I am having trouble understanding the following question:
For $u > 0$, let $T(u) = \inf\{t \geq u : X_t = X_{t−u}\}$ be the first time (after $u$) there are no arrivals in the process ${X_t, t \geq 0}$ during a period of length at least $u$.
(a) Find a constant $c$ so that the process $X′_t = X_{t+S_1} − c$ is a Poisson process.
(b) Let $T′(u) = \inf\{t \geq u : X′_t = X′_{t−u}\}$. Express $T(u)$ in terms of $S_1$ and $T′(u)$.
(c) Compare the two quantities $E[T(u)]$ and $E[T′(u)]$.
(d) Obtain $E[T(u)]$ by conditioning on the time of the first arrival.
Here is what I have known so far:
It does not matter what has happened in the process between time $0$ and time $u$. Time $t$ is a time after time $u$.
In this question, $X_t$ means the part of the process occurring between time $u$ and time $t$. Time $t$ is also the first time after time $u$ where the interval between $u$ and $t$ contains no arrivals. This interval has to be of length at least $u$.
In part a) we want to find a constant such that the part of the process between time $t$ and the first success following time $t$ (interval length $S_1$) minus some constant is a Poisson process. Would this simply be equal to $X_t$?
Thanks very much for the help.
I think this problem is a clumsy (in my hastily considered opinion) way of getting you to write down a recursion for the expected value. If the first arrival takes longer than $u,$ then $T(u)$ will be $u.$ If not, it will be the first arrival, plus something independent of the first arrival time with the same expectation value as $T(u)$ (since we just 'try again'). So we have $$ E(T(u)) = uP(S_1>u) + (E(T(u))+E(S_1\mid S_1<u))P(S_1<u),$$ which we can solve to get $$ E(T(u)) = \frac{uP(S_1>u) + E(S_1\mid S_1<u)P(S_1<u)}{1-P(S_1<u)} = \frac{u e^{-\lambda u} + \frac{u}{2}(1-e^{-\lambda u})}{e^{-\lambda u}} = u\left(1+\frac{e^{\lambda u}-1}{2}\right).$$
I'll mostly leave it to you to clean this up and translate it into the framework of the question. Some more help:
(a) $X_{t+S_1}$ is just the process starting at the first arrival, the first time where $X=1$ so letting $c=1$ should make $X_{t+S_1}-c$ into a Poisson process.
(b) $T'(u)$ is my vaguely described 'thing with the same distribution as $T(u),$" that, by the Markov property, is independent of $S_1.$ Expressing $T(u)$ in terms of $T'(u)$ and $S_1$ involves similar reasoning to my recursion equation.