Stochastic behaviour analysis of a Markov model under multi-state failures

Abstract

To solve a real problem, how to calculate the reliability of a system with time-varying failure rates in an industry system. This is a study of a system consisting of three identical units in parallel, each of which can be in two states: operational or non-operational. Under the assumptions that each unit has a constant failure and repair time rate. The system is regarded as operational if at least one of its elements is working. The system analyzed incorporating three types of failures namely partial failure, human failure and catastrophic failure, which also leads to the system failure. With the help of probability analysis, supplementary variable technique, Laplace transformations, and the Markov process theory, the transition state probabilities, availability, reliability, mean time to failure, sensitivity analysis and cost effectiveness of the system have been evaluated. Finally, some particular cases, numerical examples and graphical illustrations have also been taken to describe the model.

Appendices

Appendix 1

By the probability of the considerations and continuity arguments, we can obtain the set of differential equations governing the present mathematical model. At any time t, the probability that the system is in state S _i at time t and remains there in the interval (t, t + Δt) or/and if it is in some other state at time t then it should be transit to the state S _i in the interval (t, t + Δt) provided transition exists between the states and Δt → 0. Accordingly, the Eqs. (28) to (35) are interpreted as given below:

The probability of the system to be in state S ₀ in the interval (t, t + Δt) is given by

$$ P_{0} (t + \varDelta t)\; = (1 - 3\lambda_{1} \varDelta t)(1 - \lambda_{h} \varDelta t)(1 - \lambda_{csf} \varDelta t)(1 - \alpha_{j} \varDelta t)P_{0} (t)\; + \int\limits_{0}^{\infty } {P_{13} } (x,t)\upsilon (x)\varDelta tdx $$

$$ + \int\limits_{0}^{\infty } {P_{5j} (y,t)\phi (y)} \varDelta tdy + \int\limits_{0}^{\infty } {P_{6h} (z,t)\mu (z)\varDelta tdz} $$

$$ \begin{aligned} & \Rightarrow \mathop {\lim }\limits_{\varDelta t \to 0} \frac{{P_{0} (t + \varDelta t) - P_{0} (t)}}{\varDelta t}\; + (3\lambda {}_{1} + \lambda_{h} + \lambda_{csf} + \alpha_{j} )P_{0} (t) \\ & = \int\limits_{0}^{\infty } {P_{13} (x,t)\upsilon (x)} dx + \int\limits_{0}^{\infty } {P_{5j} (y,t)\phi (y)} dy + \int\limits_{0}^{\infty } {P_{6h} (z,t)\mu (z)} dz \\ & \Rightarrow \left[ {\frac{\partial }{\partial t} + 3\lambda_{1} + \lambda_{h} + \lambda_{csf} + \alpha_{j} } \right]P_{0} (t) = \int\limits_{0}^{\infty } {P_{13} (x,t)\upsilon (x)} dx + \int\limits_{0}^{\infty } {P_{5j} (y,t)\phi (y)} dy + \int\limits_{0}^{\infty } {P_{6h} (z,t)\mu (z)} dz \\ \end{aligned} $$

(28)

For state S ₁,

$$ \begin{aligned} P_{11} (t + \varDelta t) & \; = (1 - 2\lambda_{1} \varDelta t)(1 - \alpha_{j} \varDelta t)P_{11} (t)\; + 3\lambda_{1} \varDelta tP_{0} (t) \\ & \Rightarrow \mathop {\lim }\limits_{\varDelta t \to 0} \frac{{P_{11} (t + \varDelta t) - P_{11} (t)}}{\varDelta t} + (2\lambda_{1} + (1 - \alpha_{j} ))P_{11} (t)\; = 3\lambda_{1} P_{0} (t) \\ & \Rightarrow \left[ {\frac{\partial }{\partial t} + 2\lambda_{1} + \alpha_{j} } \right]P_{11} (t) = 3\lambda_{1} P_{0} (t) \\ \end{aligned} $$

(29)

For state S ₂,

$$ \begin{aligned} P_{12} (t + \varDelta t)\; & = (1 -\lambda_{1} \varDelta t)(1 - \alpha_{j} \varDelta t)P_{12} (t)\; +2\lambda_{1} \varDelta tP_{11} (t) \\ & \Rightarrow \mathop{\lim }\limits_{\varDelta t \to 0} \frac{{P_{12} (t + \varDelta t)- P_{12} (t)}}{\varDelta t} + (\lambda_{1} + \alpha_{j} )P_{12}(t)\; = 2\lambda_{1} P_{11} (t) \\ & \Rightarrow \left[{\frac{\partial }{\partial t} + \lambda_{1} + \alpha_{j} }\right]P_{12} (t) = 2\lambda_{1} P_{11} (t) \\ \end{aligned} $$

(30)

For state S ₃,

$$ \begin{aligned} P_{13} (x + \varDelta x,t + \varDelta t)\;& = \{ 1 - \upsilon (x)\varDelta t\} P_{13} (x,t)\; \\ &\Rightarrow \mathop {\lim }\limits_{\begin{subarray}{l} \varDelta x \to 0 \\ \varDelta t \to 0 \end{subarray} } \frac{{P_{13} (x +\varDelta x,t + \varDelta t) - P_{13} (x,t)}}{\varDelta t} +\upsilon (x)P_{13} (x,t) = 0 \\ & \Rightarrow \left[{\frac{\partial }{\partial t} + \frac{\partial }{\partial x} +\upsilon (x)} \right]P_{13} (x,t) = 0 \\ \end{aligned} $$

(31)

For state S ₄,

$$ \begin{aligned} P_{4j} (t + \varDelta t)\; & = (1 -\omega_{j} \varDelta t)P_{4j} (t)\; + \alpha_{j} \varDelta t(P_{0}(t) + P_{11} (t) + P_{12} (t)) \\ & \Rightarrow \mathop {\lim}\limits_{\varDelta t \to 0} \frac{{P_{4j} (t + \varDelta t) -P_{4j} (t)}}{\varDelta t} + \omega_{j} P_{4j} (t)\; = \alpha_{j}(P_{0} (t) + P_{11} (t) + P_{12} (t)) \\ & \Rightarrow \left[{\frac{\partial }{\partial t} + \omega_{j} } \right]P_{4j} (t) =\alpha_{j} (P_{0} (t) + P_{11} (t) + P_{12} (t)) \\ \end{aligned}$$

(32)

For state S₅

$$ \begin{aligned} P_{5j} (y + \varDelta y,t + \varDelta t)\; & = \phi (y)\varDelta tP_{5j} (y,t)\; \\ & \Rightarrow \mathop {\lim }\limits_{\begin{subarray}{l} \varDelta y \to 0 \\ \varDelta t \to 0 \end{subarray} } \frac{{P_{5j} (y + \varDelta y,t + \varDelta t) - P_{5j} (y,t)}}{\varDelta t} + \phi (y)P_{5j} (y,t) = 0 \\ & \Rightarrow \left[ {\frac{\partial }{\partial t} + \frac{\partial }{\partial y} + \phi (y)} \right]P_{5j} (y,t) = 0 \\ \end{aligned} $$

(33)

For state S₆

$$ \begin{aligned} P_{6h} (z + \varDelta z,t + \varDelta t)\; & = \{ 1 - \mu (z)\varDelta t\} P_{6h} (z,t)\; \\ & \Rightarrow \mathop {\lim }\limits_{\begin{subarray}{l} \varDelta z \to 0 \\ \varDelta t \to 0 \end{subarray} } \frac{{P_{6h} (z + \varDelta z,t + \varDelta t) - P_{6h} (z,t)}}{\varDelta t} + \mu (z)P_{6h} (z,t) = 0 \\ & \Rightarrow \left[ {\frac{\partial }{\partial t} + \frac{\partial }{\partial z} + \mu (z)} \right]P_{6h} (z,t) = 0 \\ \end{aligned} $$

(34)

For state S₇

$$ \begin{aligned} P_{7c} (t + \varDelta t)\; & = \lambda_{csf} \varDelta tP_{0} (t) \\ & \Rightarrow \mathop {\lim }\limits_{\varDelta t \to 0} \frac{{P_{7c} (t + \varDelta t) - P_{7c} (t)}}{\varDelta t}\; = \lambda_{csf} P_{0} (t) \\ & \Rightarrow \frac{\partial }{\partial t}P_{7c} (t) = \lambda_{csf} P_{0} (t) \\ \end{aligned} $$

(35)

Appendix 2

Boundary conditions of the system are obtained corresponding to transitions between the states where the transition from a state with and without elapsed repair time exists when elapsed repair times x, y and z are 0. Hence from Fig. 1, we have following boundary conditions:

$$ P_{ 1 3} \left( {0, t} \right) = \lambda_{ 1} P_{ 1 2} \left( t \right) $$

(36)

$$ P_{ 6h} \left( {0, t} \right) = \lambda_{h} P_{0} \left( t \right) $$

(37)

$$ P_{ 5j} \left( {0, t} \right) = \omega_{j} P_{ 4j} \left( t \right) $$

(38)

When system is perfectly good i.e. in initial state S ₁, then

$$ P_{0} \left( 0 \right) = 1 {\text{ and other state probabilities are zero at}}\,t = \, 0 $$

(39)

Appendix 3

Taking the Laplace transform of Eqs. (28–35) and (36–38), we get

$$ \left[ {s + 3\lambda_{1} + \lambda_{h} + \alpha_{j} + \lambda_{csf} } \right]\bar{P}_{0} (s) = 1 + \int\limits_{0}^{\infty } {\upsilon (x)\bar{P}_{13} (x,s)} dx + \int\limits_{0}^{\infty } {\phi (y)\bar{P}_{5j} } (y,s)dy + \int\limits_{0}^{\infty } {\mu (z)\bar{P}_{6h} } (z,s)dz $$

(40)

$$ \left[ {s + 2\lambda_{1} + \alpha_{j} } \right]\bar{P}_{11} (s) = 3\lambda_{1} \bar{P}_{0} (s) $$

(41)

$$ \left[ {s + \lambda_{1} + \alpha_{j} } \right]\bar{P}_{12} (s) = 2\lambda_{1} \bar{P}_{11} (s) $$

(42)

$$ \left[ {s + \frac{\partial }{\partial x} + \upsilon (x)} \right]\bar{P}_{13} (x,s) = 0 $$

(43)

$$ \left[ {s + \omega_{j} } \right]\bar{P}_{4j} (s) = \alpha_{j} \left[ {\bar{P}_{0} (s) + \bar{P}_{11} (s) + \bar{P}_{12} (s)} \right] $$

(44)

$$ \left[ {s + \frac{\partial }{\partial y} + \phi (y)} \right]\bar{P}_{5j} (y,s) = 0 $$

(45)

$$ \left[ {s + \frac{\partial }{\partial z} + \mu (z)} \right]\bar{P}_{6h} (z,s) = 0 $$

(46)

$$ s\bar{P}_{7c} (s) = \lambda_{csf} \bar{P}_{0} (s) $$

(47)

$$ \bar{P}_{13} (0,s) = \lambda_{1} \bar{P}_{12} (s) $$

(48)

$$ \, \bar{P}_{6h} (0,s) = \lambda_{h} \bar{P}_{0} (s) $$

(49)

$$ \, \bar{P}_{5j} (0,s) = \omega_{j} \bar{P}_{4j} (s) $$

(50)

Solving (40–47) with the help of (48–50), and (39) one may get various transition state probabilities as given in Sect. 4.