Reliability prediction of fault tolerant machining system with reboot and recovery delay

Abstract

The present paper deals with the reliability analysis of fault tolerant multi-component machining system having multi-warm spares and reboot provisioning. The time-to-breakdown and repair of active/spare units and server are assumed to be exponentially distributed. The reboot process and recovery delay are also counterfeited exponentially distributed. The spectral method is adapted to compute the transient state probabilities of the system states. In order to predict the transient behavior of the system, various performance measures such as reliability function, mean-time-to-failure $\left( {MTTF} \right)$, etc. have been established. To show the practicability of the developed model, we present numerical results by taking an illustration. The sensitivity of various system parameters on the reliability function and $MTTF$ has also been examined.

Appendix 1

$$\begin{aligned} {\mathbf{A}}_{{\mathbf{n}}} & = \left[ {\begin{array}{*{20}c} { - \left( {s + \sigma + n\gamma + \lambda_{c} } \right)} & {\lambda \left( {1 - c} \right)} & {\lambda \left( {1 - c} \right)} \\ 0 & { - \left( {s + \lambda + n\gamma + \alpha + \lambda_{c} } \right)} & \beta \\ 0 & \alpha & { - \left( {s + \lambda + n\gamma + \beta + \lambda_{c} } \right)} \\ \end{array} } \right] \\ {\mathbf{A}}_{{\mathbf{m}}} & = \left[ {\begin{array}{*{20}c} { - \left( {s + \sigma + m\gamma + \lambda_{c} } \right)} & {\lambda \left( {1 - c} \right)} & {\lambda \left( {1 - c} \right)} & {\lambda \left( {1 - c} \right)} \\ 0 & { - \left( {s + \lambda + m\gamma + \alpha + \mu + \lambda_{c} } \right)} & \beta & 0 \\ 0 & \alpha & { - \left( {s + \lambda + m\gamma + \beta + \lambda_{c} } \right)} & 0 \\ 0 & 0 & 0 & { - \left( {s + \lambda + m\gamma + \lambda_{c} } \right)} \\ \end{array} } \right];\quad 1 \le m \le n - 1 \\ {\mathbf{A}}_{{\mathbf{0}}} & = \left[ {\begin{array}{*{20}c} s & \lambda & \lambda & \lambda \\ 0 & { - \left( {s + \lambda + \alpha + \mu } \right)} & \beta & 0 \\ 0 & \alpha & { - \left( {s + \lambda + \beta } \right)} & 0 \\ 0 & 0 & 0 & { - \left( {s + \lambda } \right)} \\ \end{array} } \right] \\ \end{aligned}$$

$$\begin{aligned} {\mathbf{B}}_{{\mathbf{1}}} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] \hfill \\ {\mathbf{B}}_{{\mathbf{0}}} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] \hfill \\ \end{aligned}$$

$$\begin{aligned} {\mathbf{C}}_{{\mathbf{n}}} & = \left[ {\begin{array}{*{20}c} {n\gamma } & 0 & 0 \\ \sigma & {\lambda c + n\gamma C} & 0 \\ 0 & 0 & {\lambda c + n\gamma C} \\ 0 & {n\gamma \left( {1 - C} \right)} & {n\gamma \left( {1 - C} \right)} \\ \end{array} } \right] \\ {\mathbf{C}}_{{\mathbf{m}}} & = \left[ {\begin{array}{*{20}c} {m\gamma } & 0 & 0 & 0 \\ \sigma & {\lambda c + m\gamma C} & 0 & {\lambda c + m\gamma C} \\ 0 & 0 & {\lambda c + m\gamma C} & 0 \\ 0 & {m\gamma \left( {1 - C} \right)} & {m\gamma \left( {1 - C} \right)} & {m\gamma \left( {1 - C} \right)} \\ \end{array} } \right];\quad 1 \le m \le n - 1 \\ {\mathbf{C}}_{{\mathbf{1}}} & = \left[ {\begin{array}{*{20}c} {\lambda_{c} } & {\lambda_{c} } & {\lambda_{c} } & {\lambda_{c} } \\ \sigma & {\lambda c + \gamma C} & 0 & {\lambda c + \gamma C} \\ 0 & 0 & {\lambda c + \gamma C} & 0 \\ 0 & {\gamma \left( {1 - C} \right)} & {\gamma \left( {1 - C} \right)} & {\gamma \left( {1 - C} \right)} \\ \end{array} } \right] \\ \end{aligned}$$

$$\begin{aligned} {\mathbf{D}}_{{\mathbf{1}}} = \left[ {\begin{array}{*{20}c} {\lambda_{c} } & {\lambda_{c} } & {\lambda_{c} } \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right] \hfill \\ {\mathbf{D}}_{{\mathbf{0}}} = \left[ {\begin{array}{*{20}c} {\lambda_{c} } & {\lambda_{c} } & {\lambda_{c} } & {\lambda_{c} } \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] \hfill \\ \end{aligned}$$