Journal of Applied Mathematics and Stochastic Analysis

Volume 2008, Article ID 905721, 13 pages

http://dx.doi.org/10.1155/2008/905721

## 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., [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 is skilled at repairing the failed **n**-unit, whereas repairman 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 related to a security level 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 with a probability larger than .
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 with finite mean and a general repair time
distribution .
The failure-free time and the repair time are denoted by and .
We assume that is Lebesgue absolutely continuous with a
density function (in the Radon-Nikodym sense) of bounded variation on .

(ii) The **n**-unit has a constant failure rate in the operative state and a constant failure
rate in standby. Note that the inequality 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 (resp., ). The *common* repair time of any **n**-failure is denoted by with (common) repair time distribution .
In addition, we assume that has finite mean and variance.

(iii) The random variables , and 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, Note that The corresponding Fourier-Stieltjes
transforms are called *dual* transforms. Without loss of generality (see Remark 7.4), we may assume that and have density functions of bounded variation on .
Note that the bounded variation property implies that, for
instance,

(v) In order to derive the survival function of
the **T**-system, we employ a stochastic process with discrete state space and absorbing state characterized by the following exhaustive set
of mutually exclusive events.

:
the **p**-unit is operative and the **n**-unit is in warm standby at time .

:
the **n**-unit is operative and the **p**-unit is under progressive repair at time .

:
the **p**-unit is operative and the **n**-unit is under progressive repair at time .

:
the **T**-system is down at time .

Note that the
absorbing state implies that a transition of the process into state is only possible via states or ,
whereas a transition from state or into state terminates the lifetime of the system.
Therefore, the inclusion of the absorbing state into the state space of the process triggers the introduction of a so-called *stopping
time*. Consequently, we first define the non-Markovian process on a filtered probability space where the *history * satisfies the Dellacherie conditions:

(i) contains the **P**-null sets of ;(ii)for
all ;
that is, the family is right-continuous. Consider the -stopping time where is the past failure-free time of the **p**-unit
being operative at time .
We assume that the **T**-system starts functioning at some time origin in state ;
that is, let , -a.s. Thus, from onwards, is the *survival* time (lifetime) of the **T**-system.
The corresponding survival function is denoted by .
Clearly, .
A (vector) Markov characterization of the non-Markovian process with absorbing state is piecewise and conditionally defined by

(1) if (i.e., if the event occurs), where denotes the remaining failure-free time of the **p**-unit being up at time ;(2) if ,
where denotes the remaining repair time of the **p**-unit
being under progressive repair at time ;(3) if ,
where denotes the remaining repair time of the **n**-unit
being under progressive repair at time ;(4) if (the absorbing state).

The state space of the underlying Markov process, with absorbing state , is given by For , let , .

(vi) Finally, we introduce the transition measures: Note that, for instance,

#### 3. Notations

(i) The indicator (function) of an event is denoted by .

(ii) The complex plane and the real line are, respectively, denoted by and with obvious superscript notations such as and . For instance, .

(iii) We frequently use the characteristic function: Note that

Property 3.1 (see [22, Appendix]). *The function , has no zeros in .*

(iv) The Heaviside unit step function, with the unit step at , is denoted by , that is,

(v) The greatest integer function is denoted by .

(vi) The Laplace transform of any locally integrable and bounded function on is denoted by the corresponding character marked with an asterisk. For instance, Observe that Moreover, by the product rule for Lebesgue-Stieltjes integrals (see, e.g., [23, Appendix])

(vii) Let , be a bounded and continuous function. is called -integrable if exists, where . The corresponding integral, denoted by is called a Cauchy principal value in double sense.

(viii) A function , is called Hölder-continuous on if for all , there exists : The function , is called Hölder-continuous at infinity if there exists : Hölder-continuous functions with exponent are called Lipschitz-continuous.

(ix) Note that the Hölder continuity of on and at infinity is sufficient for the existence of the Cauchy-type integral:

#### 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 .
Grouping terms of and taking the absorbing state into account reveal that Taking the
definition of *directional* derivative into account, for instance, entails that for ,
and , Note that the initial condition , **P**-a.s. implies that Moreover, .
Finally, observe that (4.3)–(4.6) are consistent with the probability law and that .

#### 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 . Consequently, each Laplace transform exists for . Moreover, the integrability of the density functions and the transition functions with regard to , and 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 , and , Adding (5.1) and (5.3) yields the functional equation

#### 6. Survival Function

In order to obtain the Laplace transform of the survival function, we first remark that by (5.4) and (3.6), Inserting (resp., ) into (5.2) entails that Finally, inserting into the functional equation (5.5) reveals that Invoking the relation yields by (6.1)–(6.4) that Hence, we only have to determine .

#### 7. Methodology

In order to derive the unknown , we first eliminate the function by the substitution of. Noting that reveals that by (5.5), Dividing (7.2) by the factor , taking Property 3.1 into account, yields the boundary value equation where Equation (7.3) constitutes a -dependent Sokhotski-Plemelj problem on , solvable by the theory of sectionally holomorphic functions (see, e.g., [21]). First, we need the following property.

Lemma 7.1. *The function , is Lipschitz-continuous on and at infinity.*

*Proof. *Note that Property 3.1 implies that .
Hence, the existence of ,
and entails that Consequently, by the mean value
theorem (see, e.g., [24]), there exists a constant such that for all , Hence, is Lipschitz-continuous on **R**.

Finally, note that the Lipschitz continuity of at infinity follows from the boundedness of and (2.3).

Corollary 7.2. *The function
** is sectionally holomorphic
and regular.*

Moreover, by (7.3), Note that (7.8) is only valid for . However, by the Sokhotski-Plemelj formula (see, e.g., [21, page 36]), where On the other hand, we have by continuity . Hence, The function 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 ** where *

*Remark 7.4. *It should be noted that Property 3.1 also holds for an *arbitrary * 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 also holds for an arbitrary with finite mean and variance. Therefore,
Lemma 7.1 remains valid for arbitrary .
The requirement of a finite variance is extremely mild. In fact, the current
probability distributions of interest to statistical reliability engineering
even have moments of any order. Finally, the functional exists for an arbitrary as a Lebesgue-Stieltjes integral on and has no impact on the existence of the
integral 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 ,
where for some ,
which is called the security level. In practice, is usually large. For instance, .
Therefore, we require that the **T**-system satisfy the risk criterion .
Note that the security interval, corresponding to the security level ,
ensures a continuous operation (survival) of the **T**-system up to time with probability larger than .
See the forthcoming example.

#### 9. Deterministic Repair

As an example, we consider the particular case of
deterministic repair (replacement); that is, let ,
where is taken as time unit. Clearly, .
Furthermore, let .
Note that By Property 7.3, we
have We recall that For ,
the integrand represents a *meromorphic* function in with single pole .
Moreover, the function vanishes at infinity in .
An application of the residue theorem entails that Hence, by (9.2), Applying the inversion
technology presented in [25] yields the *exact* survival function Figure 1 displays the graph of ,
and with the security interval , .
The interval ensures a continuous operation of the **T**-system up to time with a probability of at least 90%.

#### References

- A. Birolini,
*Reliability Engineering: Theory and Practice*, Springer, Berlin, Germany, 2004. - A. Birolini,
*Quality and Reliability of Technical Systems: Theory-Practice-Management*, Springer, Berlin, Germany, 1994. View at Zentralblatt MATH - 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 - B. Gnedenko and I. A. Ushakov,
*Probabilistic Reliability Engineering*, John Wiley & Sons, New York, NY, USA, 1995. - 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 - J. A. Buzacott, “Availability of priority redundant systems,”
*IEEE Transactions on Reliability*, vol. 20, pp. 60–63, 1971. View at Google Scholar - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - F. D. Gakhov,
*Boundary Value Problems*, Pergamon Press, Oxford, UK, 1996. - 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 - P. Brêmaud,
*Point Processes and Queues*, Springer Series in Statistics, Springer, Berlin, Germany, 1981. - T. M. Apostol,
*Mathematical Analysis*, Addison-Wesley, Amsterdam, The Netherlands, 1998. - 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