1 Introduction

Reliability plays a key role in the design of engineering systems. Achieving a high or required level of reliability is often an essential requisite. Several authors including Uematsu and Nishida (1987), Jin et al. (2009), Kharoufeh et al. (2010), Kaur et al. (2013) and Kadyan et al. (2013) studied single unit reliability models under the common assumptions:

  1. (i)

    No warranty is provided for system.

  2. (ii)

    Repair of the failed unit is always possible.

But, market survey shows that a large number of products are now being sold with a warranty which indicates the high popularity of warranty among customers. The warranty assures the users that a faulty item will either be repaired or replaced at no cost and that may increase sales.

Further, repair of the failed unit does not always feasible. Sometime replacement of the failed unit is cheaper than to continue its repair. In such cases, the failed unit may be replaced by new unit after getting necessary inspection for feasibility of repair in order to avoid unnecessary expenses on repair.

While considering the above facts, here we developed a single unit system with the concept of inspection for feasibility of repair beyond warranty subject to single repair facility. The following reliability measures are obtained by using supplementary variable technique.

  1. (i)

    MTSF and Reliability;

  2. (ii)

    Steady-state availability of the system; and

  3. (iii)

    Profit analysis of the user.

2 Assumptions

  1. (1)

    The system has a single unit.

  2. (2)

    There is single repairman, who is always available with the system to do repair, inspection and replacement of the unit.

  3. (3)

    The cost of repair of the failed unit within warranty is borne by the manufacturer provided failures are not due to the negligence of users such as cracked screen, accident, misuse, physical damage, damage due to liquid and unauthorized modifications etc.

  4. (4)

    Beyond warranty, unit goes for inspection after failure.

  5. (5)

    Repairman inspects the failed unit to see the feasibility of repair or replacement.

  6. (6)

    The distribution of failure time is taken as negative exponential while the inspection, and repair times are considered as arbitrary.

3 State-specification

S 0/S 1 :

The unit is operative under warranty/beyond warranty

S 2/S 4 :

The unit is in failed state under warranty/beyond warranty

S 3 :

The failed unit is under inspection

4 Notations

λ/λ 1 :

Constant failure rate of the unit within warranty/beyond warranty

α:

Constant rate of completion of warranty

\(p/q\) :

Probability that repair is feasible/not feasible

\(\mu (x),S(x)/\mu_{1} (x),S_{1} (x)\) :

Repair rate of the unit and probability density function, for the elapsed repair time ‘x’ in warranty/beyond warranty

\(h(y),S_{2} (y)\) :

Inspection rate of the failed unit and probability density function, for the elapsed inspection time ‘y’

\(p_{0} (t)/p_{1} (t)\) :

The Probability that at time‘t’ the system is in good state within warranty/beyond warranty

\(p_{2} (x,t)\varDelta /p_{4} (x,t)\varDelta\) :

The Probability that at time‘t’ the system is in failed state within warranty/beyond warranty, the elapsed repair time lies in the interval [x,x + ∆)

\(p_{3} (y,t)\varDelta\) :

The Probability that at time‘t’ the failed unit is under inspection, the elapsed inspection time lies in the interval [y,y + ∆)

p(s):

Laplace transform of function p(t)

S(x):

\(\mu (x)\exp [ - \int_{0}^{x} {\mu (x)dx} ]\)

S 1(x):

\(\mu_{{1}} (x)\exp [ - \int_{0}^{x} {\mu_{1} (x)dx} ]\)

S 2(x):

\(h(y)\exp [ - \int_{0}^{y} {h(y)dy} ]\)

5 Formulation of mathematical model of system

Using the probabilistic arguments and limiting transitions (shown in Appendix), we have the following difference-differential equations (Cox 1962):

$$\left[ {\frac{d}{dt} + \lambda + \alpha } \right]p_{0} (t) = \int_{0}^{\infty } {\mu (x)p_{2} (x,t)dx}$$
(1)
$$\left[ {\frac{d}{dt} + \lambda_{1} } \right]p_{1} (t) = \alpha p_{0} (t) + \int_{0}^{\infty } {\mu_{1} (x)p_{4} (x,t)dx} + \int_{0}^{\infty } {qh(y)p_{3} (y,t)dy}$$
(2)
$$\left[ {\frac{\partial }{\partial t} + \frac{\partial }{\partial x} + \mu (x)} \right]p_{2} (x,t) = 0$$
(3)
$$\left[ {\frac{\partial }{\partial t} + \frac{\partial }{\partial y} + h(y)} \right]p_{3} (y,t) = 0$$
(4)
$$\left[ {\frac{\partial }{\partial t} + \frac{\partial }{\partial x} + \mu_{1} (x)} \right]p_{4} (t) = 0$$
(5)

5.1 Boundary conditions

$$p_{2} (0,t) = \lambda p_{0} (t)$$
(6)
$$p_{3} (0,t) = \lambda_{1} p_{1} (t)$$
(7)
$$p_{4} (0,t) = \int_{0}^{\infty } {ph(y)} p_{3} (y,t)dy$$
(8)

5.2 Initial conditions

$$\begin{aligned} p_{i} (0) = 0;\; {\text{when }}i \ne 0 \\ p_{i} (0) = 1;\; {\text{when }} i = 0 \end{aligned}$$
(9)

6 Model analysis

6.1 Solution of the equations

Taking Laplace transforms of Eqs. (1)–(8) and using (9), we obtain

$$\left[ {s + \lambda + \alpha } \right]p_{0} (s) = 1 + \int_{0}^{\infty } {\mu (x)p_{2} (x,s)dx}$$
(10)
$$\left[ {s + \lambda_{1} } \right]p_{1} (s) = \alpha p_{0} (s) + \int_{0}^{\infty } {\mu_{1} (x)p_{4} (x,s)dx} + \int_{0}^{\infty } {qh(y)p_{3} (y,s)dy}$$
(11)
$$\left[ {\frac{\partial }{\partial x} + s + \mu (x)} \right]p_{2} (x,s) = 0$$
(12)
$$\left[ {\frac{\partial }{\partial x} + s + h(y)} \right]p_{3} (y,s) = 0$$
(13)
$$\left[ {\frac{\partial }{\partial x} + s + \mu_{1} (x)} \right]p_{4} (x,s) = 0$$
(14)
$$p_{2} (0,s) = \lambda p_{0} (s)$$
(15)
$$p_{3} (0,s) = \lambda_{1} p_{1} (s)$$
(16)
$$p_{4} (0,s) = \int_{0}^{\infty } {ph(y)} p_{3} (y,s)dy$$
(17)

Taking integration of Eqs. (12), (13) and (14), we get the following equations

$$p_{2} (x,s) = p_{2} (0,t)\exp ( - (sx + \int_{0}^{x} {\mu (x)dx))}$$
(18)
$$p_{3} (y,s) = p_{3} (0,t)\exp ( - (sy + \int_{0}^{y} {h(y)dy))}$$
(19)

and

$$p_{4} (x,s) = p_{4} (0,t)\exp ( - (sx + \int_{0}^{x} {\mu_{1} (x)dx))}$$
(20)

Using Eqs. (15) and (18), Eq. (10) yields

$$\left[ {s + \lambda + \alpha } \right]p_{0} (s) = 1 + p_{2} (0,s)\int_{0}^{\infty } {\mu (x)\exp \left( { - \left( {\int_{0}^{x} {\mu (x)dx} } \right)} \right)dx} = 1 + \lambda p_{0} (s)S(s)$$
$$p_{0} (s) = \frac{1}{T(s)}$$
(21)

where

$$T(s) = s + \lambda + \alpha - \lambda S(s)$$
(22)
$$p_{4} (0,s) = p_{3} (0,t)\int_{0}^{\infty } {ph(y)} \exp ( - (sy + \int_{0}^{y} {h(y)dy))} dy$$
$$p_{4} (0,s) = p\lambda_{1} p_{1} (s)S_{h} (s)$$
(23)

Using Eq. (23), Eq. (20) yields

$$p_{4} (x,s) = p\lambda_{1} p_{1} (s)S_{h} (s)\exp \left( { - \left( {sx + \int_{0}^{x} {\mu_{1} (x)dx} } \right)} \right)$$
(24)

Using Eqs. (16), (19), (23) and (24), Eq. (11) yields

$$\left[ {s + \lambda_{1} } \right]p_{1} (s) = \alpha p_{0} (s) + p_{4} (0,s)\int_{0}^{\infty } {\mu_{1} (x)\exp ( - (sx + \int_{0}^{x} {\mu_{1} (x)} dx)dx} + p_{3} (0,s)\int_{0}^{\infty } {qh(y)\exp ( - (sy + \int_{0}^{y} {h(y)} dy)dy} = \alpha p_{0} (s) + p\lambda_{1} p_{1} (s)S_{1} (s)S_{h} (s) + q\lambda_{1} p_{1} (s)S_{h} (s)$$
$$p_{1} (s) = \frac{A(s)}{T(s)}$$
(25)

where

$$A(s) = \frac{\alpha }{{\left( {s + \lambda_{1} - p\lambda_{1} S_{1} (s)S_{h} (s) + q\lambda_{1} S_{h} (s)} \right)}}$$
(26)

Now, the Laplace transform of the probability that the system is in the failed state is given by

$$p_{2} (s) = \int_{0}^{\infty } {p_{2} (x,s)dx = \lambda p_{0} (s)\left( {\frac{1 - S(s)}{s}} \right)}$$
$$p_{2} (s) = \frac{\lambda B(s)}{T(s)}$$
(27)

where

$$B(s) = \left( {\frac{1 - S(s)}{s}} \right)$$
(28)

Similarly \(p_{3} (s) = \int_{0}^{\infty } {p_{3} (y,s)dy = \lambda_{1} p_{1} (s)\left( {\frac{{1 - S_{h} (s)}}{s}} \right)}\)

$$p_{3} (s) = \frac{{\left( {\lambda A(s)C(s)} \right)}}{T(s)}$$
(29)

where

$$C(s) = \left( {\frac{{1 - S_{h} (s)}}{s}} \right)$$
(30)

Similarly \(p_{4} (s) = \int_{0}^{\infty } {p_{4} (x,s)dx = p\lambda_{1} p_{1} (s)S_{h} (s)\left( {\frac{{1 - S_{1} (s)}}{s}} \right)}\)

$$p_{4} (s) = \frac{{\left( {\lambda_{1} A(s)S_{h} (s)D(s)} \right)}}{T(s)}$$
(31)

where

$$D(s) = \left( {\frac{{1 - S_{1} (s)}}{s}} \right)$$
(32)

It is worth noticing that

$$p_{0} (s) + p_{1} (s) + p_{2} (s) + p_{4} (s) = \frac{1}{s}$$
(33)

6.2 Evaluation of Laplace transforms of up and down state probabilities

Let Av(t) is the probability that the system is operating satisfactorily at time ‘t’. The Laplace transforms of Av(t) or probabilities that the system is in up state (Pup(t)) (i.e. good state) and down state (Pdown(t)) (i.e. failed state) at time ‘t’ are as follows

$$A_{v} (s)\,\,\,\,\,or\,\,\,\,\,p_{up} (s) = \,p_{0} (s) + p_{1} (s)$$
$$A_{v} (s) = \frac{{\left( {1 + A(s)} \right)}}{T(s)}$$
(34)
$$p_{down} (s) = \,p_{2} (s) + p_{3} (s) + p_{4} (s)$$
$$p_{down} (s) = \,\frac{{\left( {\lambda B(s) + \lambda_{1} A(s)C(s) + \lambda_{1} pA(s)D(s)S_{h} (s)} \right)}}{T(s)}$$
(35)

6.3 Steady-state behavior of the system

Using Abel’s Lemma in Laplace transforms, viz.

\(\mathop {\lim }\limits_{s \to 0} s\left( {A_{v} (s)} \right) = \mathop {\lim }\limits_{t \to \infty } A_{v} (t) = A_{v} (say),\) Provided the limit on the right hand side exists, the following time independent probabilities have been obtained.

$$A_{v} = \frac{1}{{\left( {1 - p\lambda_{1} S_{1}^{'} (0) - \lambda_{1} S_{h}^{'} (0)} \right)}}$$
(36)
$$P_{down} = \frac{{ - p\lambda_{1} S_{1}^{'} (0) - \lambda_{1} S_{h}^{'} (0)}}{{\left( {1 - p\lambda_{1} S_{1}^{'} (0) - \lambda_{1} S_{h}^{'} (0)} \right)}}$$
(37)

6.4 Reliability of the system (R(t))

Using the method similar to that in Sect. 5, the differential–difference equations for reliability are:

$$\left[ {\frac{d}{dt} + \lambda + \alpha } \right]p_{0} (t) = 0$$
(38)
$$\left[ {\frac{d}{dt} + \lambda_{1} } \right]p_{1} (t) = \alpha p_{0} (t)$$
(39)

Theorem 1

The reliability of the system is given by

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

The proof of the Theorem-1 is given in the Appendix.

Corollary 1

The mean time to system failure (MTSF) is

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

Proof

Calculating \(MTSF = \int_{0}^{\infty } {R(t)dt}\) implies the result ‘*’ given in the Appendix.

7 Particular results

7.1 Availability of the system

If repair and inspection times follow exponential distribution i.e. \(S(s) = \frac{\mu }{{\left( {s + \mu } \right)}},\,\,S_{1} (s) = \frac{{\mu_{1} }}{{\left( {s + \mu_{1} } \right)}}\,\,\,\,\,and\,\,\,\,S_{h} (s) = \frac{h}{{\left( {s + h} \right)}}\,\)where μ and μ 1 are constant repair rates and h is constant inspection rate. Putting these values in Eqs. (21)–(26), we get

$$p_{0} (s) = \frac{1}{I(s)}$$
(40)

where

$$I(s) = \frac{{\left( {s^{2} + s(\lambda + \alpha + \mu ) + \alpha \mu } \right)}}{{\left( {s + \mu } \right)}}$$
(41)
$$p_{1} (s) = \frac{E(s)}{I(s)}$$
(42)

where

$$E(s) = \left[ {\frac{{\alpha \left( {s + \mu_{1} } \right)\left( {s + h} \right)}}{{\left( {s + \mu_{1} } \right)\left( {s + h} \right)\left( {s + \lambda_{1}^{{}} } \right) - ph\lambda_{1} \mu_{1} - qh\lambda_{1} \left( {s + \mu_{1} } \right)}}} \right]$$
(43)
$$A_{v} (s)\,\,\,\,\,or\,\,\,\,\,p_{up} (s) = \,p_{0} (s) + p_{1} (s)$$
$$= \left[ {\frac{{\left( {s^{4} + b_{3} s^{3} + b_{2} s^{2} + b_{1} s} \right)}}{{s\left( {s^{2} + s(\lambda + \alpha + \mu ) + \alpha \mu } \right)\left( {s^{2} + sa_{1} + a_{0} } \right)}}} \right]$$
(44)

where \(\begin{aligned} b_{3} = \left( {\lambda_{1} + h + \alpha + \mu_{1} + \mu } \right),\,\,b_{2} = \left( {\lambda_{1} \mu + \mu \alpha + \alpha \mu_{1} + \mu_{1} h + \mu h + \mu \mu_{1} + \lambda_{1} h\!\,+\,h\alpha + \lambda_{1} \mu_{1} - qh\lambda_{1} } \right), \hfill \\ b_{1} = \left( {\mu \mu_{1} h + \lambda_{1} \mu h + \lambda_{1} \mu \mu_{1} - \lambda_{1} \mu qh + \alpha \mu \mu_{1} + h\alpha \mu + h\alpha \mu_{1} } \right),\,\,b_{0} = \left( {\alpha \mu \mu_{1} h} \right) \hfill \\ a_{1} = \left( {\mu_{1} + h + \lambda_{1} } \right)\,and\,\,\,a_{0} = \left( {\lambda_{1} h + \mu_{1} h + \lambda_{1} \mu_{1} - q\lambda_{1} h} \right) \hfill \\ \end{aligned}\)

Taking inverse Laplace transforms of Eq. (44), we get

$$\begin{aligned} A_{v} (t) = \frac{{b_{0} }}{{z_{1} z_{2} z_{3} z_{4} }} + \left\{ {\frac{{z_{1}^{4} + b_{3} z_{1}^{3} + b_{2} z_{1}^{2} + b_{1} z_{1} + b_{0} }}{{z_{1} \left( {z_{1} - z_{2} } \right)\left( {z_{1} - z_{3} } \right)\left( {z_{1} - z_{4} } \right)}}} \right\}\exp \left( {z_{1} t} \right) + \left\{ {\frac{{z_{2}^{4} + b_{3} z_{2}^{3} + b_{2} z_{2}^{2} + b_{1} z_{2} + b_{0} }}{{z_{2} \left( {z_{2} - z_{1} } \right)\left( {z_{2} - z_{3} } \right)\left( {z_{2} - z_{4} } \right)}}} \right\}\exp \left( {z_{2} t} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \left\{ {\frac{{z_{3}^{4} + b_{3} z_{3}^{3} + b_{2} z_{3}^{2} + b_{1} z_{3} + b_{0} }}{{z_{3} \left( {z_{3} - z_{1} } \right)\left( {z_{3} - z_{2} } \right)\left( {z_{3} - z_{4} } \right)}}} \right\}\exp \left( {z_{3} t} \right) + \left\{ {\frac{{z_{4}^{4} + b_{3} z_{4}^{3} + b_{2} z_{4}^{2} + b_{1} z_{4} + b_{0} }}{{z_{4} \left( {z_{4} - z_{1} } \right)\left( {z_{4} - z_{2} } \right)\left( {z_{4} - z_{3} } \right)}}} \right\}\exp \left( {z_{4} t} \right) \hfill \\ \end{aligned}$$
(45)

\(z_{1} \,and\,\,z_{2}\) are the roots of the equation \(\left( {s^{2} + s(\lambda + \alpha + \mu ) + \alpha \mu } \right) = 0\) and \(z_{3} ,\,\,z_{4}\) are roots of the equation \(\left( {s^{2} + sa_{1} + a_{0} } \right) = 0\) (Balagurusamy 2009)

7.2 Profit analysis of the user

Suppose that the warranty period of the system is (0, w]. Since the repairman is always available with the system, therefore beyond warranty period, it remains busy during the interval (w, t]. Let K1 be the revenue per unit time and K2 be the repair cost per unit time, then the expected profit H(t) during the interval (0, t] is given by

$$H(t) = K_{1} \int_{0}^{t} {A_{v} (t)dt - K_{2} \left( {t - w} \right)}$$
$$H(t) = K_{1} \left[ \begin{aligned} \frac{{b_{0} t}}{{z_{1} z_{2} z_{3} z_{4} }} + \left\{ {\frac{{z_{1}^{4} + b_{3} z_{1}^{3} + b_{2} z_{1}^{2} + b_{1} z_{1} + b_{0} }}{{z_{1}^{2} \left( {z_{1} - z_{2} } \right)\left( {z_{1} - z_{3} } \right)\left( {z_{1} - z_{4} } \right)}}} \right\}\exp \left( {\left( {z_{1} t} \right) - 1} \right) + \left\{ {\frac{{z_{2}^{4} + b_{3} z_{2}^{3} + b_{2} z_{2}^{2} + b_{1} z_{2} + b_{0} }}{{z_{2}^{2} \left( {z_{2} - z_{1} } \right)\left( {z_{2} - z_{3} } \right)\left( {z_{2} - z_{4} } \right)}}} \right\}\exp \left( {\left( {z_{2} t} \right) - 1} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \left\{ {\frac{{z_{3}^{4} + b_{3} z_{3}^{3} + b_{2} z_{3}^{2} + b_{1} z_{3} + b_{0} }}{{z_{3}^{2} \left( {z_{3} - z_{1} } \right)\left( {z_{3} - z_{2} } \right)\left( {z_{3} - z_{4} } \right)}}} \right\}\exp \left( {\left( {z_{3} t} \right) - 1} \right) + \left\{ {\frac{{z_{4}^{4} + b_{3} z_{4}^{3} + b_{2} z_{4}^{2} + b_{1} z_{4} + b_{0} }}{{z_{4}^{2} \left( {z_{4} - z_{1} } \right)\left( {z_{4} - z_{2} } \right)\left( {z_{4} - z_{3} } \right)}}} \right\}\exp \left( {\left( {z_{4} t} \right) - 1} \right) \hfill \\ \end{aligned} \right] - K_{2} \left( {t - w} \right)$$
(46)

8 Numerical analysis

The research of numerical analysis is given in Tables 1, 2 and 3.

Table 1 Effect of failure rates (λ and λ 1) and rate of completion of warranty (α) on Reliability (R(t))
Full size table
Table 2 Effect of repair cost (K2) on expected profit (H(t))
Full size table
Table 3 Effect of inspection rate (h) on expected profit (H(t))
Full size table

9 Interpretation and conclusion

Table 1 shows the behavior of system reliability and it indicates that reliability of the system decreases with the increase of failure rates (λ and λ 1) and rate of completion of warranty (α) with respect to time and for fixed values of other parameters. Table 2 interprets that expected profit increases with the decrease of repair cost (K 2) with respect to time. Table 3 shows the effect of inspection on expected profit and it is observed that the expected profit increases with the increase of inspection rate (h) with respect to time. Hence, on the basis of the numerical results obtained for particular values of various parameters, it is concluded that the system with inspection beyond warranty can be made profitable to use by decreasing the repair cost and increasing the inspection rate.