Assume some machine is continually working on some items. Processing a single item takes some time $X$. The machine is subject to failures. If it fails, it will suspend the processing of the single item for some time $Z$. The time between the end of one failure and the start of the next failure is denoted by $Y$. Random variables $X$, $Y$, $Z$ are positive and continuous. For an example timeline, see below.
I am interested in the effective process time $T$ of an item. Sometimes $T$ is just $X$ and sometimes it is equal to $X + Z$. Is it possible to say something about $\mathbb{E}[T]$ and $\mathrm{Var}(T)$ without making simplifying assumptions about the distributions of $X$, $Y$ or $Z$? If $Y$ is exponentially distributed, I think it should be doable, since I can examine a single processing time $X$, but now I need to include history. Does anyone have ideas on how to approach this?