Availability, MTTF and cost analysis of a system having two units in series configuration with controller

Abstract

The paper deals with the availability analysis of a system, which consists of two subsystems namely subsystem-1 and subsystem-2. Subsystem-1 is working under k-out of n: good configuration while subsystem-2 has two identical units connected in parallel configuration. A controller is attached with each subsystem for proper functioning of the system. All failure rates are constant but repairs follow general and exponential distributions. The transitional state probabilities, asymptotic behavior and some characteristics such as reliability, availability, MTTF and the cost effectiveness of the system have been evaluated with the help of supplementary variable technique, Laplace transformations and copula methodology. At last, some particular cases and numerical examples have been taken to describe the model.

Appendices

Appendix 1

Equation 1 has been obtained by limiting case of following probabilistic difference equation:

$$ \begin{aligned} P_{0} (t + \Updelta ) & = (1 - \lambda_{1} \Updelta t)(1 - \lambda_{C} \Updelta t)(1 - \lambda_{CB} \Updelta t)(1 - \lambda_{h} \Updelta t)(1 - 2\lambda \Updelta t)P_{0} (t) + \int\limits_{0}^{\infty } {\phi (x)P_{1} (x,t)} {{dx}}\Updelta t \\ + & \int\limits_{0}^{\infty } {\exp [x^{\theta } + \{ \log \phi (x)\}^{\theta } ]^{1/\theta } P_{2} } (x,t){{dx}}\Updelta t + \int\limits_{0}^{\infty } {\phi (x)P_{3} (x,t)} {{dx}}\Updelta t + \int\limits_{0}^{\infty } {\phi (x)P_{4} } (x,t){{dx}}\Updelta t \\ + & \int\limits_{0}^{\infty } {\exp [x^{\theta } + \{ \log \phi (x)\}^{\theta } ]^{1/\theta } P_{5} } (x,t){{dx}}\Updelta t + \int\limits_{0}^{\infty } {\exp [x^{\theta } + \{ \log \phi (x)\}^{\theta } ]^{1/\theta } P_{6} } (x,t){{dx}}\Updelta t \\ + & \int\limits_{0}^{\infty } {\exp [x^{\theta } + \{ \log \phi (x)\}^{\theta } ]^{1/\theta } P_{9} } (x,t){{dx}}\Updelta t \\ \end{aligned} $$

Now $\mathop {\lim }\limits_{\Updelta t \to 0\;} \;\frac{{P_{0} (t + \Updelta ) - P_{0} (t)}}{\Updelta t} + (\lambda_{1} + \lambda_{C} + \lambda_{CB} + \lambda_{h} + 2\lambda ) \cdots = \int\limits_{0}^{\infty } {\phi (x)P_{1} (x,t)} {{dx}} + \cdots P_{0} (t)$ yield Eq. 1 and similarly Eqs. 2–10 have been obtained.

Appendix 2

Solving 22–30, one can have

$$ \bar{P}_{1} (x,s) = \bar{P}_{1} (0,s)\exp \left( {( - (s + \lambda_{2} + 2\lambda + \lambda_{C} + \lambda_{CB} + \lambda_{h} } \right)e^{{ - \int\limits_{0}^{x} {\phi (x){{dx}}} }} $$

where

$$\bar{P}_{1} (0,s) = \lambda_{1} \bar{P}_{0} (s),\quad S_{\phi } (x) = \phi (x)e^{{\int\limits_{0}^{x} {\phi (x){{dx}}} }} ,\quad S_{\phi } (s) = \int\limits_{0}^{\infty } {e^{ - sx} } \phi (x)e^{{\int\limits_{0}^{x} {\phi (x){{dx}}} }} {{dx}}. $$

This implies that

$$\bar{P}_{1} (s) = \int\limits_{0}^{\infty } {\bar{P}_{1} (x,s){{dx}}} $$

. So on. Using values obtained by solving 22–30, one can get the transition state probabilities of the system.

Appendix 3

We have

$$ \begin{aligned} {{D}}(s) = s + \lambda_{1} + \lambda_{C} + \lambda_{h} + \lambda_{CB} + 2\lambda \\ - \left( {\lambda_{1} + \frac{{2\lambda \lambda_{1} S_{\phi } (s + \lambda + \lambda_{C} + \lambda_{CB} + \lambda_{h} )}}{{\left( {1 - 2\lambda S_{\phi } (s + \lambda + \lambda_{C} + \lambda_{CB} + \lambda_{h} )} \right)}}} \right)\bar{S}_{\phi } (s + \lambda_{1} + \lambda_{C} + \lambda_{h} + \lambda_{CB} + 2\lambda ) \\ - \left( {\lambda_{2} \lambda_{1} + \frac{{2\lambda \lambda_{1} S_{\phi } (s + \lambda + \lambda_{C} + \lambda_{CB} + \lambda_{h} )}}{{\left( {1 - 2\lambda S_{\phi } (s + \lambda + \lambda_{C} + \lambda_{CB} + \lambda_{h} )} \right)}}} \right)S_{{\mu_{0} }} (x) + 2\lambda S_{\phi } (s + \lambda + \lambda_{C} + \lambda_{CB} + \lambda_{h} ) \\ - 2\lambda^{2} \left( {\lambda_{1} + \frac{{2\lambda \lambda_{1} S_{\phi } (s + \lambda + \lambda_{C} + \lambda_{CB} + \lambda_{h} )}}{{\left( {1 - 2\lambda S_{\phi } (s + \lambda + \lambda_{C} + \lambda_{CB} + \lambda_{h} )} \right)}}} \right)S_{{\mu_{0} }} (x) \\ - \lambda_{h} \left( {1 + \lambda_{1} + \frac{{2\lambda \lambda_{1} S_{\phi } (s + \lambda + \lambda_{C} + \lambda_{CB} + \lambda_{h} ) + 2\lambda \lambda_{{}} }}{{\left( {1 - 2\lambda S_{\phi } (s + \lambda + \lambda_{C} + \lambda_{CB} + \lambda_{h} )} \right)}} + 2\lambda } \right)S_{{\mu_{0} }} (x) \\ - \lambda_{C} \left( {1 + \lambda_{1} + \frac{{2\lambda \lambda_{1} S_{\phi } (s + \lambda + \lambda_{C} + \lambda_{CB} + \lambda_{h} ) + 2\lambda \lambda_{{}} }}{{\left( {1 - 2\lambda S_{\phi } (s + \lambda + \lambda_{C} + \lambda_{CB} + \lambda_{h} )} \right)}} + 2\lambda } \right)S_{{\mu_{0} }} (x) \\ - \lambda_{CB} \left( {1 + \lambda_{1} + \frac{{2\lambda \lambda_{1} S_{\phi } (s + \lambda + \lambda_{C} + \lambda_{CB} + \lambda_{h} ) + 2\lambda \lambda_{{}} }}{{\left( {1 - 2\lambda S_{\phi } (s + \lambda + \lambda_{C} + \lambda_{CB} + \lambda_{h} )} \right)}} + 2\lambda } \right)S_{{\mu_{0} }} (x) \\ \end{aligned} $$

when s = 0, D(s) becomes zero and $P_{0} (0) = \mathop {\lim }\nolimits_{s \to 0} sP_{0} (s) = \mathop {\lim }\nolimits_{s \to 0} \frac{s}{{{{D}}(s)}} = \frac{1}{{{{D}}^{ '} (0)}}$ by D’L Hospital rule $S_{\phi } (s) = \int\nolimits_{0}^{\infty } {e^{ - sx} } \phi (x)e^{{\int\nolimits_{0}^{x} {\phi (x){{dx}}} }} {{dx}} = \frac{\phi }{s + \phi }$ implies that $S_{\phi } (0)\; = \frac{\phi }{0 + \phi } = 1$and so on.