝑒^{−𝑖(𝜏−𝑖𝑧)𝑓}=−𝑖𝜆𝜏−𝑖(𝜆+𝑧).(9.1) By Property 7.3, we have ℜ^∗(𝑧)=1𝜆+𝑧(1−1+𝛼𝛼(𝑧()𝑧𝑒)^{−(𝜆+𝑧)}).(9.2) We recall that 𝛼(𝑧)=lim_{𝜔→0𝜔∈𝐂−}12𝜋𝑖∫^Γ𝜑(𝑧,𝜏)𝑑𝜏𝜏−𝜔=lim^{𝜔→0𝜔∈𝐂}₋12𝜋𝑖∫^Γ−𝑖𝜆(𝜏−𝜔)⁻¹𝜏−𝑖(𝜆+𝑧)⁻¹1+𝜆_𝑠𝐄𝑟_𝑠𝛾^+𝑠(𝜏)d𝜏.(9.3) 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 𝛼(𝑧)=lim_{𝜔→0𝜔∈𝐂−}𝑖𝜆(𝜆+𝑧)𝜔−𝑖(𝜆+𝑧)𝜆+𝑧+𝜆^𝑠1−𝑒^{−(𝜆+𝑧)}=−𝜆+𝑧+𝜆_𝑠𝜆1−𝑒^{−(𝜆+𝑧)}.(9.4) Hence, by (9.2), ℜ^∗(𝑧)=1𝑧+𝜆(1+𝜆𝑧+𝜆_𝑠+𝜆−𝜆_𝑠+𝜆𝑒^{−(𝑧+𝜆}).(9.5) Applying the inversion technology presented in [25] yields the exact survival function ℜ(𝑡)=𝑒^−𝜆𝑡(1+𝜆𝜆_{𝑠[𝑡]𝑘=0}𝜆+𝜆𝜆_𝑠^𝑘(1−𝑒^{−𝜆𝑠𝑘(𝑡−𝑘)}_𝑗=0(𝜆_𝑠(𝑡−𝑘))^𝑗𝑗!)).(9.6) Figure 1 displays 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%.

Figure 1 

Graph of ℜ(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.