Electrical components failing in a Poisson process

Question

A machine has infinitely many identical components. They fail according to a Poisson process with rate $\lambda = 4$/hour. A repairman arrives at time $t$ and instantly repairs all of the broken components, but components cost more to fix the longer they have been broken. The cost to repair a component is $\$10$/hour it has been down.

Suppose that all components are functional at time $0$. What is the expected cost of the repairs at time $t$ hours?

(Example for clarification: if one component fails at time $t=3.5$ and another at time $t=4$, and the repairman arrives at time $t=5$, then the total cost would be $15+10=25$.)

My thinking for this problem so far is as follows. The expected number of failures is $λt=4t$. The expected mean downtime for these broken components should be $t/2$. Thus, the expected cost of repairs is the product $(4t)(t/2)(10 \text{ per hour})$, or $20(t^2)$. Is this line of reasoning correct? Thanks.

Answer 1

Your reasoning is sound for a Homogeneous Poisson Process.

1. over time $\tau$, there will be $\lambda\tau$ failures,
2. From the definition of the Poisson Process, the occurrences are distributed uniformly on any interval of time, giving the expected arrival time of all of them in the middle of the interval.