Table of Contents
Journal of Applied Mathematics and Stochastic Analysis
Volume 2008, Article ID 905721, 13 pages
http://dx.doi.org/10.1155/2008/905721
Research Article

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

Department of Decision Sciences, University of South Africa, P.O. Box 392, Pretoria 0003, South Africa

Received 9 June 2008; Accepted 1 September 2008

Academic Editor: Karl Sigman

Copyright © 2008 Edmond J. Vanderperre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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., [14]). 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., [620]). 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<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 variables 𝑓,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, 𝐄𝑒𝑖𝜔𝑟=0𝑒𝑖𝜔𝑥𝑑𝑅(𝑥),Im𝜔0.(2.1) Note that 𝐄𝑒𝑖𝜔𝑟=0𝑒𝑖𝜔𝑥𝑑1𝑅(𝑥),Im𝜔0.(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 𝑅𝑠 have density functions of bounded variation on [0,). Note that the bounded variation property implies that, for instance, |||𝐄𝑒𝑖𝜏𝑓|||1=𝑂||𝜏||,||𝜏||.(2.3)

(v) In order to derive the survival function of the T-system, we employ a stochastic process {𝑁𝑡,𝑡0} 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 state 𝐷 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,𝑡0} triggers the introduction of a so-called stopping time. Consequently, we first define the non-Markovian process {𝑁𝑡,𝑡0} on a filtered probability space {Ω,𝒜,𝐏,} where the history ={𝑡,𝑡0} satisfies the Dellacherie conditions:

(i)0 contains the P-null sets of 𝒜;(ii)for all 𝑡0,𝑡=𝑢>𝑡𝑢; that is, the family is right-continuous. Consider the -stopping time 𝜃=inf𝑡𝑁𝑡=𝐷𝑁0=𝐴,𝑉0=0,(2.4) 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 (). Clearly, (𝑡)=𝐏{𝜃>𝑡},𝑡0. A (vector) Markov characterization of the non-Markovian process {𝑁𝑡,𝑡0} 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 state 𝐷, is given by 𝐷(𝐴,𝑢)(𝐵,𝑥)(𝐶,𝑢,𝑦),𝑢0,𝑥0,𝑦0.(2.5) For 𝐾=𝐴,𝐵,𝐶,𝐷, let 𝑝𝐾(𝑡)=𝐏{𝑁𝑡=𝐾}, 𝑡0.

(vi) Finally, we introduce the transition measures: 𝑝𝐴𝑁(𝑡,𝑢)𝑑𝑢=𝐏𝑡=𝐴,𝑈𝑡,𝑝𝑑𝑢𝐵𝑁(𝑡,𝑥)𝑑𝑥=𝐏𝑡=𝐵,𝑋𝑡,𝑝𝑑𝑥𝐶𝑁(𝑡,𝑢,𝑦)𝑑𝑢𝑑𝑦=𝐏𝑡=𝐶,𝑈𝑡𝑑𝑢,𝑌𝑡.𝑑𝑦(2.6) Note that, for instance, 𝑝𝐶(𝑡)=00𝑑𝑢𝑑𝑦𝐏𝑁𝑡=𝐶,𝑈𝑡𝑢,𝑌𝑡=𝑦00𝑝𝐶(𝑡,𝑢,𝑦)𝑑𝑢𝑑𝑦.(2.7)

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 by 𝐂 and 𝐑 with obvious superscript notations such as 𝐂+ and 𝐂. For instance, 𝐂+={𝜔𝐂Im𝜔>0}.

(iii) We frequently use the characteristic function: 𝛾+𝑠(𝜏)=𝐄𝑒𝑖𝜏𝑟𝑠1𝑖𝜏𝐄𝑟𝑠if𝜏0,1if𝜏=0.(3.1) Note that 𝛾+𝑠1(𝜔)=𝐄𝑟𝑠0𝑒𝑖𝜔𝑥1𝑅𝑠(𝑥)𝑑𝑥,Im𝜔0.(3.2)

Property 3.1 (see [22, Appendix]). The function 1+𝜆𝑠𝐄𝑟𝑠𝛾+𝑠(𝜔), Im𝜔0, has no zeros in 𝐂+𝐑.

(iv) The Heaviside unit step function, with the unit step at 𝑡=𝑡0, is denoted by 𝐻𝑡0(), that is, 𝐻𝑡0(𝑡)=1if𝑡𝑡0>0,0if𝑡<𝑡0.(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, 𝑝𝐴(𝑧)=0𝑒𝑧𝑡𝑝𝐴(𝑡)𝑑𝑡,Re𝑧>0.(3.4) Observe that (𝑧)=1𝐄𝑒𝑧𝜃𝑧,Re𝑧>0.(3.5) Moreover, by the product rule for Lebesgue-Stieltjes integrals (see, e.g., [23, Appendix]) 𝑧𝑝𝐷(𝑧)=0𝑒𝑧𝑡𝑑𝑝𝐷(𝑡)=𝐄𝑒𝑧𝜃,Re𝑧>0.(3.6)

(vii) Let 𝜑(𝜏),𝜏𝐑, be a bounded and continuous function. 𝜑 is called Γ-integrable if lim𝑇𝜖0Γ𝑇,𝜖𝜑(𝜏)𝑑𝜏𝜏𝑢,𝑢𝐑,(3.7) exists, where Γ𝑇,𝜖=(𝑇,𝑢𝜖][𝑢+𝜖,𝑇). The corresponding integral, denoted by 12𝜋𝑖Γ𝜑(𝜏)𝑑𝜏𝜏𝑢,(3.8) is called a Cauchy principal value in double sense.

(viii) A function 𝜑(𝜏),𝜏𝐑, is called Hölder-continuous on 𝐑 if for all 𝜏1,𝜏2𝐑, there exists (𝛽,𝐴),0<𝛽1,𝐴>0 :|||𝜑𝜏2𝜏𝜑1||||||𝜏𝐴2𝜏1|||𝛽.(3.9) The function 𝜑(𝜏),𝜏𝐑, is called Hölder-continuous at infinity if there exists 𝛾>0: ||||||1𝜑(𝜏)=𝑂(||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: 12𝜋𝑖Γ𝜑(𝜏)𝑑𝜏𝜏𝜔,𝜔𝐂.(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),Δ0. Grouping terms of 𝑜(Δ) and taking the absorbing state 𝐷 into account reveal that 𝑝𝐴(𝑡+Δ,𝑢Δ)=𝑝𝐴(𝑡,𝑢)1𝜆𝑠Δ+𝑝𝐵(𝑡,0)d𝐹d𝑢(𝑢)Δ+𝑝𝐶𝑝(𝑡,𝑢,0)Δ+𝑜(Δ),𝐵(𝑡+Δ,𝑥Δ)=𝑝𝐵(𝑡,𝑥)(1𝜆Δ)+𝑝𝐴(𝑡,0)d𝑅𝑝d𝑥(𝑥)Δ+𝑜(Δ),𝐶(𝑡+Δ,𝑢Δ,𝑦Δ)=𝑝𝐶(𝑡,𝑢,𝑦)+𝜆𝑠𝑝𝐴(𝑡,𝑢)d𝑅𝑠𝑝d𝑦(𝑦)Δ+𝑜(Δ),𝐷(𝑡+Δ)=𝑝𝐷(𝑡)+𝜆0𝑝𝐵(𝑡,𝑥)𝑑𝑥Δ+0𝑝𝐶(𝑡,0,𝑦)𝑑𝑦Δ+𝑜(Δ).(4.1) Taking the definition of directional derivative into account, for instance, 𝜕𝜕𝜕𝑡𝜕𝜕𝑢𝑝𝜕𝑦𝐶(𝑡,𝑢,𝑦)=limΔ0𝑝𝐶(𝑡+Δ,𝑢Δ,𝑦Δ)𝑝𝐶(𝑡,𝑢,𝑦)Δ,(4.2) entails that for 𝑡>0,𝑢>0,𝑥>0, and 𝑦>0, 𝜆𝑠+𝜕𝜕𝜕𝑡𝑝𝜕𝑢𝐴(𝑡,𝑢)=𝑝𝐵(𝑡,0)d𝐹d𝑢(𝑢)+𝑝𝐶𝜕(𝑡,𝑢,0),(4.3)𝜆+𝜕𝜕𝑡𝑝𝜕𝑥𝐵(𝑡,𝑥)=𝑝𝐴(𝑡,0)d𝑅𝜕d𝑥(𝑥),(4.4)𝜕𝜕𝑡𝜕𝜕𝑢𝑝𝜕𝑦𝐶(𝑡,𝑢,𝑦)=𝜆𝑠𝑝𝐴(𝑡,𝑢)d𝑅𝑠dd𝑦(𝑦),(4.5)𝑝d𝑡𝐷(𝑡)=𝜆𝑝𝐵(𝑡)+0𝑝𝐶(𝑡,0,𝑦)𝑑𝑦.(4.6) Note that the initial condition 𝑁0=𝐴,𝑉0=0, P-a.s. implies that 𝑝𝐴(0,𝑢)=d𝐹d𝑢(𝑢),𝑢>0.(4.7) Moreover, 𝐏{𝜃𝑡}=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 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,Im𝜔0,Im𝜂0, and Im𝜁0, (𝜆𝑠+𝑧+𝑖𝜁)0𝑒𝑧𝑡𝐄𝑒𝑖𝜁𝑈𝑡𝟏𝑁𝑡=𝐴𝑑𝑡+𝑝𝐴(𝑧,0)=𝑝𝐵(𝑧,0)𝐄𝑒𝑖𝜁𝑓+0𝑒𝑖𝜁𝑢𝑝𝐶(𝑧,𝑢,0)𝑑𝑢+𝐄𝑒𝑖𝜁𝑓,(5.1)(𝜆+𝑧+𝑖𝜔)0𝑒𝑧𝑡𝐄𝑒𝑖𝜔𝑋𝑡𝟏𝑁𝑡=𝐵𝑑𝑡+𝑝𝐵(𝑧,0)=𝑝𝐴(𝑧,0)𝐄𝑒𝑖𝜔𝑟,(5.2)(𝑧+𝑖𝜁+𝑖𝜂)0𝑒𝑧𝑡𝐄𝑒𝑖𝜁𝑈𝑡𝑒𝑖𝜂𝑌𝑡𝟏𝑁𝑡=𝐶𝑑𝑡+0𝑒𝑖𝜁𝑢𝑝𝐶(𝑧,𝑢,0)𝑑𝑢+0𝑒𝑖𝜂𝑦𝑝𝐶(𝑧,0,𝑦)𝑑𝑦=𝜆𝑠0𝑒𝑧𝑡𝐄𝑒𝑖𝜁𝑈𝑡𝟏𝑁𝑡=𝐴𝑑𝑡𝐄𝑒𝑖𝜂𝑟𝑠,(5.3)𝑧𝑝𝐷(𝑧)=𝜆𝑝𝐵(𝑧)+0𝑒𝑖𝜂𝑦𝑝𝐶|||(𝑧,0,𝑦)𝑑𝑦𝜂=0.(5.4) Adding (5.1) and (5.3) yields the functional equation 𝜆𝑠1𝐄𝑒𝑖𝜂𝑟𝑠+𝑧+𝑖𝜁0𝑒𝑧𝑡𝐄𝑒𝑖𝜁𝑈𝑡𝟏𝑁𝑡=𝐴𝑑𝑡+𝑝𝐴(𝑧,0)𝑝𝐵(𝑧,0)𝐄𝑒𝑖𝜁𝑓+0𝑒𝑖𝜂𝑦𝑝𝐶(𝑧,0,𝑦)𝑑𝑦+(𝑧+𝑖𝜁+𝑖𝜂)0𝑒𝑧𝑡𝐄𝑒𝑖𝜁𝑈𝑡𝑒𝑖𝜂𝑌𝑡𝟏𝑁𝑡=𝐶𝑑𝑡=𝐄𝑒𝑖𝜁𝑓.(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), 𝐄𝑒𝑧𝜃=𝜆𝑝𝐵(𝑧)+0𝑒𝑖𝜂𝑦𝑝𝐶|||(𝑧,0,𝑦)𝑑𝑦𝜂=0.(6.1) Inserting 𝜔=𝑖(𝜆+𝑧) (resp., 𝜔=0) into (5.2) entails that 𝑝𝐵(𝑧,0)=𝑝𝐴(𝑧,0)𝐄𝑒(𝜆+𝑧)𝑟,(6.2)(𝑧+𝜆)𝑝𝐵(𝑧)+𝑝𝐵(𝑧,0)=𝑝𝐴(𝑧,0).(6.3) Finally, inserting 𝜁=𝑖𝑧,𝜂=0 into the functional equation (5.5) reveals that 𝐄𝑒𝑧𝑓=𝑝𝐴(𝑧,0)𝑝𝐵(𝑧,0)𝐄𝑒𝑧𝑓+0𝑒𝑖𝜂𝑦𝑝𝐶|||(𝑧,0,𝑦)𝑑𝑦𝜂=0.(6.4) Invoking the relation (𝑧)=1𝐄𝑒𝑧𝜃𝑧,Re𝑧>0,(6.5) yields by (6.1)–(6.4) that (𝑧)=1𝐄𝑒𝑧𝑓𝑧(1+𝑝𝐴(𝑧,0)𝐄𝑒(𝜆+𝑧)𝑟)+𝑝𝐴(𝑧,0)1𝐄𝑒(𝑧+𝜆)𝑟𝑧+𝜆.(6.6) Hence, we only have to determine 𝑝𝐴(𝑧,0).

7. Methodology

In order to derive the unknown 𝑝𝐴(𝑧,0), we first eliminate the function 0𝑒𝑧𝑡𝐄𝑒𝑖𝜁𝑈𝑡𝑒𝑖𝜂𝑌𝑡𝟏𝑁𝑡=𝐶𝑑𝑡,(7.1) by the substitution of𝜂=𝜏,𝜁=𝜏+𝑖𝑧,𝜏𝐑,Re𝑧>0. Noting that 𝑧+𝑖𝜁+𝑖𝜂=0 reveals that by (5.5), 𝑝𝐴(𝑧,0)1𝐄𝑒(𝑧+𝜆)𝑟𝐄𝑒𝑖(𝜏𝑖𝑧)𝑓+0𝑒𝑖𝜏𝑦𝑝𝐶(𝑧,0,𝑦))𝑑𝑦1+𝜆𝑠𝐄𝑟𝑠𝛾+𝑠(𝜏)𝑖𝜏0𝑒𝑧𝑡𝐄𝑒𝑖(𝜏𝑖𝑧)𝑈𝑡𝟏𝑁𝑡=𝐴𝑑𝑡=𝐄𝑒𝑖(𝜏𝑖𝑧)𝑓.(7.2) Dividing (7.2) by the factor 1+𝜆𝑠𝐄𝑟𝑠𝛾+𝑠(𝜏), taking Property 3.1 into account, yields the boundary value equation 𝜓+(𝑧,𝜏)𝜓(𝑧,𝜏)=(1+𝑝𝐴(𝑧,0)𝐄𝑒(𝜆+𝑧)𝑟)𝜑(𝑧,𝜏),(7.3) where 𝜓+𝜙(𝑧,𝜔)=+(𝑧,𝜔)1+𝜆𝑠𝐄𝑟𝑠𝛾+𝑠,𝜙(𝜔)+(𝑧,𝜔)=0𝑒𝑖𝜔𝑦𝑝𝐶(𝑧,0,𝑦)𝑑𝑦𝜆𝑠𝐄𝑟𝑠𝛾+𝑠(𝜔)𝑝𝐴𝜓(𝑧,0),Im𝜔0,(𝑧,𝜔)=𝑖𝜔0𝑒𝑧𝑡𝐄𝑒𝑖(𝜔𝑖𝑧)𝑈𝑡𝟏𝑁𝑡=𝐴𝑑𝑡𝑝𝐴(𝑧,0),Im𝜔0,𝜑(𝑧,𝜏)=𝐄𝑒𝑖(𝜏𝑖𝑧)𝑓1+𝜆𝑠𝐄𝑟𝑠𝛾+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),𝑅𝑒𝑧0, is Lipschitz-continuous on 𝐑 and at infinity.

Proof. Note that Property 3.1 implies that sup𝜏𝐑|1+𝜆𝑠𝐄𝑟𝑠𝛾+𝑠(𝜏)|1<. Hence, the existence of 𝐄𝑓,𝐄𝑟𝑠, and 𝐄𝑟2𝑠 entails that sup𝜏𝐑|||𝜕|||𝜕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,𝜏2𝐑, ||𝜑(𝑧,𝜏1)𝜑(𝑧,𝜏2)||||𝜏𝐾1𝜏2||.(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+𝜆𝑠𝐄𝑟𝑠𝛾+𝑠(𝜏)|1 and (2.3).

Corollary 7.2. The function 12𝜋𝑖Γ𝜑(𝑧,𝜏)𝑑𝜏𝜏𝜔,𝜔𝐂,(7.7) is sectionally holomorphic and regular.

Moreover, by (7.3), 𝜓(𝑧,𝜔)=(1+𝑝𝐴(𝑧,0)𝐄𝑒(𝑧+𝜆)𝑟)12𝜋𝑖Γ𝜑(𝑧,𝜏)𝑑𝜏𝜏𝜔,𝜔𝐂.(7.8) Note that (7.8) is only valid for 𝜔𝐂. However, by the Sokhotski-Plemelj formula (see, e.g., [21, page 36]), lim𝜔0𝜔𝐂𝜓(𝑧,𝜔)=(1+𝑝𝐴(𝑧,0)𝐄𝑒(𝑧+𝜆)𝑟)𝛼(𝑧),(7.9) where 𝛼(𝑧)=lim𝜔0𝜔𝐂12𝜋𝑖Γ𝜑(𝑧,𝜏)𝑑𝜏1𝜏𝜔=21𝜑(𝑧,0)+2𝜋𝑖Γ𝜑(𝑧,𝜏)𝑑𝜏𝜏.(7.10) On the other hand, we have by continuity lim𝜔0𝜔𝐂𝜓(𝑧,𝜔)=𝜓(𝑧,0)=𝑝𝐴(𝑧,0). Hence, 𝑝𝐴(𝑧,0)=𝛼(𝑧)1+𝛼(𝑧)𝐄𝑒(𝑧+𝜆)𝑟.(7.11) The function (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 1(𝑧)=1+𝛼(𝑧)𝐄𝑒(𝑧+𝜆)𝑟{1𝐄𝑒𝑧𝑓𝑧1𝐄𝑒(𝑧+𝜆)𝑟𝑧+𝜆𝛼(𝑧)},(7.12) where 1𝛼(𝑧)=2𝐄𝑒𝑧𝑓1+𝜆𝑠𝐄𝑟𝑠+12𝜋𝑖Γ𝐄𝑒𝑖(𝜏𝑖𝑧)𝑓1+𝜆𝑠𝐄𝑟𝑠𝛾+𝑠(𝜏)𝑑𝜏𝜏,𝛾+𝑠(𝜏)=𝐄𝑒𝑖𝜏𝑟𝑠1𝑖𝜏𝐄𝑟𝑠if𝜏0,1if𝜏=0.(7.13)

Remark 7.4. It should be noted that Property 3.1 also holds for an arbitrary 𝑅𝑠 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 |||𝜕𝛾𝜕𝜏+𝑠|||1(𝜏)𝐄𝑟𝑠0𝑥1𝑅𝑠1(𝑥)𝑑𝑥=2𝐄𝑟2𝑠𝐄𝑟𝑠<(7.14) 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 𝜎2𝑟𝑠 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)𝑥d𝑅(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 12𝜋𝑖Γ𝜑(𝑧,𝜏)𝑑𝜏𝜏𝜔,𝜔𝐶.(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=sup𝑡0(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 lim𝑡𝜏(𝑡)>𝛿0. 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()=𝑅𝑠()=𝐻𝑡0(), where 𝑡0=1 is taken as time unit. Clearly, 𝐄𝑒𝑧𝑟=𝐄𝑒𝑧𝑟𝑠=𝑒𝑧. Furthermore, let 𝐹(𝑢)=1𝑒𝜆𝑢,𝑢0. Note that 𝐄𝑒𝑖(𝜏𝑖𝑧)𝑓=𝑖𝜆𝜏𝑖(𝜆+𝑧).(9.1) By Property 7.3, we have 1(𝑧)=𝜆+𝑧(1𝛼(𝑧)1+𝛼(𝑧)𝑒(𝜆+𝑧)).(9.2) We recall that 𝛼(𝑧)=lim𝜔0𝜔𝐂12𝜋𝑖Γ𝜑(𝑧,𝜏)𝑑𝜏𝜏𝜔=lim𝜔0𝜔𝐂12𝜋𝑖Γ𝑖𝜆(𝜏𝜔)1𝜏𝑖(𝜆+𝑧)11+𝜆𝑠𝐄𝑟𝑠𝛾+𝑠(𝜏)d𝜏.(9.3) For 𝜔𝐂,Re𝑧0, 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)=lim𝜔0𝜔𝐂𝑖𝜆(𝜆+𝑧)𝜔𝑖(𝜆+𝑧)𝜆+𝑧+𝜆𝑠1𝑒(𝜆+𝑧)𝜆=𝜆+𝑧+𝜆𝑠1𝑒(𝜆+𝑧).(9.4) Hence, by (9.2), 1(𝑧)=𝜆𝑧+𝜆(1+𝑧+𝜆𝑠𝜆+𝜆𝑠𝑒+𝜆(𝑧+𝜆).(9.5) Applying the inversion technology presented in [25] yields the exact survival function (𝑡)=𝑒𝜆𝑡𝜆(1+𝜆𝑠[𝑡]𝑘=0𝜆𝜆+𝜆𝑠𝑘(1𝑒𝜆𝑠𝑘(𝑡𝑘)𝑗=0(𝜆𝑠(𝑡𝑘))𝑗𝑗!)).(9.6) Figure 1 displays the graph of (𝑡),0𝑡20,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%.

905721.fig.001
Figure 1: Graph of (𝑡),0𝑡20,𝜆=0.5, and 𝜆𝑠=0.25 with security interval [0,𝜏], 𝜏=2.259, and security level 𝛿=0.9.

References

  1. A. Birolini, Reliability Engineering: Theory and Practice, Springer, Berlin, Germany, 2004.
  2. A. Birolini, Quality and Reliability of Technical Systems: Theory-Practice-Management, Springer, Berlin, Germany, 1994. View at Zentralblatt MATH
  3. I. B. Gertsbakh, Statistical Reliability Theory, vol. 4 of Probability: Pure and Applied, Marcel Dekker, New York, NY, USA, 1989. View at Zentralblatt MATH · View at MathSciNet
  4. B. Gnedenko and I. A. Ushakov, Probabilistic Reliability Engineering, John Wiley & Sons, New York, NY, USA, 1995.
  5. M. Shaked and I. G. Shanthikumar, “Reliability and maintainability,” in Handbook in Operations Research and Management Science 2, D. P. Heyman and M. J. Sobel, Eds., North-Holland, Amsterdam, The Netherlands, 1996. View at Google Scholar
  6. J. A. Buzacott, “Availability of priority redundant systems,” IEEE Transactions on Reliability, vol. 20, pp. 60–63, 1971. View at Google Scholar
  7. B. B. Fawzi and A. G. Hawkes, “Availability of a series system with replacement and repair,” Journal of Applied Probability, vol. 27, no. 4, pp. 873–887, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. R. Gupta, “Analysis of a two-unit cold standby system with degradation and linearly increasing failure rates,” International Journal of Systems Science, vol. 22, no. 11, pp. 2329–2338, 1991. View at Google Scholar · View at MathSciNet
  9. Y. Lam and Y. L. Zhang, “Repairable consecutive-k-out-of-n: F system with Markov dependence,” Naval Research Logistics, vol. 47, no. 1, pp. 18–39, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. S. Makhanov and E. J. Vanderperre, “A note on a Markov time related to a priority system,” WSEAS Transactions on Mathematics, vol. 6, no. 9, pp. 811–816, 2007. View at Google Scholar
  11. H. Mine, “Repair priority effect on the availability of a two-unit system,” IEEE Transactions on Reliability, vol. 28, pp. 325–326, 1979. View at Google Scholar
  12. D. Montoro-Cazorla and R. Pérez-Ocòn, “A deteriorating two-system with two repair modes and sojourn times phase-type distributed,” Reliability Engineering & System Safety, vol. 91, no. 1, pp. 1–9, 2006. View at Publisher · View at Google Scholar
  13. T. Nakagawa and S. Osaki, “Stochastic behaviour of a two-unit priority standby redundant system with repair,” Microelectronics and Reliability, vol. 14, no. 3, pp. 309–313, 1975. View at Google Scholar
  14. S. Osaki, “Reliability analysis of a two-unit standby redundant system with priority,” Canadian Journal of Operations Research, vol. 8, pp. 60–62, 1970. View at Google Scholar · View at Zentralblatt MATH
  15. D.-H. Shi and L. Liu, “Availability analysis of a two-unit series system with a priority shut-off rule,” Naval Research Logistics, vol. 43, no. 7, pp. 1009–1024, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. R. Subramanian and N. Ravichandran, “A two-unit priority redundant system with preemptive resume repair,” IEEE Transactions on Reliability, vol. 29, pp. 183–184, 1980. View at Google Scholar
  17. E. J. Vanderperre, “Long-run availability of a two-unit standby system subjected to a priority rule,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 7, no. 3, pp. 355–364, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. E. J. Vanderperre and S. S. Makhanov, “Long-run availability of a priority system: a numerical approach,” Mathematical Problems in Engineering, vol. 2005, no. 1, pp. 75–85, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. E. J. Vanderperre, “A Markov time related to a priority system,” Mathematical Problems in Engineering, vol. 2006, Article ID 92613, 9 pages, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  20. R. C. M. Yam, M. J. Zuo, and Y. L. Zhang, “A method for evaluation of reliability indices for repairable circular consecutive k-out-of-n: F systems,” Reliability Engineering & System Safety, vol. 79, pp. 1–9, 2003. View at Google Scholar
  21. F. D. Gakhov, Boundary Value Problems, Pergamon Press, Oxford, UK, 1996.
  22. E. J. Vanderperre, “A Sokhotski-Plemelj problem related to a robot-safety device system,” Operations Research Letters, vol. 27, no. 2, pp. 67–71, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. P. Brêmaud, Point Processes and Queues, Springer Series in Statistics, Springer, Berlin, Germany, 1981.
  24. T. M. Apostol, Mathematical Analysis, Addison-Wesley, Amsterdam, The Netherlands, 1998.
  25. E. J. Vanderperre, “Reliability analysis of a renewable multiple cold standby system,” Operations Research Letters, vol. 32, no. 3, pp. 288–292, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet