A D-MMAP to Model a Complex Multi-state System with Loss of Units

Abstract

A complex multi-state system subject to different types of failures and preventive maintenance, with loss of units, is modelled by considering a discrete marked Markovian arrival process. The system is composed of K units, one online and the rest in cold standby. The online unit is submitted to different types of failures and when a non-repairable failure occurs the corresponding unit is removed. Several internal degradation states are considered which are observed when a random inspection occurs. This unit is subject to internal repairable failure, external shocks and preventive maintenance. If one internal repairable failure occurs, the unit goes to the repair facility for corrective repair, if a major degradation level is observed by inspection, the unit goes to preventive maintenance and when one external shock happens, this one may produce an aggravation of the internal degradation level, cumulative external damage or external extreme failure (non-repairable failure). Preventive maintenance and corrective repair times follow different distributions. The system is modelled in transient regime and relevant performance measures are obtained. All results are expressed in algorithmic and computational form and they have been implemented computationally with MATLAB and R. A numerical example shows the versatility of the model.

Appendix

The MMAP that governs the system has been introduced in Sect. 3. Some matrix blocks have been described in that section, the rest are given in this Appendix.

Matrix ${\mathbf{R}}_{p}^{k,O}$

The elements of the matrix ${\mathbf{R}}_{p}^{k,O}$ for k = 1, …, K are given by

$${\mathbf{R}}_{p}^{k,O} = \left( {\begin{array}{*{20}c} {{\mathbf{D}}_{00}^{k,O} } & {\mathbf{0}} & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ {{\mathbf{D}}_{10}^{k,O} } & {{\mathbf{D}}_{11}^{k,O} } & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{D}}_{21}^{k,O} } & {{\mathbf{D}}_{22}^{k,O} } & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ \vdots & {} & \ddots & \ddots & {} & \vdots \\ {\mathbf{0}} & {} & {} & {{\mathbf{D}}_{k - 1,k - 2}^{k,O} } & {{\mathbf{D}}_{k - 1,k - 1}^{k,O} } & {\mathbf{0}} \\ {\mathbf{0}} & \cdots & {} & {} & {{\mathbf{D}}_{k,k - 1}^{k,O} } & {{\mathbf{D}}_{kk}^{k,O} } \\ \end{array} } \right),$$

where

$${\mathbf{D}}_{00}^{k,O} = {\mathbf{H}}_{0} + {\mathbf{H}}_{2}^{\prime} I_{{\left\{ {k = 1} \right\}}} ,$$

${\mathbf{D}}_{l,l}^{k,O} \left( {i_{1} , \ldots ,i_{l} ;i_{1} , \ldots ,i_{l} } \right) = \left( {{\mathbf{H}}_{0} + {\mathbf{H}}_{2}^{\prime} I_{{\left\{ {l = k - 1} \right\}}} } \right) \,\otimes \, {\mathbf{S}}_{{i_{1} }}$, for $l = 1, \ldots ,k - 1; \, i_{s} = 1,2,3,4{\text{ for }}s = 1, \ldots ,l$,

$${\mathbf{D}}_{k,k}^{k,O} \left( {i_{1} , \ldots ,i_{k} ,;i_{1} , \ldots ,i_{k} } \right) = \left( {{\mathbf{L}} + {\mathbf{L}}^{0} {\varvec{\upgamma}}} \right) \,\otimes \, {\mathbf{S}}_{{i_{1} }} ,$$

for $i_{s} = 1,2,3,4{\text{ for }}s = 1, \ldots ,k;k \ne K, \, i_{k} = 1,k = K,$

$$\begin{array}{*{20}c} {{\mathbf{D}}_{10}^{k,O} \left( {i_{1} } \right) = {\mathbf{H}}_{0} \,\otimes \, {\mathbf{S}}_{{i_{1} }}^{0} } & ; & {i_{1} = 1,2,3,4} \\ \end{array} ;k \ge 2,$$

$$\begin{array}{*{20}c} {{\mathbf{D}}_{l,l - 1}^{k,O} \left( {i_{2} , \ldots ,i_{l} ;i_{1} , \ldots ,i_{l} } \right) = {\mathbf{H}}_{0} \,\otimes \, {\mathbf{S}}_{{i_{1} }}^{0} {\varvec{\upbeta}}^{{i_{2} }} } & ; & {l = 2, \ldots ,k - 1; \, i_{s} = 1,2,3,4{\text{ for }}s = 1, \ldots ,l} \\ \end{array} ,$$

$${\mathbf{D}}_{k,k - 1}^{k,O} \left( {i_{2} , \ldots ,i_{k} ;i_{1} , \ldots ,i_{k} } \right) = {\varvec{\upalpha}} \,\otimes \, \left( {{\mathbf{L}} + {\mathbf{L}}^{0} {\varvec{\upgamma}}} \right) \,\otimes \, {\varvec{\upomega}} \,\otimes \, {\varvec{\upeta}} \,\otimes \, {\mathbf{S}}_{{i_{1} }}^{0} {\varvec{\upbeta}}^{{i_{2} }} ,$$

for $i_{s} = 1,2,3,4{\text{ for }}s = 1, \ldots ,k;k \ne K, \, i_{k} = 1,k = K$,

$${\mathbf{D}}_{10}^{1,O} \left( 1 \right) = {\varvec{\upalpha}} \,\otimes \, \left( {{\mathbf{L}} + {\mathbf{L}}^{0} {\varvec{\upgamma}}} \right) \,\otimes \, {\varvec{\upomega}} \,\otimes \, {\varvec{\upeta}} \,\otimes \, {\mathbf{S}}_{1}^{0} .$$

Matrix ${\mathbf{R}}_{p}^{{k,B_{i} }}$

The elements of the matrix ${\mathbf{R}}_{p}^{{k,B_{i} }}$ for i = 1, 2, 3 and for k = 2, …, K are given by

$${\mathbf{R}}_{p}^{{k,B_{i} }} = \left( {\begin{array}{*{20}c} {\mathbf{0}} & {{\mathbf{D}}_{01}^{{k,B_{i} }} } & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{D}}_{11}^{{k,B_{i} }} } & {{\mathbf{D}}_{12}^{{k,B_{i} }} } & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{D}}_{22}^{{k,B_{i} }} } & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ \vdots & {} & \ddots & \ddots & {} & \vdots \\ {\mathbf{0}} & {} & {} & {{\mathbf{D}}_{k - 2,k - 2}^{{k,B_{i} }} } & {{\mathbf{D}}_{k - 2,k - 1}^{{k,B_{i} }} } & {\mathbf{0}} \\ {\mathbf{0}} & \cdots & {} & {} & {{\mathbf{D}}_{k - 1,k - 1}^{{k,B_{i} }} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & \cdots & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right),\quad {\mathbf{R}}_{p}^{{1,B_{i} }} = {\mathbf{0}}$$

$${\mathbf{D}}_{01}^{{k,B_{i} }} \left( 2 \right) = {\mathbf{H}}_{2}^{i} \,\otimes \, {\varvec{\upbeta}}^{2} ,$$

$$\begin{array}{*{20}c} {{\mathbf{D}}_{l,l + 1}^{{k,B_{i} }} \left( {i_{1} , \ldots ,i_{l} ,2;i_{1} , \ldots ,i_{l} } \right) = {\mathbf{H}}_{2}^{i} \,\otimes \, {\mathbf{S}}_{{i_{1} }} } & ; & {l = 1, \ldots ,k - 2} \\ \end{array} ;i_{s} = 1,2,3,4{\text{ for }}s = 1, \ldots ,l,$$

$$\begin{array}{*{20}c} {{\mathbf{D}}_{l,l}^{{k,B_{i} }} \left( {i_{2} , \ldots ,i_{l} ,2;i_{1} , \ldots ,i_{l} } \right) = {\mathbf{H}}_{2}^{i} \,\otimes \, {\mathbf{S}}_{{i_{1} }}^{0} {\varvec{\upbeta}}^{{i_{2} }} } & ; & {l = 1, \ldots ,k - 1} \\ \end{array} ;i_{s} = 1,2,3,4{\text{ for }}s = 1, \ldots ,l.$$

Matrix ${\mathbf{R}}_{p}^{k,C}$

The elements of the matrix ${\mathbf{R}}_{s}^{k,C}$ for k = 2, …, K are given by

$${\mathbf{R}}_{s}^{k,C} = \left( {\begin{array}{*{20}c} {{\mathbf{D}}_{00}^{k,C} } & {\mathbf{0}} & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ {{\mathbf{D}}_{10}^{k,C} } & {{\mathbf{D}}_{11}^{k,C} } & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{D}}_{21}^{k,C} } & {{\mathbf{D}}_{22}^{k,C} } & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ \vdots & {} & \ddots & \ddots & {} & \vdots \\ {\mathbf{0}} & {} & {} & {{\mathbf{D}}_{k - 1,k - 2}^{k,C} } & {{\mathbf{D}}_{k - 1,k - 1}^{k,C} } & {\mathbf{0}} \\ {\mathbf{0}} & \cdots & {} & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right)$$

${\mathbf{D}}_{00}^{k,C} = {\mathbf{H}}_{3}^{{}} I_{{\left\{ {k > 1} \right\}}}$,

$$\begin{array}{*{20}c} {{\mathbf{D}}_{l,l}^{k,C} \left( {i_{1} , \ldots ,i_{l} ;i_{1} , \ldots ,i_{l} } \right) = \left( {I_{{\left\{ {k \ne K - 1} \right\}}} {\mathbf{H}}_{3}^{{}} + I_{{\left\{ {k = K - 1} \right\}}} {\mathbf{H}}_{3}^{\prime} } \right) \,\otimes \, {\mathbf{S}}_{{i_{1} }} } & ; & {{\text{for }}l = 1, \ldots ,k - 1} \\ \end{array} ,$$

and $i_{s} = 1,2,3,4{\text{ for }}s = 1, \ldots ,l$,

$$\begin{array}{*{20}c} {{\mathbf{D}}_{l,l - 1}^{k,C} \left( {i_{2} , \ldots ,i_{l} ;i_{1} , \ldots ,i_{l} } \right) = {\mathbf{H}}_{3}^{{}} \,\otimes \, {\mathbf{S}}_{{i_{1} }}^{0} {\varvec{\upbeta}}^{{i_{2} }} } & ; & {l = 2, \ldots ,k - 1 \, ;} \\ \end{array} \, i_{s} = 1,2,3,4{\text{ for }}s = 1, \ldots ,l,$$

$$\begin{array}{*{20}c} {{\mathbf{D}}_{10}^{k,C} \left( {i_{1} } \right) = {\mathbf{H}}_{3}^{{}} \,\otimes \, {\mathbf{S}}_{{i_{1} }}^{0} } & ; & {i_{s} = 1,2,3,4{\text{ for }}s = 1, \ldots ,l} \\ \end{array} .$$

Matrix ${\mathbf{R}}_{s}^{k,C}$

The elements of the matrix ${\mathbf{R}}_{s}^{k,C}$ for k = 2, …, K are given by

${\mathbf{R}}_{s}^{k,C} = \left( {\begin{array}{*{20}c} {{\mathbf{D}}_{00}^{k,C} } & {\mathbf{0}} & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ {{\mathbf{D}}_{10}^{k,C} } & {{\mathbf{D}}_{11}^{k,C} } & {\mathbf{0}} & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{D}}_{21}^{k,C} } & {{\mathbf{D}}_{22}^{k,C} } & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ \vdots & {} & \ddots & \ddots & {} & \vdots \\ {\mathbf{0}} & {} & {} & {{\mathbf{D}}_{k - 1,k - 2}^{k,C} } & {{\mathbf{D}}_{k - 1,k - 1}^{k,C} } & {\mathbf{0}} \\ {\mathbf{0}} & \cdots & {} & \cdots & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right)$,

where

$${\mathbf{D}}_{00}^{k,C} = {\mathbf{H}}_{3}^{{}} I_{{\left\{ {k > 1} \right\}}} ;$$

${\mathbf{D}}_{l,l}^{k,C} \left( {i_{1} , \ldots ,i_{l} ;i_{1} , \ldots ,i_{l} } \right) = \left( {I_{{\left\{ {k \ne K - 1} \right\}}} {\mathbf{H}}_{3}^{{}} + I_{{\left\{ {k = K - 1} \right\}}} {\mathbf{H}}_{3}^{\prime} } \right) \,\otimes \, {\mathbf{S}}_{{i_{1} }}$,

for $l = 1, \ldots ,k - 1;i_{s} = 1,2,3,4{\text{ for }}s = 1, \ldots ,l$,

${\mathbf{D}}_{l,l - 1}^{k,C} \left( {i_{2} , \ldots ,i_{l} ;i_{1} , \ldots ,i_{l} } \right) = {\mathbf{H}}_{3}^{{}} \,\otimes \, {\mathbf{S}}_{{i_{1} }}^{0} {\varvec{\upbeta}}^{{i_{2} }}$, for $l = 2, \ldots ,k - 1; \, i_{s} = 1,2,3,4{\text{ for }}s = 1, \ldots ,l$,

${\mathbf{D}}_{10}^{k,C} \left( {i_{1} } \right) = {\mathbf{H}}_{3}^{{}} \,\otimes \, {\mathbf{S}}_{{i_{1} }}^{0}$, for $i_{s} = 1,2,3,4{\text{ for }}s = 1, \ldots ,l$.

Matrix ${\mathbf{R}}_{p}^{1,FC}$

The elements of the matrix ${\mathbf{R}}_{s}^{1,FC}$ are given by

$${\mathbf{R}}_{s}^{1,FC} = \left( {\begin{array}{*{20}c} {{\mathbf{D}}_{00}^{1,C} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right),{\text{where}}\,{\mathbf{D}}_{00}^{1,C} = {\mathbf{H}}_{3}$$