MTSF (mean time to system failure) and profit analysis of a single-unit system with inspection for feasibility of repair beyond warranty

Abstract

This paper discussed the MTSF and profit analysis of a single-unit system with inspection for feasibility of repair beyond warranty subject to a single repair facility. Any failure during warranty is rectified by the manufacturer free of cost to the users provided failures are not due to the negligence of users. Beyond warranty, unit goes under inspection after failure for feasibility of its repair or replacement. The failure time of the system follows negative exponential distribution while repair and inspection time distributions are taken as arbitrary. The expressions for reliability, MTSF, availability of the system and profit function have been determined by using supplementary variable technique. Using Abel’s lemma steady state behavior of the system has been derived. The numerical results for reliability and profit function are also obtained by taking particular values of various parameters and repair cost.

Appendices

Appendix

The derivation of Eqs. (1)–(5)

Assuming failure rates of the system are constant and repair rates are arbitrary. By applying supplementary variable technique, we develop the following differential-difference equations associated with the state transition diagram (Fig. 1) of the system at times (t + Δt), (y + Δy) and (x + Δx).

Fig. 1

State Transition Diagram of the Model

$$p_{0} (t + \varDelta t) = p_{0} (t)\left( {1 - \left( {\alpha + \lambda } \right)\varDelta t} \right) + \int_{0}^{\infty } {\mu (x)p_{2} (x,t)\varDelta tdx + o(\varDelta t),}$$

$p_{1} (t + \varDelta t) = p_{1} (t)\left( {1 - \lambda_{1} \varDelta t} \right) + \alpha p_{0} (t) + \int_{0}^{\infty } {\mu_{1} (x)p_{4} (x,t)\varDelta tdx + } \int_{0}^{\infty } {qh(y)p_{3} (y,t)\varDelta tdy + o(\varDelta t),}$ $p_{2} (x + \varDelta x,t + \varDelta t) = p_{2} (x,t)\left( {1 - \mu (x)\varDelta x} \right) + o\left( {\varDelta x,\varDelta t} \right),$

$$p_{3} (y + \varDelta y,t + \varDelta t) = p_{3} (y,t)\left( {1 - h(y)\varDelta y} \right) + o\left( {\varDelta y,\varDelta t} \right),$$

$$p_{4} (x + \varDelta x,t + \varDelta t) = p_{4} (x,t)\left( {1 - \mu_{1} (x)\varDelta x} \right) + o\left( {\varDelta x,\varDelta t} \right),$$

The proof of Theorem-1

Proof

Taking Laplace transforms of Eqs. (38) and (39), using (9), we get

$$\left[ {s + \lambda + \alpha } \right]p_{0} (s) = 1$$

$$\left[ {s + \lambda_{1} } \right]p_{0} (s) = \alpha p_{0} (s)$$

The solution can be written as

$$p_{0} (s) = \frac{1}{{\left( {s + \lambda + \alpha } \right)}}$$

$$p_{1} (s) = \frac{\alpha }{{\left( {s + \lambda + \alpha } \right)\left( {s + \lambda_{1} } \right)}}$$

$$R(s) = p_{0} (s) + p_{1} (s)$$

$$= \frac{1}{{\left( {s + \lambda + \alpha } \right)}} + \frac{\alpha }{{\left( {s + \lambda + \alpha } \right)\left( {s + \lambda_{1} } \right)}}$$

Taking inverse Laplace transform, we get

$$R(t) = \exp ( - (\lambda + \alpha )t)\left[ {\frac{{\lambda - \lambda_{1} }}{{\lambda - \lambda_{1} + \alpha }}} \right] + \exp ( - \lambda_{1} t)\left[ {\frac{\alpha }{{\lambda - \lambda_{1} + \alpha }}} \right]$$

(*)