Availability analysis of a repairable duplex system: a z-dependent Sokhotski–Plemelj problem

Abstract

We analyse the point availability of a repairable duplex system characterized by cold standby and by a priority rule. The system is attended by two (general) heterogeneous repairmen. To describe the random behaviour of the system, we introduce a stochastic process endowed with probability 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. The unique solution of the equations determines the point availability of the system. Computational results for the point availability are derived by a numerical solution of an appropriate integral equation.

Appendix

For direct reference, we propose to state some particular properties of sectionally holomorphic functions and their ramifications for the solution of some boundary value problems on the real line. See Gakhov (1994, pp. 1–360), Roos (1996, pp. 118–242) for proofs and details. Let $\varphi(\tau)$ be a function satisfying the Hölder condition on R and infinity. In addition let

$$ \mathcal{L}^+(u):=\lim_{\mathop{\scriptstyle w \rightarrow u}\limits_{\scriptstyle w \in{\bf C}^+}} \frac{1}{2\pi i} \int\limits_\Upgamma \varphi(\tau)\frac{\rm{d}\tau}{\tau-w}, u \in {\bf R}, $$

$$ \mathcal{L}^-(u):=\lim_{\mathop{\scriptstyle w \rightarrow u}\limits_{\scriptstyle w \in{\bf C}^-}} \frac{1}{2\pi i} \int\limits_\Upgamma \varphi(\tau)\frac{\rm{d}\tau}{\tau-w}, u \in {\bf R}. $$

we have

$$ \mathcal{L}^+(u)=\frac{1}{2}\varphi(u)+\frac{1}{2\pi i} \int\limits_\Upgamma \varphi(\tau)\frac{\rm{d}\tau}{\tau-u}, $$

(27)

$$ \mathcal{L}^-(u)=-\frac{1}{2}\varphi(u)+\frac{1}{2\pi i} \int\limits_\Upgamma \varphi(\tau)\frac{\rm{d}\tau}{\tau-u}. $$

(28)

Hence, for $u \in {\bf R}$

$$ \mathcal{L}^+(u)-\mathcal{L}^-(u)=\varphi(u), $$

(29)

$$ \frac{\mathcal{L}^-(u)+\mathcal{L}^-(u)}{2}=\frac{1}{2\pi i} \int\limits_\Upgamma \varphi(\tau)\frac{\rm{d}\tau}{\tau-u}. $$

(30)

The relations (27)–(30) are called the Sokhotski–Plemelj formulas on the real line. The functions $\mathcal{L}^+(u), \mathcal{L}^-(u)$ are continuous on R and infinity. The function $\varphi(\tau)$ has a unique decomposition, and the resulting boundary value Eq. (29) has a unique regular solution

$$ \frac{1}{2\pi i} \int\limits_\Upgamma \varphi(\tau)\frac{\rm{d}\tau}{\tau-w} $$

valid for all $w \in {\bf C}$ and the Cauchy-type integral generates a regular sectionally holomorphic function in C cut along the real line.

Furthermore,

$$ \begin{aligned} \mathcal{L}^+(w)&=\int\limits_\Upgamma \varphi(\tau)\frac{\rm{d}\tau}{\tau-w}, w \in {\bf C}^+, \\ \mathcal{L}^-(w)&=\int\limits_\Upgamma \varphi(\tau)\frac{\rm{d}\tau}{\tau-w}, w \in {\bf C}^-. \end{aligned} $$