# Hazard function of a simple system of components

I have been asked to complete these tasks, and a few more, but I am stuck somehow:

Figure 6 is the simple system above.

For task number 1: [DONE] I started by recognizing that the total survival time for the network must be $$T=min(max(min(T_1,T_2),T_3),T_4)$$ for the system above. This led to some calculations and I think I´ve found the survival function to be:

$$S_{T}(t)=(1-(1-S_{T1}(t)S_{T2}(t))(1-S_{T3}(t)))S_{T4}(t)$$ where I used $$S_{Ti}(t)=\exp(-\int_{0}^{t}\lambda_{Ti}(x)dx)$$ for $$i=1,...,4$$. So far so good and I´ve found the expected value.

For task number 2: [EDIT: DONE, see EDIT below] I´m having trouble either finding the hazard function or plotting it in R. I´m new to R, so I´m guessing that is the reason to why I´m stuck. Any insight in how to to acutally find the function and then plotting it according to the task would be very helpful!

I have tried to find it using $$\lambda_{T}(t)=-\frac{\partial}{\partial t}logS_T(t)$$ but the expressions are obviously a little bit "annoying" when you plug in the explicit expressions for each $$S_{Ti}(t)$$.

EDIT: I found the hazard function analytically using $$\lambda_{T}(t)=\frac{-\frac{\partial}{\partial t}S_{T}(t)}{S_{T}(t)}$$ where $$-\frac{\partial}{\partial t}S_{T}(t)=f_{T}(t)$$ and plotted it successfully i R (hopefully)!

For task number 3: [EDIT:] I did what Ben suggested below and found the probability using the integral $$Pr(T_{4}

For brevity, I will refer to your components and system using only the subscripts for the component number (and I recommend you use the same practice). For this kind of problem you should try to solve as much as you can analytically before using numerical solutions to evaluate particular integrals. The first thing to do is to derive the survival functions for the individual components, which are:

$$\begin{matrix} S_1(t) = \exp ( - \tfrac{2}{3} t^{3/2}) & & & & & S_2(t) = \exp ( - \tfrac{2}{3} t^{3/2}) \quad \quad \quad \quad \quad \ \ \\[12pt] S_3(t) = \exp ( - \tfrac{4}{3} t^{3/2}) & & & & & S_4(t) = \exp ( - t + \tfrac{1}{3} - \tfrac{1}{3} \exp(-3t) ). \\[6pt] \end{matrix}$$

Then, using the form you derived in your first step, you can get the survival function for the system and put this in its simplest form. Take the derivative of this function and use this to get the hazard function for the system and put that in its simplest form. That will complete steps 1-2 of your problem.

For your final task, you have two errors. Your first error is that your expression is not the time-to-failure for the subsystem composed of components 1-3 (this is not the minimum of the times-to-failure for the components). Your second error is that the events in your expression are not independent, and so their joint probability cannot be decomposed by multiplication of the marginal probabilities. So it does not surprise me that you are getting a nonsensical answer here. To solve this part, let $$T_{123}$$ denote the time-to-failure for the subsystem composed of components 1-3 and then use the law of total probability (conditioning on $$T_4$$) to get an expression for the probability of interest. This expression will be an integral and you should simplify the integrand as much as you an before using numerical methods to evaluation.

The hazard function for the system in this problem has a closed-form expression, and the only part of the problem that requires numerical integration is the last part of the problem. For these types of problems you should always do as much as you can analytically before turning to numerical methods, and you should simplify each step as much as possible to avoid over-complicating your expressions.

• Hi again Ben, I have a new task on the same system, but now, component $T_3 \sim Weibull(1/3,\mu)$ and component $T_3 \sim Weibull(1/3,\lambda)$, where $\mu$ and $\lambda$ are both positive. The task is to find these parameters so they maximize the expected life length of the network under the constraint $\mu + \lambda = 3$. I derived a new survival function for the "new" complete network, and expressed this survival function in terms of only $\mu$ using $\lambda = 3-mu$. But I don´t know how to do this. Any suggestions? Oct 24 '21 at 13:16
• Can I split the network in two seperate sub-networks, one being only component $T_4$ and one being the rest, then try to optimize the expected life length of $T_4$, obtain a value for $\lambda$ and use that to find both $\mu$ and then the maximized life length of the total network using these values? Oct 24 '21 at 13:23
• As you say, the constraint gives $\lambda = 3-\mu$, so you really only have one unknown parameter after substitution. Consequently, your optimisation must consider all components affected by that parameter.
– Ben
Oct 24 '21 at 20:47
• Thank you once again! I made it through! Oct 24 '21 at 21:40