A machine receives messages according to a Poisson process of rate $\lambda$ messages per second. If it does not receive a message for $\theta$ seconds, then it is programmed to stop functioning. Evaluate mean value $E(T)$ of the total time $T$ the machine is working.
Attempt. If $N$ is the total amount of messages the machine receives while working, then for $N=n$, the time working is $T=W_1+\ldots+W_n+\theta$, where $W_i$ are the intermediate times between the arrivals of the $n$ messages (each of the exponential distibution, parameter $\lambda$). Somehow we should take into account that the intermediate times above are $<\theta$ and also that $W_{n+1}>\theta$, in order to evaluate $E(T|N=n)$ and then $E(T)=E(E(T|N))$.
Thanks in advance for the help.