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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%.

Standby provides a powerful tool to enhance the
reliability, availability, quality, and safety of operational plants (see,
e.g., [

Cold or warm standby systems, subjected to priority
rules, have received considerable attention in previous literature (see, e.g.,
[

In order to determine the survival function of the

Consider the

(i) The

(ii) The

(iii) The random variables

(iv) Characteristic functions are formulated in
terms of a complex transform variable. For instance,

(v) In order to derive the survival function of
the

Note that the
absorbing state

for
all

The state space of the underlying Markov process, with
absorbing state

(vi) Finally, we introduce the transition measures:

(i) The indicator (function) of an event

(ii) The complex plane and the real line are,
respectively, denoted by

(iii) We frequently use the characteristic function:

The function

(iv) The Heaviside unit step function, with the
unit step at

(v) The greatest integer function is denoted by

(vi) The Laplace transform of any locally
integrable and bounded function on

(vii) Let

(viii) A function

(ix) Note that the Hölder continuity of

In order to derive a system of differential equations,
we observe the random behavior of the

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

Applying a Laplace-Fourier transform technique to
(

In order to obtain the Laplace transform of the
survival function, we first remark that by (

In order to derive the unknown

The function

Note that Property

Finally, note that the Lipschitz continuity of

The function

Moreover, by (

The Laplace transform of the survival function is
given by

It should be noted that Property

Along with the survival function of the

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

Graph of