Suppose I have a non-negative random variable $X$ with finite mean $\mu$. Let's for example assume it denotes demand, and we denote by $X(t)$ demand in period $t$, demand in different periods are iid. Now suppose, you get $q<\mu$ units each period as part of a subscription. This reduces your demand $X$ by $q$. Intuitively one would think the remaining demand is $Y=(X-q)^+$.

However in certain periods, it could happen, that $X(t)< q$. Hence the remaining demand in $t$ is 0, and the demand of the next period gets reduced by $q-X(t)$.

Is there any way to express the distribution of $Y$ given the distribution of $X$? Or the variance? For the mean e.g. it seems obvious that the mean of $Y$ is $\mu-q$. If you have any literature to which this is similar, I would be very grateful. Example of the shift and censoring

What I've tried so far:

Let Z denote the green part. $Z(t) = (Z(t-1)+q-X(t))^+$, which is the waiting time distribution in a GI/D/1 queue. I tried to find something similar for $Y$>

$K(t+1) = X(t+1)-q-(K(t))^- = X(t+1)-q-Z(t)$

$Y(t+1) = K(t+1)^+ = (X(t+1)-q-Z(t))^+$

I don't now what to do next.


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