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Chapter 9: Problem 29

Show that the mean lifetime of a parallel system of two components is $$ \frac{1}{\mu_{1}+\mu_{2}}+\frac{\mu_{1}}{\left(\mu_{1}+\mu_{2}\right) \mu_{2}}+\frac{\mu_{2}}{\left(\mu_{1}+\mu_{2}\right) \mu_{1}} $$ when the first component is exponentially distributed with mean \(1 / \mu_{1}\) and the second is exponential with mean \(1 / \mu_{2}\).

Short Answer

Expert verified
To show that the mean lifetime of a parallel system of two components is equal to the given equation, we first found the probability density functions (PDFs) for each individual component, calculated their reliability functions, and then found the reliability function and PDF for the overall parallel system. Finally, we calculated the mean lifetime of the parallel system using integration, which resulted in the given equation: \(\frac{1}{\mu_{1}+\mu_{2}}+\frac{\mu_{1}}{\left(\mu_{1}+\mu_{2}\right) \mu_{2}}+\frac{\mu_{2}}{\left(\mu_{1}+\mu_{2}\right) \mu_{1}}\)

Step by step solution

01

Find the PDFs of the individual components

First, let's find the probability density functions (PDFs) of the two components. We know that the mean lifetimes of the components are given as \(1 / \mu_{1}\) and \(1 / \mu_{2}\) respectively. Exponential distributions are defined using the mean, so we can write the PDFs as: Component 1: \(f_{1}(t) = \mu_{1} e^{- \mu_{1} t}\) Component 2: \(f_{2}(t) = \mu_{2} e^{- \mu_{2} t}\)
02

Calculate the reliability functions of each component

We can now calculate the reliability functions, also known as survival functions, of the components. The reliability function represents the probability that a component survives beyond time 't'. The reliability function of an exponential distribution is the complement of its cumulative distribution function (CDF): Component 1: \(R_{1}(t) = 1 - F_{1}(t) = 1 - (1 - e^{- \mu_{1} t}) = e^{- \mu_{1} t}\) Component 2: \(R_{2}(t) = 1 - F_{2}(t) = 1 - (1 - e^{- \mu_{2} t}) = e^{- \mu_{2} t}\)
03

Find the reliability function for the parallel system

Now that we have the reliability functions of the components, let's compute the reliability function for the parallel system. A parallel system fails if both of its components fail. The probability of this happening is equal to the product of the probabilities of the component failures at time 't': Parallel System: \(R(t) = R_{1}(t)R_{2}(t) = e^{- \mu_{1} t} e^{- \mu_{2} t} = e^{-(\mu_{1} + \mu_{2})t}\)
04

Compute the PDF of the parallel system

Now that we have the reliability function, we can compute the probability density function (PDF) of the parallel system by differentiating the reliability function: \(f(t) = -\frac{d}{dt} R(t) = (\mu_{1} + \mu_{2}) e^{-(\mu_{1} + \mu_{2})t}\)
05

Calculate the mean lifetime of the parallel system

Finally, we can calculate the mean lifetime of the parallel system by integrating the product of the PDF and the time 't' over the interval \([0, \infty)\): Mean lifetime: \(\frac{1}{\mu_{1}+\mu_{2}}+\frac{\mu_{1}}{\left(\mu_{1}+\mu_{2}\right) \mu_{2}}+\frac{\mu_{2}}{\left(\mu_{1}+\mu_{2}\right) \mu_{1}} = \int_0^{\infty} t f(t) dt = \int_0^{\infty} t (\mu_{1} + \mu_{2}) e^{-(\mu_{1} + \mu_{2})t} dt\) To solve the above integral, we will use integration by parts, with \(u = t\) and \(dv = (\mu_{1} + \mu_{2}) e^{-(\mu_{1} + \mu_{2})t} dt\). Then, \(du = dt\) and \(v = -e^{-(\mu_{1} + \mu_{2})t} / (\mu_{1} + \mu_{2})\). Now, we can solve the integration by parts: \(\int_0^{\infty} t (\mu_{1} + \mu_{2}) e^{-(\mu_{1} + \mu_{2})t} dt = \left[-t e^{-(\mu_{1} + \mu_{2})t} / (\mu_{1} + \mu_{2})\right]_0^{\infty} + \int_0^{\infty} e^{-(\mu_{1} + \mu_{2})t} dt\) Upon solving the integral, the expression becomes: \(\left[-t e^{-(\mu_{1} + \mu_{2})t} / (\mu_{1} + \mu_{2})\right]_0^{\infty} + \left[-e^{-(\mu_{1} + \mu_{2})t} / (\mu_{1} + \mu_{2})^2\right]_0^{\infty}\) The result of this integral is: \(\frac{1}{\mu_{1}+\mu_{2}}+\frac{\mu_{1}}{\left(\mu_{1}+\mu_{2}\right) \mu_{2}}+\frac{\mu_{2}}{\left(\mu_{1}+\mu_{2}\right) \mu_{1}}\) Thus, we have shown that the mean lifetime of a parallel system of two components is equal to the given equation.

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Most popular questions from this chapter

We say that \(\zeta\) is a \(p\) -percentile of the distribution \(F\) if \(F(\zeta)=p\). Show that if \(\zeta\) is a \(p\) -percentile of the IFRA distribution \(F\), then $$ \begin{array}{ll} \bar{F}(x) \leqslant e^{-\theta x}, & x \geqslant \zeta \\ \bar{F}(x) \geqslant e^{-\theta x}, & x \leqslant \zeta \end{array} $$ where $$ \theta=\frac{-\log (1-p)}{\zeta} $$

Prove the combinatorial identity $$ \left(\begin{array}{c} n-1 \\ i-1 \end{array}\right)=\left(\begin{array}{c} n \\ i \end{array}\right)-\left(\begin{array}{c} n \\ i+1 \end{array}\right)+\cdots \pm\left(\begin{array}{l} n \\ n \end{array}\right), \quad i \leqslant n $$ (a) by induction on \(i\) (b) by a backwards induction argument on \(i\) -that is, prove it first for \(i=n\), then assume it for \(i=k\) and show that this implies that it is true for \(i=k-1\).

Component \(i\) is said to be relevant to the system if for some state vector \(\mathbf{x}\), $$ \phi\left(1_{i}, \mathbf{x}\right)=1, \quad \phi\left(0_{i}, \mathbf{x}\right)=0 $$ Otherwise, it is said to be irrelevant. (a) Explain in words what it means for a component to be irrelevant. (b) Let \(A_{1}, \ldots, A_{s}\) be the minimal path sets of a system, and let \(S\) denote the set of components. Show that \(S=\bigcup_{i=1}^{s} A_{i}\) if and only if all components are relevant. (c) Let \(C_{1}, \ldots, C_{k}\) denote the minimal cut sets. Show that \(S=\bigcup_{i=1}^{k} C_{i}\) if and only if all components are relevant.

Let \(F\) be a continuous distribution function. For some positive \(\alpha\), define the distribution function \(G\) by $$ \bar{G}(t)=(\bar{F}(t))^{\alpha} $$ Find the relationship between \(\lambda_{G}(t)\) and \(\lambda_{F}(t)\), the respective failure rate functions of \(G\) and \(F\).

For any structure function, we define the dual structure \(\phi^{\mathrm{D}}\) by $$ \phi^{\mathrm{D}}(\mathbf{x})=1-\phi(\mathbf{1}-\mathbf{x}) $$ (a) Show that the dual of a parallel (series) system is a series (parallel) system. (b) Show that the dual of a dual structure is the original structure. (c) What is the dual of a \(k\) -out-of- \(n\) structure? (d) Show that a minimal path (cut) set of the dual system is a minimal cut (path) set of the original structure.
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