Value at Risk Associated with Maintenance of a Repairable System

Abstract

This paper presents the derivation of the probability distribution of maintenance cost of a repairable system modeled as an alternative renewal process and subjected to an age-based replacement policy. The key idea of the paper is to formulate a renewal equation for computing the characteristic function of the maintenance cost incurred in a fixed time interval. Then, the Fourier transform of the characteristic function leads to the complete probability distribution of cost. This approach also enables the derivation of probability distributions of the down time and the number of failures in a given time period. The distribution of the cost can be used to evaluate the value at risk (VaR) and other measures needed for the financial planning of a maintenance program.

Appendix

12.1.1 Computation of $F_{\phi } \left( {\Upomega ,k} \right)$ and $G_{\phi } \left( {\Upomega ,t} \right)$

Using the law of total expectation by conditioning on $X_{1} f_{\phi } \left( {\omega ,k} \right)$ is obtained as

$$\begin{aligned} f_{\phi } \left( {\omega ,k} \right) & = \sum\limits_{j = 1}^{{\rm{min}\left\{ {k,t_{p} } \right\}}} {{\rm E}\left[ {e^{{i\omega C_{1} }} \left| {X_{1} = j,Y_{1} = k - j} \right.} \right]} \\ & \quad P\left\{ {X_{1} = j,Y_{1} = k - j} \right\} \\ \end{aligned} $$

(12.18)

Note that the cost incurred in the first renewthat the cost incurred in the first renewal cycle is equal to $c_{D} Y_{1} + c_{R} \left( {X_{1} } \right),$ where

$$ c_{R} \left( {X_{1} } \right) = \left\{ {\begin{array}{*{20}c} {c_{{CM}} ,} \\ {c_{{PM}} ,} \\ \end{array} \begin{array}{*{20}c} {{\text{if}}} & {X_{1} < t_{p} ,} \\ {{\text{if}}} & {X_{1} = t_{p} ,} \\ \end{array} } \right.$$

(12.19)

is the repair cost. Furthermore, X1 and Y1 are independent of each other. Then Eq. (12.18) gives

$$ f_{\phi } \left( {\omega ,k} \right) = \sum\limits_{{j = 1}}^{{\text {min}\left\{ {k,t_{p} } \right\}}} {e^{{i\omega \left[ {c_{D} \left( {k - j} \right) + c_{R} (j)} \right]}} } f_{X} \left( j \right)f_{Y} \left( {k - j} \right) $$

(12.20)

Similarly, noting that $C\left( t \right) = c_{D} \left( {t - X_{1} } \right) + c_{R} \left( {X_{1} } \right)$ if $X_{1} < t < T$ and $C\left( t \right) = 0$ if $X_{1} > t,G_{\phi } \left( {\omega ,t} \right)$ is obtained as

$$\begin{aligned} G_{\phi } \left( {\omega ,t} \right) &= \sum\limits_{{j = 1}}^{{\text {min}\left\{ {t,t_{p} } \right\}}} {{\rm E}\left[ {e^{{i\omega C\left( t \right)}} \left| {X_{1} = j,Y_{1} > t - j} \right.} \right]} P\left\{ {X_{1} = j,Y_{1} > t - j} \right\} \\ &\quad+ {\rm E}\left[ {e^{{i\omega C\left( t \right)}} \left| {X_{1} > t} \right.} \right]P\left\{ {X_{1} > t} \right\} \\ &= \sum\limits_{{j = 1}}^{{\text {min}\left\{ {t,t_{p} } \right\}}} {e^{{i\omega \left[ {c_{D} \left( {t - j} \right) + c_{R} (j)} \right]}} } f_{X} (j)\bar{F}_{Y} \left( {t - j} \right) + \bar{F}_{X} \left( t \right) \\ \end{aligned} $$

(12.21)